^^.3
TRANSACTIONS
OP THE
MOYAIL imiSH ACABEMir.
VOL. XII.
^
KT.,
A
^. <i:i^, /^
THE
TRANSACTIONS
OP THE
ROYAL IRISH ACADEMY.
VOL. XII.
DUBLIN: '
PRINTEKS TO THE ROYAL IRISH ACADEMT.
1815.
THE ACADEMY desire it to be under stoody that, as a
body, they are not answerable for any opinion, representation of
facts, or train of reasoning, which may appear in the following
Papers. The authors of the, several essays are alp?ie responsible
for their contents.
ERRATil.
SCIENCE.
I'age Line.
48 1 Dele full point after p
11 Dele p. after 19" ,28 78 _ 5 Read He derives by, &p. S8 — 11 ror5=:ireadi=l
POLITE LITERATURE. (51—2 After the word :press add a comma, and dele tUe comma after the wonl everi/
oq 11 For non una read nulla
88—1 For from meaning read from the meaning «9 — 27 (Last line of the poetry )/or Far read Ian 97 — 8 For these pursuits ; read their pursuits ; 11 For to rectify read to purify.
DIRECTIONS TO THE BINDER.
In Science-Tho Water Spout to be placed opposite page S9- ll-__-l>late, relative to Uack Horizon Glass opposite S-
CONTENTS.
SCIENCE.
Page
I. AN Explanation of the Method of Adjustment of the
JRac.k Horizon Glass of Hadleys Quadrant, by two near objects; also a Description of a Projected Ad- dition to the Quadrant for refecting that Adjustment, according to the Method of Mr. Blair. By the Rev. James Little. - - .'3
II. Two proofs of the Binomial Theorem^ by the Rev.
Samuel Vince, A. M. F. R. S. Flumian Professor of Astronomy and Experimental Philosophy, in the Uni- versity of Cambridge. — Communicated by the Rev. J. Brinkley - - 31
III. On certain Properties of Numbers, by the Rev. Samuel Vince, A. M. F. R. S. and Plumian Professor of As- stronojny in the University of Cambridge. — Communi- cated by the Rev. J. Brinkley - 31
IV. An Account of a very remarkable Water Spout, which appeared at Ramsgate, July Kith, 1810, — by the Rev, S. Vince, A. M. F. R. S. Plumian Professor of Astro- nonu) in the University of Cambridge. Communicated
by the Rev. J. Brinkley - - -29
b . . '
/ IV
Pag«5
V. An Account of Observations made at the Observatory of Trinity College, Dublin, zmth an Astronomical Circle^ eight feet in diameter, which appear to point
out an Annual Parallax in certain fixed Stars.
Also a Catalogue of l^orth Polar distances of forty - seven principal fixed Stars, from recent observations, and a comparison thereof with those of the same Stars, obtained by other Instruments, and by the same InstrU' ment, at a former period. By John Brinkley, JJ. D. M. R. I. A. F. R. S. and Andrews' Professor of As- tronomy in the University of Dublin - *S5
VI. Analytical Investigations respecting Astronomical Re- fractions, and the application thereof to the formation of convenient Tables, together with the results of ob- servations of Circumpolar Stars, tending to illustrate the Theory of Refractions. By John Brinkley, D. D. M. R. I. A. F. R. S. and Andi-ews' Professor of As- tronomy in the Univcrify of Dublin - 77
VII. Appendix to the Account of Observations tnade at the Observatory of Trinity College, Dublin, which appear to point out an Annual Parallax in certain fixed Stars, i}c. ^-c. By the Rev. J. Brinkley, D. D. F. R. S. M. R. I. A. and Andrews' Professor of Astronomy in
the University of Dublin - J19
* The folio of this page, and the seven subsequent ones, are, by an error of the press, duplicates of the folios of the eight preceding pngcs.
POLITE LITERATURE.
Page
I. AN Essay on the subject proposed by the Royal Irish Aca-
demyy viz. " Whether, and how far, the Pursuits of Scientific and Polite Literature, assist, or obstruct each other." A prize Essay. By William Phelan, Esq. A. B. T. C. D. - - 3
II. An Essay on the subject proposed by the Royal Irish
Academy viz. " on the Influence of Fictitious History on Modern Manners." A prize Essay. By Miss Har- riet Kiernan - - 61
III. An Essay on the question proposed by the Royal Irish Academy, viz. " on the Influence of Habit, considered in conjunction with the Love of Novelty." A prize Essay. By Andrew Carmichael, Esq. M. R. I. A. 99
IV. An Essay on the Invention of Alphabetic Wi'iting.
By Andrew Carmichael, Esq. M. R. I. A. 168
SCIENCE.
VOL. xir.
'■--So
' .wDiVju. AN EXPLANATION
OF THE METHOD OF ADJUSTMENT
OF THE OF
HADLEY'S QUADRANT,
BY TWO NEAR OBJECTS :
^I'P Ui.. ALSO A DESCRIPTION
OF A PROJECTED ADDITION TO THE QUADRANT^.,' /
50H REFLECTING THAT ADJUSTMENT ACCOKDING TO THE METHOD OF
MR. BLATR,,
BY THE REV. JAMES LITTLE.
■^ca^^id^O^^^
,Rcad, January 28th, 1811.
How desirable as well as difficult it is, to adjust on every occasion the Back Horizon Glass of Hadley's Quadrant with necessary precision, is declared by the many different con- trivances which have been suggested for that purpose ; and this I hope will procure an indulgent approbation of the pre-^ sent, as well as the future, attempts that may be made for that end, till it shall be accomplished in every manner de- sirable. The mode of its adjustment, by two near objects,, has been described by the late Rev. Mr. Ludlam in his trea- tise on the quadrant ; and it may b}' this be accurately per- formed, if executed with due and intelligent attention to the. » B 2
requisite circumstances : but as neither Mr. Ludlam, nor any other person that I know, has explained the grounds of the directions he has given ; and as these directions will pro- bably be applied in an unskilful and negligent manner, un- less it be generally understood and impressed of what im- portance they are : as moreover this is the method, at least the most generally practicable, of adjusting the back hor". glass, as well as of trying the accuracy of the construc- tion of the quadrant for effectiug it in Mr. Blair's method ; and is also subservient to the contrivance hereafter men- tioned for accomplishing it in the same Avay ; it is necessary, before I proceed to the description of it, to state the prin- ciples on which Mr. Ludlams judicious instructions are founded.
He directs that the back horizon glass may be adjusted at i-ight angles to the index glass, by the means of two near objects, such as two lines sustaining plummets in water, or two candles *, &c. lying in the plane of the quadrant placed horizontal, and in a line joining the objects equidist- ant from the quad'.; one of them being before, and the other behind the observer; by reversing the instrument by 'turning it half round in its own plane, and shifting its posi- tion laterally on cither sidc^ till the images of the two objects are seen, through the back sight vane, to coincide, when each of them alternately is viewed by the observer, by di-
• WTien plummets are used, they must bo placed at opposite doors or windows against the light of the sky: and if candles be employed, their light should he seen tlirough a small slit in a screen placed before each.
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rect vision after a half turn of the instrument; the index fixed at 0, and the back horizon glass shifted, till the images are brought to coincide, whichsoever of them be viewed directly. The quadrant is to be supported on a moveable stand, on the points of two erected pins fixed on the stand, inserted into two conical holes made in the middle of the heads of the screw pins in the back of the instrument, which fasten the central pins supporting the index and the back horizon glass j the placing the respective glasses alternately on these points, in the manner represented in fig. ]., will veveree the quadrant, by giving it just a semicircular mo- tion in its own plane. The manner of performing this ad- justment has been fully described b}' Mr. Liicllam, to whom I refer; but as I have seen no demonstration of its accuracy, 1 give the following proof of it ; assuming the established optical principles.
Let A P Q (fig. 1.), be the octant, fixed on two points ^mder the centres of the index glass A, and the back hori- zon glass B, or any two other fixed points ; and let C and c be the two candles or objects by which the glass B is to be adjusted. The image of the object C will be seen by the eye E, looking through the sight vane, coincident with the object c, when the stand of the quadrant is properly placed, by the ray E B, parallel to A C, if the glasses are at right angles. Let the quadrant now be turned half round, and placed on the points in the position a p q ; and if the back horizon glass is properly adjusted, then the eye -looking
6 *^; "
' through the vane at e, will see the reflected image of tiie object c coincident with the object C : because in these dif- ferent positions of the quadrant, the incident rays become the reflected ones, and vice versa ; and the index glass in the 2ad position a, will be parallel to the same, as it was in the 1st position A ; as also the horizon glass b to B.
But if the speculums A and B were not rightly adjusted at right angles to each other, the reflected ray B E \a the former position of the quadrant would not he parallel to the incident ray C A, but these rays would make an angle, equal, (suppose) to £ B M (or E B N) ; and consequently this B M (or B NJ is the reflected ray, by which, and in the direction of which, the image of the object C is seen : then the object c must be placed at m (or n) in order to coin- cide with the image of C, which appears only in the direc- tion of M B (or N S), f,et them coincide in m ; and let the quadrant no^v be turned half round, and put into the position a p q ; in which the glasses a and A, and b and B are parallel : the angle of incidence is now one half of the angle m a d greater than half c a A or CAB by the angle c a m; so that the reflected ray ad will fall without the angle cab', and will therefore either fall quite without the hori- zon glass b, or at least at a distance from its centre : in the former case the image of c would not be seen by the eye at c at all, unless the index glass were so long, and the object c so near, that a ray in g could fall on it in an angle so much less than the half of /« a d, as that the reflected ray
g h would fall on the glass b, and be again reflected to the eye at e.*
But if the rays forming the ijnage are reflected from the middle part only of the mirrors, the image of m orn could not be seen to coincide with C by the eye at e ; for if the incident ray were different from c «, as suppose m a, the re-
* Int)rder to understand the theory of the reflection of the rays forming the image seen in the back horizon glass, the following circumstances are to be considered :
1st, Because the speculum b is parallel to B, whatever inclination B has, which di" verts the image of the object C from the point c to m, the same inclination 6 also has, tending to divert the image of c to o, let the reflected rays g b or ad fall where they will on the speculum, or with whatever inclination.
2d, If the mirrors were at right angles, the rays m a, m g would be reflected from the mirror b, in a direction parallel to themselves, i. e. the ray m a falling on the point d ill the mirror ft, would be reflected in d o parallel to m a; and the ray m g falling on h, in the mirror b, would be reflected in h i parallel to m g : but when the mirrors are inclined to each 6ther, the rays d o , and h i will decline from such parallelism in an angle equal to c am double the inclination of the mirrors.
3d, When the angle cam exceeds the angle a m g subtended by half the length of the mirror a, by a difference equal to, or exceeding the angle, subtended by half the length of the horizon glass at the middle of the index glass ; all the rays reflected from the latter will fall without the horizon glass, and not be reflected by it ; but when the angle c a m is less than this, some of the rays incident on the index mirror a, will fall from it on the mirror b, and be again reflected : and since the object c or m is so near that there is a considerable diffierence in the incidences of the rays diverging from it on the mirror a, from the point a to g, (the greatest difi'ereuce equal to a »i g), and the same in the reflections from the mirror 6, the image of the object c placed at m, may be seen in different places by some of the rays diverging from this mirror.
4th, The diffierence of the incidences of the jays m a and mg is that of their reflec- tions in a d and g h ; and the difference of the incidei»ces of a d and g h will be that of their reflections from the mirror b ; and the angular motion of the speculum b will be half of either these or those, in order to its reflecting one of these rays in the same di- rection in which the other liad been reflected.
W'
8
fleeted raj would be difFerent from a h', i. e. it would not fall on the speculum in the point b, nor consequently be seen< (by the eye at e), to coincide with C ; but would fall without a b as at d, and would be reflected in do; in which direc- tion the image would be seen, and would be painted in the bottom of the eye, in a different place from that of the di- rect image of C; so that these images would be divaricated ; and it would be necessary to make them unite, by givino- such a motion to the little mirror, as would have made the first reflected ray B E parallel to the incident ray C A , by which the first image would be transferred from vi to c, and the second image from o to the eye at e.
In the same manner it may be shewn, that if the second reflected ray tended to any point N on the other side of the line B E, from an inclination of the speculum B on the other side, there would be a divarication of the images to the eye at e, till such inclination of the glass was removed. It also appears that the objects C and c, by which the ad- justment is made, may be placed very near the instrument, provided the reflection be made from the middle part only of the glasses, especially of the index-glass, the incidence of rays being difFerent in different parts of it; for unless the sight vane or eye hole for the little mirror, be large, so as that tiie eye could shift across the vane the axis of vision, which ought to be fixed, or the hole be very near the mirror; the image, if reflected from a part of it distant from the middle, would yet not appear coincident with the object seen
9 .
directly through the middle, so as to prevent the separation of the images ; because when the rays which form both imafes, cross one another, and proceed in different direc- tions, though they should even cross in the same point in the mirror, yet they will penetrate the eye diverging, and form different images on the retina. But if the image may be seen by reflection from any part of the index-glass a, the angle of incidence of a ray m g, (of the near object c re- moved to m,) falling on that glass at a point g distant from its middle point a, will be less than that of a ray incident on the point a, by the angle a m g: (for if the line a g were produced, the external angle at g would be equal to the angle at a and also to a m g together ; and therefore as much as the external angle at g is encreased above that at a, as falling toward a perpendicular from m to the line ag (El. 1. 32. cor.) the internal one, or the angle made with the mirror, is diminished ;) but if the incidences on a and g were equal, the reflections would be so too ; i. e. both m a and m g, and also ad and gh would be parallel ; which is the case when the object is very remote, the angle a m g then vanishing. Also since by reflection from any number of plane mirrors, the direction of the rays is changed, but not their inclination to each other, the ang. am g made by the rays incident on the index-glass, will be likewise the measure of their divergence reflected from the horizon glass. * If therefore the glasses
* The angle b a d h equal to c a m, and the angle made by e J and o d is equal to either ; and au angular motion of the speculum 6 equal to half of any of these
VOL. XII. C
10
are uncovered, the eye may see the image of w, (by rays in*- cident on a and g in the mirror a, and on d and h in the mirror 6,) in different places, whose angular distance is amg; and if the sight vane or hole at e be of any breadth, the second reflected image may be seen in two extreme places, whose distance will be as near to that anglie as the breadth of the vane and of the horizon glass will allow, and it may be also seen coincident with the direct image C, because the unsilvered part of the glass b extends- across its whole breadth, so that in whatever part of it the reflected image appears, the sight may be directed through that part to C. *
angles would make do issue parallel to be;- But if tHe speculums were uncovered, thedif. ference only of the angles amg and cam would require to be corrected by an angular motion of the speculum b,. which would be half of this difference ; and this being done the image of m would be seen in the direction ft e by the ray m g, wliile the same image' would be visible at an angular distance equal to a m g, by the ray m a; so that the image of c or m might be- seen in different places under the same inclination of the glasses; i. e. the adjustment would be uncertain..
* To shew that what is here stated is applicable to observations made with the quadrant, let d (fig, 2.) be a luminous body, from which light falls on the mirror a b with an angle of incidence «f/ e : its image will be visible to an eye at e in the direction e/, when the angle efc is equal to c/rf; Let the mirror be turned on its axis/, carrying the perpendicular/ c with it : when this has arrived to the position/g-, the angle of incidence will be encreased hy cfg; and the angle of reflection must be augmented by the same, so as now to be equal to dfg: if therefore the image is to be seen still in the point /i and no other point in the speculum, the eye must be placedat A ; when g-/A will be equal to g f d; in which case the angle efh will be equal to twice tjie angular motion af i of the speculum, or of its perpendicular c/, which is the same ; i. e. ef h will be equal to twice cfg. If the eye may be shifted, from the place h to a different place, as n, by looking through a hole or vane, whose breadth is equal to the interval h n, the imag»^ of the object d may be seen by reflection from the mirror i A: in a different place or di-
11
From this it appears, that to adjust the horizon glass pro- perly by two near objects, the face of both mirrors should be covered, except the middle parts only, or means must be used to view the images by those rays only, which are inci- dent on the middle of both mirrors But if according to Mr. Ludlam's direction, the object C be seen directly through the middle of the glass ft, and if the image of no other part of the glass a, but its middle part also, can be seen by re- flection from the middle of the mirror b; then no rays inci- dent on any other part g of the index glass could be seen to coincide with the object C. Suppose this to be effected as Mr. Ludlam directs, by covering the index glass with a piece of card-paper, equal in size to itself, and lying close to it, having a black line marked on the middle perpendicular to the plane of the instrument : and the whole card to be made visible in the horizon glass h, and the black line to appear in
rectioB n m, visible iu the mirror not in the place/, but in m, by a ray dm, reflected in w» n, making an angle with the former line of vision /A equal to the angle/ rf in ; and as the eye shifts along the interval h n, carrying with it the axis of vision through the dif- ferent points in that interval, tlie line of direction of the image, or its visible place will also shift through the interval/ m in the mirror with an angular motion tinally equal to the angle/ rf m. Hence the place of a very near object seen by reflection from a mirror through a vane, also very close to the mirror as in the back observation for this adjust- ment, may be very inaccurately determined, unless it be seen only in that place or spot in the mirror from which spot the image had been reflected in a reversed position of the quadrant in the adjustment. When the object d is so remote, that the angle/ d m be- comes insensibly small, tben the apparent place of the image will be the same, in what- ever part of / m in the mirror it is seen reflected from : but when the object is near, since the axis of vision cannot be fixed by contracting the eye-hole to a point, the images must be seen iu the same place iu the mirror.
c 2
appear in the middle oft, through which the object C is seen directly. As the whole card covering the mirror a, is seen equidistant from the extremities of b, every point in the surface of a, and consequently every ray reflected from such point, must be in the same manner seen to preserve their re- lative positions, and as the picture of a seen in b, should occupy nearly its whole surface ; the extremities of a, or any rays reflected from such extremities, could not be seen in the centre of b; but if the objects C and c, being small, could not subtend at the eye so great a space as the whole mirror a, the image of c would cover but a small part of the image of a ; and if that image proceeded by reflection not from the centre, but the extremity of a, it would be visible in the extremity of the image of a as seen in b; i. e. at a distance from the centre of b, (and consequently remote from the image of C ;) if it were seen in the centre of 6, it must be reflected from the centre of a; but if the whole surface of a were not apparently coincident with that of b, this might not be the case.
Hence appears the justness of Mr. Ludlam's direction^ that the centres of both mirrors should be seen to coincide in the horizon glass with the object seen directly ; for the images can appear thus, in both positions of the quadrant coincident only under a certain and invariable position of the specula, though their whole surfaces were uncevered ; it is hard however to distinguish by the eye what is the
IS
middle part ofi the back horizon glass. * By the glass herein- after proposed to be used for Mr. Blair's adjustment, instead of the polished edge of the index glass, the beam of light is reflected to the eye undivided, which Avill allow the axis of vision to pass through the axis of the back horizon glass ; as it ought to do, whether for adjustment of this glass, or for taking angles ; and as the axis of vision cannot be the same
* Tliis glass lies so obliqueto the eye, that I thiak it yet remains to be enquired what is to be considered as its middle part, whether the middle of the fore or back surface or the middle of its substance, or lastly that ppiut in the Siinie, -which is the vertex of tiie angle made by the incident ray with the sanre refracted by its fore surface after re- flection from its silvered surface* . It would appear to me of litlle moment, which of the two beams of light, proceeding, singly from the middle of the index-glass, and re_ fleeted double from the two surfaces of the horizon glass, be chosen for adjustment as the fixed axis of vision, (for both cannot be iudiscriminatejy used, as emerging from dif- ferent parts of the glass,) provided the reflected image be seen only by the same beam, , issuing from the same part of the horizon glass in all reversals of the quadrant ; were it not that the axis of vision ought to pass through the middle or axis of the glass, for the convenience of direct as well as reflex vision ; according to which the reflected ray cannot, in the oblique position of the glass, impinge on the middle of either sur- face ; but must be made (by turning the instrument in its plane, and placing the sight vane properly,) to fall on its fore surface between the middle of it, and the edge next the eye, if the reflection is to be made from the fore surface; and between the middle and the remote edge, if the image reflected from the back surface is to be seen. The proper place for reflection in the designed axis of vision, may be marked on the face of the glass, by sticking to it a fine waxed thread; and thea the black line on the card before mentioned, covering the face of the index^lass, (or such another thread fixed along the middle of it,) must be made to coincide with this thread in every posi- tion of the quadrant for this adjustment; and as two images of the line will appear from the two surfaces, one only of them must be invariably used : the card to be removed im. order to view the objects, when the line on it is made coincident with the thread.
