342 
DR. J. C. FIELDS ON THE FOUNDATIONS OF THE 
therefore, obtained on adding the lowest exponent in the series 6 , to the order of 
coincidence of the function {z, u) with the branch in question. If, then, the order 
of coincidence of the function Yl{z,u) with the branch —Pj = 0 is ^ the series 
Bg can involve no negative exponent. If the order of coincidence of the function 
\\{z,u) with tlie branch w —= 0 is >—1 the lowest exponent in the series 
6 , must be > — 1 . Now the coefficient of in the reduced form of H(2, m), as 
given in ( 8 ), is 
2 0 .,.( 10 ) 
5=1 
a function which is evidently of rational character for the value of the variable z in 
question. If, then, the orders of coincidence of the function H (2, u) with the 
branches u — V^ = 0 , —P„ = 0 are greater than tlie corresponding numbers in the 
set Ml— 1 , 1 , the lowest exponent in the principal coefficient is > —I and must 
therefore be ^ 0 , because of the rational character of the coefficient for the value 
z = a (or 0 = 00). We shall say of a function of 2 that it is integral with regard to 
the- element z—a (or I/2) if its expansion in powers of the element involves no 
negative exponents. The principal coefficient in a function H (2, u) of rational 
character for the value z =■ a (or 2; = co) is then integral with regard to the element 
z—a (or 1/2) if the orders of coincidence of the function with the branches of the 
corresponding cycles are greater than the numbers /xj — 1 , ..1 respectively. 
Otherwise stated, the principal coefficient in a function H (2, vi) of rational character 
for the value z = a (or 2: = co) must be integral with regard to the element z—a (or 
1/2) if the orders of coincidence of the function with the branches of the several 
cycles do not fall short of the numbers 
1 + 
fx^— 1 + 
I'r 
( 11 ) 
respectively. A set of orders of coincidence which do not fall short of the numbers 
given in (ll) we call a set of adjoint orders of coincidence, and, if a function possess 
such a set of orders of coincidence, we say that it is adjoint for the value of the 
variable 2 in question. The theorem which we have just proved may then be briefly 
stated as follows :—If a function H (2, ti) of rational character for the value z = a (or 
2 = GO) is adjoint for this value of the variable its principal coefficient must be 
integral with regard to the element z—a (or I/2). This theorem, so far as it has 
reference to the value 2 = co, is evidently also embodied in the statement that the 
degree in (2, m) of the principal term in a function H (2, \C) of rational character for 
the value 2 = 00 must be ^ n—I if the function is adjoint for this value of 2. 
If a function H (2, w) of rational character for the value z = a (or 2 = 00) is 
conditioned for this value of the variable by a certain set of adjoint orders of 
coincidence /x'l, nj, and if for a single one of these orders of coincidence we 
have f, ^ fj-s , fhe principal coefficient, already integral, will in the general function so 
