212 PHILOSOPHICAL TRANSACTIONS. [ANNO 1/62. 



parallax expressed by p. But the angle aEx may be taken to be equal to 



aox; and as the foregoing form gives us the angles xEb, xec. we are to add gei 

 or 5" to xEb, and we shall have aEb. In general, let the distance of the stars be 

 d, and let the observed distance at e be d ; then will v> = d -{- p, and therefore 

 the whole parallax of the annual orbit will be expressed by — — — - — = p. 



Suppose the two stars now to differ only in latitude, one being in the ecliptic, 

 the other, for instance, 5" north, when seen at o. This case may also be re- 

 solved by the former; for imagine the stars b, c, fig. 7, to be elevated at rect- 

 angles above the plane of the figure, so that aob, or aoc, may make an angle 

 of 5" at o: then, instead of the lines oabc, eg, Eb, ec, ep, imagine them all 

 to be planes at rectangles to the figure; and it will appear, that the parallax of 

 the stars in longitude must be the same as if the small star had been without 

 latitude. And since the stars b, c, by the motion of the earth from o to e, will 

 not change their latitude, we shall have the following construction for finding 

 the distance of the stars ab, ac, at e, and from thence the parallax p. Let the 

 triangle ab(Z, fig. 10, represent the situation of the stars; ab is the subtense of 

 5", that being the angle under which they are supposed to be seen at o. The 



quantity b(Z by the former theorem is found p, which is the partial parallax 



that would have been seen by the earth's moving from o to e, had both stars 

 been in the ecliptic; but on account of the difference in latitude it will now be 

 represented by «(3, the hypothenuse of the triangle ab@: therefore, in general, 



putting ab = d, and a(3 = d, we have — = p. Hence d being 



taken by observation, and d, m, and m, given, we obtain the total parallax. 



If the situation of the stars differs in longitude as well as latitude, we may- 

 resolve this case by the following method. Let the triangle abfi, fig. 1 1, repre- 

 sent the situation of the stars, ab = d being their distance seen at o, a(3 = d 

 their distance seen at e. That the change Z'|3 which is produced by the earth's 



motion will be truly expressed by p, may be proved as before, by supposing 



the star a to have been placed at a. Now let the angle of position baa, be taken 

 by a micrometer,* or by any other method that may be thought sufficiently ex- 

 act; then, by solving the triangle aba, we shall have the longitudinal and latitu- 

 dinal differences aa and ba of the two stars. Put ax = x, ba = y, and it will 



be x 4- bQ = an. whence d = V \{x -\- p) 2 + mi~\ ; and — — — — — - 



urn = p. 



* The position of a line passing through the two stars, with the parallel of declination of the 

 largest of them, may be had by the micrometer I invented for this purpose in the year J 779, of which 

 a description has been given in a former paper; whence, by spherical trigonometry, we easily deduce 

 their position bax fig. 1 1, with regard to the ecliptic— Orig. 