14 •
for both these, since the incidence of the rays from the middle, and of those from the normal edge of the index glass on the horizon glass, is different ; so the position of the back sight vane, and the position and direction of a tele- scope, (if one be used,) must be altered for these different purposes. The vane may, without moving its support, have its position changed, by having the eye hole.made in a little 'moveable plate fastened on the support ; but a complicated motion would be requisite for the telescope, to place it in the best manner for each of the above intentions. If it is expected to answer by only a circular motion of its upright stand, changing its direction, without moving it from its place, the stand should be placed as near as possible to the back horizon glass ; for the farther it is removed from it, the more distant in one of its two positions will its axis be from the axis of that glass.
To ascertain the direction of the sight and of the telescope in making an observation by the edge of the .index-glass, or of the glass here to be proposed for. the same purpose, let a moveable rule or square, perpendicular to the face of the quadrant, be applied to the farther side of the quadrant opposite to the back horizon glass; and when the direct and reflected images are brought to unite, as the eye looks through the axis of the glass, let the rule be shifted, till its edge is made to appear in the place of their coincidence. ^If then a mark be made on the side of the quadrant at the edge of the rule, a line drawn from the mark through the axis of
'15
the horizon glass, will point out the axis of visfon and direc- tion of the telescope. If the position of the latter be wrong, the observations will be erroneous, unless Mr. Hadley's cor- rection be applied. *
If the object which bj the eye at JE is seen in m, i. e. the object c removed to m, were to be brought to appear to the eye at e to coincide with C, by giving the mirror h an angular motion sufficient for thisj such motion would be too great ; for then the incident ray ma, and the reflected ray 6 €\ would not be parallel, nor consequently the glasses perpendicular: only half this motioH must be given, and then the stand changed, or the object w moved to c, till the object and image are made to unite; (it being the same in effect, whe*- ther the stand be moved toward the object, or line joining the objects, or the object toward the stand) ; and then the qua^ drant must be turned half round to its first position, and the images brought half way together by turning the horizon glass and united as before: this- to be repeated atevery semirevolu- tion, so often as necessary, till the adjustment of the horizon glass is perfected'.
When the objects C &cc are very distant, a small removal of the quadrant to the right or left of a line joining the objects^
* Whether the eye, which- is itself a telescope, and with a large aperture, ever re- quires a correction of this sort, when it looks through a sight vane,, is not questioned ; nor whether it views any thing obliquely ; i. e. whether its axis be always the axis of its vision ; but enough is said here to shew the errors that may arise in some cases, from looking through an eytj hole or vane of too great magnitude; and these errors would not be corrected by using, a telescope, unless Mr. Hadley's correction, (in hia- ith corollary,) were applied.
16
will make no sensible difference in the angle of incidence and reflection of the rays, nor consequently alter the place of the images, as would be the case if the objects were near.
If the quadrant, instead of being turned half round from the position A F Q to ap q^ were to be so inverted, that tlie in- dex and horizon glasses A and B should be placed on the lines E c, C e, the adjustment could not be made, unless the ob- jects were so remote, that the interval between the glasses would make an insensible angle at either of the objects, and that any little motion on either side of a line joining the objects, which might accidentally be given in reversing the quadrant, would cause likewise only an imperceptible divari- cation of the images. For if the quadrant were to be turned upside down, and so that the centres of the mirror would fall on the lines C c, E c as before, the centre A on D, and B on F ; then the angle of incidence of a ray falling from C on Dy would be different from that of a ray from C on A ; it would therefore not be reflected to F ; so that it would be necessary to turn the instrument in its plane, in order to make the image of C be visible in the horizon glass ; by which the glasses in the 2d. position would not be parallel to them- selves as they were in the 1st. nor is there any certain posi- tion in which they could be placed, as this will depend on the distance of the objects. So that the horizon glass cannot be adjusted by reversing the face of the octant, unless the objects by which this is to be done, are so far removed, that the distance between the glasses subtends at them an imper-
17
ceptible angle'; which Mr. Ludlam says will be, when thej are removed at least half a mile oft": * and for the same rea- sons, the adjustment cannot be made by the observer's turn-; ing himself half round with the instrument, without revers- ing it, unless the objects arc at a distance as great as this, if it be not fixed on the same points, as above directed ; by which alone the parallelism of the glasses is preserved, and also the same incidences and reflections, which are only ex- changed one for the other by a half turn of the instrument ; so that when the horizon glass is rightly adjusted, the direct and reflected images are reciprocally visible and coincident.
By this mode of adjusting the back horizon glass, by placing the quadrant on two fixed points between two near objects, a contrivance is made practicable, of using with full advan- tage the excellent method proposed by Mr. Blair of adjust- ing it at all times, by placing it parallel to a reflecting plane perpendicular to the index glass : for ascertaining which per- pendicularity, the above mode of adjustment is necessary; as Avithout knowing and making allowance for any deviation from it, in all observations taken, they would all be erroneous ; wdiich circumstance, as also this adjustment being the test of the accuracy of the addition, which I am to propose to the furniture of the quadrant, is the reason why 1 have been so diffuse in the explanation of this method.
The reflecting plane Mr. Blair proposed to be formed of
t This depends on the magnifying power of the telescope, and the snjallness of the ang. it will render disccrnable.
VOL. XII. D
18
the lower edge of the index glass itself, by grinding and po- lishing this edge perpendicular to the plane of the glass. The adjustment would be thus rendered admirably easy and certain, if the edge of the glass be formed perfectly plane and truly at right angles to it's face ; were it not that this edge is neces- sarily ,so narrow, as not to afford a sufficient field of view to the observer, for distinguishing the object by which the ad- justment is to be made : for the rays fall on the edge of the mirror so obliquely (making an angle with the plane of the edge, of no more than about 21 or 22 degrees, and forming oii the back horizon glass an image equal in breadth, on its oblique surface, to the edge), that if the index glass were so great as half an inch in thickness, its edge would subtend at the eye near the horizon glass an optic angle of about 85 mi- nutes ; and if its thickness be, as usual, 1th of an inch, it would take in a field of only about 20 minutes; which is too small to distinguish with ease the terrestrial objects to be viewed, though it would serve with difficulty for adjustment by the contact of the edges of the direct and reflected imaires of the sun or moon : this however it would do with all facility, if the thicknesses of the index and horizon glasses were such, and so proportioned to each other, that the image of the former might be reflected from the fore and back surfaces of the back horizon glass, single, so as to form one image of double breadth, by the double reflection : for which purpose the buck horizon glass must be very thin, and the index glass too thick; as otherwise the image from the under surface of
19
the former would emerge at a distance from that reflected from its face ; and the interval would to the naked eye ap- pear like a shaded list, preventing the contact of the images observed from being seen by the double reflection, and con- fining the field to one of the images emerging from one sur- face of the glass ; which will be as contracted as above stated.* However, as it will always be easier and more
* This will readily ai)pear on inspection of Fig. 3 : in which A is the index glass, and B the back horizon glass, placed at right angles to each other ; each glass being Jth ^ of an inch in thickness : on which a beam of light a b, proceeding from a remote object S, is incident on the edge of the mirror A, in an angle with the plane of the edge of about 22 degrees, being the complement of the angle of incidence on the same ; which in the quadrant is generally about at least 68 degrees: from which edge it is reflected to the glass B, and reflected again'from both surfaces of the same; the extreme rays a and h of the beam of light, being throughout its progress, distinguished by the same letters ; and those reflected from the back surface marked a 2, and h 2 : their course (as the fig. itself will shew), is traced with sufficient exactness; from which it appears, that the beam of light a b, contracted by reflection from the mirror A to the li^Xh part of an inch in breadth, preserves the same dimension till it enters the eye ; both in the beam *■, reflected from the anterior surface of the glass B, and in the beam 2 reflected from its back surface: for though this latter is diff"used when it has penetrated the surface of the glass, it is again contracted on emerging from it ; and is, as reflected from both sur- faces, become a double and divided beam, the interval between both its parts being al- most the thickness of each of them, which is equal to the sine of 22 degrees to a radius jth of an inch : and if the thickness of the index glass were to that of the horizon glass, as the sine of the refraction of the rays to its cosine, the interval between the beams would be equal to the breadth of either. To fill up the vacuity of the reflected light in this interval, by making the beams x and z issue contiguous, the thickness of the index glass must be to tliat of the horizon glass, as double the sine of refraction, to the cosine : tliis may be made evident as follows.
Let tiie beam of light a b (fig. 4.), be reflected from the edge of the index glass A t»
D 2
20
pleasant to adjust by Mr. Blair's method, when the eyo takes in a sufficient field of view ; and moreover as not every where a quadrant can be procured, furnished with an index
tlie horizon glass D, iu tlie same manner, and with the same incidences, refractions and reflections as in fig. 3 ; on the mirror C it will occupy a space / /, equal to the breadth of the edge of the mirror A ; and will cover the equal space It r, on the back surface of the mirror B ; after reflection from which, it will be refracted in the surface Ik, enier,r;ing in the beam z; the several rays iu this beam issuing at distances from / toward k, equal to the distances of their first incidence from i toward 1; the last ray b i emerging coincident with the ray I a: so that if the beam x did not fill the space i I, the beam s would not fill the space k I, but would leave an interval next to /equal to the deficiency toward i. Let the line p r be drawn perpendicular to the mirror, bisecting the line / 1, and the angle of incidence and reflection / r i, and parallel to c # the cosine of the angle of re- fraction n I s, which angle is equal to c s I. In the similar triangles s c I, r p i, the side r p, the thickness of the mirror B, is to p i half the thickness of the mirror ^, as « c the cosine of the angle of refraction, to c i or s n the sine of the same ; so that when the thick- ness of the mirror B, is p r the cosine of refraction, the thickness of the mirror A must be double of p i the sine of the same angle. Now to make the index mirror of so great thickness may produce a small inaccuracy, when angular distances are to be taken bc- ,tween very near objects, at which a small part of the length of this mirror would subtend a perceptible angle; for the thicker the glass is, and the greater the complement of the angle observed, the greater intervals on its surface will there be between the places of iucidence and emergence of the rays forming the reflected images; which will therefore be seen, sometimes by rays issuing from the middle of this mirror, and sometimes by rays distant from the same : from which variation I have above stated the errors that may arise: and because every minutia in the construction or use of this admirable instrument is deserving attention, it may be worth while to shew the manner in which this hap- pens.
Let I G (fig. .5.) be the index glass, in its position when the index is at o, and //the ttorizon glass at right angles to it ; its adjustment being made by the reflected image of an object .V, seen by the eye at E to coincide with another opposite object visible in the direction E //. The image of S is conveyed to the eye by the ray S A refracted in A C,
21
mirror, whose edge is grouiid accaratel}' at right angles to its plane, and the edge also set up perpendicular to theplaneof the instrument; (for which the purchaser must generally rely on
reflected from C to B, thence refracterl again in B 11, and reflected by the mirror 7/' in // E parallel to S A. Since the mirror //, and the axis of vision E H are fixed, the ray a fj K also fixed, in all observations taken; and every object must be seen by rays nl- timately coincident with B H. Suppose it be required to find the angulaV distance of anotlier object s, from the object seen directly in the line E H; and that for this pur- pose, and to make the image of s appear in E H, the index is moved to the position " i g, through half the angular distance SAs of the objects, (the lines SA and EH being supposed the same, and the interval AH to be accounted for) ; then the inwe will be seen by the ray s a, inflected, as before traced, in the lines a c, c b, h H, and HE', and the thickness of the index glass being moderate, there will be an interval between the place of incidence on it of the rays S A and* a, so small as to be imper- ceptible, and to occasion no error. But if the thickness of this mirror were great, as AD ox ad, and the rays to be reflected from D and d; the image of S would be visible by the ray J? j», proceeding in ;> D, D B, B H, and f/\E ; and the image of s by the ray t e, e d, d b, b H, H E. So tlrat when the adjustment was made, by the ray Rj) incident at p ; the object s wouhl afterward be seen, and the angle s A S measured, by the r>y t e, incident at e, considerably distant from p. If the object s was very remote, the rays saand fe would be as it were parallel, and their incidences and course the same ; but if the object s were near, as at r, then the incidences would diflcr, and the error of observation be equal to tlie angle era, so much the greater, as the object is nearer, or ■ as the compIeme«it of the aDgu^dr distance observed is greater: and the same will be the caacin the fore as well as in the back observation; which latter may be made as true as the former, if the line of direction of the sight be accurately fixed, by a long eye-tube, or telescope rightly placed, and if the other requisites above mentioned be observed.
Thus though in observing remote objects, and for naulical uses, no iHconvenience will arise from the thickness of tlieindex glass ; (which if it be duly proportioned, as here stated, to that of the horizon glass, and its edge truly formed, is doubtless the best and surest mechanical organ for adjusting the latter); and though no error can hence arise in performing the above described adjustment ; wherein the position of tiie index glass
22
the maker ; and few artists can be furnished with the exqui- site apparatus, which must be employed for effecting tiic former); I think it may be to many desirable to have an
to the object is not changed, nor consequently the incidences of the rays on it ; yet in observing very near objects, as the height or angular distances of buildings, offsets in surveys, bearings^ &c. a .great thickness of the index glass will produce a variable error, which though trifling, is unsatisfactory in an instrument, whose general excellence would make one wish it to be exempt from even the smallest imperfection. And this error can be diminished only by choosing such a certain position for the index glass with respect to its centre of motion, as would cause a part of the field of view to be lost in measuring angles but little exceeding 90°, when the rays fall very obliquely on the index glass, and when also the error eucreases, as does the complement of the observed angle to 180 degrees. For the point e (in fig. 5.), can be made to approach to the point j», only as the triangle e d h, which is of given dimensions, shifts toward the mirror H, by its angle b advancing toward it in the line h H, the triangle being moved parallel to itself; by w hich the point h would fall beyond the end of the mirror i g at g, and the field would be contracted. But the face of the mirror et, and consequently the triangle e b d, will be elevated, by advancing toward H, more or less, as the centre of motion of the in- dex is placed farther from, or nearer to, the line p B the face of the index glass : so that the point of incidence of the ray t e cannot fall nearer to that of the ray R p, with- out causing a part of the field to be lost, and this where it is most contracted.
It has been made evident here, that if the thickness of die index and horizon glasses be equal, or as formerly in use, there will be an interval between the beams of light re- flected from the opposite surfaces of the Jatfer ; and in this interval the reflected image is not visible to the observer; who can only see there the object directly through the glass: and if he is to view both images coincident, he can only do so in the space filled by the beams ; as in x or 3 fig. 3 ; for if he attempted to make the extremities of the images to coincide at the internal edge of either of the reflected beams, he could not hold the quadrant steady enough to keep them there ; for which purpose it would require to be absolutely immoveable.
On these accounts, if advantage is to be taken of the double reflection, (without which the narrowness of either beam of light, and the evanescence of the reflected
2S
easy contrivance to be substituted, where required, instead of this operation on the edge of the mirror ; and which can be executed with httle additional labour by any instrument- maker ; so as to afford a sufficient field of view, with a ca- pacity for accurate adjustment.
This I have effected by the contrivance of a second small index mirror; requiring only one plane surface, and fixed on the index at right angles to the great mirror ; being totally free and detached from the index mirror, and capable of every adjustment for itself, without interfering with, or impeding any motion requisite for that purpose for the index glass, or altering its position. The following description of such a one, Avhich I have made, will shew that it is a very simple ajid easily fabricated addition to the quadrant.
image in their interval of separation, will make the observation witli the naked eye un- certain and troublesome) ; it is neccsary that the light from both surfaces of the glass, should be contiguous, having no interval ; which can only be effected, either by making the index glass almost one quarter of an inch thick, or by reducing the thickness of the horizon glass to less than t'jth of an inch; or by such a mutual compensation of both, as would still leave one as much too thick as the other would be too thin, for the uses above stated: and though this may be remedied, while yet the glasses remain of thcit- due and proper dimensions, by using a telescope, whose aperture is large enough to take in botti beams of the reflected light v/itlt the interval of their separation ; yet in ordinary quadrants, of simpler construction and more moderate price, not designed to be furnished with a telescope, or mirror with a polished edge, I cannot but think that an easy and cheap substitution for both, would, if found to answer, be very useful ; as securing at all times the advantages of Mr. Blair's invention for the back observation, (at least for taking altitudes), to those navigators, who do not furnish themselves with a more perfect and expensive instrument ; as well as to those who on land desire, in sur- veys, to ascertain large angular distances by the quadrant ; or by an artificial horizoa
24
The ichnographical plan and position of both the mirrors is represented in fig. u. as they are fixed on the head of the index.
jB is the great mirror, and b is the cock supporting it, with its case ; D the wing or adjusting lever of the cock ; /and g the screws for erecting it perpendicular to the plane of the instrument in the usual manner; and e e are the steady pins in the index fastening the cock ; d d are two pins on which the edge of the mirror rests.
A is a round brass plate, with a milled edge of the same size with the head of the index, and with the fig, both being three inches in diameter, and screwed fast to it, concentric with the index, by the four screws s ss s screwed into the in- dex. C is the little mirror screwed to the plate by the screw h passing through the Aving IL of its cock c : it is erected and fastened perpendicular by means of the screws h and i, in the same manner as the mirror B is by the screws y and g: m is a steady pin fastened in the cock c, inserted into a hole in the plate A ', and k another strong steady pin rivetted in the plate, the upper part of which, being cylindrical passes up through a hole in the strong bar or wing E of the cock c, which hole it exactly fills, but allowing the cock to be elevated or depressed a little for adjustment of the mirror, without any angular mo-
to take altitudes or angles exceeding 45 degrees, to find the latitude, &c., for whicli the back horizon glass must be used. It is also desirable for the interests of science and navigation, that quadrauts of snfRcieot performance should be made capable of being fabricated in different places.
25
tion about the centre of the index. Thus by the screws A and if and the steady pins k and m, the little mirror is made erect arid fixed on the plate A : it is also set at right angs. to the index mirror B, by loosening a little the screws s s s s, and turn- ing the whole plate A by its milled edge, round its centre on either side, so far as necessary ; and when this is found to be accurately effected, the screws sss-s- are to be again made fast; when the little mirror will be perpendicular both to the plane of the instrument, and also to that of the great mirrors and cannot,^ without suffering violence, alter its position..
This circular motion of the plate A, and of the little mirror fastened to it, is permitted, without communicating any mo- tion or even contact of it, to the index-glass, its cock, steady pins, or screws, by the following contrivance.
Through the plate A are cut long holes or slrts, formed as represented in the fig. concentric with it, at the places of the screws s s s s, f, and g, and also at the pins d d and e e . these slits are made just so wide, as that the screws and pins will not touch them, and that the heads of the screws will, when screwed down, press upon the edges of the slits : the slits at e g are not represented in the fig. to avoid confu- sion : the slits at s and g should be so long as to allow the plate A to turn through the space about -~\.\i of an inch on each side of the screws fixed erect; and the slits atdd, e e^ and f, may be shorter, according as they lie nearer the cen-r tre, each slit bounded within the same sector of a circle.,
VOL. XII^ £
20
Through these sHts in the plate A, the screws s s s s, f, and ^ g are inserted, and all except g fastened in the head of the index i in which latter, the pins e e penetrate also through the wing D of the cock of the index-glass, to steady it.
