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 dilferent 
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 - 
13 n' 
8 n 
m' 
M 
( 9 ) 
e'V 
m 
sin {13 (n't -f- s') 
— 8(nt -fa) — 
whence 
sin (ni -f s — 0) = sin (nt - f- a — 0) -f 
m 
( 9 ) 
1 a M a 1 e n f 
1 3 n — 8 n * ft ' a! * m ’ <£> 
cos (n t -f s — 6) . sin { 13 (n! t + s) — 8 (n t + s) — 3 rar' — & — 6} 
And 
<p = d> - 
n 
( 9 ) 
m! a M a 
13 n 1 — 8 n ’ 
/x a : 
m 
. e 3 f. cos { 13 (n t -f s') 
— 8 (n t -f- i) — 3s?'-^ — 
The product of these expressions gives for the latitude of Venus 
a M (9 «' 
. sin (n t- f- s — 0) -{- 
m 
1 3 n' — 8 n ’ fj. ' a' ’ m 
. e z f. sin { 13 in' t -f- s) 
— 9 (n t + s) — 3 vr — 0'} 
where & has the same value which 6 had in the investigation for the Earth. 
i ^ 
The perturbation in latitude is therefore , „ . — . ~ . — — - e 3 f . sin 
{13 (n't -|- a') — - 9 (n t + s) — 3 us — and similarly for the other terms. 
Comparing this with the term in (65) it will readily be seen that we have only to 
TYt 7Z CL 
multiply the expression of (66) by — and to put 9 (nt-\-s) — 1 3(n't-{-s) 
instead of 8 (n t + a) — 12 (n t + s), and the perturbation of Venus in latitude 
will be found. Thus it becomes 
