520 
NOTE ON A THEOREM OE JACOBIS, 
[220 
d dr 
d/tf' 
_ __ rcos22/ (_) -r(^) + 
'dy\ 2 n 2 a s _ dii 
dt dt 
d 
.dt, 
dt{ ri 0082 ÿ 
d 
where 
il = m 
dtfdt) + ^ aosysmy 
vV 2 + r' 2 — 2rr' cos H 
r 2 dr 
_ cZfl 
dr ’ 
_di2 
~ dy ’ 
r cos H 
or, since cos JT = cos y cos (v — v'), and the Sun is considered as moving in a circular 
orbit (i.e. r' = a , v=n't), we have 
il = m' 
r cos y cos (r — nt)\ 
Vr 2 + a' 2 — 2m' cos y cos (v — n't) 
so that il is a function of r, v, y and of t, which last quantity enters only in the 
combination v — rit. Hence the complete differential coefficient of il is 
d (il) _ d'il , di1 
dt dt U dv ’ 
d'il 
where —denotes, as usual, the differential coefficient in regard to the time, in so 
far as it enters through the coordinates r, v, y of the disturbed body. 
We have, as usual, 
d dr 2 4- r 2 (cos 2 y dv 2 + dy 2 ) _ d'il 
dt dt 2 dt ’ 
and, from the foregoing equation, 
d'O _ d (il) , dii 
dt dt 71 dv 
d{&) /d 
dt + dt 
+ n' T , (r 2 cos 2 y^-l; 
dv 
dt 
hence, substituting this value, transposing, and integrating, we have 
'dr- 
'dv\ 2 (dy 
dv 
+ r 2 •] COS 2 y \ -jj_ ) + ( ^7 ) f — n 'r 2 COS 2 y = il + C, 
dt 
which is Jacobi’s equation expressed in terms of the coordinates r, v, y.