and the Methods of calculating their Residts. 421 



and substitute d and a for the arbitrary quantities u and v used 

 in the transformation of the sum of three squares, we shall 

 have {a b' - a' by + {ad - a' cf + (be' — b' cf = [ (« b' - 

 a' b) cos d + (ac 1 — a'c) sin d cos a — (be' — b' c) sin d sin a] 2 

 + [ (a c' — a' c) sin a + (b c' — b' c) cos a] 2 , and the expres- 

 sion which forms the first part of the equation (2) is thus re- 

 duced to the sum of two squares. 



The angles d and a by which this is effected may be con- 

 sidered, the first as the declination (or latitude), the second as 

 the right ascension (or longitude) of a point of the sphere of 

 the heavens, which may be easily demonstrated to be the point 

 in which the great circles passing through the true and appa- 

 rent places, respectively, of the bodies, intersect each other. 

 For in the expressions by which d and a have been deter- 

 mined, the last parts of the expressions [1] of a, b, c } a', b\ c' 

 vanish ; so that we have 



{G sin d = sin ?r sin D - sin w' sin 8 



G cos d . cos a = sin it cos D cos A — sin n' cos 8 cos a 

 G cos d . sin a = sin % cos D sin A — sin m' cos 8 sin a 



In these equations is contained the condition that the three 

 points concerned in it, viz. the two true places of the bodies 

 and the point determined by d and a, are situated in a great 

 circle ; this condition may be reduced to the form in which it 

 is usually represented, by eliminating G,'sin 7r, sin w', which is 

 done by multiplying the three equations respectively by 



sin (as— A) tang d . . tang D . tang d . 



— —, H V sin A S- Sin a > V cos A 



cos a 7 cos 2 cos d ' cos d 



+ — — - cos a, and we shall have 



COS d 



(15) o = tang I sin (A — a) — tang D sin (a — a) + 



tang dsin (a — A) the usual form of the condition above men- 

 tioned. But as we have likewise 



G sin d = sin tt . A'.sinD' — sin^r'. A- sin 8' 



G cos^. cos a = sin7r. A '. cos D' cos A' — sin 71' A cos 8'. cos a' 



G cos d. sin a = sin7r. A'.cosD' cos A'— sin it' . A cos 8'. sin a 



And as these equations have the same form as the preceding 

 ones, the point determined by d and a is likewise situated in 

 the great circle passing through the apparent places. Sub- 

 stituting for a, b, Cy a', b', d their expressions in [1] we obtain 



a b' — a' b = cos 8 . cos D sin (a — A) — G . r cos <p' cos d . 



sin (/x — a) 



a c' — a' c = cos d . sin D sin a — cos D . sin 8 sin A — 

 G [r cos <$' sin d sin /x — r sin <p' cos d . sin a] 



be'- 



