IN THE MOTIONS OF THE EARTH AND VENUS. 
119 
posed to be inclined at a small angle to the plane of x y. We have remarked 
that, in the development of R for Venus as the disturbed body, the sign of f 
will be changed: and as the term of R on which the perturbation in latitude 
depends is a multiple of odd powers of f, the sign for Venus will be different 
from that for the Earth. Besides this there will be no difference, except that 
a m n is to be substituted for a! m n'. Proceeding then as in (65), and con¬ 
sidering the effect of the first term of (63), we find 
0 = 0 - 
n 
13 n [ — bn 
m' 
a 
a' 
( 9 ) 
M a' 
m 
e —qf- • s ^ n (13 (n't -f- s') 
— 8(nt +s) —3 nr'- 0'— 0} 
whence 
7Z 77l) CL 
sin (nt + &-0) = sin (nt + s - 0 ) + f3n > _ % n • ~ • ^ 
M (9 V e n f 
m <p ^ 
cos (n t 4- £ — 0) . sin {13 (n't -f-s') — 8 (n t -f- s) — 3 & — 0' — 0} 
And 
<p = <f> 
n 
( 9 ) 
m' a a' 
13 n —bn 
a 
m 
. e z f. cos {13 (n t -f s') 
- 8 (n t -f- s) — 3 zs — 0' — 0} 
The product of these expressions gives for the latitude of Venus 
n 
m 
<S.sm(»/ + £ -0) + - |3B ,_ 8)! . ^ . a , 
— 9 (n t + s) — ?> m — 0'} 
-_( 9 ) , 
a M a' 
m 
e z f. sin {13 (n't -f- s') 
where 0’ has the same value which 0 had in the investigation for the Earth. 
, ( 9 ) 
71 tv n M n f 
The perturbation in latitude is therefore r^- 7 — 5 — . — . — . - e ' 3 f . sin 
{13 (n t + 0 — 9 (nt + s) — 3 ■nr' — 0'}, and similarly for the other terms. 
Comparing this with the term in (65) it will readily be seen that we have only to 
7fl* d 
multiply the expression of (66) by- -7—,, and to put 9 (nt-\-s) — \3(n t-\-i) 
7YI 71 CL 
instead of 8 (n t -f- s) — 12 (n t + s'), and the perturbation of Venus in latitude 
will be found. Thus it becomes 
