364 
DR. J. C. FIELDS ON THE FOUNDATIONS OF THE 
Any rational function of {z, u) can be represented in the form 
H (2, u) = 2 
K 
{z, u) 
{z-a^Y'^ 
+ V{z,u), 
(59) 
where P (2, u) is a polynomial in (2, u), where the summation is extended to a finite 
number of v.alues z — only, and where any numerator (2, u) is a polynomial in 
(2, u) of degree i^—1 in 2. The polynomials are here, of course, assumed to be reduced 
in u. 
We shall first assume that the basis (r) involves no positive orders of coincidence 
for finite values of the variable 2, no such restriction, however, being made for the 
value 2 = CO. Writing t/*' = — o-/*', we may say of a function built on the basis (t) 
that, for a finite value 2 = a^, it becomes infinite to orders which do not exceed the 
respective numbers of the corresponding set 0-/*^ ..., while for the value 2 = 00 
its orders of coincidence do not fall short of the numbers respectively. 
Here the numbers are zero or positive, whereas the numbers may be positive, 
zero, or negative. The orders of coincidence furnished Ijy the basis (f) for finite 
values of the variable 2 are in this case plainly all adjoint. 
Suppose H (2, u) in ( 59 ) to be the general rational function of (2, u) conditioned by 
the partial basis (t)' here in question. The polynomial P (2, u) is evidently arbitrary. 
Furthermore we may, in the summation on the right-hand side of the formula, take 
for the greatest of the integers 
+ s = 1 , 
( 60 ) 
To show this we note that the orders of coincidence of the numerator (2, u) with 
the branches of the respective cycles corresponding to the value z = must not fall 
short of the numbers 
h —s = 1,2,.,., r^. 
( 61 ) 
These numbers, however, would be simultaneously greater than the corresponding 
numbers /x/*' if we should give a value greater than the greatest of the integers in 
( 60 ), and the numerator (p^"'‘^{z,u) would therefore, by a theorem proved in § 3 , be 
divisible by the factor z — a^. 
Choosing then for the greatest of the integers in ( 60 ), we readily see that the 
orders of coincidence in ( 61 ) are not simultaneously greater than the corresponding 
numbers and that therefore the general numerator (2, u) is not divisible by 
the factor z — a^. We see at the same time that the orders of coincidence in (Ol) 
are adjoint relatively to the value 2 = a^. As a consequence, the number of the 
conditions to which we must subject the otherwise unconditioned constants in the 
general function of the type (2, u) in imposing on it the orders of coincidence 
