VOL. LXXII.] PHILOSOPHICAL TRANSACTIONS. 2 J I 



parallax p = ■ X '_* — = l^OgOQ. If the stars are of the 3d and 24th magni- 



tude, the partial parallax will be - — — = ^~ p = — p; and if by observation, p 



1 x 3 x 24 



is found to be a 10th of a second, the whole parallax will come out ■ — - _ * 



= 0".3428. 



It will be necessary to examine some different situations. Suppose the stars, 

 being still 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 still be expressed by 

 the same algebraic form, and one of the maxima will still lie at o, the other at 

 e; but the whole effect will be divided into 2 parts, which will be in proportion 

 to each other as radius — sine to radius + sine of the stars distance from the 

 nearest conjunction or opposition. 



When the stars are any where out of the ecliptic, situated so as to appear in 

 one line oabc at rectangles to oe, the maximum of parallax will still be expressed 

 by '" ~ M p ; but there will arise another additional parallax in the conjunction 



* MM 



and opposition, which will be to that which is found go" before or after the sun., 

 as the sine (s) of the latitude of the stars seen at o is to radius (r) ; and the ef- 

 fect of this parallax will be divided into 2 parts; half of it lying on one side of 

 the large star, the other half on the other side of it. This latter parallax also 

 will be compounded with the former, so that the distance of the stars in the con- 

 junction and opposition will then be represented by the diagonal of a parallel- 

 ogram, of which the two semi-parallaxes are the sides; a general expression for 



ss 



which will be p X V (- h l): for the stars will apparently describe 2 



ellipses in the heavens, whose transverse axes will be to each other in the ratio of 

 M to in (fig. 8), and A.a, b£, cc, vd, will be cotemporary situations. Now, if bo. 

 be drawn parallel to ac, and the parallelogram bqvo. completed, we shall have 

 bo. = ^ca — ±ca = ice = \p 3 or semi-parallax Q0° before or after the sun, and 

 bZ> may be resolved into, or is compounded of, bo. and bq; but bq = {bd — \bd 

 = the semi-parallax in the conjunction or opposition. We also have R : s :: bo. : 

 [,q — p.; therefore the distance bZ> or vd = */ [(|) 2 + (~) 2 ] ; and by substitut- 

 ing the value of p into this expression we obtain ~ p X \/( 1- l), as above. 



When the stars are in the pole of the ecliptic, bq will become equal to bo., and 



.... m — m 



hb will be .707 1p — — • 



Hitherto we have supposed the stars to be all in one line oabc; let them now 



be at some distance, suppose 5" from each other, and let them first be both in 



the ecliptic. This case is resolvable into the first; for imagine the star a, fig. 9, 



to stand at x, and in that situation the stars x, b, c, will be in one line, and their 



e e 2 



