62 
MR. LUBBOCK’S RESEARCHES 
When the latitude is reckoned from the plane of the orbit of the planet P, the 
following- terms must be added, in the general case where i is not equal to zero, 
to that expression ; 
“ <77 T 77 -— o taiw& 3>1 sin (2 n t — n ,t + 2 e — e, - v) 
2 (n — n t ) (3 n — n t ) a ( ~ 
+ 
2 (n — n t ) {n + n y ) a°~ 
- tan i i> 3jl sin (n t t + e t — y y ) 
2 (2 « — 2n t ) {An — 2 n t ) a 
- tan 1 1» 3)2 sin (3 n t — 2 n t t + 3 e — 2 s t — v) 
4 (2 n — 2 ra,) a~ 
t 
- tan i 6 3>2 sin (w t — 2 n/ + e — 2 + v) 
2 (3 « — 3 «,) (5 n — 3 ra,) a y 2 
- tan « 6 3>3 sin (4 n t — 3 n t t -f 4 £ — 3 e y — v) 
+ ” — tan i b„ » sin (2 n t — 3 n.t + 2 s — 3 £ + v) 
2 (n - 3 m ( ) (3 n - 3 «,) a ( 2 3,3 v J ' ’ 
All the equations hitherto given apply to the case of an inferior disturbed by 
a superior planet, or when a t > a, in order to render them applicable to the 
case when «,< a it is necessary to write a instead of a t in the denominator 
of the terms multiplied by b x , and « 3 instead of af in the denominator of the 
terms multiplied by b n n , in the disturbing function R , but the expressions for 
the quantities q are not the same in this case. 
It will I think be admitted that the expressions which occur in the theory 
of the disturbances of the planets are more simple in terms of the quan- 
tities of which the general symbol is b, than in terms of the partial differen- 
tial coefficients of the quantity called A in the notation of the Mecanique 
Celeste. The development of the disturbing function R in terms of the 
differential coefficients &c. admits of reductions, so that it may be 
expressed in terms of the differential coefficients of A with respect to a only. 
In this state it has been left by Laplace as may be seen, vol. ii. p. 12, but 
the coefficients of the terms multiplied by the squares and products of the 
eccentricities may be expressed veiy simply in terms of the quantities of 
which the general symbol is b, by means of reductions, of which two exam- 