^ The edge of the index-glass D rests on the pins d d, which project only so ftir above the surface of the plate J, as to keep the nHr^ and the wing D of its cock clear of it, so as that the plate can turn about under both without touching tliem : and the bar or wing E of the cock c lies about |th of an inch above the wing D, so as to be quite clear of it, and permit adjustment by raising or depressing the wing. It may be supposed that the cock c and its wing E must at first be so formed that the mirror C, when fastened to them, will be nearly at right angles to the mirror J3, when the bar E lies parallel to the index mirror, and the screws, &c. are in the middle of the slits in the plate A ; that a small metion of the plate A on either side, will suffice for an exact adjustment of the mirror C,
There is a round hole in the middle of the plate A, a good deal wider than the head of the pin, about which the index turns ; and the plate is made to turn concentric with the pin by a little ring or sock'.'t R, brazed in the hole in the plate, or by an annular ledge formed on it projecting downward below the under surface of the plate about TTth part of an inch. The outside of this projecting ring is to be exactly fitted into a circular groove or cavity formed in the index, so
27
far distant from the head of the pin, that the ring will not touch it, nor affect its motion or position.
Tlie cock c and its cap are formed with an indenture in. them, (as in the fig.) at the end of the index-gUiss, in order that the mirror C may He nearer to it, which will allow the little mirror to be made broader, Avithout enlarging the brass plate A and the head of the index ; while the part of the cock not indented may, as well as its wing -E, be mad^ so massive and strong, as not to be bent and strained easily by any accident : a notch is cut in the aide of the bar E at the screw/, to allow this screw to be turned, without touching the bar: thus both the mirrors may be adjusted independent on each other.
The little mirror C requires not to be silvered on the back, and consequently its opposite surfaces need not be parallel, so that it may be made of a piece of well polished and plane looking glass, but the polish must be taken away from its back surface by grinding it on a plate with fine emery and water ; and the surface thus made rough should be smeared over with a feather dipt in oil of turpentine mixed with lampblack, to prevent all reflection from that surface.
The addition of this mirror to the index adds no trouble to the business of fixing the index-glass: the extra work re- quired is that of the little mirror and of the plate d ; and the fabrication of this plate will be greatly facilitated, if it be cast from a model, in which the slits and perforations, and
E 2
28
the central annular ledge, (the former to be of a size a little less than requisite in the. plate when finished,) are already made in their proper places. The plates cast from this model will require no measurement nor piercing, and only want to be filed, turned and polished.
The mirror C is to be adjusted by making it, by a fore ob- servation, parallel to the back horizon glass, by turning with the left hand the plate A by its milled edge, till the images of the object viewed are seen to coincide ; and then fastening the plate A to the index by the screws s s s 5 : it follows, that the back horizon glass must itself be for this purpose previ- ously adjusted at right angles to the index-glass, in the way before mentioned, Avhich may always on land be easily per- formed. And I should imagine that even atsea, when the sea is calm, it might perhaps be practicable, by fixing up two papers, with a black line drawn vertically on each, on the adja- cent sides of two masts of the ship ; by which lines, the qua- drant fixed on two points of a moveable stand, placed on deck between the two masts, may have the back horizon glass ad- justed as above, in order by this means to try at any time whe- ther the little mirror C has its adjustment altered i which, however, must be very unlikely to happen, especially if the contiguous sides of the plate A and of the index be not po- lished, lest the plate should slide on the index. At other times the back horizon glass is to be adjusted by the mirror C, supposed to be itself right in position ; and indeed considering that this mirror is not, as the horizon glasses are, rested on the
39
points of two pins like a lever, and that these glasses are moved and secured, not by the outer edges of the circular plates of their frames, but by the small axes of those plates, very near the centres of their motion ; whereas the mirror C is fastened firmly on a broad plate, screwed tight near its margin by 4 screws to the head of the index, it is not easy to conceive how it can alter its adjustment ; though from the above mode commonly in use for fixing the horizon glasses, it is not unaccountable why they should frequently do so, not being fastened by the margins of their frames. These frames, however, are sometimes moved by endless screws playing ia their ratched edges ; and this construction when well executed, is much better than the former.
Read 21st May, 1810.
Two Proofs of the BmOMIAL THEOREM, hj the REV. SAMUEL VINCEy A.M.F.R.S. Plumian Professor of Astronomy and experimental Philosophy, in the University of Cambridge
Eead May, 1810
When n is a whole positive number, it is proved bjr common algebra, that
(Ifo;) " = 1 + w* + n. -==ii?^ + - - -n. "-=i "idtlx'+ &c.
Now if this be not true when w is a fraction, let the general co-efficient be C + w. ~- "-^^ .r'. Then the quan- tity C must vanish when w=l,2, oo. Now as C is
expressed in terms of w and given co-efficients, it must alv^ays be of the same form whatever n is, and, as it must vanish Avhen w— I, 2, oo, it must be represented by ^^ ^ n ix n 2
X n — 00—^^(7/ — an~\+ &c.) where r is infinite j this
therefore must be the value of C. But when n is a fraction, this value of C becomes infinite, which it cannot be, and as no other value of C can enter in addition, but this, the gene-
ral value of the co-efficient of x can be no other than
n. n — 1 n — r-\- X
3
ITi-EM ALITER.
Let n and s be indefinitely great, whole, positive numbers, 80 that -7- may represent any fraction ; then by common algebra (i + ^)-= i+nsx + ns^-f^'+ScciP')
{i + x)" = J+nx 4. n.^x^+ Stc. (P).
Now it is proved in my Fluxions, that (i+a)— may be represented by a series of the form
1 + cf.r 4. ijr + &c. (P)— , where the form of the series in respect to x is the same as that of the above series ; we have therefore only to consider what is the relation of the corres- ponding co-efficients. Now the series (P') and (P) are ex- actly of the same form in every respect, the factor ns in the former being represented by n in the latter. If therefore we perform the same operation on these two series, the re- sults must have the same form, and whatever change maj'
take place on ns in (P'), the same must take place on ji in
(P). If therefore we extract the s Root of (P') and (P), the forms of the two series expressing the roots must be the same, and the roots be deduced by the same rule. Now the
33
reiluction of ( p') to (P) is made by writing for the quantity jis in (P') that quantity divided by the root s to be ex- tracted, or writing 71 for ns ; the reduction is therefore made simply by the root s to be extracted, dividing ns by s, and
writing the quotient for m; hence we extract the s root of (P) by the same rule, that is, by writing for n in (PJ, n divided by s ; hence
n
It matters not liow the s" root of the series of the form 1 + ax -\- bx^+ &c. can be extracted, or whether we should have been able to accomphsh it if we had not known that the series (P*) and {p) are represented by {i+cc)"' and (i+.r)". By whatever process the / root of (P') is extracted, whether discoverable or not, by the same process the s" root of (P) will be extracted. The Binomial Theorem shews the series
(P) to be the s" root of (P'), which is all we want to ascer- tain.
VOL. XII.
On certain Properties of Numbers, by the REV. SAMUEL VINCE, A M. F. R. S. and Plumian Professor of Astro- nomy, in the University of Cambridge. An extract of a letter to the Rev J. Brinkky, D. D. F. R. S. M. R. I. A. and An- drews' Professor of Astronomy in the University of Dublin.
Ramsgate, June 26, 1810.
EuLER in his Introductio in Analysim Infnitorum, in the chapter de partitione Numerorum, has shown, that bj a com- bination of the numbers in each of the Geometric Series 1, 2, 4, 8, &c. and 1, 3, 9, 27, &c. all the natural numbers 1, 2, 3, 4, &c. may be formed, as far as the sum of each series goes. This he has proved, from assuming the products of an indefinite
8 4 8
number of factors(l+jr )(l + a' )(1+^ )^l + x ) &c. in the
first instance, and( 0? +1+.T ){^ t + I +x j(^ x +1+0:' ^&c. in the second; shewing that in each case, such products may
8 3
be represented by a series containing the terms i +x+x +x
4
+x + &c. the indices of which must necessarily arise from the combination of the indices in the assumed factors. But
35 ^
the property now stated may be otherwise proved in a very simple manner immediately from the expression for the sum of each series, I have also added the rules for filling up the intervals of the terms, which Euler has not given ; and shew- ed under what circumstances, other series will have the same property.
First, for the seines 1, 2, 4, 8, &c.
The sum of 1+2+4+8+ 2 ""^ "~l-S; hence,
8+1=2 the next term. The difference, then, between the
sum 5' of w terms, and the next term 2," is 1 ; therefore the sum S of n lerms, carries on within 1 of the next term. If therefore you can for n terms, make up all the natura,! num- bers to their sum, you make them all up to the number next
less than the next term S ; and by adding all those numbers
K
to 2 , you get all the numbers to the number next less than «+i
2 . If therefore the rule be true for n terms, it must be
true for n+l terms. Now if we take two terms 1, 2, we get 1+2=3, that is, we get all the numbers as far as the sum of the two numbers, and within 1 of the next term. But, as pro- ved above, if the rule be true for 2 terms, it must be true for
3 terms; if true for 3 terms, it must be true for 4 terms ; and so on; hence, the rule is true in general.
56
Secondly, for the Series 1 , 3, 9, 27, &c.
The sum of 1+3 + 9+27+ s'~L^zzS ; hence,
tl
25+1=3 the next term. The difference, then, between the sum 5 of w terms and the next term 3 is S+i ; therefore the sun»
S subtracted from the next term 3", leaves 5+ 1] that is, it brings jou back to the number next greater than the sum S. If therefore j-ou can for 7i terms make up all the numbers to
S, the same numbers subtracted from the n+i" term will bring you back to S+i, the number next greater than ■S';
thus you fill up all the numbers in the interval between the n term and the w + l' term ; and if the same numbers be added to the w-|-l" term, you make up all the numbers as far as the sum of w+1 terms ; if therefore the rule be true for n terms, it must be true for n+i terms. Now if we take two terms 1, 3, we have 3 — 1=2, 3+1=4, and 4 subtracted from the next term 9> leaves 5 the next number greater than the sum , of two terms. But, as proved above, if the rule be true for 2 terms, it must be true for 3 terms ; if true for 3 terms, it must be true for 4 terms ; and so on ; hence, the rule is true in general.
The intervals of the first series may be filled up by the following Rule.
o
Let A be any number, and 2" the term next less than A. Take ^^ next less than A—2"; 2* next less than ^—2"— 2';
37
2* next less than A — 2" — 2*^ — 2 , and so on till there be no
remainder; and-then 2"+2'"+S*-f2'+ &c. = A.
In the second series, all the numbers in the general interval
from 3— 3"~i 3"^! &c. i to 3"+3""'+3'""' + &c.
+1, including those terms, may be made up by the
following Rule.
After 3" for the j^rs^ term put — 3"'"' for 3""' times, then
cyphers as often, and then + 3°'"' as often.
For the second term put — 3""" for 3"~ times, then cyphers
as often, and then +3""^ as often ; this to be continued three times.
For the tJiird term put — S""" for 3" ' times, then cyphers
as often, and then 4-3""' as often ; this to be continued nine times.
For the fourth term put — 3"~ for 3"~ times, then cyphers
as often, and then +3"~ as often; this to be continued ifz2?en;y- seven times.
In general, for the r ' term put — 3"""^ for 3"~' times, then
cyphers as often, and then +3""^ as often ; this to be con-
r— 1
tinned 3 times.
Proceed thus through all the terms, and you will fill up all
the numbers.
But besides these two series, there are many others which
have the same property ; of these, the two first terms must
58
necessarily be 1,2, or 1, S, or the interval between the two first terms cannot be filled up. The series must also have this further property, that the sum preceding any term (P) must reach at least half way from (P) to the next term (Q), or to (Q — i). The following series have this property.
1, 2, o, 10, 17, &c.
1, 2, 7 17, 33, &c.
1, 3, 6, 10, 15, &c.
1, 3, 9, 19, 33, &c. and many others ; but the series which requires the smallest number of terms to fill up the interval from 1 to any given number, is, 1, S, 9, 27, 81, &c.
An Account of a -oery remarkable WATER SPOUT, which ap'- peared at Ramsgate, July l6, 1810, a little before 3 o'clock in the afternoon, just after a Thunder Storm; by the REV. S. VINCE, A.M.F. R. S. Plumian Professor of Astronomy and Experimental Philosophy at Cambridge.
IN the annexed figure, L M N" represents a cloud, in which there first appeared a figure in the form jP G, resembling an huge serpent; this immediately stretched itself out in an hori- zontal direction AM B; at J5 it turned at right angles down- ward in the direction J5 C to the sea D E, the sea immedi- ately under it rising up in a cylindrical form v w x y io meet it. The horizontal part, (which was straight), I judged to be about 3 or 400 yards long, and the perpendicular part B C in the proportion now represented, the greatest diameter of which I estimated to be about o or 6 feet. It was attended with an hissing noise, and continued about 5 minutes, when it almost instantaneously disappeared, every part of it at the same time dissolving as it were into air, the water in the sea then ceasing to rise up. Water Spouts are an electrical phe- nomenon, lightning being sometimes seen to play in them» Perhaps this, which appears to be of a very singular form (for I have never seen such a one described), maj be thus account- ed for. If the cloud L M N, and the air at B were charged
40
with different powers, the spout might take the horizontal di- rection MB; and if the air at B, and the sea immediately under it were also charged with different poAvers, the spout might take a perpendicular direction downward, and the sea rise up to meet it. The spout could not be water in its li- quid state, for water in that state projected from the cloud, must necessarily have descended in a curve; and further, had it been water in that state, whpn the supply from the cloud ceased, from the ceasing of the cause, it would have disap- peared gradually from the cloud, shortening till it vanished at the sea; whereas it vanished altogether almost instantane- ously. From all the circumstances attending the spout, it ap- pears that it was nothing but part of the clouddrawnout in a very condensed state, for although the cloud was very black, the spout was much blacker, the part in the cloud appearing very distinctly in the cloud itself. On this supposition we may account for the sudden disap- pearance of the spout; since, by the operation of the elec- tric power, the watery vapour might be resolved into its two constituent airs, and thus disappear almost in an instant. All water spouts, as they are produced by the same cause, Ave may conclude to be of the same nature, that is, a very condensed watery vapour. They have, perhaps, been con- sidered as water, from the torrents of rain which frequently attend them, so as to render it difficult to distinguish that from the spout; and also from the rising up of the sea where they fall, the effect being such as might arise from the falling of such abody of water as the spout has been supposed to be.
Re/rac?u)n H
Jn Account of Observations made at theObsevvatory of TRINITY" COLLEGE, DUBLIN, with an Astronomical Circle, eight feet in diameter, which appear to point out an annual parallax in certain fixed stars.
Also a Catalogue of North Polar distances of forty-seven prin- cipal fixed stars, from recent observations, and a compariso*i thereof with those of the same stars, obtained by other instru- ments, and by the same instrument, at a former period. By JOHN BRINKLEY, D. D. M. R. 1. A. F. R. S. and ANDREWS' Professor of Astronomy, in the University of Dublin.
Read May 9, 1 814.
To prove the motion of the earth about the sun, bj actual observation of change of distance from some of the fixed stars, at different times of the year, has long been an object of research. Soon after the Copernican System be* came generally adopted, and while Astronomy was yet in an imperfect state, tliis was considered in some measure ne- cessary to establish the truth of that system. Afterwards the discoveries in physical astronomy made this enquiry, as far as the above motive was concerned, less interesting.
YOL. XII. ft
34
Modern astronomers have looked to this object principally with a view of ascertaining whether any apparent annual motion of the fixed stars, from this cause, existed necessary to be noticed, in computing the mean place from the observed.
Dr. Bradley, by his celebrated observations, which led him to the discovery of the aberration of light, first esta- blished that as to certain stars no parallax existed capable of being noticed. His observations were made with an in- strument, that, for observations near the zenith, has not since been surpassed.
Since his time it seems to have been generally allowed, that the annual parallax of every fixed star was too small to be noticed, till lately M. Piazzi, of Palermo, conceived that his observations pointed out a parallax in certain stars. An account of his conclusions is given in the Conn, des Temps, 1808, together with an account of some observa- tions made at Rome on a Lyrs.
My observations, by the eight feet circle, which com- menced in 1808, have pointed out also a parallax in a Lyra, but considerably less than that observed by M. Piazzi. It is only with respect to this star and Arcturus that our conclu- sions agree in pointing out a parallax.* My observations
* I can only refer to the account of M. Piazzi's observations given in the Con. des Temps. 1808, p. 4'32. In which it is mentioned that the observations themselves are to be found in the 10th vol. of the Italian Society. By the account in the Con. des Temps, it appears that M. Piazzi observed in Procyon, a parallax of declination, such
35
tend to point out a parallax in a. Lyroe, a, Aquilae, Arcturus, « Cygni, and « Ophiuchi, and some others. M. Piazzi considers c& Aquilas as having no discernible parallax, whereas my ob- servations tend to point out that a Aquilae has a greater parallax than any other star that I have observed. Besides this discordance between the results of my observations and those of M. Piazzi, it is to be noticed that other results ob- tained by instruments executed by the first artist, and by observers justly celebrated, do not accord with mine in point- ing out a parallax. Itis therefore with great diffidence that I offer my results to the Academy. These results tend to prove that the parallax (the angle subtended at the star by the diameter of the earth's orbit) of a Ljtsb, by 152 ob- servations amounts to 2" ; of » Aquil<E, by 96 observations = 5", 5 ; of Arcturus, by 92 observations = 2", 2 ; of « Cyg-
as would result from a double parallax of 20" whereas by my observations I find no indication of parallax in this star. Indeed from tlie account given in the Con. des Temps, by M. Delambre, the results of M. Piazzi, as to parallax, cannot be consi- dered as entitled to much confidence, and in consequence M. Delambre may appear justified in making the following remark.
" Malgre les efforts dont nous venons rendre corapte, il ne parait pas que nous " ayons encore rien de bien certain sur la distance des etoiles. Cette connaisance est « peut etre du norabre de celles qui nous seront toujours refusees."
No one should consider this as reflecting any censure on the observations of a person who has rendered such excellent services to astronomy. It should only be understood that the instrument used by M. Piazzi was inadequate to the research.
36
ni, by 47 observations = 2'', i.* The results as to other stars I shall reserve till my observations have been more numerous.
I shall endeavour to make such remarks respecting these results, that the Academy may form a proper judgment how far they serve to- establish the important conclusion, which I conceive may be derived from them.
For this purpose it is necessary to give some account of -the instrument by which the observations have been made.
The superiority of circles over quadrants, and the advan- tages derived from the micrometer microscope, are so well known, as to make it quite unnecessary, that they should be -Stated here.
About the year 1788, soon after the late celebrated artist Mr. Ramsden had strenuously recommended those improve- ments, which his mechanical skill had rendered so very prac- ticable, the Provost and Senior Fellows of Trinity College, Dublin, by the advice of my predecessor, Dr. Usher, di-
* Perhaps it may be objected to me that what I call parallax, should be called double parallax, since properly the parallax is the diflference between the observed and mean place. In this way the
parallax of a, Lyrx = 1",0 of a, Aquilae = 2,7 of Arcturus = 1,1 of a Cygni = 1,0
But I believe that, generally, when the parallax of a fixed star is spoken of, the greatest change of place is intended, which is equal to the angle subtended by the dia- meter of the earth's orbit at a fixed star.
^ 37
reeled a circle, ten feet in diameter, to be made for this ob- servatory. Mr. Ramsden protracted for many years the exe- cution of the instrument. After beginning one of 10 feet diameter, he afterwards rejected it for one of 9 feet, which was actually divided. The latter he also rejected, and at his death he left unfinished our present instrument of 8 feet dia- meter. This was finished by his successor, Mr. Berge, and placed in the observatory, about the middle of 1808. The long period which elapsed, while the instrument was ex- pected, will be always a subject of regret to myself. On some future occasion I may, perhaps, lay a detailed account of this instrument before the Academy. At present I shall on-ly mention such particulars, as may be necessary to render intelligible the method of making observations with it, and the degree of accuracy to be expected from it.
The circle is supported in a frame, which frame turns on a vertical axis. The upper part of the frame is of cast iron, turns in a collar, and is connected with the lower part of the frame by four hollow brass cylindrical pillars. The lower part of the frame, which is also of cast iron, terminates in a pivot of steel, which turns in a socket of bell-metal. This socket is moveable south and north, by one screw, and east and west, by another, for the purpose of adjusting the vertical axis.
