274 
Mr. Herschel on the 
Eliminating from these three equations, r and r' , we get 
the relation required between <p and <p', viz. 
/ 2 sin. 0 Z 
COS. <p = 7 — j- 
“ i — (2 — e ) e a . cos. <p 
This equation will however be reduced to a much more con- 
venient form for our present purpose by a transformation, viz. 
tan. = 
T cos. <p' cos. <p - 
_ (i — e z ) z . cos. <p z 
sin. < p z 
that is simply 
— tan. q>' = ( l — e 2 ). cot. <p 
the sign — being prefixed in extracting the root, became in 
the ellipse C P and C D lie in different quadrants. Now in 
the case before us we have 
(p = 7r — <r and c p f = 9 ; so that 
tan. 6 = - — ( l —e 2 ). cotan. {tt — <r) 
But (l — <? 2 ) = sin. X 9 , and by equation (i) — tan. 0 = sin. /. 
cotan. (o — /). Hence substituting, we get 
sin. X 3 . tan. ( o — l) = sin. /. tan. (tt — <r) ; (2) 
This equation gives at once the value of o the sun’s longi- 
tude, and therefore (by consulting an ephemeris) the time of 
year sought, by an exceedingly simple process adapted to 
logarithmic computation. 
The actual advantage or disadvantage in point of situation 
in the case of any particular star, is expressed by the magni- 
tude of the whole change in the angle of position produced 
by parallax, i. e. by the angle subtended by the ellipse at the 
small star, or, (calling the distance of the latter from its 
centre on the original supposition of the whole effect being 
small, by the expression P = ^ . sin. ( <?> ' — <p ). Now if we 
