V.] THE VARIATION. '21 



The coefficient of cos 2i/ in the first of these expressions, and that of 

 sin 2t/> in the second, are respectively 



4 (n - n'Y a., 4n (n n') b a + 3 . . a, - - n'-, 



a? 2 



and - 4 (n n')- b, + in(n n') a., + - ' 2 , 



and these are evidently reduced to zero by giving a.,, b., the values pre- 

 viously found, if we substitute for /u./a :i the approximate value n* + -n'*. To 



find the more correct value of ju./a 3 , equate to zero the constant term in 

 the first expression ; 



2 (n - n'Y a? -n*-2(n- n'Y b: + ^(i+- a*} - - n H + - n* b., = 0, 



a \ 2 " / 2 2 



that is 



= n- + J n H - 2 ( - nj a; + 2 (n - nj b; - 1 n'*b, 



= ri + ^ n H + 2(n- nj I (2m, + m, 2 ) a.; - j- m, 4 1 . 



Hence we see that p/a? dift'ers from ri* + -n'- only in terms of the 



fourth order, if we consider TO, a quantity of the first order and conse- 

 quently a a , b., quantities of the second order. Hence also by taking 



in the multiplier of a 2 , when we equate to zero the coefficient of cos2//, 

 we only neglected a quantity of the sixth order in m 1( and the error 

 in the resulting values of a.,, b, is of that order. 



We see that the substitution just made in our equations leaves out- 

 standing terms of the fourth order in cos 4i// and sin 4t/>. In order to get 

 rid of these we must add terms of this form to the assumed values of 

 1/r and 0, respectively. Suppose that 



- = -[!+, cos 2t/f + a t cos 4</], 



/ Ow 



6 = nt + e + fe, sin 2/ + b t sin 4/>, 

 where, as we shall find, a t and 6 4 are small quantities of the fourth order. 



