348 PROCEEDINGS OF THE AMERICAN ACADEMY. 



(18) ~J~ Z *.t,0 Z /i,l-l,0 Z k,l,d — 0. 



v ' dx x x ' 



We have seen how to solve this system. Any one of the n solutions 

 that we have obtained may be used as the first approximation. This 

 approximation having been chosen, we insert it in the right side of (17) 

 and obtain the following relations for the first correction : 



j -i i=m j—e, 



d 1 r k \\' -AJ 



-r z i,i,l z k,i-l,l z &,i,1 ■— 2L Zj b k i Z '.j,Q- 



ax x x i . , * ,c J ' 



The right side is now a known function of x ; and we have, conse- 

 quently, a system of non-homogeneous linear differential equations to 

 solve for z ktli \. Having determined this first correction, it is inserted in 

 the right side of (17), and the resulting equations are solved for the 

 second correction. This process is repeated again and again, the relation 

 connecting the qih. and the (q + l)th correction being : 



n a\ d I r * X \V J 



\ l y ) dx**' '' g+l ~ x **• /-1, ?+1 ~~ x Zk ' l ' 9+i ~ ^ -^ *• l Z ' J ' q ' 



i—l j—1 



Each equation (19) may be written: 



i—l j—1 



whence : 

 (20) 



r r /* t=m J=e ' i ■ ~i 



**, /, t+i = a: r * / *~ ~* k z k, i-i, t +i dx + I x~ r * 2 2 b *!i 9 V. t dx • 



Now writing out the value of ? i|Mpf+1 in the same way, and substitut- 

 ing it in the first integral of (20), we have: 



X X 



**, i. g+i = * k J - dxj x' 1 "** z kt ,_ 2> 9+1 dx 



c i,l C fc,l—\ 



XX X 



/, r* i—m j=e,- n i—m j=ej -, 



