368 
DE. J. C. FIELDS ON THE EOUNDATIONS OF THE 
If in the product (74) we take for the function '^{z,u) the general integral 
rational function of degree i — 2 in 2 , a7id if in this product we equate to 0 
independently of tiie values of the arbitrary constants S the principal residue relative 
to the value 2 = 00 we evidently also in this case obtain the necessary and sufficient 
conditions that the function '^( 2 ,'?/) maybe built on the basis (f). For the conditions 
so arrived at include among them the conditions obtained on equating to 0 the 
principal residue relative to the value 2 = 00 in the product (72). These conditions, 
however, are necessary and sufficient in order that, for the value 2 = co, the integral 
rational function ’F ( 2 , u) of degree ^ —2 in 2 should have orders of coincidence which 
are complementary adjoint to the order 2 to the orders of coincidence ..., TrJ^\ 
They thei’efore involve the reduction of the degree of Sk ( 2 , w) in 2 from i — 2 to M. 
On taking, then, for 'F ( 2 , u) in the product (74) the general integral rational function 
of {z,u) of degree i — 2 in 2 and equating to 0 independently of the values of the 
arbitrary constants ^ the principal residue relative to the value 2 = 00 in the product 
we impose on the coefficients of the function 'F ( 2 , u) the necessary and sufficient 
conditions in order that it may be built on the basis (f). 
The general integral rational function ^{z,u), of degree i—2 in 2 , conditioned by 
equating to 0 the principal residue in the product (74), independently of the values of 
the arbitrary constants S, is readily seen to be the most general rational function built 
on the basis (f). For the orders of coincidence furnished by the basis (f) for finite 
values of the variable 2 are liere all adjoint, and therefore the general rational 
function of ( 2 , u) built on the basis (t) must be integral. Also i was chosen suffi¬ 
ciently large so that a rational function conditioned for 2 = 00 by the orders of 
coincidence could not be of degree in 2 greater than i—2. We might 
here again recall the limitations imposed on our choice of the integer i in the 
preceding argument:—It was taken ^ M -t 2 and also so large that terms involving 
2 “* and higher powers of I /2 in the coefficients were not conditioned by the partial 
bases (t)^"^^ and (t)''^^ At the same time we required i to be sufficiently large to 
serve for each of the integers in the products (70), the least values eligible for these 
integers being severally dependent on the degree M of "F ( 2 , u) in 2 . 
§6. The 7i{i—l) coefficients c_q n-t regarded as linear expressions in the arbitrary 
constants ^ may or may not be linearly independent of one another. We shall 
suppose that just X of them are linearly independent of one another, the remaining 
w(^ —1) — X coefficients being linearly expressible in terms of these. Indicating such X 
linearly independent coefficients by the notation we shall assume that the 
n{i—l) coefficients are all expressed linearly in terms of these X coefficients. 
The principal residue relative to the value 2 = co in the product (74) will then be an 
expression bilinear in Ci,...,Ca and the coefficients of '^{z,u). In this expression 
equating to 0 the multiplier of each of the quantities c^, ..., c,v, we impose on the 
constant coefficients of ’F ( 2 , u) conditions not greater in number than X. The 
function “F ( 2 , u) so conditioned is built on the basis (t), for with 'F ( 2 , u) so conditioned 
