IN PHYSICAL ASTRONOMY. 
333 
therefore, 
ancos\Jrsm\ {*/«+' (a DJ* t— a nsmxfr cos\ 
hr = - — 
mi - e% y 
which is the equation X of the Mecanique Celeste, vol. i. p. 258. 
2 r 2 d A' r'*d A 2 
Multiplying the equation of p. 1 1 4, 1. 2, by - ^ , and integrating, ^ 2 
-}- 2 Jr ' 2 ( = K 2 , if h 2 = h 0 2 — 2 Jr ' 2 (^) d h being variable, and h G the 
value of h at a given epoch, h d h = — r ' 2 d X v , r 2 d A' = h d t , and making a.' 
the independent variable instead of t, 2 r' d r' d A' = d h d t + h d 2 t 
x d 2 r r'dr'd 2 ^ 
r d* 3 ~ 
d t 3 
r 2 dA' 2 ^ 
“T" T 
• -v , 
N /d 2 r"\ r'dr' 
r \d£ 2 / hdi 
dt 
df n r\l + s 3 ) 
2r' 2 dr' 2 dA r' 2 dA' 2 
^ d^ 3 
d* 2 ~ 
I nr'l 
s (d R\ (d R\ 
+ r (d?) -Hd7) = ° 
d*A 
dA' 
2 H~ r ' 
r'(l + 3 2 )' 
3 r f ,/diE\ /d£\ 1 (dR\dr \ _ 
A 2 ( 1 +s 2 ) t “ A 2 | r VdrV ~ S \ d s) r VdW dA'J “ U 
d 2 s 
dA^ 2 
-H s {(X + , 2) (^f) - ,(%f) - ( d af) a4} =0 
If all the constants in the elliptic integrals are supposed to vary, subject to 
the condition that they still satisfy these differential equations, and that the 
• d v d s 
form of the first differential coefficients remains unaltered, 
i 
d. 
dA' 
/X cos 
h? 
d s 
1 d ^ 
— e sin (X' — zs) > , ^ = tan t cos (X' — v ) 
(1 +s qi dA' 
— 2 {(1 + s 2 )* + e cos (X' — zs)} {h 2 sin i d / + cos t h d h} 
+ h 2 cos i cos ( X ' — 73-) d e -f- h 2 cos i e sin (X' — w) d rs = 0 
^ f s tain cos (a' — v) 
\ (1 + s 2 )* 
— h 2 cos i sin (X' — tz) d e + h 2 cos i e cos ( X' — w) d zs 
cos i s r 2 f . , . / d R\ , (d R\ (d R\ /dA] , . 
— ( TT7jT { 0 + (dl) - r ( d?) ~ ( dv) U') j d x 
— e sin (/,' — ®) ] { h 2 sin t d t + cos t h d h } 
