[FI=Lps] EXISTENCE OF THE POWER-SERIES 
Let us consider a sum of the type (19.). Among the exponents 
c,c,....c) may appear fractions. Suppose e to be the least common 
multiple of the denominators of all such fractions, assumed to be 
represented in their reduced form. The sum in (19.) may then be 
written in the form 
m my My Mp 
21. u=aze+a'ge +a"ge +....+a®&-Dge 
where m < M1 < Ma < .... < Mmx-1 Here the elements of the sum 
have been successively obtained by a process described in the pre- 
ceding pages. Among the sums arrived at by this process may 
readily be seen to be included also all those sums derived from (21.) 
1 
1 1 
by replacing in it ge by e ge wheree is any eth root of unity. To see 
this let us recall the precise conditions subject to which the successive 
elements #1,....u#in the sum (19.) are chosen. 
I. The functions in (20.) must present a successively increasing 
sequence of orders. 
II. For the exponent c® in u;.,;=az* must be taken 
ies ee he A 
5 Fe  c® c@ cf) c@ co ci) co) c® | 
4 ; DRE BEC a OR ae el 
11 2 S n | 
WHerE CCE ne GON recta: c® are the orders of the coefficients in 
the equation 
Ba: F(z, dW +t2+.... +; +;) =0. 
Now on assuming that these two conditions are satisfied by the 
successive elements in the sum (21.) it is evident that condition I. is 
also satisfied by the successive elements in the sum 
m my ™e—-1 MR] 
24. w. Sae™se+aeMizge +... +aR De ge » 
1 1 
derived from (21.) on substituting e ge for ge in this sum, since this 
substitution in the functions (20.) plainly does not alter their orders. 
Furthermore the condition II. is also satisfied by the successive 
elements in the sum (24), for the substitution in the elements 2... . 14; 
1 1 
in the equation (23.) of eze for se evidently does not alter the 
orders c®, c,,....c of the coefficients of this equation in v;, so 
that the condition If. requires for the (¢+1)th element in the sum 
(24.) the same order c“) which is possessed by the element #;:1 = 
a) << 2 age’ in the sum (21.). This order however is that which 
is actually possessed by the (i+1)th element a e”i ge’ in the sum 
(24.). The conditions I. and II. are then satisfied by the sum (24.) 
where € is any one of the eth roots of unity. We have however 
already seen that the number of sums of the type (19.) can never 
exceed 7 the amplitude of the original equation F(z,v)=o. It follows 
therefore that we must have 
25: BRUT: 
_We now see that in the sequence of increasing orders c, c’,...... 
c,®)....of the elements %,%,....w; ....in the sum (19.) the increase 
