388 Mr. J. R. Wilton 



obtained is as follows : — 



7T-/<2 / oo, \ oo, cc oo 



0= — +C(1+ 2n 2 AJM + S nV+ SA, 2 m(m + rc)A m A m+nj 



o = c 



2rA r + 2 I m(.m + r)A«A m+r l+A r + I (2n + r)A re A Ji+r 



?' — 1 00 oc 



+ U rA„A,._ M + X A„ % m(m + n + r) A m A m+n + r 



CO 00 



+ z A n 2 w(m + 7i— r)A OT A TO+re _ r , 



for all values of r from 1 to infinity. It is understood that 

 in the last summation m + n—r is positive. 



These equations must be solved by a process of successive 

 approximation. The equations, to any desired order, may 

 be written down from the above general expression, or by 

 means of a rule derived from it. The rule, which is some- 

 what complicated, is as follows : — 



The equation obtained by equating the coefficient of cos.pf 

 to zero is made up of terms : 



(1) C multiplied by 



2pA ;3 +2(^ + l)A 1 A p+1 + 4:(;9 + 2)AoA /J+2 +&c. 



(2) A,. 



(3) All terms made up of the product of two coefficients 

 the sum or difference of whose subscripts is p ; and the 

 numerical multiplier of any term is the sum of the subscripts 

 unless the term is a square, in which case it must be halved. 



(4) All terms made up of the product of three coefficients 

 whose subscripts are such that the sum of two of them minus 

 the third is ±p ; and the numerical multiplier of any term 

 is the product of the two numbers whose difference is p and 

 whose sum is the sum of the subscripts of the coefficients 

 forming the term. If, however, one of the terms contains 

 the square of a coefficient, and is such that twice the sub- 

 script of this term minus the subscript of the remaining 

 coefficient is ±p, the term is to be halved, but not otherwise. 



As an example of the use of the rule, the equation for 

 which p = 2 is here written down to the twelfth order. It is 



= C(4A 2 + 6A 1 A 3 + 16A 2 A 4 + 30A 3 A 5 + 48A 4 A 6 + 70A 5 A 7 ) 

 + A 2 + A 1 2 -i-3A 1 2 A 2 + 4A 1 A 3 + 8A 1 A,A 3 + 4A 1 2 A 4 + 4A 2 3 



4- 6 A 2 A 4 -f 15A 1 A 3 A 4 + 15 A 4 A 2 A 5 + 15A 2 A 3 2 + 8A 3 A 5 

 + 24A 1 A 4 A 5 4-24A 1 A 3 A 6 + 24A 2 A 4 2 + 12A 2 2 A 6 + 12A 3 2 A 4 

 4- 10A 4 A- 6 + 35 A 2 A, A 6 + SSA^A, 4 35A 2 A 5 2 

 -f 35A 2 A 3 A 7 + 35A 3 A 4 A 5 + 12A 5 A 7 + 6A 6 2 . 





