Variation of the Semi-axis Major of the Moon's Orbit. 339 



In the Lunar Theory terms of the order w 3 , and of the nature of 

 those under consideration, may arise not only in the third but in 

 every succeeding approximation, and hence it becomes absolutely 

 necessary to seek some mode of proof which admits of unlimited 

 extension, and I have therefore endeavoured to modify the proof 

 given by Poisson, so as to include all terms of the order wz 3 , how- 

 ever far the approximation be pushed, and without introducing 

 any peculiar system of constants. 



For this purpose it is necessary to suppose the disturbing func- 

 tion R 



dR dR 



S= nt d7 = -d7 =l * ; 



and to define 8 f, 8 a, 8 c, &c., to represent, not the total variation 

 of the quantities , a, c, e, &c., as in Poisson's paper, but that por- 

 tion only which consists of arguments corresponding to those in- 

 equalities which are not depressed by integration, as will be pre- 

 sently explained. The effect of the secular inequalities of the con- 

 stants c, 03 and , and also that of all the other inequalities of which 

 the arguments are independent of c, are supposed to be already 

 included in the quantities 



As the constant c in Poisson's notation always accompanies n t, 

 all the arguments in the development of R may be properly repre- 

 sented by an expression of the form 



* (n t + c) 4- j m t + 1 1 + |8, 



where i andj are whole numbers or zero, I a certain multiple of m 2 

 depending upon the secular variation of the angles c, &>, a, and /3 a 

 quantity rigorously constant, which accompanies 1 1, but which I 

 shall in future omit to write down. Whenever i = without,/ be- 

 coming = at the same time, the corresponding inequality in the 

 expressions for the elliptic constants is necessarily a multiple of m at 

 least, and only when i andj are both equal to zero the correspond- 

 ing inequality in each of those expressions may no longer be mul- 

 tiplied by m. 



d a = [a, c] -r- d t 4 [a, a>] T d t + &c. 

 Z2 * 



