346 
DE. J. C. FIELDS ON THE FOUNDATIONS OF THE 
finite. The quotient H {z, u)/^ {z, u) is then a function of rational character for the 
value z — a (or ^ = go ) whose orders of coincidence for this value of the variable are 
Ti, and yet in its product by the function {z^u) the principal term is not 
integral in character. If then the set of orders of coincidence fi, ...,T„of the 
function d' (2, '?i) Ije not complementary adjoint to the set of orders of coincidence 
Ti, ..., the principal term in the product of ^ (z, w) by the general function ^{z^u), 
which possesses the latter set of orders of coincidence, is not integral in character 
with regard to the element z—a (or I/2). It follows, therefore, that the sufficient, as 
well as the necessary condition, in order that the orders of coincidence of a function 
d'-( 2 , u) of rational character for tlie value z = a (or 2 = co) shall be complementary 
adjoint to a given set of orders of coincidence ti, ..., is contained in the statement 
that the coefficient of the principal term in the product of (2, u) by (2, u) is 
integral with regard to the element z — a (or I/2), where ^{z,u) is the general 
function 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, ..., r,.. 
In what precedes we have assumed that the orders of coincidence fi, ..., of fh® 
function d^ (2, u) are all finite. Suppose now that certain of these orders of coincidence 
are infinite, and that nevertheless the set is not complementary adjoint to the set of 
orders of coincidence tj, As before, let ^{z,u) be the general function of 
rational character for the value z = a (or z = cc) conditioned by the set of orders of 
coincidence ti, ..., r^. Construct a function dk' (2, u) of rational character for the value 
z = a (or 2 = co) which possesses for tliis value of the variable a set of orders of 
coincidence which is complementary adjoint to the set ti, all of its orders of 
coincidence being at the same time finite, and each one of them difierent from the 
corresponding order of coincidence in the set tj, ..., t,.. It is evident that the orders 
of coincidence of the function 
■T" (2, u) = "T (2, u) + d'' (2, u) 
for the value of the variable 2 in question are all finite, and that they constitute a 
set which is not complementary adjoint to the set ti, By what we have 
already seen, then, the principal coefficient in the product <I> (2, u) (2, u) will not be 
integral. The principal coefficient in the product (2, u) d'' (2, u), however, is integral, 
since d^' (2, u) has orders of coincidence which are complementary adjoint to those of 
the set Ti, ..., r^. It follows that the principal coefficient in the product (2, u) d' (2, u) 
is not integral. If then the orders of coincidence of a function (2, u) of rational 
character for the value 2 = a (or 2 = co) be not complementary adjoint to the orders 
of coincidence tj, it follows that in the product (2, u) d'(2, m) the principal 
coefficient is not integral, where (2, u) is the general function of rational character 
for the value z — a (or 2 = cx>) conditioned by the set of orders of coincidence 
Tj, The necessary and sufficient condition then that a function "T (2, w) of 
