IN PHYSICAL ASTRONOMY. 
335 
If x = 
r cos X 
(1 + s 2 )*’ 
d s 2 
y = 
rsinX 
(1 +s 2 ) v ’ 
rs 
(1 + s 9 )* 
d ’' , + rr?(nr? +d ^) 
dt 3 
T + ^ + 2 ^ d /l> — 0 
d 2 r 
d? 
1 + s 2 d ^ r 3 ^ \dr/ 
r 3 dx' 
1 + s a ‘ d t 
dt 
, /. d 22\ 
+ Cav) = 0 
r 2 d 2 s + 2rdrds- f r sd f + r 2 sdX V2 
(1 +S 2 ) 
d* 2 
+ (1 + s 2 ) 2 (^jf) = O 
Of which equations the integrals are 
I u. cos 
l + 5 » • d X v _ h d t, r — A a ^ + 
$r»{ (1 
+ «9 2 ) 2 + e cos 
(x' - «) j 
s — tan i sin (X v — v ) 
If d t — \J- r d y, and v be taken for the independent variable 
a + r + ~ {‘i/AR + r (^f) j = 0 
r = a { 1 — e' cos (y — a)} in the elliptic motion. 
e' is accented for the present in order to distinguish it from e. 
If the constants in the elliptic integrals are supposed to vary, subject to the 
condition that they still satisfy these differential equations, and that the form 
of the first differential coefficient -7- remains unaltered. 
Cl If 7 
rdudfl , f~a 1 , rd 3 r rdr 3 rdrd?t 
d ' = !7^ + Vi: lrd "- aK=Jf-Tr 
d 2 r . , a r 2 /d R\ dr da 
d^ - a + r + — ) - 2^5 - 0 
(l — e' cos (y — a)) d a — a cos (y — a) d e' — a e' sin (y — a) d a = 0 
d sin (y — a) d a + « sin (y — a) d e' — a e' cos (u — a) d a 
2 x 
MDCCCXXX. 
