STARS. 



or four times as far off as one of the firft. Tliefe principles, 

 which he premifes as pollukta, hare fo great a probability 

 in their favour, that they will hardly be objefted to by thofe 

 who are the leaft acquainted with the doftrine of chances. 

 See Mr. Michell's Inquiry, &c. already cited. Phil. Tranf. 

 vol. Ivii. p. 234. 236, 237. 240. And Dr. Halley on the 

 Number, Order, and Light of the Fixed Stars. Phil. 

 Tranf. vol. xxxi. or Abr. vol. vi. p. 148. 



Accordingly, let O E (PIaU\X. yljlronomy. Jig. 9.) be 

 the whole diameter of the earth's annual orbit ; and let a, b, c, 

 be three ftars fituated in the ecliptic, in fuch a manner, 

 that they may be feen all in one line O abc, when the earth 

 is at O. Let the line O j^<: be perpendicular to O E, and 

 draw P E parallel to <: O : then, if O a, ab, b c, are equal 

 to each other, a will be a ftar of the firll magnitude, b of 

 the fecond, and c of the third. Let us now fuppofe the 

 angle O a E, or parallax of the whole orbit of the earth, to 



When the flars are any where out of the ecliptic, fituated 

 fo as to appear in one line Oabc perpendicular to O E, the 



maximum of parallax will ftill be exprelled by — ^^ P ; 



M m 

 but there will arife another additional parallax in the con- 

 junftion and oppofition, which will be to th.u which is found 

 90" before or after the fun, as the fine (S) of the latitude 

 of the ilars feen at O is to the radius (R) ; and the effeft 

 of this parallax will be divided into two parts ; half of it 

 lying on one fide of the large liar, the other half on the 

 other fide of it. This latter parallax, moreover, will be 

 compounded with the former, fo that the diftance of the 

 itars in the cor.junftion and oppofition will then be repre- 

 fented by the diagonal of a parallelogram, of which the two 

 femi-parallaxes are the fides ; a general expreffion for which 



be i" of a degree ; then we have P E a = O a E = 1": and ^^'^^ ^^ 

 becaufe very fmall angles, having the fame fubtenfe O E, 

 may be taken to be in the inverfe ratio of the lines O a, 

 Ob, Oc, &c. we fhall have ObE= i", OcE = \", &c. 

 Now when the earth is removed to E, we (hall have P E ^ = 

 EbO = i", andPEa-PEi5=aEi = i": ;. e. the ftars 

 a, b, will appear to be i" diftant. We alfo have P Ec = E<rO 

 = ;", and PEa- PEf = aE,: = 4"; i.e. the ftars a, ::, 

 will appear to be >" diftant, when the earth is at E. Now, 

 fince we have i E P = i", and <r E P = -!", therefore bEP 

 ~cEP = bEc = :^"- 5" = i-"; ;, e. the ftars b, c, will 

 appear to be only ^" removed from each other, when the 

 earth is at E. Whence we may deduce the following ex- 

 preffion, to denote the parallax that will become vifible in 

 the change of diftance between the two ftars, by the removal 

 of the earth from one extreme of its orbit to the other. Let 

 P exprefs the total parallax of a fixed ftar of the firft mag- 

 nitude, M the magnitude of the largeft of the two ftars, m 

 the magnitude of the fmalleft, and p the partial parallax to 

 be obferved by the change in the diftance of a double ftar ; 



then will p = --:^ P ; and /, being found by obferva- 





M 



+ ' 



for the ftars will ap- 



SS 



2mM ^1 ^ RR 



parcntly defcribe two ellipfes in the heavens, whofe tranf- 

 verfe axes will be to each other in the ratio of M to m 

 i^S- '°-) ^"'i A J, B b, C c, D (/, will be contemporary 

 fituations. Now, if i Q be drawn parallel to A C, and 

 the parallelogram b gB Q be completed, we (hall have b Q 

 = ^CA — ^ca— ^Cc — i/, or femi-parallax 90° before 

 or after the fun, and Bi may be refolved into, or is com- 

 pounded of, b Q and b q ; but b q = l^H — ^b d = the 

 femi-parallax in the coiijunclion or oppofition. We alfo 



have R : S : 



:^Q:^? = J|; 



therefore the diftance B b (or 



D </J = \/ Jl\ -f- ^ ; and by fubftituting the value 



of ^ into this expreffion, we obtain 



tion, will give us P := 



pM> 



-M 



E. gr. Suppofe a ftar of the firft magnitude ftiould have 

 a fmall ftar of the twelfth magnitude near it : then will the 



partial parallax we are to expeft to fee be , or 



12 — I 



44th of the total parallax of a fixed ftar of the firft magni- 

 tude ; and if we ftiould, by obfervation, find the partial 

 parallax between two fuch ftars to amount to 1", we (hall 



I X I X 1 2 



have the total parallax P = = i".o909. If the 



ftars are of the third and twenty-fourth magnitudes, the par- 



24. — ^ 21 



tJal parallax will be — = — P : and if, by obfervation, 



3 X 24 72 



p is found to be a tenth of a fecond, the whole parallax will 



I X ^ X 24. 

 come out 5 ~ =©".3428. 



*4 ~ 3 



Farther, fuppofe the ftars, being ftill in the ecliptic, to 

 appear in one line, when the earth is in any other part of its 

 orbit between O and E ; then will the parallax ((ill be cx- 

 prefled by the fame algebraic formula, and one of the maxima 

 will ftill lie at O, the other at E : but the whole efleft will 

 be divided into two parts, which will be in proportion to 

 each other as radius — fine to radius -f- fine of the ftars' 

 diftance from the nearcit coniuodion or oppofition. 



Vol. XXXin. 



^RR+'' 



as above. When the ftars are in the pole of the ecliptic, 

 b q will become equal to b Q, and B b will be .7071 P 



-j-j — . Again, let the ftars be at fome diftance, e. g. 5", 



from each other, and let them both be in the ecliptic. This 

 cafe is refolvable into the firft ; for imagine the ftar a 

 (fg. 1 1 .) to ftand at .r, and in that fituation the ftars x, b, c, 



will be in one line, and their parallax exprefl'ed by -^ — P. 



But the angle aE x may be taken to be equal to a O x ; 

 and as the foregoing formula gives us the angles .r E b, 

 X E c, we are to add a E x or j" to xEb, and we (hall have 

 aEb. In general, let the diftance of the ftars be d, and 

 let the obferved diftance at E be D ; then will D = //-)- /, 

 and therefore the whole parallax of the annual orbit will be 



_.. DMm-JMm 

 t'xpreflcd by ry- — = "• 



'^ ' m — M 



Suppofe the two ftars now to differ only in latitude, one 

 being in the ecliptic, the other, e.g. 5" north, when feen at 

 O. This cafe may alfo be refolved by the foimer ; for 

 imagine the ftars b, c, {Jig. 9.) to be elevated at right angles 

 above the plane of the figure, fo that aO b, or aO c, may 

 make an angle of 5" at,0 ; then inftcad of the lines Ojbc, 

 Ea, Eb, Ef, E P, imagine them all to be planes at right 

 angles to the figure ; and it will appear, that the parallax 

 of the ftari in longitude inuft be the Came as if the (iiiall ftar 

 had been without latitude. And fince the H.vs b, c, by the 

 motion of the earth from O to E, will not cliaiii;e their lati- 

 tude, we (hall have Uic following conllxuttiuu tor finding 

 5 C the 



