THEOEY OF ALGEBKAlC FUNCTIONS OF ONE VARIABLE. 
343 
conditioned have a constant term which is 0. This we can readily see from the 
form of representation for the function given in (8). If, namely, we suppose 
tt—Pj = 0, u — Vy, — 0 to be the branches of the cycle of order v^, the series 
$ 1 , ..., dy, in (8) will be conjugate series in which the lowest exponent is > — 1, and in 
each of which the constant term is the same and unconditioned by the orders of 
coincidence here in question. The constant term in the principal coefficient of the 
partial sum 
6iQi{z,u) + ...+0y,Qy,{z,u), .(12) 
on the right-hand side of (8), is then arbitrary, and this is, therefore, also the case for 
the constant term in the total sum on the right-hand side of (8), since the series 
Oy.+i, 0„ are determined independently of the series 0i, 
The theorem just stated, together with the theorem preceding, may be included in 
the one statement:—-In the general function H {z, u) of rational character for the 
value z = a (or 2 = co) and conditioned for this value by a set of adjoint orders of 
coincidence ij.\, ...,yw',., the principal coefficient must be integral with regard to the 
element z — a (or ijz), and, furthermore, will involve an arbitrary constant term unless 
yu'i, ..., n'y are simultaneously greater than the corresponding numbers in the set 
Ml j • ••5 Mr • 
Let H ( 2 , u) be the general function of rational character for the value z = a (or 
2 = 00 )^ which is conditioned by a set of orders of coincidence 
i+n'i, ■’•yi + n'r, .( 13 ) 
where -i is a positive or negative integer or 0, and where /x'j, ..., constitute a set of 
adjoint orders of coincidence which, however, are not simultaneously greater than the 
numbers ni, The function (2 — H (2, ti), or 2*11 (2, w), is then evidently the 
general function of rational character for the value z = a (or 2 =co), which is 
conditioned by the set of adjoint orders of coincidence ij.\, ..., /x',. , and its principal 
coefficient, by the theorem last stated, must therefore be integral in the element z — a 
(or 1/2) and involve an arbitrary constant. It follows that the lowest term in the 
principal coefficient of the general function H (2, u), conditioned by the set of orders of 
coincidence (13), is a{z—a)\ or az~\ where a is an arbitrary constant. 
Evidently any set of orders of coincidence corresponding to the value z = a (or 
2 = go) can be written in the form (13), so that we may also state the last theorem 
as follows:—^The lowest term in the principal coefficient of the general function 
H(2, u) of rational character for the value z = a (or z = and conditioned by a 
given set of orders of coincidence for this value of 2, is a (2 —a)\ or az~\ where a is an 
arbitrary constant, and where i is the greatest integer whose subtraction from each 
number of the set leaves a set of adjoint orders of coincidence. 
From the form (8) it is readily seen that we can construct a function 11(2, ti) of 
rational character for the value z = a (or z =^), which possesses an arbitrarily 
