4-22 Prof. Bessel's Additions to the Theory of Eclipses, fyc. 



be' — b' c = cos 8 sin D cos « — cos D sin 8 cos A — 

 G \r cos <p' sin d . cos \l — r sin <p' cos d . cos a] 

 and consequently, 



(ab'—a'b) cos d+ (ad— a! c) s\nd.cosa—(b'c'—b c) sin dsina 



== — sin 8 cos D sin d sin ( A — a) + cos 8 sin D sin d x 



sin (« — a) + cos 8 cos D cos d sin («- A) — G.rcos $'sin (ft — a) 



and adding the product of equation [15] by cos 8 cos D sin d 



= cos 8 sin (a — A) — G . r cos p' sin (^— «) 



We likewise obtain 



(a C 1 — a' c) sin a -f (b c ! — b' c) cos a = cos 8 sin D cos (a — a) 

 — sin 8 cos D cos (A — a) + G [r sin <p' cos d — r cos $' sin d x 

 cos (ju, — a) ]. 



The second part of equation [2], viz. (a' sin g + a sin R) 3 

 + (V sin g + b sin R) 2 + (c' sin £ + c sin R) 2 is more conveni- 

 ent for calculation, if represented in its irrational form. It is 

 the square of A • A' sin 2 = sin g \/ (A' 2 — sin R 2 ) + 



sin R s/ ( A 2 — sin g-) where 



A ' = a* -f Z> 2 + c 2 = 1 — 2 r sin 7r . cos y + r 2 sin 7r 2 

 A n = a' 2 + b' 2 + c n = 1 — 2 /• sin w'. cos y' + r 3 sin 7r' 2 



cos y and cos y' being written for 



sin <p' sin 8 + cos <J>' cos 8 cos (/*—«) and 

 sin <J>' sin D + cos <£' cos D cos (f*. — A). If we denote, 

 therefore, 



-v/[cos g- — 2 r sin 7T cos y + r~ sin tt] by A 

 //[cos R 2 — 2 ;• sin w' cos y' + ?- 2 sin 7r'-] by A' 

 the required part is (A' sin §> + A sin R) 2 . 



[13.] The equation [2] becomes by substituting these trans- 

 formations of its several parts : 



. . /A.' sin + X sin R\ 2 



(16)... (^ § ) 



i cos D cos 2 sin (Vi— A) . . . . \ 2 



= 1 -^T" g r cos 4 sin G*~ fl ) \ 



i sin 3 cos D cos (A — a) — cos 2 sin D cos (a. — a) , . , -, 



+ I g t — r (sin f cos d — 



cos <p' sin c? cos (/x— a) f 



Ithas consequently induced the form £ 2 = (P— m) 9 + (Q— ») a , 

 ^//£ s«?H£ which takes place for the case of occultations of 

 fixed stars. The difference between the general equation and 

 the particular case consists in this, that in the former there is, 

 instead of the constant k, a variable one dependent on the place 



of 



