IN THE MOTIONS OF THE EARTH AND VENUS. 
117 
66 . If we put for sin {12 («' t -f- s') — 8 (n t + s) — 3 — 0} its equivalent 
cos (3 w' + 2 &) . sin {12 (n t -f- s') — 8 (n t -f- s) + ()} — sin (3 w' -{- 2 0) . eos 
{12 (//1 -f- s') — 8 (nt -f- s) + and similarly for the other terms, we find for 
the whole coefficient of sin {12 (n't -f- s') — 8 (n t + s) -f- #)}, 
( 10 ) , ( 11 ) 
+ e ' 2 e/ 2 . M . cos (2 to- 7 -f- ra- -f- 2 0 ) -f- e e 2 f 2 . M . cos (ot' -[■ 2 w -f* 2 0 ) 
( 12 ) ( 10 ) 
-f- c 3 / 2 . M . cos (3 sr + 2 0 ) + 2 t?'/ 4 . N . cos (V + 4 0 ) 
and for the whole coefficient of cos {12 {rt t + s') — 8 (w t + s) -{- 0 }, 
On performing the calculations, the inequality is found to be 
~f- 0 ',0086 . sin { 8 (^ 2 - / —{- s) — 12 (n t - f- £) — 0} 
-f- 0",0060 . cos {8 (nt -j- s) — 12 ( n t -I - s) — 0 } 
or + 0",0105 . sin {8 (n t + s) — 12 (n 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 
