THEORY OF ALGEBRAIC FUNCTIONS OF ONE VARIABLE. 
365 
given in ( 6 l) is obtained on substituting these orders of coincidence for the symbols 
n'g in ( 25 ). For the number of the conditions in question we thus obtain 
ni, 
- 2 0 -, 
s = 1 
(*)„ (*) i V 
s = 1 
M, 
(k) 
(*) 
(62) 
Subtracting this number from the total number ni^ of the arbitrary constant 
coefficients in the unconditioned function of the form {z, u), we obtain the 
expression 
1 = 
2 
S = 1 
! 2 
s = 1 
. . (63) 
for the number of the arbitrary constants involved in the numerator {z, u) of an 
element of the summation on the right-hand side of ( 59 ). This numerator then we 
can write in the form 
= 2 .( 64 ) 
s = 1 
where the coefficients are arbitrary constants, while the functions {z, u) 
are linearly independent and possess orders of coincidence for the value 0 = which 
do not fall short of the numbers given in ( 61 ). It is evident that the summation in 
( 59 ) is to be extended not only to all those finite values of the variable 2 to which 
negative elements in the basis (r) correspond, hut also to all those values 2 = for 
which the corresponding numbers •••5 are not all 0, even if the 
corresponding elements in the basis (t) are all 0. 
In order that a rational function H (2, u) should be built on the basis (t), it is 
necessary and sufficient that it should be simultaneously representable in the two 
forms ( 48 ) and ( 59 ). Identifying the representation of the function H (2, u) given in 
(59) with the representation given in ( 48 ), we have 
2 2 
K 5 = 1 
(2-a,)'* 
l(X> 
-t 2 
5 = 1 
(i) 
-P(2,«) + 2-’((f 
(65) 
Here V {z,u) evidently identifies itself with that part of the sum — 2 (I/2, w) 
s = 1 
which is integral in (2, u) and the conditions to which the constants ^ are subjected, 
because of the identity ( 65 ), are obtained on developing in powers of I/2 the 
coefficients of the several powers of u on the left-hand side of the identity and 
equating to 0 the aggregate coefficient of z~Hi''~^ for the values g = 1 , 2 , ..., i— \ ; 
t = 1 , 2 , ...,n, since 0 is the coefficient of the corresponding term on the right-hand 
side of the identity. If for q^i we equate the coefficient of z~HC'~'' on the left- 
hand side of the identity to the corresponding coefficient on the right-hand side, we 
so determine an otherwise unconditioned coefficient of the expression 2~'((I/2,-2^)) in 
terms of the constants 8 . The coefficient of on the left-hand side of the 
