THEORY OF ALGEBRAIC FUNCTIONS OF ONE VARIABLE. 
361 
cannot be 0 independently of the values of the constants where (l/z, 
is a specific function of the type We shall suppose that (1/z, 7^) 
actually contains a term and that 7i”“* is the highest power of u which 
appears in the function, while 2 “®' is the highest power of I /2 which actually presents 
itself in the coefficient of this power of u. For the second factor in the product (54) 
we shall take the single term The principal residue of the product is 
then evidently and is not 0 independently of the value of To equate 
to 0 the principal residue in a product of a type (54) then imposes a condition on the 
constants 
Represent the first factor of the product (53) in the form given in (47) and equate 
to 0 the principal residue in the product for arl^itrary values of the constants 
We thus subject the constants to l-j, conditions. The individual conditions are 
obtained on equating to 0 the principal residues in the products 
n 7—1 
2 2 
1 = 1 7 = -7+2 
a, 
7-1,«—U 
s = 1, 2, 
(55) 
That the L conditions so imposed on the constants are linearly independent of 
one another is readily shown. For suppose that there is a linear equation connecting 
the principal residues of the products (55), regarded as linear expressions in the 
constants and suppose in this equation that the multipliers are di, d 2 , 
respectively. Constructing the function 
we see that we should have the principal residue equal to 0 in a product of the type 
( 54 ) independently of the values of the constants a^_i f_i. This, however, we have seen 
to be impossible. It follows that the conditions to which we subject the constants 
on equating to 0 the principal residues in the products (55) are linearly 
independent of one another. On equating to 0 the principal residue in the product 
( 53 ), for arbitrary values of the Z* constants involved in the first factor, we then 
impose on the constants just linearly independent conditions. These 
conditions are all necessary in order that the functioji 
"F ( 2 , u) 
n 7-1 
2 2 
7 = 1 q= -7+2 
should have orders of coincidence for the value 2 = co which do not fall short of the 
numbers ..., respectively. To prove that these conditions are also sufficient 
we only have to show that is the total number of the conditions to which we must 
subject the constants ay_i in order that the function F ( 2 , u) may have the orders 
of coincidence ..., for the value 2 = co. 
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VOL. CCXII.-A. 
