IN THE MOTIONS OF THE EARTH AND VENUS. 
117 
66. If we put for sin { 12 (ri t + s') — 8 (n t -j- s) — 3 v*’ — 6} its equivalent 
cos (3 vs + 2 6) . sin {12 ( ri t -j- z) — 8 [n t + g) + 0} — sin (3 «/ + 2 0) . cos 
{12 (w ' t -f- g') — 8 («/ -f- a) + 0}, and similarly for the other terms, we find for 
the whole coefficient of sin { 12 (n't + s') — 8 (w t + s) -f- 6)}, 
ft! Wl I & (* /o /».-) /o ' \ o /1\ 
— — 5- . — . . — < e 3 / 2 • M . cos (3 w + 2 6) 
1 3 n' — 8 n (x j m [_ J v 1 ' 
( 10 ) ( 11 ) 
+ e 2 ef 2 . M .cos (2s/ + ® + 2 0) + e'e 2 / 2 .M . cos (ri + 2 or + 2 0) 
( 12 ) ( 10 ) 
+ e 3 / 2 . M . cos (3 -a + 2 0) + 2 <?'/ 4 . N . cos (ri -f 4 0) 
(ii) 1 
+ 2 e/ 4 . N . cos (©• + 4 0) > 
and for the whole coefficient of cos { 12 (ri t + z) — 8 (n t + e) + ^}, 
«' m l a' f ,, _ T (®) . 
• — • ~e . — i e 3 / 2 . M . sin (3 sr + 2 &) 
1 \3 n — % n fj. J m J v 1 ' 
(1°) . (ii) 
-f- e' 2 ef 2 . M . sin (2 zr + ® H - 2 6) + e e 2 f 2 • M . sin (ri -f- 2 w + 2 0) 
+ e 3 / 2 . M ( \ sin (3 sr + 2 6) + 2 e'/ 4 . N ( \ sin (?/ + 4 6) 
+ 2 e/ 4 . N ( \ sin (nr + 4 6) 1 
On performing the calculations, the inequality is found to be 
-{- (y',0086 . sin {8 (n t -j- s) — 12 (n t - j- s ) — 
+ 0",0060 . cos {8 (nt + s) — 12 (ri t + s') — 0} 
or + 0",0105 . sin {8 (n t + g) - 12 (ri t + s') — 39° 29'} 
which is too small to be sensible in any observations. 
PART III. 
PERTURBATIONS OF VENUS DEPENDING ON THE SAME ARGUMENTS. 
67- If we consider Venus as disturbed by the Earth, and take the orbit of 
Venus for the plane xy, the term involving cos (13 — 8) in the expansion of R 
