210 PHILOSOPHICAL TRANSACTIONS. [ANNO J 782. 



In fig. 7, let oe be the whole diameter of the earth's annual orbit; and let a, 

 A, c, be 3 stars situated in the ecliptic, in such a manner that they may be seen 

 all in one line oabc, when the earth is at o. Let the line oabc be perpendicular 

 to oe, and draw pe parallel to co. Then, if oa, ab, be, are equal to each other, 

 a will be a star of the 1st magnitude, b of the 2d, and c of the 3d. Let us 

 now suppose the angle o«e, or parallax of the whole orbit of the earth, to be 

 l"of a degree: then we have pea = or/E = \" : and, because very small angles, 

 having the same subtense oe, may be taken to be in the inverse ratio of the 

 lines oa, ob, oc, &c. we shall have o£e = ■}/, oce = $", &c* Now, when the 

 earth is removed to e, we shall have pe6 = e6o = -J-", and pe«. — pe6 = «eZ> 

 = ±"; that is, the stars a, b, will appear to be $? distant. We also have pec = 

 eco = 4-", and PEa — pec = «ec = •§-"; that is, the stars a, c, will appear to be 

 %" distant, when the earth is at e. Now, since we have 6ep = \" , and cep = 

 -l", therefore 6ep — cep = bEc = \" — \" = 4"; that is, the stars b, c, will ap- 

 pear to be only £.* removed from each other, when the earth is at e. 



From what has been said, we may gather the following general expression, to 

 denote the parallax that will become visible in the change of distance between 

 the two stars, by the removal of the earth from one extreme of its orbit to the 

 other. Let p express the total parallax of a fixed star of the first magnitude, 

 M the magnitude of the larger of the two stars, m the magnitude of the smaller, -f- 

 and p the partial parallax to be observed by the change in the distance of a double 

 star; then will p = — ^— p; and/>, being found by observation, will give us p = 



i " MW . An example or two will explain this sufficiently. Suppose a star of the 

 first magnitude should have a small star of the 12th magnitude near it; then will 

 the partial parallax we are to expect to see be - p; or -}~±- of the total paral- 



lax of a fixed star of the first magnitude; and if we should, by observation, find 

 the partial parallax between 2 such stars to amount to \", we shall have the total 



* This proves what I have before remarked on the parallax of y Draconis ; for that star, (admitting 

 it to be a star of between the 2d and 3d magnitude, which ought to be ascertained by experiments, 

 as mentioned in the note above) by the postulata, will have its place assigned somewhere between b 

 and c, and therefore its parallax will be between \ and ^ of the parallax of a star of the first magni- 

 tude. And if Dr. Bradley thought that he should have perceived a parallax in y Draconis, if at most 

 it had amounted to 2", it follows, that the angle oue may nearly amount to i or 5" for any thing we 

 can conclude to the contrary from those observations. — Orig. 



t Asm and m are here taken to express the relative distances of the stars, in measures whereof 

 the distance of the nearer star is taken as unity, those who diink the postulata on which diese esti- 

 mations are built cannot be granted, may still use the following formulae, if instead of the magni- 

 tudes m, m, they put their own estimations of the relative distances of the stars, according to any 

 other mediod whatever they may Uiink it most eligible to adopt; for the apparent magnitude of stars 

 is here only proposed as the most probable means we have of forming any conjectures about their rela- 

 tive distances. — Orig. 



