THEORY OF ALGEBRAIC FUNCTIONS OF ONE VARIABLE. 
369 
the principal residue relative to 2 = co in the product (74) is 0 independently of the 
values of the arbitrary constants S. The general rational function of ( 2 , u) built on 
the basis (f) must then involve at least n arbitrary coefficients, since the 
general integral rational function ^{z,u) of degree ^ —2 in 2 has n{i—l) arbitrary 
constant coefficients. 
Indicating by n{i—l)—\' the actual number of arbitrary coefficients involved 
in the general rational function ( 2 , u) built on the basis (r), we have 
n (^ — 1 ) —X' ^ n (^ — 1 ) —X, and therefore X' ^ X. Let us now consider the product 
I 2 'c'_,.„_,2-%”-Vd^(2,^).(75) 
t = l q = l 
in which the n (^-l) coefficients c'_y are arbitrary constants, while 'ir{z,u) is a 
specific function of degree in 2 not greater than i—2. The principal residue in this 
product cannot be 0 independently of the values of the constants For if 
be the highest power of u which appears in d' ( 2 , u), and if the term 
(a^_i 7 ^ 0 ) actually presents itself in the function, the principal 
residue in the product 
. d'(2, U) 
is evidently c'_^ (_], which can only be 0 for = 0. 
Let 'Ll ( 2 , m), ..., "Fp ( 2 , w) be p linearly independent integral rational functions of 
degree in 2 not greater than i—2. If in each of the products 
2 2 . ^^( 2 ,^ 4 ); s = 1, 2, ..., g,.(76) 
t = i ? = 1 
we equate the principal residue to 0, we impose p independent conditions on the 
constants c'_g n-f For suppose the principal residues in the p products, regarded as 
expressions linear in the arbitrary constants to be linearly connected, and 
suppose (7i,...,c?p to be the respective multipliers in the relation existing between 
them. On constructing the function 
^ ( 2 , u) = ( 2 , w) +... + dpd^p ( 2 , u), 
the principal residue relative to 2 = co in the product (75) would be 0 independently 
of the values of the constants and this we have seen to be impossible. It 
follows that we impose p independent conditions on the arbitrary constants 
when we equate to 0 the principal residue relative to the value 2 = 00 in each of the p 
products (76). If, then, in the product (75) the integral rational function ^{z,u), of 
degree i—2 in 2 , involves a certain number of arbitrary coefficients, we impose just 
this number of conditions on the constants on equating to 0 the principal 
residue relative to 2 = 00 in the product (75). This means that we connect the 
constants c'_g n-t by this number of independent linear equations. 
3 B 
VOL. CCXII.-A. 