The axis cf the circle, a double cone four feet in length, is supported an Ys which are themselves supported by strong bars of brass attached to the cylindrical pillars. The pres- sure of the weight of the circle and its axis is relieved by an
38
ingenious application of friction wheels and the lever. There is also an ingenious contrivance for adjusting the axis hori- zontal.
The circle of brass is divided into intervals of 5 minutes, which intervals are subdivided by micrometer microscopes into seconds and parts of a second as usual.
There are three microscopes. One called the bottom mi- croscope, opposite the lowest part of the circle : a second opposite the left extremity of the horizontal diameter, and a third opposite the right extremity of the horizontal dia- meter.
The frame carrying the circle turns on the vertical axis with the greatest steadiness. The circle also turns on the horizontal axis with equal steadiness.
The vertical axis of the instrument is adjusted by a, plumb line. The plumb line which performs this adjust- ment is about 10 feet long, and is suspended from a point about 8 inclies from the centre of the top of the frame, and passes over a point below, 8 feet from the point of sus- pension. By help of this point which is moveable by a screw, and by the moveable socket below, the axis of the instrument is made vertical. The adjustment of this axis, as to the north and south positions is, it is evident, of the most es- sential consequence to the exactness of the zenith distance of the object observed. It is likewise evident that, from the great interval between the upper and lower parts of the in- strument, the temperatures above and below must occasion-
39
ally differ, and thence the relative positions of the point of suspension, and of the point below, be changed. To obviate this inconvenience, which would be fatal to the accuracy of the observations, the point of suspension is on a compound bar formed of bars of brass and steel, and the point below is also placed on a similar compound bar. By this the dis- tance of the plumb line from the vertical axis remains always the same. This contrivance appears to answer in a very satis- factory manner. *
The axis of the instrument being adjusted vertical, and the plane of the circle in the meridian, and facing the east ; let b, I, r, be the zenith distances of a star as shewn by the bottom, left and right hand microscopes respectively. AVhen the plane of the circle is in the meridian, and facing the west, let U, I', r\ be the zenith distances of the same star as shewn by the respective microscopes. Then the true zenith
distance = g or-j-V— ^ — I 3— A And the cor- rection of the mean of the three microscopes
The accuracy of the result of an observation is affected
* The circular instrument in plate 8 of Professor Vince's Practical Astronomy, may be referred to for our circular instrument. Except that in our instrument there is np azimuth circle. The plumb line is suspended in the position represented, but the com- pound bars are not represented. Also the pillar F is perforated for the insertion of the bottom microscope. The horizontal microscopes are not represented, and there is no microscope at n.
40
1. By the error of the mean of Uie six readings arising from inaccuracy in the divisions and error of eccentricity.
2. By inexact adjustment of tiie vertical axis. S. By error of reading off,
4. By error of bisection of the star.
5. By error from change of temperature affecting the parts of the instrument.
From my examination of this instrument, I have reason to conckide there is no sensible error of eccentricity, and that as far as the divisions aro concerned, the mean of the six readings can never occasion a greater error than 1", and an error of this amount will take place ordy in very few parts of the circle. A comparison of the results determined by the bottom, and by the two horizontal microscopes : also a com- parison of the corrections of the mean of the microscopes determined by stars at different distances from the zenith, seem to leave no doubt on this point.
It may appear an imperfection in this instrument that we cannot avail ourselves of a microscope at the highest part of the circie. The reading off a microscope so placed would be highly inconvenient, and on account of the circumstances of the instrument, the use of it might be attended with some danger, both to the observer and instrument. However it does not appear that the accuracy of the results would be materially affected by such an addition.
By means of the plumb line the vertical axis can in ge- neral be adjusted with great precision ; but this operation is
41
oftentimes very troublesome, and, it is to be feared, sometimes cannot be performed with the desired accurac3^ The plumb line passing near the plane of the circle*, no screen can be used. Hence even a slight agitation of the air occasions a slow motion in the plumb line ; and the observer, without much tedious precaution, may be deceived as to this essen- tial adjustment. From hence, doubtless, have arisen greater discordances in the observations than would have otherwise taken place. This imperfection arises principally from the situation of the plumb line, it would not be difficult to give it another situation in which it might be safely screened from the agitation of the air, and were the instrument Avithin a convenient distance from the maker, I should endeavour to have this alteration effected.
Notwithstanding this imperfection, I think I may safely pronounce that as far as errors of observation are concerned a-^mean of 10 observations (five the face of the circle being east, and five the face being west) will give the zenith dis- tance exact to much less than one second, and that a mean of 20 observations, according to a very high degree of probabi- lity cannot induce an error of nearly half a second as far as errors of observation are concerned.
* An adjvistment of the microscopes was intended by means of the plumb line, and four gold dots placed on the limb of the circle. This made it necessary that the plumb line should pass very near the plane of the circle. But the method of observing thence resultuig I found inferior both in accuracy and conveniencs to that in which the plumb line is only used for the adjustment of the vertical axis.
VOL. XII. 11
42 .
Ill errors of observation I include errors of adjustment in the vertical axis, errors of bisection of the star, errors of read- ing off, and also errors arising from changes of temperature in the instrument.
From what has been said, the principal circumstances rela- tive to the astronomical circle at our observatory will be readily comprehended.
The angle to be obtained by each observation is the exact zenith distance of the object, from which, the zenith distance of the pole having been previously determined, the polar dis- tance or declination of the object is known. The zenith dis- tance can only be had by the assistance of a plumb line or spirit level. The inaccuracies and inconveniencies to which both these instruments are liable have long been known, and as the zenith distance is not necessary for finding the polar distance, it has been, sometime ago, proposed to find the polar distance without a reference to the zenith point, by simply observing the arches of the meridian intercepted be- tween the object, the polar distance of which is required, and stars, the polar distances of which are known, or can be ob- tained by help of circumpolar stars.
Mr. Troughton, whose fame as an artist is justly so cele- brated, has made several circles with this view, and has lately made a mural circle of 6 feet in diameter for the Royal Ob- servatory at Green\vich. The recent construction of this instrument, in which Mr. Troughton has availed himself of his long experience and the latest improvements ; the cir-
43
cumstance of tlie telescope admitting of being shifted to dif- ferent parts of the circle i the number of microscopes used, and the firmness of their position in the pier; and the great knowledge, skill and assiduity of Mr. Pond, the Astronomer Roj-al, all promise very important results for Astronomy.
There is, however, one circumstance to be noticed, respect- ing this method of observing, of some importance, and which bears particularly on tlie question, whether any of the fixed stars have a visible parallax.
In the mode of ascertaining the north polar distance of an object by the mural circle, it is requisite to know the north polar distances of certain fixed stars with the changes from precession, effects of the semi annual equation, aberration, nur tation and refraction. If some of these stars be also affected by annual parallax, it is obvious that if no notice be taken of this, the north polar distance required will be inexact. It may be said that the same correction being obtained by dif- ferent stars, will shew that no such annual parallax exists so as to be sensible. This point certainly might be ascertained in this manner b}' a sufficiently numerous set of observations. Still, as it may be suggested, that many of the fixed stars may have small sensible annual parallaxes, observations, which will point out only the difl'crences of these, are not so proper as those by which the whole quantity would be pointed out. This may perhaps be better understoood by considering what would have taken place had this mode ol
observing been used before the discovery of the aberration
II 2
44
of light. It would have been extremely difficult to have separated the compound results, and assigned the proper quantity to each star, as Bradley was enabled to do at once, as to his observations by his zenith sector.
The observations I am about to state point out changes of zenith distance in certain stars at difterent seasons of the year, which changes are explained by annual parallax, and after long and anxious consideration I have not been able to assign any other cause.
45
a Lyras near Opposition.
|
Time of IFace of Observation Circle |
Mean Zen. Dist. Jan. 1, 1811. |
Mult. for Paral. |
Time of Observation |
Face of Mean Zen. Dist. Circle Jan. 1, 1811. |
Mult. for Paral. |
||
|
1808, July 28 Aug. 21 23 |
W E E |
14 46 19,93 20,41 21,58 |
+ 1 ,77 ,55 ,53 |
1811, Julyl7 20 21 |
E W E |
14 46 20,80 19,05 21,28 |
+ ,84 ,82 ,82 |
|
2i 1809, June 17 Julys |
W E W |
18,03 21,49 17,38 |
,52 ,87 ,87 1 |
22 23 26 |
W E W |
18,22 17,56 17,41 |
,81 ,80 ,78 |
|
8 13 14 |
W E E |
19,93 21,03 18,31 |
,86 ,85 ,84 |
29 31 Aug. 1 |
E E W |
18,02 19,44 19,48 |
,76 ,75 ,74 |
|
15 .18 ^ ,19 |
E- E W |
17,86 IS, 46 18,74 |
,84 ,83 ,83 |
3 4 5 |
E W E |
20,94 18,40 21,81 |
,72 ,71 ,70 |
|
20 23 24 |
E E W |
17,43 19,18 19,29 |
,82 ,80 ,80 |
6 1812, June 28 Julys |
W W |
18,04 18,79 18,10 |
,69 ,88 ,88 |
|
26 Aug. 4 S |
W W W |
21,79 19,76 19,15 |
,79 ,71 ,67 |
Aug. 6 7 8 |
E W W |
13,42 18,07 18,44 |
,69 ,68 ,67 |
|
9 E 1810, July 1 , E ... 8i E |
20,73 19,67 19,69 |
,67 ,S8 ,86 |
11 24 25 |
E E W |
16,25 20,28 19,69 |
,65 ,52 ,51 |
|
|
'{■'' 9 j W 15 E 24 W |
19,22 18,26 18,27 |
,86 ,84 ,80 |
27 1813, July 2 3 |
E E w |
19,59 20,42 21,11 |
.48 ,88 ,88 |
|
|
- . 26 -; ' .,,27 30 |
E E W |
21,25" 17,.32 19,59 |
',78' ,77 ,75 |
4 7 |
E W E |
21,30 20,57 19,44 |
,88 ,87 ,87 |
|
' 18U, July 2 ; 7 9 |
" W E |
19,20 18,96 19,43 |
,88^ ,87 ,87 |
10 14 : 15 |
E W |
20,22 19,63 j 18,81 |
,86 ,85 ,85 |
|
j^, 16 |
W -*- W |
19,49 - 20,20 18,13 |
,87' ,85 ,84 |
' 20 23 |
E- W |
'' 16,21 ---■ 18,68 |
,82 ,80 |
46 tt Lyras near Conjunction.
|
Time of observation |
Face of Circle |
Mean Zen. Dist. Jan. 1. 1811. |
Mult. for Paral. |
Time of observation |
f Mean Zen. Dist. Circle! J^"- 1.1811 1 |
Mult. for Paral. |
|
|
1808, Oct. 22 Nov. 2 6 |
W E E |
14 46 20,61 22,59 . 22,57 |
,33 ,48 ,53 |
1811, Dec. 10 11 13 |
W E W |
14 46 21,53 20,43 17,66 |
,83 ,83 ,85 |
|
Dec. \' 5 19 |
EW EW EW |
18,99 21,74 20,91 |
,80 ,81 ,85 |
28 29 1812, Jan. 14 |
E W w |
22,14 22,62 20,32 |
,88 ,85 ,83 |
|
22 1809, Jan. 22 30 |
EW E W |
20,33 22,19 22,65 |
,88 ,80 ,73 |
19 20 21 |
E E W |
21,25 21,48 20,68 |
,81 ,80 ,60 |
|
Dec. 7 1810, Jan. 22 23 |
E E W |
19,53 22,20 20,04 i |
,82 ,79 ,78 |
29 Dec. 2 8 |
E W E |
22,95 20,85 21,58 |
,72 ,79 ,82 |
|
Feb. 4. 9 13 |
E W E |
21,30 20,81 18,49 |
,68 .63 ,59 |
10' 11 14 |
W E W |
22,19 21,19 20.46 |
,83 ,84 ,85 |
|
IS 1811, Jan. 10 11 |
E E W |
18,53 20,14 18,71 |
,52 ,84 ,86 |
15 22 31 |
E E |
19,74 21,70 20,07 |
,85 ,88 ,88 |
|
12 15 16 |
E W W |
21,65 22 89 21,62 |
,85 ,83 ,83 |
1813, Jan. 4 6 Nov. 28 |
E W E |
21, Ol. 19,97 19)48 |
,87 ,86 ,76 |
|
17 20 21 |
E W E |
19,70 19,71 21,73 |
,82 ,80 1 ,80 |
Dec. 6 9 14 |
W W E |
21,64 22,90 20,11 |
,81 ,83 ,85 |
|
22 23 Nov. 19 |
W E W • W E |
22.55 22,86 20,76 |
,79 ,79 ,68 |
20 21 26 |
W W W |
21,63 ■^ 20,20 19,85 |
,87 ,87 ,87 |
|
Dec. 2 5 |
20,79 23,24 |
,79 ,81 |
28 29 30 1814, Jan. 3 4 |
E W E W W |
21,55 21,36 20.17 22,40 22,84 |
,88 ,88 ,88 ,88 ,88 |
4^
«e Lyras near 6 o'clock in the Evening.
|
Time of Observatioa |
Face of Circle |
Mean Zen. Dist. Jan. 1. 1811. |
Mult. ! for \ Paral. ' i |
Time of : ^Y Observation j.?^,^ |
Mean Zen. DisU Jan. 1. 1811. |
Mult. for Paral. |
|
|
ItJlO, Aug.28 Sept. 5 |
E W E |
14 iC iy,?7 17,79 20,00 |
-|-,4G + ,35 1 + .32 |
I811,0ct.ll ^2 16 |
E W E |
14 46 20,38 18,91 .20,03 |
—,17 —,17 —.24 |
|
8 16 Oct. 1 |
E \V E |
19,98 21,21 20,61 |
+ .31 i +,20 ■ —,03 |
1813, Sep. 29 Oct. 1 3 |
W E W |
18,09 19,85 20,21 |
—,00 —,03 -,07 |
|
2 5 6 |
E. W ' W |
20,56 18,84 21,53 |
—,06 —,10 — 11 1 |
6 7 8 |
E W E |
21,03 19,67 22,95 |
-,ll — ,13 -,14 |
|
7 8 15 1811, Oct. 9 |
E E E W |
20,64 20,16 22,82 22,7S |
-,13 —,14 -.23 -,14 |
11 14 15 18 |
W E E W |
17,48 20,35 20,85 20,8g |
-,17 —,22 —,23 -,27 |
The first column points out the clay of observation. The second shews the position of the face of the circle. The third the mean zenith distance, Jan. 1, 1811, to which the observations have been reduced a? being a middle epoch between the observations. The last Column is the multiplier of the semiannual parallax to obtain the parallax in zenith distance at each observation. Tlie product is to be applied, according to the sign, to the zenith distance in the third column, to obtain the mean zenith distance..
4a
Let p. represent the semiannual parallax of « Lyras.
Mean Zen. Dist.
a / //
Then by the first 20 observations near opposition J* 46 19,51 -\-,76p
fay next 20 .... 19,07-l-,82p
by next 25 .... 19,26 + ,75p
Mean of 65 observations near opposition 14 46 19,28 + ,78;?
By first 20 observations near conjunction - 14 46 20,84 — ,72p
by next 20 - - - - 21,24— ,8Ip
by next 21 . . - - - 2l,0l_,85p
Mean of 61 observations near conjunction 14 46 21,03 — ,79^
Hence 19",28p.+,78p = '2l"fi3—,79P' orp = 1 ,1 and the parallax of the annual orbit for «■ Lyrae = 2",2.
Thus the mean zenith distance of « Lyr^, Jan. 1, 1811, by 126 observations = 14 46 20,15
By the 26 observations, near 6 o'clock in the evening, the mean zenith distance, Jan. 1, 1811=14 4b' 20", 'i 5 — , 0 op.
If the above conclusion respecting the parallax of a. Lyra? be 'not admitted, some explanation of the differences of the zenith distances must be sought for.
First, it cannot arise from errors of observation, compre- hending error of adjustment in the vertical axis, error of bi- section of the star, and errors of reading off. These errors by their nature are corrected by taking a mean of repeated ob-
49,
servations, and an inspection of the result of each observation will s-hew that it is impossible a mean of 60 observations can. be aftected by a greater error of observation than a very small fraction of a second- It occurred that the mean of the observations made and r^ad off in day -light might differ from the mean, of the ob- servations made near midnight. It soon however was satis- factorily ascertained that the differences could not arise froni! this cause.
, Secondly, the difference cannot arise from errors of divi- sion, for in fact the same divisions are used as to the same, star. The correction of the mean of the microscopes, obtained by observations of different stars, which have been used to deduce the observed zenith distance of a Lyrae, although, affected by errors of division, occasions no error in the result,, because care has been taken that the numbers of observations East and West should be nearly equal. The zenith distances corrected for the mean of the microscopes have been put down merely to shew the consistency of the observations.. The means of the zenith distances at each time of the year,, depend only on the observations of a, Lyr» itself..
Thirdly, it cannot arise from uncertainty in the changes of refraction. This star is too near the zenith for^any materiuK uncertainty of this kind.* -
* These observations have been calculated by Bradley's refractions. Had they been ? calculated- by the French Tables, (which I have used for a, Aquilae and Arcturus) ; the parallax would have been about two seconds. This alteration arises from the differ- - VOL. XII.. I
Fourthly, it cannot arise fpom aay uncertainty in the max- imum of aberration of light; whether we take the maximum at 30" or 20i" Because when « Lyrae passes the meridian near noon and midnightj the aberration is very small, and therefore not affected by a small error in the maximum.
But it is necessary to compute with precision according to the sun's longitude at the time of the passage, as the aberra- tion changes rapidly at these times. The semi annual equa- tion is nearly the same at these times, and therefore no error from thence. The precession or any small uncertainty in the quantity of proper motion can occasion no en'or;
When indeed « Lyrce passes near 6 o'clock, then an un- certainty in the maximum of aberration may affect, the con- clusion, because the aberration in declination is nearly a max- imum, and therefore in this enquiry it is of some consequence to know the maximum of aberration.
Hence the observations of » Lyr^ near quadrature aie less proper for this enquiry, and have accordingly been less aU tended to. Those that have been made are however, very* consistent with the observations made near noon and mid-' night.
The only solution, perhaps that we have left, unless we admit of parallax, is, that in different degrees of tempera-
ent laws of change of denaity frpip change of temperature in Bradley's, aad in the French Tables.^ — In « Lyrae.it is scarcely wortli notice, but is considerable in » Aquilte. In, that, star the parallax comes out legs by the French, than by Bradley's Refractions.
61
ture, the figure of the instrument changes, and gives dif-^ ferent results for the same star. This cannot be the case.
For 1st. with respect to several stars, the results are the same when the means of the thermometer differ by many degrees. Thus,
« Polaris.
14 observations mean therm. 55,2 give seconds in zen. dist. - 5,79
23 do. mean therm. 39 give do. - 6,27
II do. mean therm. 411 give do; - 6,02
a Polaris S. P.
23 observations mean therm. 59,8 give seconds in zen. dist. - 25,26
21 do. mean therm. 39,2 do. . 25,65
The above are even computed by Bradley's formula for refraction which certainly gives the change of refraction from change of temperature too great.
Arcturus.
1 8 observations mean therm~. 59 give seconds in zen. dist. - ^4,68
20 do. do. 64 do. - 3*i99
23 do. do. 40 do. - 35,23
These are also computed by Bradley's refractions. Secondly, if the figure of the instrument changed in different degrees of temperature, the zenith distance of a.
I 2
AS-
star, determined by the bottom microscope onl}'^ would not preserve, in different temperatures, the same relation to the zenith distance determined by the mean of the three micros- copes. No alteration however is observed to take place. Thus for a Lyrae the following are the corrections to be ap- plied to the zenith distance by the three microscopes to give llie zenith distance by the bottom microscope only.
|
Correction. |
||
|
Summer |
1811 |
— 0"40 |
|
Winter |
1811 |
—0,26 |
|
Summer |
1812 |
—0,46 |
|
Winter |
1812 |
—0,07 |
|
Summer |
1813 |
—0,03 |
|
Autumn |
1813 |
—0,24 |
This indicates no change of figure, and the same is ob- served with respect to other stars. In a. Aquilae, for in- stance, the quantity to be applied to the mean of the three microscopes to give the result by the bottom microscope
only = + 1 nearly, and no material change occurs in dif- ferent temperatures. *
Besides if this parallax arise from some deception, it ought to appear in all stars sufficiently near the zenith, so as not to be afteeted by uncertainty in change of refraction.
