THEOEY OF ALGEBRAIC FUNCTIONS OF ONE VARIABLE. 
353 
principal residue relative to the value z — a\\\ the product (2, u) d' (2, u) we obtain 
the necessary and sufficient conditions that the rational function d' (2, u) may have 
orders of coincidence for the value z — a which are complementary adjoint to the 
orders of coincidence tj, If ^^(2,n.) is an integral rational function, these 
conditions are evidently all obtained on equating to 0 the principal residue relative to 
the value z = a in the product 
0 ^^'^ {z, U) 
{z—aY 
(2, 11) 
( 26 ) 
This, however, is equivalent to equating to 0 in this product the principal residue 
relative to the value 2 = co ^ since in the principal coefficient of the product the sum 
of the residues must be 0 and the only residues which could here present themselves 
would have to correspond to the values z = a and 2 = go . The necessary and sufficient 
conditions, then, that d' (2, u) should have orders of coincidence for the value 2 = a, 
which are complementary adjoint to the orders of coincidence tj, ..., are obtained 
on equating to 0 the principal residue relative to the value 2 = co in the product ( 26 ). 
Supposing the integral rational function T' (2, u) to have a definite degree M, and 
representing the first factor of the product ( 26 ) in the form 
{z — CcY ( = lr=l 
(27) 
we see that, on choosing y sufficiently large, the residue relative to the value 2 = co of 
the principal coefficient in the product ( 26 ) will be the same as the residue of the 
principal coefficient in the product 
i Y y_„._, 2 -V-‘.'I'( 2 ,M).( 28 ) 
< = 1 )• = ! 
The vanishing of the principal residue in the product ( 28 ), independently of the 
values of the arbitrary parameters involved in the expression of the coefficients 
then gives the necessary and sufficient conditions in order that the function 
(2, u) may have orders of coincidence for the value z = a which are complementary 
adjoint to the orders of coincidence ti, ...,Ty, the integer y being supposed to be 
chosen sufficiently large. 
If the orders of coincidence ti, ..., were all adjoint the index i in ( 26 ) would be 0 
and the function (2, u) would not exist. In this case the orders of coincidence of 
(2, u) would simply have to be 0, or positive, in order that they might be comple¬ 
mentary adjoint to the orders of coincidence tj, ..., and that is already the case for 
the function (2, u) since it is integral, and because we are here assuming the 
fundamental equation (l) to be integral. We might remark that where we have 
occasion later on in this paper to make explicit use of the results just obtained the 
orders of coincidence ti, ..., will be none of them positive. On writing 
Tj —^ ^ 1 ? • • • > 
VOL. CCXII. — A. 2 Z 
