THEOEY OF ALGEBKAIC FUNCTIONS OF ONE VAEIABLE. 
345 
rational function of (2, u) can be constructed possessing precisely the orders of 
coincidence here in question and having a principal coefficient which is not integral 
with regard to the element z — a (or 1/2). 
We say of two sets of orders of coincidence ti, and fj, corresponding 
to a value z = a (or z = co)^ that they are complementary adjoint to each other if 
they satisfy the inequalities 
Ti + Ti > Ml—IH , •••, Vr + Tr ^ Mr—1+ -.(15) 
»'l I'r 
When they satisfy the inequalities 
1 • 1 
Ti+Ti ^ ^ + Ml—1 H-, Tr + T^ ^ t + Mr—1 + - ; .... (16) 
Z/J V., 
they are said to be complementary adjoint to the order i. If the sets of orders of 
coincidence of two functions for a given value of the variable 2 are complementary 
adjoint, we say also that the functions are complementary adjoint to each other for 
the value of the variable in question. The orders of coincidence of the product of 
the two functions are evidently obtained on adding the corresponding orders of 
coincidence of the functions. If the functions <I> (2, u) and 4 ^ (2, u) are complementary 
adjoint for the value z — a (or 2 = 00) their product is adjoint for the value of the 
variable in question, and the coefficient of the principal term in the product must 
therefore be integral with regard to the element z — a (or I/2). When we speak of 
the principal term in a product it is, of course, to be understood that we mean the 
principal term in the product expressed in its reduced form. 
In order that a function 'k (2, iC) of rational character for the value 2 = a (or z — cc) 
shall have orders of coincidence which are complementary adjoint to a given set of 
orders of coincidence ti, ...,Tr, the necessary and sufficient condition is that the 
coefficient of the principal term in the product (2, tt) "k (2, m) shall be Integral with 
regard to the element z — a (or I/2), where <I> (2, w) represents the most general 
function of (2, u) of rational character for the value z — a (or 2 = 00 ) whose orders of 
coincidence with the branches of the corresponding cycles do not fall short of the 
numbers tj, ..., t,. respectively. That this is a necessary condition has been seen in 
what just precedes. That it is a sufficient condition may be proved as follows :— 
Suppose "k (2, m) to be a specific function of (2, v)j of rational character for the value 
z — a (or 2 = Go)j and suppose its orders of coincidence for this value of the variable 
to be Tj, ...,f,.. Furthermore, suppose this set of orders of coincidence not to be 
complementary adjoint to the set tj, ...jT^. The numbers ri + fj, ...,T,. + f,., then do 
not constitute a set of adjoint orders of coincidence, and we can therefore construct a 
function H (2, u) of rational character for the value z = a (or z — cc) which possesses 
precisely this set of orders of coincidence and whose principal term is not integral in 
regard to the element z — a (or I/2). 
For the moment we shall suppose that all of the orders of coincidence fj, ..., f,., are 
2 Y 
VOL. CCXII.-A. 
