THEORY OF ALGEBRAIC FUNCTIONS OF ONE VARIABLE. 
357 
( 39 ) gives the number of the conditions imposed on the coefficients of the general 
function V\,_x{ljz,u) by the orders of coincidence i for all n branches, so long as X 
has been chosen at least as large as the greatest degree of a coefficient in a rational 
function E, (2, u) which is consistent with the orders of coincidence i here in question. 
For i positive this is evident. For negative i = —j it is plain that the coefficients of 
the general function V\>_x{\jz,u), already conditioned by the orders of coincidence —j 
for all n branches, must be subjected to nj further conditions if we would increase 
its orders of coincidence to 0 for all n branches. These nj conditions are counted in 
the expression ( 40 ), which gives the number of the conditions required by the orders 
of coincidence 0 for all n branches. Subtracting nj then from this expression, we 
obtain, for the number of the conditions imposed on the coefficients of the general 
function R_;^(l/2, u) by the negative orders of coincidence % = —j for all n branches, 
tlie expression given in ( 39 ). 
Indicate by any set of orders of coincidence with the branches of the 
cycles corresponding to the value 2 = 00, and take the Integer ^ equal to or less 
than the least of these. We have in ( 39 ) an expression for the number of the 
conditions imposed on the coefficients of the general function E_;^(l/2, u) by the 
orders of coincidence i for all 7 i branches, where we assume that X has been chosen at 
least as large as the greatest degree of a coefficient in a rational function It (2, u) 
which is consistent with the orders of coincidence i here in question. To obtain 
tlie number of the conditions imposed on the coefficients of the general function 
(1/2, u) by the set of orders of coincidence ..., we must evidently add 
to the expression ( 39 ) the number represented by the sum 
2 
(cc) 
S = 1 
This gives us for the number of the conditions imposed on the coefficients of the 
general function E_^(l/2,i^) by the set of orders of coincidence the 
expression 
nX + 
(00) (») 
s = 1 
S = 1 
(00) 
( 41 ) 
We have derived this formula on assuming that X has been chosen at least as large 
as the greatest degree of a coefficient in a rational function E (2, u) which is 
consistent with the orders of coincidence i for all n branches. It is now evident, 
however, that the formula holds so long as X is not less than the greatest degree X' in 
2 which a coefficient in a rational function E (2, ii) can have consistently with the 
possession by the function of the orders of coincidence For among 
the conditions whose number is given in ( 41 ) are included the w(X—X') conditions 
which make the terms of degree >X' in the coefficients of the powers of ?c vanish. 
Under the general rational function ^^),(l/z,u), with coefficients of degree X in 2, 
is evidently included the general rational function of degree X in (2, u). To pass 
