360 
DR. J. C. FIELDS ON THE FOUNDATIONS OF THE 
necessary and sufficient that the constant coefficient of ^ should be 0 in the 
reduced form of the product 
It d'(0,u.).(49) 
with the coefficients of the powers of expanded in powers of ijz. In order that the 
orders of coincidence of a rational function d' {z,u) for the value z = 00 may he com¬ 
plementary adjoint to the order 2 to the orders of coincidence it is 
necessary and sufficient tliat tlie principal residue, relative to the value 2 = co ^ in the 
product (49), should he 0 . This Ave have seen in § 2 . In order, then, that the orders 
of coincidence of the function d' {z, u) for the value z = cc should not fall short of the 
orders of coincidence defined by the equalities 
(cc) 
, s = 1 , 
(50) 
it is necessary and sufficient that the principal residue in the product ( 49 ) should 
be 0 . Among the conditions imposed on the constants in the function dk (z, u) by the 
orders of coincidence here in question are included those obtained on equating to 0 the 
principal residue relative to 0 = 00 in the product 
j d^ (0, m).(51) 
To tlie function "T (0, n) we shall iioav give the form 
■T (0, u) 
n i-1 
t = 1 (J = -J + 2 
(52) 
We shall assume that the integer { has been chosen so large that terms involving 
0“* and higher poAvers of I/0 are unconditioned by either of the sets of orders of 
coincidence ..., or Furthermore, we shall assume, Avhere 
this is not already implied, that i has been chosen so large that a rational function of 
(0, u), conditioned by either of these sets of orders of coincidence, cannot involve 
a poAAmr of 0 higher than 0*"h The function {l/z,u) is then of the type 
Pdffi,+ 2 ( 1/2,'co¬ 
write the product ( 51 ) in the form 
n 
2 
t=i 
i-1 
z 
q = -i + 2 
a 
s-i, i-n 
(53) 
Avith the constants as yet arbitrary. Noav equate to 0 the principal residue 
in this product. We thus subject the constants to independent conditions, 
that is to say, Ave subject these constants to as many conditions as there are arbitrary 
constants involved in (I/2, "(O- To see this Ave note first that the principal 
residue in a product 
\0 / i =1 1 q = —i+2 
(54) 
