VTI.] CORRECTION OF APPROXIMATE SOLUTIONS. 31 



sn 



_ 

 at at at 



v , ^di.dS'e , dm 



i 2 -~ -T + 2v -j- 

 at at dt 



We see that their value is known when 87, 8'0 are determined. 



Now S'Z, S'0 may be determined as follows. 



Let X = p a + 'Zpf cos i\jj, Y Sg ; sin i^, 



where i takes all positive integral values ; and assume 



S'l = a a + Sa ; cos ty, S'# = SZ>j sin i\jj. 



Then substituting and equating coefficients, the constant term gives 



p - 3c = 0, 

 and the terms in i\ji give 



p f - ret; -2(l+n') ibf - 3ca ; = 0, 



q { - rb f -2(l+n')ia,i = ; 



the second of these may be written 



2(1 + it,') -4(1+ nj a, - 2 (1 + n'} ib, = ; 

 subtract from the first and we have 



#-2(1+ n') q ! - a, [i 2 - 4 ( 1 + nj + 3c] =0, 



or 



and 



We see that ; , b { will be of the same order of small quantities as p t , 

 q t , in general. And therefore the coefficients of the terms of X', Y' will 

 be of order higher than those of X, Y. Proceed then to determine further 

 corrections 8"l, S"6 satisfying the equations 



