Sb THE ROYAL SOCIETY OF CANADA 
rt 
from one order to the next must be = ig and also that in the suc- 
cessively increasing sequence of orders ©, Co,..... Cy co. Fa of the 
functions in (20.) the difference between two successive orders must 
Le =<. Beom this will ferthecmarc fallow thar ea eee 
function 
26. F(z,m+....+Up ) 
will increase indefinitely with the indefinite increase of k. This how- 
ever will not be the case for the function F’,(z,u). To see this it is 
only necessary to recall the fact that we can find integral rational* 
functions g(z, v), h(z,v) and D(z) such that 
27. g(z,v) F’y (2,v) +h(z, v) F(z, v) = D(z) 
.dentically. The order of D(z) we shall indicate by d. For v in the 
identity we shall now substitute 
U=U+uU+....+Uz, 
where a sufficient number of elements are taken torensure that F(z, u) 
has an order > d. The order of the product (z,u)F(z,u) will then 
also be > d, for the order of k(z,u) can evidently not be negative. 
From 
28. g(2, u) FP’, (2, wu) +h(z, u) F(z, uw) =D(z) 
then follows that the order of g(z, uw) F’, (z, wv) must be precisely equal 
to d and that therefore the order of F’,(z,u) must be=d. When the 
order of F(z, 2) is>d then the order of F’, (2, 2) is =d. 
Now consider the equation 
29. F(Z, Ui+u2+....+u,+%,)=0 
with reference to the orders c®), c,....c® of the coefficients of the 
several powers of v, and with reference to the number 
k k k k k k () __ .( 
30. ®=Max.{ & 4, GE, © GE, Co On |, 
1 2 Ss nN J 
The amplitude of the equation (29.) will be 1 if c—c{” is 
greater than any of the other numbers on the right of (30.). Bearing 
in mind that the numbers c™, c™,..... c® are none of them 
negative we see that c®)—c(*) will certainly be greater than any of 
the other mondes parentheses if it is greater than each of the 
The 
k k 
Go. ee 2 ne and therefore if it is > © 
3 n 2 
number c% —c(%) will consequently be greater than any of the other 
numbers here in question if c® > 2c and therefore quite surely 
when c > 2d. If then c% > 2d the amplitude of the equation 
(29.) is 1. The method of proving the existence of a power-series 
which satisfies an equation of amplitude 1 is well-known. For the 
sake of completing our proof however we reproduce it here. 
numbers 
* If the polynomials which serve as coefficients in /(z,v) are replaced by integral adenine in 2 
we evidently still have an identity of the form (27.) in which however the coefficients of the powers of v in 
2 (2, v) and & (z, v) are integral power-series in 2. RE 