• The difference between the mean of the three microscopes and the bottom aakros^ cope is no where greater than for a Atjuilse.
.53
Capella, /3 Tauri Procyon, Polaris, above and below the pole, y Draconis, (3, ^, ti, Ursae majoris and other stars do not shew changes of zenith distance similar to what appear as to a Lyrae, Arcturus, «, Aquilaj, * Cygni, and a, Ophiuchi.
The mean zenith distances from a number of observations of the pole star above and below the pole are given, as in- stances where no changes of zenith distance are noticed. * Also the results as to « Aquiloe, Arcturus and « Cygni. The results as to other stars which seem to have a sensible pa- rallax will be given when the observations are more nu- merous.
If parallax be not admitted, it must appear very remarka- ble that in no stars have annual changes of zenith distance been observed by this instrument that cannot be explained by a parallax. It might be expected that in some stars the changes would have been quite opposite to the changes from parallax.
It may perhaps be suggested that there may be some un- known peculiarity in my mode of observing, that would ex- plain these appearances of parallax. In answer to this it is
* In the observations of the pole stat, each zenith distance is the result of observa- tions made before and after the meridian passage of the star, the instrument having been reversed in the interval. This has sometimes been done for other stars, but not often. Tlie value of this instrument may be considered as much enhanced from being capable of being used at a small distance on each side of the meridian, by noting the time of observation.
54
only necessary to mention that many of the observations have been made by my son, Mr. John Brinkley, A. B. and comparing the lesuUs of our observations no differences are observed.
a Polaris in the Spring,
|
Time of Observation. |
Face of Circle. |
Mean Zen. Dist. Jan. 1, 181J. |
Mult. for Paral. |
Time of Observation. |
^'^*:? Mean Zen. Dist. Circle J«"-'. '811. |
Mult. for ParaL |
|
|
1809, Mar. 3 Apr. 10 22 |
E\V EW EW |
o 34 54'44','51 46,84 44,70 |
+ ,86 ,98 ,97 |
1810, Apr. 24 26 May 27 29 |
EW EW EW EW |
34 54 46,35 44,27 45.65 44,98 |
+ ,91 ,91 ,56 ,54 |
|
23 May 9 10 |
EW EW EW |
46,45 46.33 43,77 |
,93 ,80 ,80 |
30 1811, Mar. 27 Apr. 22 |
EW EW EW |
44,15 45,90 46,11 |
,53 ,99 ,93 |
|
14 22 23 |
EW EW EW |
43,77 45,09 45,67 |
,74 ,65 ,64 |
1814, Feb. 2 7 9 |
EW W W |
45,63 47,17 44,54 |
,50 ,58 ,61 |
|
27 1810, Mar. 5 |
EW EW |
44-,50 44.27 |
,62 ,88 |
16 24 |
EW 45,05 E 44,80 |
,69 |
Themean of 23 gives mean zenith distance = 34° 54' 45", 24.+,7f>p.
The greatest zenith distance of the pole star when above the pole as affected by parallax, is on Oct. 4. and the least on April 2. Here as well as in the results which follow, the refraction has been com- puted by the French tables.
< ' 55 Polaris in the Autumn.
|
Time of Observation. |
Face of Circle. |
Mean Zen. Dist. Jan. 1, Jsn- |
Mult, for |
Time of |
Face of Circle |
Mean Zen. Dist. |
Mult. |
|
|
Paral- |
Observation. |
Ja«. 1.1811. jp-j |
||||||
|
1809, Oct 5 7 22 |
EW EW EW |
3:4. 54. 43,22. 45,01 44,50 ^ |
,99 ,99 ,94. |
1811, Oct. 16 22 23 |
E W W |
34r54^4*,27 45,42 45,38 |
_ ,97 ,94 ,9* |
• |
|
26 29 Nov. J |
EW EW EW |
46,75 47,04. 43,70 |
,91 ,94 ,88 |
27 Nov. * Id' |
w E ; W |
46,42 45.*7- 46,07 |
,91 ,85 ,71 |
|
|
6 1.4. 17 |
EW EW |
42,71 43,67 |
,84 ,77 .73 |
20 29 1812, Oct. a) |
E W : W |
45i,*9 ♦7,4^ 45,32 |
,69 ,95 |
|
|
EW |
47,06 |
|||||||
|
la 1.9 81 |
EW EW EW |
45,57 45,09 46,4,5 |
,72 ,71 ,69 |
34 25 26 |
; E w > E |
46, 8» 45,W |
,94 ,9* ,92 |
■ |
|
29 Dec 3. 7 |
EW EW EW |
44,95 46,70 46,97 |
,58 ,51 .44 |
27' Nov. 3 |
E w E |
45i,19i 44(,5* 47,65« |
,92 ,91 ,87 |
- |
|
10 12 1810, Nov. 6 |
EW EW EW |
44,57 47,44 44,58 |
,39 ,36 ,84 |
5 6 7 |
E w E |
46,84 45,44. 44,93 |
,85 ,84 ,83 |
|
|
26 Dec. 1 |
EW EW |
43,85 45,64 |
,60 ,53 |
8 |
W |
44,61 |
,82 |
Mean of 39 observations gives mean zenith distance = 34° 54' 45", 51 — 79p. Comparing the two last sets of observations, viz. 23 in spring and 39 in autumn, we have 45",24+,76p. = 45", 51— ,79/> ov p = 0' ,17.
From which may be inferred that « Polaris has no sensible- parallax.
56
cc Polaris S. P. in the Spring.
|
Time of Observation |
Face of Circle |
Mean Zen. Dist. Jan. 1, 1811. |
Mult. for Paral. |
Time of Observation |
Face of Circle |
Mean Zen. Dist Jan. 1, ISII. |
Mult. for Paral. |
|
1809, Apr. 14 20 23 |
EW EW EW |
38 18 48,55 49,89 49,81 |
,96 ,94 93 |
1810, Ap. 19 26 27 |
EW EW EW |
38 18 47,54 47,83 48,36 |
,94 ,90 ,89 |
|
May 9 10 14 |
EW EW EW |
44.,81 46,48 46,96 |
,80 ,78 .74 |
28 30 May 2 |
EW EW EW |
46,45 45,61 45,34 |
,88 ,87 ,86 |
|
18 22 23 |
EW EW EW |
47,58 45,44 47,08 |
,70 ,67 ,66 |
5 1813, May 5 9 |
EW W E |
48,76 48,09 46,67 |
,83 ,83 ,79 |
|
24 June 4 15 |
EW EW EW |
45,68 46,38 46,16 |
,65 ,47 ,30 |
16 19 20 |
W E W |
49,20 48,03 47,29 |
,72 ,68 ,67 |
|
17 25 July 10 |
EW EW EW |
45,77 45,74 46,21 |
,26 ,15 + ,09 |
21 26 28 29 June 1 |
E W E W E |
47,92 48.71 47,00 48,58 46,89 |
,66 ,59 ,56 ,55 ,52 |
O Jf II
Mean of 32 gives mean zenith distance = 38 18 47, 21 — ,68/>.
67 - '• . oc Polaris S. P. in the Autumn.
|
Time of Observation |
Face of Circle |
Mean Zen. Dist. Jan. 1, 1811. |
Mult. for Paral. |
Time of Observation |
Face of Circle |
Mean Zen. Dist. Jan, 1, 1811. |
Mult. for Paral. |
|
1809, Aug. 25 Sep. 30 Oct. 5 |
EW EW EW |
;/ / // 3a 18 46,97 48,11 44,78 |
+ ,78 ,99 ,99 |
1811, Nov, 3 5 6 |
E W W |
38 IS 45,48 47,53 47,50 |
+ .87 ,8* ,85 |
|
24 28 31 |
EW EW i;w |
46,20 4.S,98 47,95 |
,93 ,91 ,88 |
11 18 22 |
E W E |
47,70 49,49 46,27 |
,80 .72 ,66 |
|
Nov. 16 17 18 |
EW EW EW |
45,25 46,51 45,42 |
,73 ,72 |
1812, Oct. 14 15 20 |
W E W |
45,77 47,45 46,57 |
.98 ,98 ,-95 |
|
19 21 23 |
EW EW EW |
47,14 45.79 46,90 |
,71 ,69 ,66 |
21 23 25 |
E W W |
46,90 48,10 47,30 |
,95 ,94 ,93 |
|
Dec. 1 11 1810, Nov. 16 |
FW EW EW |
46,75 46,98 47,35 |
,53 .37 ,72 |
27 28 29 |
E W E |
46,07 45,75 47,87 |
,91 ,91 ,90 |
|
1811, Oc. 15 16 19 |
W E W |
46.31 46.07 47,31 |
,98 ,98 ,96 |
Nov. 2 3 4 |
E W E |
45,46 46.66 47,05 |
,87 ,87 ,86 |
|
2i 25 Nor. 1 |
W E W |
46,25 45,35 46,99 |
,93 ,93 ,88 |
5 W 6 E 7 W |
45,65 45,83 47,68 |
,8S .85 ,84 |
The mean of 42 observations gives mean zenith distance = 3&M8'.46",76-[-,84p. A comparison of the mean of the observations in spring and of the mean of these in Autumn gives 4?",21 — ,68p = 46,76 + ,84p. or p = 0'',30 from which also I infer that the parallax of » Polaris (if any) is too small to require to be noticed.
VOL. XII.
58
Aquilae.
|
Time of Observation |
t Face of Mean Zen. Dist. Circle Jan. 1, 1811. |
Mult. for Paral. |
Time of Observation |
1 Face ofMean Zen. Dist. Circle Jan. 1, 1811. |
Mult. for Paral. |
||
|
1809, July 20 Aug. 21 22 |
E E W |
45 0 32,48 27,44. 27,99 |
+ 0,48 0,32 0,32 |
1811, Aug. 6 10 16 |
W W W |
45 d 28,89 30,46 31,57 |
+ ,42 .40 ,36 |
|
23 24 27 |
E W W |
29,60 29,88 28,35 |
0,31 0,.S0 0,28 |
19 20 22 |
E W E |
28,83 28,68 31,99 |
,34 ,34 ,32 |
|
28 1810, July 30 Aug. 26 |
W w E |
29,07 30,90 28,82 |
0,27 0,45 0,29 |
25 27 31 |
W E W |
31,80 28,69 31,03 |
,30 ,28 ,25 |
|
1811, July 14. 16 20 |
E W W |
28,80 28,99 29,18 |
0,50 0,49 0,48 |
Sep. 1 1812, Aug, 6 7 |
E E W |
29, U 28,29 29,29 |
,24 ,42 ,42 |
|
21 22 23 |
E W E |
31,48 30,77 27,84 |
0,48 0,48 0,48 |
8 9 16 |
W E W |
29,88 32,15 29,54 |
,41 ,40 ,36 |
|
26 29 31 1 Aug. 3 |
W E W E |
S0,63 30,34 30,81 31,38 |
0,47 0.46 0,46 0,44 |
24 25 26 Sep. 5 |
E W E W |
28,13 27,07 28,79 31,02 |
,.'?0 ,29 ,28 ,21 |
The mean of the above 38 observations gives mean zen. dist. »= 45° 0' 29^74 +,38/>. The effect of annual parallax as to « Aquilse makes the zenith distance greatest, Dec. 29, and least June 29.
59 X Aquilffi.
|
Time of Observation |
Faee of Circle |
Mean Zen. D ist. Jan. 1, 1811. |
Mult. for Paral. |
Time of Observation |
Face of C ircle |
Mean Zen. Dist. Jan. 1, 1811. |
Mult. foV Paral. |
|
1808, Nov. 29 Dec. 19 22 |
W EW E\V |
45 0 32,49 34,22 32,97 |
0,44 0,51 0,52 |
1811, Dec. 13 18 21 |
W E W |
o 45 6 31,40 32,63 33,36 |
0,50 0,53 0,52 |
|
1809, Jan. 30 Feb. 11 16 |
E E E |
30,63 32,42 30,73 |
0,44 0,37 0,33 |
29 1812, Jan. 4 8 |
E W E |
31,33 32,56 29,84 |
0,52 0,51 0,50 |
|
1810, Feb. 4 13 18 |
E W E |
29,17 31,58 31,58 |
0,41 0,35 ,038 |
21 29 30 |
W E W |
29,68 32,10 32,22 |
0,48 0,44 0,44 |
|
Mar. 10 13 1811, Jan. 2Y |
VV E E |
33,26 31,06 30,53 |
0,16 0,13 0.45 |
1813, Jan. 20 25 Feb. 3 |
E W E |
31,08 33,61 32,00 |
0,48 0,46 0,41 |
|
28 Feb. 3 13 |
W W E |
32,99 32,92 .31,80 |
0,45 '0,42 0,35 |
6 9 15 |
W E W |
32,61 31,75 32,01 |
0,40 0,38 0,34 |
|
19 23 24 Dec. 11 |
W E W E |
3i,l6 32,29 33,55 29,44 |
0,31 0,28 0,27 0,50 |
19 20 21 22 |
E 29,78 W 31.98 E 31,07 W i 32,65 |
0,31 0,30 0,29 0,29 |
The mean of the above 38 observations gives the mean zen. dist, 45° 0' 3r',87 — ,40;?. Hence by comparing the preceding set of observations with these, we have 29^74 + ,38jP = 3r',87 — ,40p orp = 2",73.
Hence the parallax of « Aqui]se = 5",5. Tiie refractions in the
k2
60
French tables have been used in the above. Had Bradley's refractions been used, the parallax would have come out considerably greater. The value of p is less exact, on ac- count of the smallness of its co-efficients.
A mean of 20 observations near six o'clock in the evening,
gives mean zenith distance = 4o°.0'. 30",64 — ,\p.
The mean of the above 76 = 4fl.O. 30,80 — ,01^?.*
* M. Delambre in his remarks on M. Piazzi's observations, proposes to examine the effects of the parallaxes in changing the right ascensions. This confirmation would be very satisfactory, and might be readily attained, were some stars so much affected by parallax as M. Piazzi has supposed. But if the parallaxes be so small as my observations tend to point out, no expectation of this kind could be entertained as to « Lyrae, Arcturus, and a, Cygni.
As to y Aquilae, the right ascensions in March and September would differ by about — of a second of time, and, under the circumstances of the case, it would require attention to detect this quantity, but it might b*e done. If this difference exist it ought to be allowed for in computing the apparent from the mean right asceasion.
61
Arcturus.
|
Time of observation |
Face of Circle |
Mean Zen. Dist. Jan. 1. 181 1. |
Mult. for Paral. |
Time of observation |
Face of Circle |
Mean Zen. Dist. Jan. 1, 1811 |
Mult. for Paral. |
|
1808, Oct. 27 31 Nov.l |
E W W |
33 12 5i',86 52,88 56,15 |
,54 ,55 ,56 |
1811, Oct. 16 18 25 |
W W E |
33 12 55,72 55,57 55,02 |
,7? ^ ,48 ,53 |
|
11 27 29 |
EW EW EW |
53,55 52,93 54,51 |
,60 ,61 ,61 |
26 Nov.l 3 |
W w E |
55,01 56,21 52,72 |
,53 ,56 ,57 |
|
Dec 10 13 U |
EW EW EW |
53,14 52,50 52,74 |
.59 ,59 ,59 |
18 19 22 |
w E E |
55,87 55,71 56,67 |
,61 ,61 ,61 |
|
1810, Sept. 6 iO 21 |
W W E |
52,56 57,61 53,98 |
,12 ,16 ,27 |
29 Dec. 1 1813, Oct. 14 |
W w E |
54,49 53,94 56,02 |
,61 ,61 ,46 |
|
Nov. 5 6 16 |
E W W |
53,75 55,48 56,62 |
,57 ,58 ,60 |
19 31 Nov. 2 |
E W E |
55,16 56,52 54.27 |
,49 ,56 ,57 |
|
22 26 Dec. 1 |
E W E |
55,42 54,86 55,97 |
,61 ,61 ,61 |
3 11 12 |
W E W |
57,61 .55,23 55,90 |
,57 ,60 ,60 |
|
10 1811, Oct. 11 12 |
E W E |
53,57 56,74 £5,84 |
,59 ,44 ,45 |
14 Dec. 14 16 |
W E W |
57,50 56,32 57,69 |
,60 ,58 ,57 |
In deducing these results from the observations of Arcturus, the annual change of N. P. D. has been taken = +16",81. The annual proper motion of Arcturus may be considered in some measure uncertain, and it may be thought that the conclusi>)n respecting parallax will be affected thereby. But this is not the case. Let the annual variation in JV. P. D. = I8",81+e. *
Then the mean of the above 42 observations gives the mean zenith distance = 33' 12' 55", 11 — ,54/> — ,4:e.
* The annual variation in N. P. D. of Arcturus seems by my observations to be at least + 19",1. But the interval since they commenced is too short to speak with much conSdence.
|
' |
Arcturus. |
||||||
|
Time of observation |
Face of Circle |
Mean Zen. Dist. Jan. 1, 1811. |
Mult. for Paral. ; |
Time of observation |
Face of Circle |
yiean. Zen. Dist. Jan. I, 1811. |
Mult. for Paral. |
|
1809 April 20 28 May 14 |
E EW EW |
33 12 53,83 54,85 53,70 |
+ ,52 ,56 ,60 |
1811 May 19 26 29 |
E E W |
3°3 12 56,07 52,35 53,79 |
+ ,61 ,61 ,61 |
|
21 June 25 1810 AprU 25 |
W EW E |
54,64 51,80 53,27 |
,61 ,52 .53 |
June 9 12 17 |
E W W |
52,83 53,89 54,71 |
,60 ,59 ,56 |
|
26 27 28 |
W E W |
56.24 51,90 53,76 |
,55 ,55 ,56 |
18 1813, May 11 16 |
W E W |
52,89 53.32 56,07 |
,56 ,60 ,61 |
|
30 May 2 ^ 4 |
E W E |
51,48 52,22 55,26 |
,57 ,57 ,57 |
20 28 29 |
E W E |
54,16 54,95 52,87 |
,61 ,61 ,61 |
|
5 6 29 |
W E W |
53,10 53,34 52,41 |
,58 ,58 ,61 |
30 June 2 4 |
W E W |
55,48 55.52 52,69 |
,61 ,01 ,60 |
|
31 1811, May 11 16 |
E W E |
54,27 56,77 53,11 |
,61 ,60 ,61 |
5 8 |
E W - |
53,81 53,17 |
,60 ,59 |
From the above 35 observations the mean zenith distance = 33" 12' 53" ,80+,59p — ,35e.
Hence 55", ll—,54p—,45e = 53,80 rf-,59p— ,35e
p= 1",1— ,09e. e cannot be so great as half a second, and therefore ,09e is too small to be noticed. Therefore from these results the parallax of Arcturus = 2",2;
And by 77 observations, the mean zenith distance = 33''.12' 54",45+,05j) — ,4e. By 15 observations in July and August, mean zen. dist. = 33° 12' 54", 50 + ,2p + ,68e.
63
Cygni.
|
Time of Observation |
Face of Circle |
Mean Zen. Dist. Jan. 1.1812. |
Mult. for Paral. |
Time of Observation |
Face of Circle |
Mean Zen. Dist Jan. 1. 1812. |
Mult. for Paral. |
|
l«10, Mar. 9 10 17 |
W E W |
8 46 26,21 26,31 23,91 |
,59 ,58 ,49 |
1813, Jan. 9 10 |
E W |
0 26,78 24,44 |
,8« ,88 |
|
18 181 I.Jan. 28 Feb. 3 |
W W w |
24,77 21,77 23,59 |
,48 ,89 ,87 |
n 19 25 |
E E W |
24,03 24,50 26,53 |
,88 ,88 ,90 |
|
23 24 28 |
E W E |
26,42 25,68 25,35 |
,74 ,73 ,70 |
Feb. 4 & 6 |
E W E |
25,81 26,46 22,83 |
,90 ,86 ,86 |
|
Mar. 12 14. 1813, Jan. 8 |
W E W |
26,22 25,64 25,55 |
,56 ,53 ,88 |
1813, Dec. 26 27 28 1814, Jan. 4 |
W E W w |
23,48 27,29 24,40 23,81 |
,81 81 ,82 |
The above 24 observations give the mean zen. dist. 8. 46' 25",07— ,76i>.
