362 
DR. J. C. FIELDS ON THE FOUNDATIONS OF THE 
For the number of the conditions imposed on the coefficients of the function 'F {z, u) 
by the set of orders of coincidence we derive from ( 41 ) the expression 
7 i{i- 2 )+ 2 - 4 .^ 
s=] S=1\ 
(oo)/ "S 
From the equalities ( 50 ) we have 
' s 
s=l s=l s=l 
(»)/ ‘ s 
= 0 , 
and by the aid of this equality the expression preceding can evidently be written in 
the form 
ni- 2 + i 2 
S = 1 
S = 1 
D)_l I_ L ) y D) 
This then is an expression for the total number of the conditions to which we must 
sul^ject the coefficients in order that the function 'F (z, u) may have the orders 
of coincidence This is, however, also the expression for given in ( 45 ). 
The total number of the conditions which we must impose on the coefficients of the 
function F (z, u) in order that it may have the orders of coincidence is 
therefore ly, and the conditions themselves are all obtained on equating to 0 the 
principal residue in the product ( 51 ) for arbitrary values of the constants involved 
in the factor (l/z, u). The necessary and sufficient conditions then in order that 
the function F (z, ii) may have for tlie value z = go orders of coincidence which are 
complementary adjoint to the order 2 to the orders of coincidence are 
obtained on equating to 0 the principal residue in the product ( 5 l), where the 
function (l/z, n) is conditioned by the set of orders of coincidence 
On taking in particular for the function F (z, ti) the integral polynomial form 
F(z,n)= 2 2 _iZ^-V-k.( 56 ) 
t-\ ri = \- 
it must evidently still hold true that the necessary and sufficient conditions in order 
that the function F (z, w) may have for the value z = oo orders of coincidence which 
are complementary adjoint to the order 2 to the orders of coincidence 
are obtained on equating to 0 the principal residue in the product ( 5 l). It is to 
be borne in mind that throughout the preceding argument we have assumed the 
integer i to be chosen sufficiently large for our purpose. We assumed, namely, that 
it was chosen large enough at least to ensure that the coefficient of a term involving 
z“^ or a higher power of l/z was not conditioned by either of the sets of orders of 
coincidence ..., or f/*', ..., and at the same time we assumed that the 
possession of either of these sets of orders of coincidence by a function was incom¬ 
patible with the presence in the function of a term involving z to a higher power than 
z'“^. If F (z, u) is a polynomial of assigned degree M in z the necessary and sufficient 
