356 
DR. J. C. RIELDS ON THE FOUNDATIONS OP THE 
It is evident that the statement just made holds not only for —l) but 
also for any value of the integer X so long as it is at least as great as the greatest 
degree X' in 2 which a coefficient of a power of u in the reduced form of a rational 
function of {z,u) can have consistently with adjointness for the value We 
see, namely, that among the conditions whose number is given in ( 37 ) are included the 
n (X—X') conditions, which dispose of the terms of degree >X' in the coefficients of the 
powers of u. 
Let us now denote by i an integer which is at least as great as the greatest of the 
integers [/w/"”'], ..., and impose on the general function II_a (l/z,the order of 
coincidence i wltli each of the branches of the fundamental equation corresponding to 
the value 2 = oo. The orders of coincidence i here in question are evidently adjoint 
and over and above the conditions requisite to adjointness, wliose number is given in 
( 37 ), impose on the coefficients of the function II_a(]/2, m) further conditions, whose 
number is given by the sum 
»■(» 
.9 = 1 
(CC) ) 
(a) 
( 38 ) 
The total numlier of the conditions here imposed on the coefficients of the general 
function II_a(1/2,-m) by the orders of coincidence i is therefore 
71 ( 7 +X)—-g- 2 
s =1 
( 39 ) 
We shall now assume not only that X has been chosen at least as large as the 
greatest degree of a coefficient in a rational function E, (2, u) which is consistent with 
adjointness relative to the value 2 = 00 on the part of the function, but also, where 
this is not already implied, that it has been chosen at least as large as the greatest 
degree of a coefficient which is consistent with orders of coincidence relative to the 
value 2 = GO , which are none of tliem negative. Let us assume for the moment, too, 
that we have chosen i positive—what is not necessarily implied for all cases in what 
precedes. Now impose on the coefficients of the general function R_;y(l/2,«) first 
the conditions required by a set of orders of coincidence for the value 2 = co ^ each one 
of which is 0. Thereafter imposing on the coefficients the 7 ii further conditions 
rec|uired by a set of orders of coincidence, each one of which is i, we arrive at the total 
number of the conditions given in ( 39 ). Substracting 7 ii then from this number we 
obtain the expression 
7 l\ 
2 
s = 1 
(00) 
1 + 
( 40 ) 
for the total number of the conditions imposed on the coefficients of the general 
function by a set of orders of coincidence for the value 2 = 00^ each one 
of which has the value 0 . 
From ( 40 ) we see, where i is an integer positive or negative, that the expression 