64
« Cygni.
|
Time of Observation |
Face of Circle |
Mean Zen- Dist. Jan. 1, 1812- |
Mult. |
Face of Circle |
Mean Zen. Dist. Jan. 1, 1812. |
Mult. for Paral. |
|
|
for Paral. |
Time of Observation |
||||||
|
181 1, July 26 28 Aug. 3 |
W E E |
8 46 22,77 24,49 24.24 |
+ /,90 ,90 ,89 |
181 I.Sep. 5 1812, Aug. 24 25 |
E E W |
S 46 26,63 24,03 26,09 |
+ .67 ,77 .76 |
|
10 13 16 |
,85 ,83 .82 |
26 27 Sep. 5 |
E W W |
23.98 22,93 24,81 |
,75 .74 .67 |
||
|
w E W |
23,08 24,31 22, 1 a |
||||||
|
19 20 22 |
E W E |
22,93 21,03 24,90 |
,80 .80 -,•39 |
7 10 11 |
E W E |
23,87 22,52 23,67 |
.64 .61 ,60 |
|
25 27 31 |
W E W |
21,22 22,54 21,59 |
,76 ,74 ,72 |
12 W Oct. 1 E |
23,46 23,78 |
,59 .33 |
The above 23 observations give the mean zcn. dist. = 8" 46' 23",48 + ,74p.
Hence 25",07— ,76p = 23",48 + ,75p Therefore p = 1",06 And the parallax of « Cygni = 2",1.
65
REMARKS.
If the results deduced from the preceding observations should be admitted, it follows that the brightest fixed stars are not so near to us as some others. « Aquilae, which is far exceeded in splendor by « Lyras and Arcturus is only at half the distance of the two latter. However extraordinary this may appear, it results from observations that appear to me fully adequate for the conclusion.
My observations on a. Lyroe were commenced with the view of examining the question of parallax ; but the results of the observations of a A-quilae forced themselves as it were on my notice. This star would not on any account have been selected for the investigation. The effect of the annual pa- rallax in declination is only about half the whole parallax; The star itself has not that splendid appearance that would lead us to suppose it as near as many others. Also its 2enith distance in this latitude being so much as 45', some uncer- tainty in so delicate an enquiry might be apprehended from refraction.
My conclusions may be considered as deriving little or no support from the results of the observations of M. Piazzi.
According to him (as appears from the Conn, des Temps 3808) the double parallax of a Lyrai is nearly five seconds, according to me only two seconds.
VOL. XII. . L
66
According to him the double parallax of Arcturus is less than that of a Lyras (the quantity is not stated in Conn, des Tenips.) according to me two seconds.
According to him « Aquil* has no sensible parallax; ac- cording to me the double parallax is five seconds and an half.
According to him Procj'on has a considerable double pa- rallax amounting to about QOf' ; according to my observa- tions it has no sensible parallax.
According to him Sirius has a considerable parallax. This star in this latitude is too much affected by refraction to af- ford any satisfactory conclusion.
The small changes of zenith distances which I find in a Lyras, in Arcturus and in a Cygni, and from which I con- clude the parallax of each, will, it is not doubted, make astronomers hesitate as to the degree of confidence with which they will receive them. It is not pretended that these quantities can be ascertained to the tenth of a second ; but by continuing the observations, it appears to nic, that I shall at last arrive at that degree of exactness. There seem to be no sources of errors in making these observations, which will not disappear by taking a mean of a great number of observations. However, until my conclusions are supported by other instruments, it is not likely that I shall impress astronomers with the same confidence which I myself pos- sess as to the results.
6?)
The astronomer royal, Mr. Pond, observing with the nevv mural circle, made by Mr. Troughton, has not hitherto con- firmed my results, although he finds indications of parallax in « Lyrje and a. Aquilas. * I had felt such confidence in my re- sults that 1 did not doubt that one of the first services that would be rendered to astronomy, by the Greenwich mural cir- cle, would be the confirmation of the existence of annual pa- rallaxes in certain stars. But, allowing the greatest accu- mcy in the observer, and excellence in the instrument, I conceive a very probable account has been given, why this has not yet taken place. Many of the stars, even of the second magnitude, such as Polaris, y Draconis, &c. may be affected by a parallax in declination, amounting to a fraction of a second. Were we certain that the standard stars were not affected by parallax, or had we ascertained the quantity, if any, then the method of observing by the mural circle would be far preferable to the methods of ob- serving in which the plumb line is used.-)-
* Phil. Trans. 1813, part 2. f I can feelingly bear testimony to the great superiority of the mural circle over our instrument, as to the convenience of the observer, and the consequent facility of multi- plying observations. In the mural circle no care is necessary but in making and reading off the observations. In our circle the previous examination of the plumb line is often a very tedious and sometimes unsatisfactory operation. Many observations have been lost thereby, a serious inconvenience in a climate ill adapted to astronomical observa- tions. The calm weather which we so often experience during a high state of the barometer, both in summer and winter, is generally unfavourable to the astronomer, be-
L 2
68
The same number of observations that I have given might have been completed in a smaller space of time, but un- favourable skies, necessary interruptions, and the expecta- tion of having my results confirmed by other instruments have made the earlier observations less numerous than they otherwise would have been. It soon appeared that increas- ing the number of observations would not materially change the results that I had already deduced. However the con- sistency of the observations in the several years may with some add weight to the conclusions.
My future exertions shall be directed in making such ob- servations as may serve to throw further light on this subject.
If I should meet with any circumstances that shall appear to me to invalidate the conclusions 1 have now ventured to make, I shall cheerfully communicate them, I shall be fully satisfied with the consciousness of having, to the utmost, ex- erted myself, as my duty led me, in the examination of this important question.
ing attended with a cloudy atmosphere. Clear skies oftener prevail during high winds. These circumstances are much against the use of the plumb line.
69
|
MEAN NORTH POLAR DISTANCES OF FORTY-SEVEN PRIN- CIPAL FIXED STARS, JAN. 1, 1813. |
|||||||
|
Names of Stars. |
No. of Obs. |
By Ref. in French Tables. Co-lat. 36° 36' 46" 5 N. P. D. Jan. 1, 1813. |
Ref Brad. Tab. Co-lat. 45 ',8 N.P.D |
G |
P |
D |
|
|
* Polaris * $ Ursae min. * /S Cephei |
36 38 21 |
1 41 21,77 15 4 49,45 20 15 31,41 |
2^71 49,26 31,14 |
+ 6^08 — 0,31 — 0,44 |
ii |
+ 0,14 |
|
|
* a Ursae maj. * a Cephei jS Ursa; maj. |
10 9 18 |
27 14 30,88 28 12 13,90 32 37 4,72 |
30,29 13,30 4,07 |
+ 1.17 — 0,83 |
— 0,29 |
||
|
f Ursae maj. * a Cassiopeae ^ Ursae maj. |
19 8 8 |
33 1 22,05 34 29 22,59 34 54 42,68 |
21,44 21,91 41,01 |
-f 0,80 |
|||
|
* y Ursae maj. » y Draconis • n Ursae maj. |
10 27 20 |
85 15 56,22 38 29 3,70 39 44 58,37 |
55,53 3,00 57,61 |
— 0,26 + 0,65 + 0,27 |
+ 0,28 |
||
|
* a Pcrsei * Capella * a Cygni |
10 30 22 |
40 48 51,36 44 12 20,71 45 22 58,34 |
50,62 19,90 57,52 |
+ 2,05 + 0,57 — 0,60 |
+ 2,34 — 0,42 |
+ 0,48 — 1,90 |
|
|
* « Lyrae * Castor * Pollux |
51 10 10 |
51 23 0,84 57 42 47,54 61 31 50,07 |
59,93 46,64 55,12 |
-- 0,53 - - 0,09 -- 1,23 |
+ 2,21 — 0,48 + 0,91 |
+ 0,11 — 2,40 — 0,55 |
|
|
* 0 Tauri * a AndromedoB * a Cor. bor. |
18 10 19 |
61 33 4V,22 61 56 30,32 62 38 55,51 |
43,19 29,31 54,44 |
+ 0,47 + 0,30 4-0,99 |
+ 0,37 • + 2,44 — 0,09 |
— 2,28 + 0.81 — 2,34 |
|
|
* a Arietis * Arcturus * Aldebaran |
9 20 20 |
67 25 36,76 69 50 19,33 73 52 35,98 |
35,82 18,19 34,62 |
+ 0,67 -f. 0,89 -- 0,74 |
+ 0,41 — 1,90 — 0,79 |
-t- 1,35 — 1,09 — 1,64 |
70
Names of Stars.
* /S Leonis
* a Herculis
* » Pegasi
* Pegasi
* Regulus
* a Ophiuchi
;}
Aquilse
* a, Orionis
* a Serpentjs
* Procyon
Ceti
a Aquarii
K Ilydrse
Rigel
Spica Virg. 1 a. Capricorn.
2 a. Capricorn. ■2 a, Librae
Sirius Antares
No.
of
Obs.
18
10,
15
10 20
25
10 30 10
18 18 16
10 10 12
10
13
9
10 10 10 10
Ref. by Frencli
Tables,
Co-lat. 26° 36' 46",5
N PD Jan. 1, 1813.
74 22 56,44
75 23 14,64. 75 47 52,80
75 51 21,18 77 7 23,06 77 17 40,49
79 50 1,34 81 36 59,85 84 3 5,22
82 38 15,94 82 58 38,81 84 18 15,33
86 39 2,04 91 13 21,75 97 51 10,99
98 25 34,27 100 10 51,33 103 4 36,09
103 6 52,03
105 15 22r59
103 28 4,27
116 0 16,77
Brad.
Ref.
Co lat,
45",8
y.p.D
55,22 13,22 51,71
20,00 21,79 39,19
0,11
58,54 3,79
14,69 37,40 .13,87
0,74
19,96
9,39
32,67 49,23 34,09
49,68 20,02
2,47 13,77
4- 2,0^ — 0,18 + 0,82 — 3,17 + 0,00 — 1,04
+ 1,00 + 0,91 4- 0,90 — 0,57 _ 0,03 — 2,76
+ 0,5.5 — 0,98 + 0,12 — 2,67 -f 0,.30 — 0,53
+ 1.03 + 1,86 + 0,49
+ 0,01 + 1,68 + 1.91
+ 1,18 + 2,07 + 1,36
+ 2,64 + 2,67 - 1,77 + 2,86
+ 0,53 + 1,14 — 0,41
— 0,46 + 0,49 + 1,45
— 0,61
— 0,74
— 2,16
+ 0,57
— 2,23 + 2,04
D
+ 3,33 — 2,26 + 0,19
— 0,46 + 0,42
— 1,25
— 2,55
— 2,53
— 3,31
|
1,35 |
|
|
— |
1,49 |
|
+ |
0,19 |
|
—. |
0,67 |
|
+ |
0,90 |
|
+ |
0,82 |
|
__ |
0,41 |
|
~- |
0,60 |
|
— |
3,04 |
+ 0,89 + 1,41 — 2,11 + 1,55
71
I find by above 500 observations of circumpolar stars the latitude of the observatory of Trinity College, Dublin, 53" 23' 13",5 using the French tables of refractions published in 1806. Or 5.T 5J3' 14",2 using Bradley's refractions.
In the preceding catalogue the third column shews the mean north polar distance, Jan. 1, 1813, the refractions ha- ving been computed by the French tables, to which tables I give the preference for reasons assigned in the paper which follows this.
The fourth column shews the seconds of the north polar distances, as computed by Bradley's tables.
It appeared to rae on several accounts of much impor- tance, to compare observations made nearly at the same time by different instrun^ents. The mural circle at the royal observatory, Greenwich, and the circle at the observatory of Trinity College, Dublin, may be ranked amongst the best instruments that have been constructed. As soon therefore as I was informed that the Greenwich circle was in use, I determined to repeat my observations of the principal fixed stars, and the present catalogue is the result of observations in the latter part of the year 1812 and in the year 1813.
To institute a comparison between the north polar dis- tances deduced by Mr. Pond and myself, it is necessary that the same tables of refraction should be used by each. Therefore as Mr. Pond has used the tables of Bradley, I also computed my observations by the tables of Bradley, and the result of the comparison of the observations is found
72
ill the column G. The quantity in G is to be applied to the fourth column to give the north polar distances by the Greenwich mural circle.
The 30 stars marked * are those which Mr. Pond uses as standard stars ; the north polar distances of which he has determined by a great number of observations in 1812 and 1813. (vid. Phil. Tran. 1813, part 2.) Now among these 30 stars there arc 24 in which the results do not differ by 1", four in which the differences exceed 1", but do not amount to 2", and two in which the differences exceed 2", but do not amount to 2"i. This is highly creditable to the divisions of our circle. In the Greenwich circle the errors of divisions, if any, will entirely disappear in a mean of a great number of observations, in consequence of the teles- cope being moveable. And in fact in this way Mr. Pond has ascertained that the errors of division of the Greenwich circle are too small to be noticed. (Phil. Tran. 1813, p. 281.) In our instrument the effect of the errors of division in the mean of the six readings of the microscopes, cannot be made to disappear. The above comparison shews satisfactorily that no material error can arise from thence.
For the stars not marked * the comparison has been made with the north polar distances given in the Phil. Tran. 1813, part 1. The differences as to these low stars are greater, and may probably be attributed partly to the uncer- tainty of refraction, and partly to the use of Bradley's tables. . In Dr. Bradley's formula for refraction the effect of the
IS
change of temperature ou the quantity of refraction is taken too great. This appears certain by the direct experiments of T.Mayer, Dalton and Gai-Lussac on the expansion of air at different temperatures. It also appeared evident to nie by observations of low stars in different temj^eratures. The consequence of which is, that even supposing the utmost accuracy in the instruments and in the observations, the zenith distances of stars will appear greater in winter than in summer, and the more so the greater the zenith distance.
The column P shews the quantity to be applied to the fourth column to obtain the north polar distances according to M. Piazzi, at Palermo. His north polar distances given in the Conn des Temps, 1812, having been reduced to Jan. 1, 1813, and also reduced to what they would have been ac- cording to Bradley's refractions. I do not know the exact date of these observations, but I suppose them recent. I believe also that M. Piazzi takes the mean refraction at 45' = 57",4 and makes the same allowance for changes in the thermometer as Dr. Bradley. If so, the correction to be applied to the north polar distances, as determined by M. Piazzi, to give what would have resulted from the use of Bradley's refractions = — 0",69— 0",5 (tan. N,P.D,—55\ 53% This quantity has been applied accordingly.
The column D is the difference between my results in 1809 and 1813. The quantities according to their signs are to be applied to the results in column 4, to give what would have resulted from the observations in 1809- In makiug
VOL. XII. M
74
this comparison, the annual motions in north polar distance, as given in the last catalogue of Dr. Maskelyne, have been used. These certainly are in several instances inaccurate from the proper motions used, and to this may be attributed some of the differences between 1809 and 1813, but it is by no means a sufficient explanation as to others. In the case of /S Leonis, particularly, there appears a difference that I cannot attempt to account for. Considerable differences be tween the results of observations of the same star when se- parated by several years have, however, been before ob- served in several instances, and yet remain to be accounted for. A comparison of the means of the results of the ob- servations of Dr. Hamilton, at Armagh, M. Piazzi, at Pa- lermo, and Mr. Pond, at Westbury made about the. sfime period, (Phil. Trans. 1806) and of the present results of the Greenwich, and of our instrument, furnishes a striking in- stance. A comparison some years hence of the present results
and of new ones obtained by the, same instruments will pyo-
. uiJ ca ii 4*, V^ a= *^t 4-
bably clear up this pomt. ., ,^
■It mav also be remarked that the observations in 1809
were computed by Bradley's refractions, and also no, '^tt^^-
tion was paid to the circumstance of parallax. The resujts
of 1813 are from, observatibns made when the zenith dia-
tjance^ from the, effects of parallax were greatest and! least.
...jflence also perhaps may be explained part of the differences
k) cQlunrin JP.
75
In computing my observations I have used max. aberra- tion of light = 20",00 Lunar nut. inN,P.D.=^S" ,'iS sin. {AR — Long. moon*)s node)
+ 1,22 sin. (^R+Long. moon's node) Solar nut. in N.F.D. = 0",48 sin. (2 Long. sun—^H.)
M 2
Ajialytkal investigations respecting ASTRONOMICAL RE- FRACTIONS and the application thereof to the formation of s convenient TABLES together with the results of observations of circumpolar Stars, tending to illustrate the Theory of Re" fractions.
Bij JOHN BRINKLEY, D. D. M. R. I. A.. F. R.S. andi ANDREWS' Professor of Astronomy, in the University of Dublin.
Read May 9, 1814;
A BRIEF detail will explain the objects of this paper.- M. Le Comte Laplace first shewed that the fluxional expres- sion for refraction may be integrated by approximation, as = far as about 74° from the sjenith, without a knowledge of the variation of density in the atmosphere. *
T. Simpson had deduced by the princi plies of the 8th sec-- fion of the first book of Newton's Principia, the fluxional* expression for refraction, by considering a particle of light: as a body acted on by a force tending to the centre of the earth .-|- He and others since deduced the integral on the hypothesis, that the density of the atmosphere decreased.
* M^c. c^. Lit. JO. c 1. toni. 4; f Math, Dissertations, p. 51, &&
78
ijfiilR)rhir>\^-TlS^mV^^ lo^ftti of the integralis that used by Bradley. ■ •■"'•' >•'■->» JVivy*;. :>..» i.-.^ ^
Laplace uses the sahfie' liietbod of oblkfnTng the fluxional equation as Simpson had done, and then proceeds to investi- gate tho laws of reflection and refraction. He deriveb y an analytical process . the conclusions, which Newton had de (luced in the 14th section of the first book of the Principia. Laplace next dtrrivGs his fundamental fluxional expression for refraction which he shews may be integrated as far as 74° from the zenith, without a knowledge of the variation of density in the atmosphere.
In this paper the same fluxional expression, that Laplace obtained, is deduced by a very short method, and by using the common principle of the given ratio of the sines of in- cidence arid refraction. Besides the simplicity of the inves- tigation it has the advantage of avoiding hypothetic prin- ciples lespecting the rays of light.
The integration of the fluxional expression is also obtained by a method that may be considered as entitled to notice. If the surface of the earth were a plane, then whatever the law of variation of the densities of the different strata of air parallel thereto might be, the refractiou for any zenith distance would be simply found from the knowledge of the refractive force at the surface, by tbe constant ratio of the sines of in- cidence and refraction. By the method given this part is separated from the rest, and the effect of the spherical form of the atmosphere is shewn. The formula for refraction
, 79
consists of two parts, one the refraction that would take place were the earth a plane, the other the effect due to the spherical form. The latter at 80° zenith distance amounts only to about 12", and at 40° zenith distance is in- sensible.
It is shewn that at 80' 4o' the error of the formula deduced cannot amount to half a second, whatever be the variation of density in the atmosphere.
;As the approximate formula for refraction as far as about 74° from ,the zenith is independent of the law of variation of dejnsily, it follows that, whatever law be assumed, the same conclusion ought to be deduced as far as about 74°. This is shewu from direct investigation by assuming different laws of variation of density ; which beside affording some conclu- sions useful in our euauAfipS P», this subject, may be consi-
,^^pi;ed..api.u.terestiug., ^ .,
,,i.,Xl^(^ rpsult? Qf th^ .ex4>eriment8 of M. M. Biot & Arago ,pr; the r<Efi;£^cUy.e.fprce. of ,^ir, and, of Mr. Da,ltou and M. Gai-Lussac on it,lieteflects of the change of temperature on the density of air^re applied, aqd a general expression for refrac- tion .at ,a,ny. zenith distance Jess than about 80° obtained,
J which is entirely independent of astronomical observations..-
From this general ex pressioii , I have .formed two tables, by help of which th.Q refijaction at any zenith distance less than 80° may be calculated with much convenience.
