370 
DR. J. C. FIELDS ON THE FOUNDATIONS OF THE 
Suppose the function {z, u) to be the general rational function built on the basis 
(f). It then involves just l)—X' arbitrary coefficients. Equating to 0 the 
principal residue relative to the value 2 = 00 in the product (75) independently of the 
values of the arbitrary coefficients in force the constants 
c'_g_n-t ft) satisfy this many independent linear equations. These w (^ —l)—X'linear 
equations must then be satisfied by the coefficients in the fii‘st factor of the 
product ( 74 ) independently of the values of the constants 8. For independently of 
the values of the arbitrary constants 8 involved in the coefficients in (74) the 
principal residue relative to the value « = go in the product is 0 when ( 0 , m) is the 
general rational function built on the basis (f). Regarded as linear expressions in 
the constants (i, then n{i—l) of the n{i—l) coefficients in (74) are linearly 
expressible in terms of the remaining W coefficients. It follows that the number of 
the coefficients which are linearly independent of one another is ^ X'. The 
number of these coefficients which are actually independent of one another is, however, 
X. We therefore have X ^ X'. We have, however, already found X' ^ X. We derive 
X = X'. The number of the arbitrary coefficients involved in the general rational 
function built on tlie basis (t) is then ^i(^—l)—X, and this is also precisely the 
number of the coefficients c_g „_t wliich are linearly expressil)le in terms of the 
remaining X coefficients. 
Employing the notation N; to designate the number of the arbitrary coefficients 
involved in the expression of the general rational function built on the basis (t), the 
number of the coefficients which are linearly independent of one another is just 
n(4 — 1 )—N?. This, then, is precisely the number of the conditions which we impose 
on the arbitrary constants 8 when we equate to 0 the n{i—l) coefficients fhe 
identity (66). The subsistence of the identity (65) therefore imposes k(^ —l)— 
conditions on the constants 8, these being the conditions which are necessary and 
sufficient in order that a rational function representable in either of the forms (48) 
or ( 59 ) should at the same time be representable in the other form also, it being 
understood that the functions {z,ii) which appear in the summation in (59) have 
the special forms given by formula (64). 
Referring to formulae (45) and (63) we obtain for the total number of the constants 
8 here in question the expression 
= ni—'L 
: = 1 
2 
.s = 1 
(77) 
since = — v/"! Subtracting 7i (^—l) —from this expression, we obtain 
+ — 
n- 
N" 
s = 1 
(«),, (k)i iV 
JS ~F *2 ^ 
V 
.s = 1 
1 + 
(0 
. . (78) 
for the number of the constants 8 which remain arbitrary after the forms (48) and 
( 59 ) have been identified. For the moment we shall indicate the expression (78) by 
