THEORY OF ALGEBRAIC FUNCTIONS OF ONE VARIABLE. 
347 
rational character for the value z = a (or 2 = oo) should have, for this value of the 
variable 2, a set of orders of coincidence complementary adjoint to a given set of 
orders of coincidence ti, is that the principal coefficient in the product 
^{z,u) ^{z,u) should be integral with regard to the element z — a (or I/2), where 
$ (2, u) is the general function of rational character for the value z = a (or 2 = co) 
conditioned by the set of orders of coincidence tj, t,.. 
Without detriment to the truth of the statement just made the expression 
of rational character for the value z = a {or 2 = 00) employed with reference to the 
functions {z, u) and (2, u) can, in connection with either or both of these functions, 
be replaced by the expression rational function of (2, u). 
If in the product of a function 4 ^ (2, u) by the function (2, m) the coefficient of 
the principal term is integral with regard to the element z — a, the residue of tliis 
coefficient for the value z — a is of course zero. Conversely, however, if the residue 
of the principal coefficient for the value z =■ a vanislies in the product of a function 
' 4 ' (2, xi) by the general function (2, ui) whose orders of coincidence for this value of 
the variable 2 do not fall short of the numl)ers tj, ..., t,. respectively, it follows that 
the principal coefficient in question must be integral with regard to the element z — a. 
For if the function d>' (2, u) is included under the general function (2, n) conditioned 
by the orders of coincidence tj, and if the principal coefficient in the product 
<h' (2, u) ’F (2, 11) actually contains a negative power (2—«)“*, then also is the residue 
relative to the value z = a in the principal coefficient of the prodiict (2 —(2, w) 
{z,u) different from 0 , while the function (2 —d>'(2, ■?<.) is evidently included 
under the general function (2, u) above conditioned by the orders of coincidence 
Ti, ..., T^. If then the residue of the principal coefficient for the value z — a vanishes 
in the product of a function ^{z,u) by the general function (2,'?i), it follows that 
the principal coefficient in question must be integral with regard to the element z — a. 
We may then say, in the case of a finite value z = a, that the necessary and 
sufficient condition in order that a function "F (2, u) should be complementary adjoint 
to a set of orders of coincidence t^, ..., for the value of 2 in question, is contained in 
the statement that the residue relative to the value z = a in the principal coefficient 
of the product <I> (2, u) ^ (2, n) should vanish, where <I> (2, u) represents the most general 
function oi‘ {z,u) of rational character for z =a whose orders of coincidence with the 
branches of the corresponding cycles do not fall short of the numbers ti, ...,Tr 
respectively. 
In like manner for the value 2 = go we may evidently say that the necessary and 
sufficient condition in order that a function ’F (2, xi) should be complementary adjoint 
to a set of orders of coincidence ti, ..., r,., corresponding to the value of the variable m 
question, is contained in the statement that the constant coefficient of the element 
27 .U'^ in the principal term of the product d> (2, u) ’k (2, u) should vanish, where (2, xi) 
represents the most general function of (2, u) of rational character tor the value 
2 = 00, whose orders of coincidence with the branches of the corresponding cycles do 
2 Y 2 