From a, comparison of the co-latitude determined by stars
■ftf'M;.'^teiPyi^» iPifi^:^,^,;^'^^ s^ni,^^ ^determined by star^im(^re re- ihno'i ' /d fl^vi.^ ahtikoijiibiitii atli wiai 'lo
80
mote, I find, by o25 observations of circumpolar stars, the refraction at 45°, (Bar. 29, 60 inches and Therm. 50".)
= 5?",42 .
The same by the French Tables - - = 57,57
The same resulting from the direct experiments on the refractive force of air, applied to the formula. - - = 57,67
The quantity in the French tables was ascertained from the resuhs ot the observations of M. M. Piazzi & Delambre, applied to Laplace's formula by Delambre himself.
My result from the number of observations, from the care ■used in making them, and from the excellence of my instru- ment, seems entitled to as much confidence as can be given to a conclusion derived from observations of circumpolar stars, and there is no difference worthy of notice between my result and that of Delambre. But from the nature of the direct experiments on the refractive force of air, the results seem capable of greater exactness than can be derived from observations of circuiri polar stars, and therefore strictly perhaps ^ve ought to adopt the result so deduced. However the quantity in the French tables is so nearly equal to this that no inconvenience can arise in the nicest researches in astronomy from adopting these tables.
It is of much importance that the same tables of refraction should be used by astronomers, and it will afford satisfaction to the author of thjs paper, should it in any manner conduce to this desirable end. It cannot be doubted but that sooner or later the refractions as given by the French tables as far as
81
80^, or a very slight modification thereof will be generally used by astronomers.
The form of the French tables may not be generally adopt- ed, others more convenient perhaps may be derived. The new form given in this paper ^vill serve as a check in the use of the French tables, and may be thought more con- venient than these for observations of the- sun, moon and planets.
Below 80° zenith distance, a knowledge of the law of va- riation of density is absolutely necessary for computing the quantity of refraction. As this cannot be had, all tables for these zenith distances must be in a manner empirical. The French tables are less so than any others, from the method used by Laplace. But the quantity of refraction varies so much from some unexplained cause, the heights of the barometer and thermometer remaining the same, that observations below 80° can be of little use. This irregularity IS very manifest at 80° 45* in the observed refractions of Capella below the pole. Sixty-five observed refractions of this star are given, and compared with those computed from the formula.
Forty-two observed refractions of x Lyrac below the pole, (zen. dist. 87° 4^',) are also given. In these the irregu- larities of refraction are very considerable. The mean of the observed refractions serves for shewing that refraction is greater than would result from a density decreasing uni- formly, and less than would result from a uniform tem-
VOL. XII. N
82 "
perature. The mean also serves as a criterion of the accu- racy of the French and of other tables at this zenith distance.
Investigation of the Jluxional equation for refraction.
Let V RP T be the path of a ray of light refracted at P and R, and let CO be perpendicular to TP produced.
(Fig.)
Let the apparent zenith distance HVR =6*
C V the radius of the earth = a
CR = r' CP = r The density of the air at P = § The density at the surface V =(f) The height of an uniform atmosphere at F = I Let m : 1 represent the ratio of the sine of incidence to the sine of refraction, when light passes from a vacuum into air of the same density as that in VR.
k' : 1 the same ratio for air of the density of that in PR, and k : 1 the same ratio for air of the density of that in TP Then it readily appears that sin. VRC : sin. CRP:: k' :m sm. C PR : sm. CPT ::k:k'
• The same quantities are denoted by the same' letters which Laplace has used (chap. I. liv. 10. torn. 4. M6c, c^l.)
83 Consequently
asin.CVR=r's\n. VRC =^ — sin. CRP=.-^ s\n.RPC
m tn
= -^ sin- OPC.
m
am
Hence sin. OPC =-,;;: %\r\. $. (i)
This equation is evidently true, whatever be the cumber of points of refraction between P and F, and therefore is true when FRP is a continued curve as in atmospherical refrac- tion.
The refraction H, that takes place between P and V = the inclination of the lines P T and R V. Hence
By equation (1) OC = j- sin. 0.
The refractive force of air is as its density, and the refrac- tive force in TP is also as >f * — 1, (vid. Newton's Optics, book 3, Prop. 10. Horsley's edition, vol. 4, p. 171.)
Therefore let 6 ^ = A;* — i, h being a constant quantity Then k=VT±Tf and m == ^/ T+T(f)
Hence OC=a sin. 6 ^'+^(>>
^^ l+bf — 'LL sin. d 1 + 4 (p))
and OP =r ' , ^ —-^
Therefore Ji' = g-
2{l + fi,) r v/l + *f- — 8in.«9(l+6(f)) _ (2)
N 3
d4
This is Jyaplace's fundamental equation (3) vid. Mec. Cel. torn, 4, \). 244. b here corresponding to -^ iu Laplace's formula.
2. The integral of this equation from g = (f ) to § = 0 gives the atmospherical refraction required. It is obvious tliat to obtain the complete integral, it is necessary to know the relation between r and §, or the law of diminution of the density of the atmosphere. This is at present unknown ; but notwithstanding, we can approximate sufficiently to the value of R for all values of ^ less than about 80°.
From the zenith to 74° zenith distance the result is the same whether we approximate to the integral, without know- ing the relation of r and §, or whether we assume any given relation, and reduce equation (2) to a convenient form for fipding the integral.
Also by assuming two certain laws of variation of density we may obtain two integrals, one of which must give the re- fraction greater than the truth, and the other less. We find that as far as 80° 45', * these refractions do not differ by one second, therefore a mean of the two must always give the refraction true within half a second so far from the zenith.
* The apparent zenith distance of the bright star, Capella, when below the pole, is in this latitude =i80''45', and having made many observations of this star S. P, I have taken that zenith distance as a limit.
85
Jipproximate integration of the Fluxional Equation.
3. Let Q represent the refraction that would take place if the surface of the earth were a plane, and the different strata of air parallel thereto, in which case the ratio of a to r would be the ratio of equality. Therefore equation (2)
— pi sin. e ^ I ^b (p)
feecomes Q = ^Trb7T7T^f^^^^^T(^)^^7T=^ ^ (3)
Hence R ^ J^V^/-izr_(JL±^'^) ) ^"lll-
r ■v/ I -f i p _ /i ^- i (p) \ ^ sin. « 9^
Let-^=1~. W
r
:w ^, • _ Qji— «)
Then JR — -j======-— ———.-— -^ rr— (5)
'1 ^bf~n-\-b{f)\ sin. ^9
or jR = Q (1 — s) (1 — s tan. ■= ^) = Q — -^^ neglecting the
second and higher powers of s, also §, (§) and their powers* It is obvious that for the part of the atmosphere which makes the refraction sensible, 5 must be very small.
By equat. (3)
Q = — -J /> p. tan. 0 neglecting §>, (§) and their powers.
• Hence ii = Q + f^rh~ nearly. (6)
Now/^s= gs— /gs = is—f~~ <by equat. 4.)
86
Let p = the pressure of a column of superincumbent air of a given base, at the distance r from the centre. Then the. pressure of a particle of air being measured by its magni- tude, density and gravity, supposing tlie gravity at the sur- face represented by unity
f'ra*
Ilence R = Q + (§8+^) l^-t Constant, when /i = 0, Q and s = 0 and p = I (§),
Therefore constant = _*illi^® = — {m^-i) itan.s Anereiore constant — ^^ ^^^ , ^ — ^ „ cos. ' e
<Jonsequently the whole fluent from g- = (|) to ^ = 0 is R == Q — (m'— I) itan.i bccausc w is nearly = unity
M ^ Q — ijUzilUJ^Hd. (7)
a COS. * 9
This expression as will be shewn farther on can be easily reduced to that of Laplace (M6c. c61. torn. 4. p. 268.) But it remains to shew how far from the zenith it can be used with- out inducing an error greater than a small fraction of a second.
4. The principal part Q of this expression is, it is evident, the deviation of a ray of light refracted at a given incidence 6 from air of the density (g) into a vacuum, and hence is en- tirely independent of the variation of density in the atmos- phere. When mis known Q is known. The method of find- ing m will be considered hereafter.
87
The seconds in the latter part of the expression = (;„— 1 ) tan. 9 ^ rp^ compute this quantity it is necessary ta
a COS. * 9 stti. I *■ A ^ ,1
know 7/?, / and a but not with much precision.
If we take & = 80' and use, for the present, round numbers,
^ I • 1 r>r\f\c> „J ^ Smiles I (m — I) Itan.Q ■, .,,
lakmff w = 1,0003 and— = -rr— = -— -, i rr-^—r, = 14
o ' a 4000 800 a cos. • d sm. \"
nearly. The terms which have been neglected, must obvi- ously be much less. The limit may be thus computed.
Let the equations (S) and (5) of the last article be ex- panded, neglecting products of three dimensions of s, f and (^) and we shall obtain
(0-f))
Now of the terms that compose the factor of ^ ''''"', the
'^ 2 COS. » r
first 5 has already been considered and found not to produce in integrating a quantity greater than a few seconds, as far as ^=80° ; therefore after integration, the 2d and 4th on account of the smallness of b (§) and b § must be quite insensible; but
the third— i^ s ' tan. ' 6, will produce a term f _ ^f^"'^""''? =«
S fbs^ tan. ^6 ^ 3 p 6 s'stan. ^ g .
T7os7^ if 2co:i. » e
The law of decrease of the density of the atmosphere is
between that which a uniform temperature gives, and that of
the density decreasing uniformly, as will be shewn further
■on. The true value of the above integral, wiU therefore he
8&
between the values deduced from an uniform temperature and an uniform density.
(1) For an uniform temperature. The density on this hy^ pothesis is as the compressing force, and we have the well known equation
( — -1 ) -f
^ = (^) c ^ where c = 2,7 12S &c.
or f = (f) c
/_ as
as
^ ssc= — — sc — -re + «» trom s = o Therefore from s = o to s == i and from g = {§) io § = o
—a
^_j2illi«i^=Hllll^- il having taken c ~= o on
»/ 4co«. »8 '2 COS.' 9 a* =»
account of jts extreme smallness, it being = — __ —
I \ oOO
V 2,7128^
whence the term in question produces a quantity in seconds=
S I - {m—\)tan. ^9
(I " CO*. * 9 MM. 1
//
Taking tf = 80° 45', — and m as before
this quantity = 2",60 Taking fi = 74'
It a= 0", 16 a quantity not requiring notice.
89
(2) If the density of the air decrease uniformly, it wiU be proved that
s == llLzJL ^ IL nearly if) «
R^»o^ /• ^fl'^'t""-'^ _ /• 3?btan.3Q f (f)—p) Y I' ncnLt,J T^~Q y~~ COS.' 6 ^ if) ^ a'
= [from ? = 0 to ^ = a] 'M^tlx^ = [ia seconds]
2 (j«— 1 ) ; » fare. 3 fl
a ' COS. * 9 si)i. l"
Taking ^ = 80° 45' this quantity = r',73. Consequently the true value of fUhlI1^<^ is between 2",60 and i",73
c/ 4 COS. * 80° 45' ' "
and therefore the mean cannot err quite half a second from the truth, and so the following formula may be considered as giving the refraction as far as 80° 45' true to less than half a second, viz.
Refraction = Q - S'!!ii}lL^±± + lin-zllLl''±lI, (7)
a COS. « e sin. 1" 2 a * cos. * 8 sin. V ^' ^
The third term is insensible when ^ is less than 74° and the second and third insensible when 0 is less than 40°
It is evident that the two first terms must be derived from assuming 0711/ law of variation of density, and then investi- gating the quantity of refraction as far as these terms. The following investigations in different hypotheses of density may be considered useful.
VOL. XII. o
V 0 + %) .UHi UiH
90
Hypothesis of uniform density.
5 Let CR be the radius of the uniform atmosphere, the height of which is / (vid. Fig.)
^ = angle of incidence at the point R; t = VRC^ then
ref. iR') =^ O' — t, and -~ sin. 6 = sin. t = ^^- (1)
Hence am sin. ^ = (a+0 sin. (/ + E) (2)
but supposing the surface of the earth a plane
ms'm.6 = sin. (^ + Q) (3)
Hence sin. ^ + E) = ""' ^'+J^ (4)
a
making Z, ^ and R to vary, in order to apply Taylor's
Theorem.
'^ ' Byequat. (4)
O+R) cos. (t+R) = -_ ~/ I ^» sin. (^+Q)
By equat. (I)
f G0S.4 ^ ""' i' . sin. ^
a(l+__)
Hence computing R+R + 8cc. making R= Q, t==6 -^o and then — = -^, we have by Taylor's Theorem
R = Q _ I- ( tan. (^+Q) — tan. ^ ) + &c. (5)
But tan. (^ + Q) = tan. & + -^^ + &c.
|
—I |
|
|
s=— — — — |
-) T |
91
Also making m and Q vary in equation (3) We get by help of Taylor's Theorem Q = Qn — 1) tan. 6 &c.
Hence substituting in equal. (5)
li == Q (_w— ) tan. ^^ ^^^ found before in art. 3.
^' a COS. ^ 9
Hypothesis of density decreasing uniformly.
Q. 13y the density decreasing uniformly is understood, that the density is as the distance from the highest part of the atmosphere. It is obvious that in this hypothesis, not taking into consideration the variation of gravity, the height of the atmosphere will be double of that of an uniform atmosphere of an uniform gravity. And it is also obvious that the effect of the variation of gravity can be but small. Lest however there should be any doubt on this head, it will be safer to investigate the height of the atmosphere on this hypothesis, gravity being supposed to vary.
Let this height = t
the pressure at any height z =p
the pressure at the surface = (p) a, /, g &c. as before.
Then p == 7^-7x1-} the gravity at the surface being re- presented by unity.
o 2
92
On this hypothesis.
- Therefore^ = ^^,1^ and by integration, /. = Mil. + qiL h. log. ia+z-) + -7^, + comt.
Hence this integral from z == I' to z = o gives
The right hand side of this equation being expanded ac- ' cording to the powers of — there results
ip) = (f) (^ - £- &^-)
but (p) = (?) ;
Hence is easily deduced I' = ^l -h y^ nearly
Having obtained I' we immediately deduce by equal, (i) the relation between § and r on this hypothesis,
Whence -^ = 1 + ^-^ (^ + -^) or regarding
only one dimension ot — '— =1 ^^ x a ^ '
11 or _1_ = (l±h-\ 6(r)a 5 being introduced to form the
r V1+6CpJ/
factor 6 f .
93
''•Let l + ig= X, 1 + i (f ) = (*) and jii^ = / Then equat. (2) of art. 1 gives ]^ _ — XX sin. 6 (x) ^
2 {x) -^ x\/ x--(llflL sin.' S
2/- 3
— XX Sin. t*
(x)^^
-i£=iiW ; 2/-1
This by integration gives
R = L_ (Circ. Arc. rad. 1 and sin. = (^)««.« ) +
constant. When R = 0, ? = (e) Therefore constant = ^yr-^ 6.
Hence the integral from ^ = (f) to ^ = o gives
13 _ ^ & — -L- CCirc. Arc. rad. 1 and sin, =
( i+Mp))-' or nearly
""•! — = sin. (^ — (2/— 1) R)
This is equivalent to Simpson's Rule, page 58, Math. Dissert.
94
By the well known analogy between the sum and difF. of the sines of two arcs and the tangents of the i sum, and T difF, the equat. (3) gives
Tan. ^ R = i-( Mzzlj b (J) ta„. {J - -£=1 R) (« orR=i-(,)tan.(*-(j|l,-i-(R) =
From equation (5) we may obtain the same conclusions as in art. 3. '
For if the surface of tlie earth were a plane, equation (5) would become
Q = {m — 1) tan. (^+ \ Q) nearly
Also because R and Q are very nearly equal at all zenith distances less than 80°. By equat. (4)
R = (m— 1) tan. (^ + iQ— /Q).
From this equation it readily appears that
R= (m-l)tan. (^+i-Q) ^\IS—
Therefore R = Q (^-i)^J°"-fl, as before in art. 3.
^ a COS. * S
^ ■-.'■. .
^ ^The formula used by Bradley \% R=.k tan. (9 — nR). He determined n from the comparison of the horizontal refraction, and the refraction at a given altitude. This would be exact if the density of the atmosphere decreased uniformly. But k and thence n may be determined by direct experiments on the refractive force of air, and also by observations of circumpolar stars at zenith distances not greater than 80°. With these values oik and n the refractions at the horizon and low altitudes may be computed, and are not found to agree with observations, therefore the density of the atmosphere does not decrease uniformly.
95
7. Remark. This last conclusion might have been very easily deduced from equat. (6) art. 3; but the above investi- ' gation has been used for the sake of deriving the formulas of Simpson and Bradley.
By equat. (4) art. 3 s = 1 — —
Therefore, by equat. (2) art. 6,s= IfizLLx H. Hence by equat. (6) art. 3.
Q b I (p) tan. 9 __ q (m — \)ltan 9
^ '2 a COS. ^9 ^ a cos. > 9 '
Hypothesis of an uniform temperature.
8. By the equat. (6) art\ 5 we; also derive the same Cbh- clusion on the hypothesis of an uniform temperature, in which case, as has been stated art- 4*
—as '•: '; { v^ , ^ as
f = (f) c ' or /J = — f- (i) G ^ Hence by equation (6) art. 5.
/' As
^ 0 c - — f'''^=(froms=otos=l)
2 COS. ^ 9
before.
lU)itanV-'iJ}^ _, ,. ^ _ n — fcl>iifl! as
96
Reduction of the formula for refraction to one convenient for computation. — Comparison with Laplace's formula.
9. From the equation which takes place, supposing the surface of the earth a plane. Viz. m sin. 6 = sin. C^'+Q) We obtain, making m constant, m sin. ^ = Q cos. (^-HQ)
0 = Q cos. (^ + Q)— Q ' sin; (^+Q) Hence making Q = 0 and then w = m — 1 we have by Taylor's theorem
Q = (m— 1) tan. 6 + JH^JH- tan. ^ ^ + &c.
taking m— 1 = ,0003 and 6 = 80'. 45'
(in — 1) ' tan. ^ 9 n't 1
^Tin. 1" '
the following terms are therefore insensible. Hence substituting in equat. (7) art. 4. We obtain for all values of 6 less than about 80°.45'
R == (>»— 1) t<^n. fl {m—\)ltan.Q , 5 (m—\) I ' tan. ^ 6
Im. V a COS.'' 9 sin. 1'' 2 a- cos. ^ 9 sin, 1"
(ni—\) ' tan. 3 ^ /-JN
+ 2«"». 1''
The two last terms are insensible except when 0 is
nearly 80'.
10. The formula of Laplace (p. 268. tom. 4. Mec. celest.)
I
in seconds of a degree = -^i—[^i A + ——g >
in whicli « =: -l^M- = i^^^'>
But '^''^'^ = "''~^
Therefore expanding "^^^ by the powers of m — 1
« = (w— 1) — i (m— 1) ' &c. substituting this value for ctin Laplace's formula. „o,r^,
-n_/? (»i — 1 ) tan, fl (w — 1) I toM. & (m — I) « <flK. 3 9 ,
* "~ wi. 1' a COS.* 9 sin. t" ' 2 «». 1" ' ■■* ^
same as equation (i) article 9, excepting the term there in- troduced to make the formula applicable as far as ^ = 80 ".45/
Value of ^li, and of — -Tables of Refraction.
11. The refractive force of air being assumed proportional to its density, the value of tn is variable, and its changes are known by the variations of the barometer and thermometer.
Let m be the value of m when the height of the baro- meter = 2^,60 inches, and the height of Farenheit's thermo- meter =?= 50°. Let also b represent the height of the baro- raeter, and t the height of the thermometer corresponding
t^P ^"" .ftw bauo] XI r .iifi i! ao-fo't'o7ijo
j^. It appears by the results of the experiments of Dal ton and Gay Lussac, tliat a column of air denoted by unity at the temperature of 32' of Farenheit becomes l,37Ji at the voi. XII. r
98
temperature of boiling water. In fact this agrees nearly with Mayer's conclusions made long before. It becomes there- fore for t degrees of the thermomeicr = 1 +,002083 (J — 3'2) It is not probable that the ratio of expansion is sensibly changed at different heights of the barometer within the limits of its usual variation. That is the ratio of the volum&s at 32° and 212° is the same when the barometer is 58t inches as when 30i inches.
