MR. LUBBOCK’S RESEARCHES 
3l>2 
This being - the case, no imaginary angles are introduced, if the quantities 
c and g are rational. This theory, which does not seem to be limited by the 
direction of the moon’s motion, and which maybe extended without difficulty, 
already embraces the terms which are included in the secular inequalities, 
and which are derived from the constant part of R carried to the order of the 
squares of the eccentricities. Generally when the method of the variation of 
constants is employed to determine any inequalities, the development of R 
must be carried one degree further, as regards the eccentricities, than the de- 
gree which is required of the inequalities sought. 
The equation for determining the coefficients of the expression for the reci- 
procal of the radius vector is. 
d 2 . r- 
2 dU 
d 2 . r 3 S — 
r 
d t- 
3 d 2 . i 
+ 
( 4 )‘ 
2 d t- 
d < 2 
£■ + ft + 2 
2 f iR + r (i£) = 
pay- j 
( 1 + 3 e°- ( 
1 + i) 
\ r > 1 
l V 
8/ 
J 2 V 8 / 
— y |2r 0 r 1 + e 2 (r ;J + r 4 )r 2 + e / 2 (r 6 + r 7 )r 5 | lcos2f + &c. 
— 3 e°- { 2 r l r, + 2 r 0 r 3 + 2 r 0 r 4 } 
Vn being the coefficient corresponding to the w tb argument in the development 
of r5 y. The development of r 3 1 — is easily deduced from that of r § 7 - given 
in the Phil. Trans. 1 832, Part I. p. 3, and that of (r & y) from that of 5 
p. 4. If C n is that part of the coefficient of the n th argument in the development 
of the quantity r 3 ^ — — y ^r& yj which is independent of r n , with a contrary 
sign; 
X ' = ^Y (’ + ^ e2 ) + ^ + j{ 2r ° r ‘ + 62(7-3 + r ^ r * + e ' 2(r e + r i) r ^\ 
r, = 
— 3e 2 {2 r l ro + 2 r 0 r 3 +2r 0 r 4 } 
y ( 1 + f e *) ( 2 r ° + e? r ») + y { (r -> + r s) r, + 2 r 0 r 2 J 
— 6 { r o 2 + y + — y- + } 
