THEORY OF ALGEBRAIC FUNCTIONS OF ONE VARIABLE. 
359 
that X has been chosen sufficiently large. Subtracting this expression from n(x + ^), 
the total number of the constant coefficients in the general function (I/2, u), we 
obtain the expression 
L = ni— 2 T, 
s = 1 
(®) (») . -y 
; -r 2 ^ 
s = I 
M, 
(») 
1 + 
(cd) 
(x) 
. ( 45 ) 
for the total number of the arbitrary constants involved in the function u), 
conditioned by the set of orders of coincidence Dropping the now 
superfluous suffix, —X, we may say that the expression ( 45 ) gives the numlier of the 
arbitrary constants involved in the general rational function (l/z, u), conditioned 
by the orders of coincidence where the index (i) still implies that the 
coefficients of the rational function expanded in powers of l/z involve no powers as 
high as (1/2)*. 
In the representation 
= + .( 46 ) 
of the general rational function of (z, u) conditioned by the set of orders of coincidence 
corresponding to the value 2 = go we shall find it convenient to write 
R»(i,d = - s .(17) 
\2/ s=i \Z / 
SO as to bring into evidence the arbitrary constants involved in the element 
(1/2, li). The/oo functions (Ps''\\lz,u) are specific linearly independent functions 
of the form implied by the index {i) and possessing for the value 2 = 00 orders of 
coincidence which do not fall short of the orders of coincidence ri"\ •••, respec¬ 
tively. The number it is to be borne in mind, depends not alone on the orders of 
coincidence here in cpiestion, but also on the particular value chosen for 
the integer i. It is, also, not to be forgotten that i is taken so large that terms 
involving powers of I/2 higher than (1/2)*“^ are not conditioned* by the orders of 
coincidence The general rational function of (2, u), conditioned by the 
orders of coincidence we shall then represent in the form 
®(b“) =-id-'"A“’(i.«)+^-((b«)).(48) 
where the number L is given by the expression in ( 45 ) and where the constant 
coefficients in ((A, u)) are all arbitrary. 
In order that a rational function 'k (2, u) may be complementary adjoint to the 
general function R(l/2, m) here in question, for the value z = ^, we know it is 
* When we here say that a term is not conditioned by the orders of coincidence ..., we mean 
that it already possesses orders of coincidence at least as great as these. 