The increase of height in the barometer from the expan- sion of mercury? by increase of temperature may be consi- dered To^o - for every degree of the thermometer.
Hence m — 1 : m' — 1 : : density of air bar. b and therm, t : density bar. 29,60 and therm. 50° : : b x j i __ (^_50) x ,0001)
X 1,0375 : 29,60 x ( 1 + ,002083 (f— 32)) Thereforewi— l = (m'— l)x 2^ x (1— (^— 50),000l)x
1,0375 ,
1+,0020«3 (<— 32)
The quantity (m' — 1) may be deduced from the experiments of Biot and Arago (Mem. Inst. tom. 7) ^^l^o have most carefully repeated the experiments of Ilawksby, or who ra- ther by a different process, have accurately determined the refractive force of air. They have found when the height of barom. = 0,76 metre and centesimal therm. = 0 that is barom. 29,93 inches and F. thermometer = 32° that w— 1 = ,0002946.
99
rj , , ,0002946 X 29,60 nnrtoono
Hence m— 1 = i,ooi8 x 1,0375 x 29193 = .0002803
And J:±=L = 57',82.
The height of an uniform atmosphere is not affected by the variation of the barometer, and therefore, if t represent the height of an uniform atmosphere, the thermometer be- ing 50°
7 J, l-{- ,00208 (t— 32)
I — IX j-^3j^
v a very accurate value of — is not required. If we take
I' = 5 miles and the semidiameter of the earth = 4000 miles — = ,00125. This in fact is sufficiently accurate.
But it will be more exact to take ^ == 5,095 miles and the semidiameter of the earth = 3979 miles, and then — =
^j22L = ,00128. 3979
It is evident that the third and fourth terms of the value of
the refraction in equat. (1) art. 9- cannot be sensibly affected
by the variation of ;«, and therefore its mean value may be
used as to these terms.
Hence substituting for '^^, and — in equat. (1) art. 9.
Ave have
Refractions -^-oo^^'-^j:^ x (l_,000l{^-50)) ^-^^^ x
57^82 tan. ^--A_ x 0",0739 ,^, + 0",000238 ^^ +
0^0080 tan. ' 6.
p2
ICO
It is worthy of notice that the second term is independent of the thermometer, this circumstance enables us to put the three last terms into a very convenient table, the a.rguments of which are the zenith distance and height of the baro- meter.
12. The above expression for atmospheric refraction is en- tirely independent on astronomical observations.
The French tables are derived from observations of cir- cumpolar stars. By these tables the refraction at 45° = 57",57 when the barometer shews 29j60 and Farenheit's ther- mometer-50° Hence by equat. (1) art. 9« ; :,•!;; i^\;U; 57",57= ^^^^ (1-— ) = -^(0,99744).
' ' • sill. i. ^ a ' sin.V \ '''•^ • /
Therefore '4^ = 57^72.
sin. \
Ey c25 observations of circumpolar stars made by myself with the ei«ht feet astronomical circle (vid. art. 14.) I de-
duce '^ = 57'',56.
;j •;; sin. 1'/ ' '
• Thus the value of 'i=^ by the French tables is between the values resulting from direct experiment and from my observa- tions. 1 am inclined to give the preference to the result from direct experiment for reasons afterwards mentioned. But the difference between this result, and that from the French tables is so small that no inconvenience can occur in adopting the French tables. Thus, bar. 29,60 inches, and Farenheit's therm. 60"-
101
Zenith distance.
45 50 60
70
74
Refraction deduced from the . experiment.
57.7 08,7 99,7
157,3 198,6
Refraction by
the French
Tables.
57^6 68,6 99,4
157,0 198,2
.y\
Therefore, as it is of considerable importance, particularly with a view of comparing observations made in different places, that the same refractions should be generally used, no objection, I apprehend, can be made to the general adoption as far as about 80° of the French refractions which are now so well known,
13. Perhaps the following tables deduced from the above formula, may be considered rather more convenient in many instances than the French tables; they will certainly furnish a useful check. The advantage they afford is derived from the faciiity with which the computation can be made by help of tables of logarithms and of logarithmic tangents to four or five places of figures, such as are in the " tables requisite to be used with the nautical ephemeris." By these the log. tan- i>ent of the zenith distance can be taken out at once, and the inconvenience of proportioning for the minutes of zenith dis- tance avoided, which is greater than the new inconvenience occasioned by the second table. Hence the tables here given
102
may be considered more convenient for observations of the sun, moon, and planets.
In computing these tables 57",72 was substituted in the above formula instead of 57",82, and therefore the refraction deduced from these tables will agree with those deduced by the French tables.
103
TABLES FOR RETRACTION.
Table 1.
Table 2, Barometer.
|
Far. ! Therm. o I |
Logarithms. |
Far. 1 I'herm. o 1 |
Logarithms. |
Far. Therm. o |
Logarithms |
|
10 11 12 |
0.3283 0.3273 0.3263 |
34 35 36 |
0.3048 0.3039 0.3030 |
58 59 60 |
0.2827 0.2818 0.2809 |
|
13 14 15 |
0.3253 0.324-3 0.3233 |
37 38 39 |
0.3020 0.3011 0.3001 |
61 62 63 |
0.2800 0.2791 0.2782 |
|
16 17 18 |
0.3223 0.3213 0.3203 |
40 41 42 |
0.2992 0.2983 0.2974 |
64 65 66 |
0.2773 0.2764 0.2755 |
|
19 20 21 |
0.3193 0.3133 0.3173 |
43 44 45 |
0.2965 0.2956 0.2946 |
67 68 69 |
0.2746 0.2737 0.2728 |
|
22 2+ |
0.3163 0.3154. 0.3144 |
46 47 48 |
0.2937 0.2928 0.2919 |
70 71 72 |
0.2720 0.2711 0.2703 |
|
25 26 27 |
0.3134 0.3124 0.3114 |
49 ,50 51 |
0.2910 0.2900 0.2891 |
73 74 75 |
0.2694 0.2685 0.2677 |
|
28 29 30 |
0.3105 0.3095 0.3086 |
52 53 54 |
0.2881 0.2872 0.2863 |
76 77 78 |
0.2668 0.2660 0.2652 |
|
31 32 33 |
0.3076 0.3067 0.3058 |
55 56 57 |
0 2854 0.2845 0.2836 |
79 80 81 |
0.2644 0.2636 0.2627 -L . — |
|
Z. D. |
28,50 |
29,00 |
29,50 |
30.00 |
30,50 |
|
o |
'■ |
H |
// |
" |
ii |
|
80 |
10,5 |
10,7 |
10,9 |
11,1 |
11,* |
|
79 |
8,1 |
8,3 |
8,5 |
8,7 |
8,9 |
|
78 |
6,3 |
6,4 |
6,6 |
6,7 |
6,9 |
|
77 |
5.1 |
5,2 |
5,3 |
5,4 |
5,8 |
|
76 |
4,1 |
4,2 |
4,3 |
4,4 |
4,5 |
|
75 |
34 |
3,4 |
3,5 |
3,6 |
3,7 |
|
74 |
3,0 |
3,0 |
3,1 |
3.1 |
3,2 |
|
73 |
2,5 |
2,5 |
2,6 |
2,6 |
2,6 |
|
72 |
2,1 |
2,1 |
2,2 |
2,2 |
2,2 i |
|
71 |
',8 |
1,8 |
1,9 |
1,9 |
1,9 |
|
70 |
1,5 |
1,5 |
1,5 |
1,6 |
1,6 |
|
69 68 |
1,3 |
1,3 |
1.3 |
1,4 |
1,4 |
|
1.2 |
1,2 |
1,2 |
1,2 |
1,2 |
|
|
67 |
1,0 |
1,0 |
|||
|
66 |
0,9 |
0,9 |
|||
|
65 |
0,8 |
0,3 |
|||
|
64 |
0,7 |
0,7 |
|||
|
63 |
0,6 |
0,<5 |
|||
|
62 |
0,6 |
0,6 |
|||
|
61 |
0,5 |
0,5 |
|||
|
60 |
0,5 |
\ |
0,.5 |
||
|
58 |
0,4 |
0,4 1 |
|||
|
56 |
0,3 |
0,3 |
|||
|
54 |
0,3 |
0,3 |
|||
|
52 |
0,2 |
0,2 |
|||
|
50 |
0,2 |
0,2 |
|||
|
45 |
0,2 |
0,2 |
|||
|
40 |
0,1 |
0,1 |
|||
|
30 |
0,0 |
0,0 |
|||
|
0 |
0,0 |
0,0 |
Logarithm in Tab. I. + log. barom. + log. tan. zenith dist. = log. ap- proximate refraction.
i\ppr. ref. — Number Tab. 2. = refraction.
Example. Zenith dist. 71°. 26', barom. 29,76 inches and therm. 4S°.
Log. Tab. 1
Log. barom. -'^
Log. tan. 7 1 ".26
Ref. 173,4 =:2'.53",4
Log. approx. ref. 175"4 - 2.2439
0.2965 1.4736 0.4-738
Appi'. ref. 175' ',4 Tab. 2. 2, 0
104
The Co-latitude of the Observatory of Trinity College, Dublin, deduced from Observations of Circumpolar Stars, by different Tables of Refraction.— Observed Refractions of Capella^ be- low the Pole.
14. Comparisons of the Co-latitude as determined by stars near to, and remote from the pole, serve for a criterion of the accuracy of the tables of refraction used.
In the following table the co-latitude is determined by four different methods of computing the refraction.
1. In column A, by the formula 56", 9 tan. (^— 3, 2 ref.)
bar. 500
^ 29,6 ^ ^50 + therm. '
2. In column B, by the formula d&',9 tan. (^—3 ref.) >;
bar. 400
29,6 350 + therm.
3. In column C, by the preceding tables, which give the same results as the French tables.
4. In column D, by the value of "^ = 57",82 as de-
' •' sin. 1
duced from experiment.
The second formula is Bradley's.
The first formula is what appeared to me by my observa- tions in 1 809, to give the refraction at low altitudes more ex- actly than Bradley's formula, and also to give the effects of the changes of temperature more exactly.
105
But both these formulae must be considered empirical. W<i are entirely unacquainted with the law of variation of den- sity at different heights, and therefore as has been shewn we cannot deduce from theory a formula of refraction that will serve much below 80°. It has been shewn indeed, art. 6. that if the density decrease uniformly, the refraction may be expressed by a similar formula, and that above 80° the re- fraction will not be sensibly changed by any law of variation of density ; but then if 56",9 be the constant quantity, the co-efficient of refraction must be 4,l4,* that is the mean ref. = 56",9 tan. (^ — 4,14 ref.) Therefore the two formula used in columns A and B are certainly inexact for all zenith dis- tances less than about 80°. For greater zenith distances, the first formula will perhaps be found as exact as any other now known, at least as far as 87° 40'. But I do not attach much importance to it. I had deduced it before I was so well convinced as I am at present of the little value of ob- servations near the horizon, and I may add of the impossi- bility of investigating an exact formula.
The mean of column C gives 36'. 36'. 46",54 for the co- latitude of the observatory or 53 23 13,46 for the lati- tude, which I conceive cannot possibly err | of a second from the truth.
* For if -^—jn ss 56",9 m— 1 -: ,0002758, and therefore ^(m-i) T =^ *'**
vid. art. 6. equat. (5)'
VOL. XII. t Q
106
The co-latitudes are each determined by a mean of the number of observations of each star above and below the pole as annexed.
|
Names of Obs. Circumpolar i above Stars Pole. |
Obs. 1 Co-lat. below 1 Pole ' A |
Co-lat. B |
Co-lat. C |
Co-lat. D |
||
|
Polaris ^ Ursae min. /S Cephei |
62 20 10 |
74 ! 36 . 36 . 45,65 18 j 46,18 10 ! 45,43 |
30.36.45,71 46,42 45,64 |
36*. 36. 46" 19 46,77 46,37 |
36°. 36, 46,26 46.85 46,46 |
|
|
a UrssB maj. a Cephei j3 Ursae maj. |
10 10 21 |
8 9 21 |
46,91 45,56 45,49 |
47,19 45,71 45,^95 |
' 47.33 46,42 46,62 |
47,44 46,53 46,75 |
|
! Ursae maj. 24 a Cassiopeae 21 f Ursae maj. 8 |
23 45,21 r 23 45,90 10 45,10 |
45,52 45.81 45,30 |
46,22 46,64 46,00 |
46,36 46,79 46,15 |
||
|
7 Ursae maj. 7 Draconis « Ursae maj, a Persei |
18 32 10 10 ■ |
21 32 10 10 |
45,81 45,93 44,90 44,53 |
46,18 46,69 .. . 4.5,40 44,35 |
46,85 47.08 46,20 46,29 |
46,90 47,27 46,40 46,51 |
|
Mean 1 2.56 |
269 36 . 36 . 4J,58 |
36.36.45,84 1 36.36.46,54 |
By 226 observations in 1808 and 1809 I had deduced for column A 36°. 36'. 45",65 B 45 ,85
-i C 46 ,54
15, Let c r: the correction of 57",82, that is, let
w'— 1
an. I"
= 57",82-i-c,
then by comparing the co-latitudes in column D determined by Po- laris, jS Ursae mi noris and j8 Gephei with the same determined by the other ten stars, we have
107
36°. S6'. 46",52 + ,82 c = 56. 36. 46,71 f- 1,56 c. Tlie co-efficients of c are o+)tained from the tangents of the re- spective zenith distances.
0" 19
This equation gives c ^ ^T = r 0",26
and therefore T~-- = 67^56'. By which, the mean refrac-
tion at 45° = 4=i- f 1 ~ ^Jl) = 57",42
Now from the number of observations used, it cannot be doubled that the above conclusion is free from the errors of observation. The only error by which it can reasonably be supposed affected, is that arising from errors of division.
It is difficult to state the limit of error from hence arising, but it wdl readily appear that much dependence cannot be had on a correction so small as that which I have deduced. For each star or each co-latitude, 12 points of the circle are used so that the quantity 36°. 3&. 46",52, the mean of the results of the three first stars is affected by the mean error of 36* points of divisions of the circle. This mean error must certainly be very small. Yet it is not improbable that it may amount at least to 0",15. ,
The error of the quantity S6°. 36'- 46",7l must be smaller, being only affected by the mean error of 120 points, yet it is not improbable it may amount to 0",04 and so the whole quantity 0",19, the numerator of the value of c, will be ac- counted for,
Q 2
108
Tims it ap|>ears that observations of circumpolar stars are not adapted for obtaining extreme accuracy* and that tli^ quantity of mean refraction at 45" so determined cannot rea- sonably be depended on to less than a quarter of a second.
The direct experiment for determining the refractive force of ait may be made independently of the divisions of an in- strument. The whole quantity of refraction is ascertained, instead of the differences of refractions as in circumpolar stars. There are also other sources of accuracy by which the result may berendered very exact.
For the above reasons, the determination -7— n;= 57",82 or
' ««. 1" '
the mean refraction at 45" (bar. 29, t>0 and therm. 60) = 57,67
appear to me more to be relied on.
16. In deducing the above value of ^^^-^ from the observations of circumpolar stars, 1 only used such stars as were less than 80" from the zenith when below the pole.
It is well known to those conversant in observations made with good instruments that near the horizon an irregularity in refraction hitherto unexplained shews itself. This com- mencing even at less zenith distances than 80°, is at first very small, but increases to a very considerable irregularity as we approach the horizon.
The bright star Capella being within the limits of this irre- gularity has not been used for the co-latitude. A considerable number of observations of this star below the pole have how- ever beenvmade by me, which may serve for two purposes.
104
(1) To shew the effects of the abovementioned irregularity ot refraction, by which it appears that at zenith distances not greater even than 80°, no use can be made of observations for the nicer purposes of astronomy.
(2) As it is reasonable to suppose this unexplained irregu- lariTiy* will disappear from a mean of a great number of ob- servations, this star, which is just at the limit where the quantity of refraction ceases to be independent of the vari- ation of density, may also serve as a criterion of the exactness of the value of "'"~;. or of the quantity of mean refraction.
The refraction observed and the refraction computed by the formula in Art. 11. are placed by the side of each other, and also the correction of the computed refraction to give the ob- served refraction. This correction is often far beyond the limit of the error of observation, and is to be attributed to the above- mentioned irregularity of refraction.
* The hypothesis upon which refractions are computed is that the different strata of air / are concentrical with the earth's surface, circumEtances may he easily imagined to affect this hypothesis, with respect to low stars.
lie
■yfhs'.viicfractions of Capelta below the Pole.
|
Time of Observations. |
Bar. |
Ther. int. |
Conijjut. llelrac. |
Observed Kefrac. |
Corr. comp. |
Time of Observation. |
Bar. |
Ther. int. |
Comput. Kefrac. |
Observed llefrac. |
Corr. comp. ref. |
|
I SOS, July 28 Aug. 1 1 23 |
29,50 29,51 29,97- |
63 61 67 -; |
i S0,3 31,9 32,6 |
5 28,8 29,1 31,3 |
— \",5 -2,8 — 1,3 |
1811, Jan. 23 27 28 |
30,33 29,40 29,32 |
32 27 24i |
6 s"4 5 56,3 57,5 |
/ // . » 5,2 5 59,S 55,3 |
+ 1,8 + 3,5 — 2,2 |
|
24. SO Nov. 23 |
29,9 .S ■ 29.16 29,84 |
66 62i 42 |
33.4 20,8 49,7 |
33,9 26,2 42,4 |
+ 0,5 1 — 0,6 — 73 |
July 1 3 6 |
29,64 29,49 29,78 |
• 64i 54i 61i |
30,7 36,5 3t,4 |
31,6 42,9 43,7 |
+ t),9 + 6,4 + 9,3 |
|
Dec. 4. 21 1 809, Jan. 20 |
29,77 29,30 29,31 |
44 31 30 |
47,4. 52,1 52,7 |
43,5 47,8 48,3 |
— 3,9 -4,3 — 4,4 |
9 14 16 |
29,8 1 29,42 29,46 |
64i 581 57i |
32,5 32,4 34,6 |
35,8 32,6 36,2 |
+ 3,3 + 0.2 + 2,6 |
|
22 May 29 June 14- |
29,33 29,^0 29,70 |
27 54 54 |
55,6 36,7 38,9 |
48,5 4:;,o 41,6 |
— 7,1 + 2,7 |
17 20 21 |
29,46 29,80 29,73 |
58 63t 64 |
33,3 33,2 32,1 |
34,7 35,3 29,4 |
+ L4 + 2,1 -2,7 |
|
15 17 July 8 |
29,72 29,61 29,90 |
55 56 63 |
38,4 S6,5 34,6 |
38,8 38,6 39,1 |
+ 0,4 + 2,1 + 4,5 + 1,2 + *.6, — 3,3 |
22 23 26, |
29,78 29,83 30,00 28,67 2.S,S7 29.75 |
61 62 65i |
34,8 34,5 34,1 |
36,6 40,1 37,5 |
+ 1,8 + 5,6 + 3-4 |
|
10 15 17 |
29,97 29,88 29,80 |
63 62i 55i - |
35,5 34,7 38,9 |
36,7 39,3 35,6 |
Dec. 9 9 13 |
40i 38 40 |
37,2 41,4 50,3 |
34,0 37,0 *7,1 |
— 3,2 — 4,4 — 3,2 |
||
|
18 19 23 |
29,89 20,92 29,71 |
571 60 60 |
38,4 37,1 34,7 |
41,0 38,3 35,5 |
+ 2,6 + 1,2 + 0,8 |
18 29,15 29 29.84 1812, Jan. 4 ! 29,20 |
451 30i 29i |
39,0 68,8 51,9 |
34,1 54,2 43,7 |
— 4,9 — 4,6 -8,2 |
|
|
Aug. 22 24 18 10, Jan. 20 |
29,19 29,16 29,83 |
53 55 58i |