THEORY OF ALGEBRAIC FUNCTIONS OF ONE VARIABLE. 
351 
We may write (2, n) in the form 
^ (Z, U) = + {{Z-CC, U)), .(19) 
where the notation <j)^'^{z,u) indicates a polynomial in {z,h) of degree i—l in 2, and 
where by the notation ((2 —a,'?^)) we designate a polynomial in u in which tlie 
coefficients expanded in powers of (2 —n) present no negative exponents. Here, since 
(2, u) is to be of integral algela’aic character for the value 2 = a, the orders of 
coincidence of the function {2, u) with the n branches of the equation (l), 
corresponding to the value 2 = a, must each l)e ^ i. On assuming, as we are free to 
do, that (p^"^{z,u) is not divisible by the factor z — a, we are forced to take for i the 
greatest of the r integers [mi], [mJj for this is evidently the greatest value which 
we can give to i without forcing tlie function (2, u) to be divisible by the factor 
z—a. Orders of coincidence, namely, which are simultaneously greater than the 
numbers ,ui, /jl^ require divisibility by z — a. 
The orders of coincidence ^ which we here require from the function (2, u) are 
adjoint, and the number of the conditions which they impose on the otherwise 
arbitrary constant coefficients of the function is evidently obtained on substituting i 
for each of the r numbers iul\, ..., p!,. in the expression (I8). This gives us 
A. + 7'ii— ^ (/U,— 1H-')j'„ 
S = ] \ vj 
for the number of the conditions to which we subject the ni coefficients of the 
otherwise unconditioned function <p‘'''\z, u). For the number of the arbitrary constants 
involved in the expression of the conditioned function (2, u) here in question we 
then have 
= 2 .(20) 
and we can write 
(2, u) 
Ia 
s = 1 
(z, u), 
( 21 ) 
where the quantities are arbitrary constants, and where the functions 0/'* (2, u) 
are specific linearly independent functions. 
Now we have seen that we impose on the coefficients of the general integral 
rational function "F (2, u) the conditions necessary and sufficient for adjointness 
relative to the value z — a on equating to 0 the residue relative to this value in the 
coefficient of the principal term in the product d> (2, %C). 'F (2, u\ This, however, from 
(19), is evidently equivalent to equating to 0 the residue relative to the value z = a 
in the coefficient of the principal term in the product 
(g. u) 
{z-aY 
■F (2, u) = 
V jz, u) 
s = i {z-aY 
. ^ (2, u). 
