80 BIRKHOFF AND LARGER 



1 " 



~r 2 [5 ih ] k \lhs — >"hs} [&sj\ 



\ k fc,«=l 



Equating the coefficients of X° we have 



ct Cj { (y,— y { ) o® + (6,-,- ha) 5 r; } = 0, 



from which it follows that a,} = when i 4= j- Again, equating the 



1 

 coefficients of - we have 

 X 



(7~ H) «# + (hi- b u ) c#+ *$' ^ 0, 



from which it is seen, upon setting i — j that a# ' = 0, namely that 

 °m — constant. The relation shows on the other hand that a\f = 

 when i =£ j. Equating to zero successively the coefficient of each 



individual power of - it is found in the same way that o\f = when 



A 



* ^ j, oft — constant for /j, = 1, 2, . . . ,(k — 1). It follows that 

 = (c,-c 7 -5i,-[l]ifc_i) +— k E~ l (an) E, 



A 



where the coefficients of [lk-_i are constants. 



Now for any choice of the set of series c,- it is clearly possible to 

 choose a set Cj such that c ; c, [lk--i = 1, j = 1,2,.. .n. The formal 

 solutions S(x) and T{x) corresponding respectively to these values of 

 Cj and Cj are closely related. They are called associated formal solu- 

 tions and satisfy the relation T(x) <S(.t) = I + —% E~ x (o^) E. 



A 



Since in particular c, may be chosen as c ; - = 1 it is seen that there 

 exists a formal solution having the form 



(37) S(x) = («*r.K*)+W [ 8 ..] )< 



In accordance with the definition above the associated solution is 

 given by the set of c/ s which satisfy the relations c ; = 1. It is 



