THEORY OF ALGEBRAIC FUNCTIONS OF ONE VARIABLE. 
363 
conditions that it may have the orders of coincidence for the value 
z = (X) are obtained as above on equating to 0 the principal residue in the product 
( 5 l), where the integer i is subject to the conditions ah’eady specified, but where it is 
well to bear in mind that the formula ( 56 ) implies that i has been taken ^ M + 2. 
§ 5 . The number of the cycles into which the branches of the equation (l) group 
themselves for a finite value z — we shall indicate by the symbol and the orders 
of these cycles we shall designate by respectively. For the corresponding 
numbers in connection with the value z = o:> we have already employed the symbols 
t'o, and So in general, numbers associated with the value 2 = will 
be designated by an index or suffix /<:, and those associated with the value 2; = co by 
an index or suffix cxd. For example, adjointness relative to a value 2 = is defined 
by the orders of coincidence 
Ml 
1 + 
'a 
(k) ’ 
M, 
-1 + 
M) 
When we speak of a set of orders of coincidence for a given value of the varialjle 2 it 
will always be understood, of course, that these are Integral multiples of the corre¬ 
sponding numbers 
A set of orders of coincidence corresponding to a value z — we shall designate by 
the notation Assigning a system of sets of orders of coincidence for all 
values of the variable 2, the value 2 = go included, we shall designate such system by 
the notation (t). We shall liere understand that all but a finite number of the orders 
of coincidence involved in a system (t) have the value 0. Such a system (r) we shall 
call a Basis of Coincidences for the building of a rational function, or, more Iwiefly, 
we shall simply refer to it as the hasis (t). A rational function of (2, u) we shall say 
is built on the basis (r) if its orders of coincidence for the different values of 2 in no 
case fall short of the corresponding orders of coincidence given by the basis. We 
shall say of two bases (t) and (t) that they are convplementary to each other when 
for finite values z = the corresponding orders of coincidence furnished by the 
bases are connected by the relations 
(k) I - (k) 
= + s = 1,2 
{k) 
V 
• 5 ' K ? 
(57) 
while for the value 2 = 00 
the orders of coincidence are connected by the relations 
'-I-T 
= M, 
(co) 
+ 1 + 
(qo) : 
1 , 2 , 
( 58 ) 
By the notation (t)' we shall designate that part of the basis (t) which has reference 
to finite values of the variable 2 , and by (t)^“^ we shall mean that part of the basis (t) 
which refers to the value 2; = 00. We shall then speak of a rational function of ( 2 , u) 
which is conditioned by the partial basis (t)' or by the partial basis (t)^“\ 
3 A 2 
