IN PHYSICAL ASTRONOMY. 
331 
+ mTTT* + r '(^) “ s (^f) - 0 
r'd¥-r' 2 d X' 2 
d? t r '(i + S 2)§ 
r' 2 d 2 s + 2r'dr'ds — r' 2 sdx' 2 , /diJ\ v /d Z2\ 
d?— + 0 + * 2 ) (dr) - r 9 \i?) = 0 
d 2 .r' 2 (l + s 2 ) 
2d£ 2 
+ -t + 2/diJ +) '(^)=0 
r'(l + s 2 ) 1 
Making X the independent variable instead of t, 
d 2 r' dr'd 2 2 vo dx' 2 
j?- 2 w- - r .ts + 
dx* 
-'4 
d^ 2 ' r'(l + s 2 ) 4 
= h 2 — 2 d x', A being a constant 
= -»•" (df) - 1 *' 
. yo j s dx' 2 y. dx' 2 d 2 £ 
4r dr 
d*.i 
{-d^ + ?■} { 1 - I /-' 2 (if) d> ' } - *» (1 + s .)i 
-iH d 3 #)-(TO)-H d al)TO} = o 
{to» + s }{ 1 -|/-' 2 (to)^'} 
+ f{o + ^)(to)-'-'(to)-(to)to}=o 
When the disturbing force is neglected 
l 
d 2 . 
d.r' 2 dx' r 1 
J ' ,a y ~ km + S 2)i 
V- 
n d * s i 
- °> d~T~ 2 + 5 
d*' — dx' 
of which equations the integrals are, 
r' 2 d ?' = hdt, y = {(1 + s 2 f + ecos (X s - ar) j 
s = tan / sin (X.' — v). 
If dt = \/l r' d v, and v be taken for the independent variable 
x dV v d 2 t v , dx' 2 
r -r dr tt , — r 2 + 
dt 2 
d t 3 
dt 2 ~ r'(l + s 2 ) 4 
