348 
DR. j. C. FIELDS ON THE FOUNDATIONS OF THE 
not fall short of the numbers ti, respectively. The vanishing of the constant 
coetlicient of the element in the principal term of the product (z, u) (z, u) 
then gives the condition that the function z^d^ (z, u) should have a set of orders of 
coincidence which is complementary adjoint to the set of orders of coincidence ti, 
Tills, therefore, is the condition that the function (z, u) should have orders of 
coincidence which are complementary adjoint to the order 2 to the orders of 
coincidence tj, ..., t^. Also the vanishing of the coefficient of the element zm"“^ in the 
principal term of the product (z, u) z^’F (z, u) is equivalent to the vanishing of the 
coefficient of the element in the principal term of the product (z, n)d^ {'Z,u). 
The vanisliing of the residue relative to the value z = co in the coefficient of the 
principal term in tlie product d> (z, n) d' (z, u) consequently gives the necessary and 
sufficient condition that the function "T (z, u) should have a set of orders of coincidence 
for the value z = go whicli Is complementary adjoint to the order 2 to the set of 
orders of coincidence tj, where ffi (z, ??.) is the most general function of {z,u) 
of rational character for the value z = oo whose orders of coincidence with tlie 
branches of the several cycles do not fall short of the numbers tj, ..., t,. respectively. 
If tlien (z, u) represents the most general function of (z, u) of rational character for 
the value z — a (orz=co) whose orders of coincidence witli the branches of the 
corresponding cycles do not fall short of the numbers tj, respectively, the 
vanishing of the residue, for the value of the variable z in question, in tlie coefficient 
of the principal term in the product (z, ?/.) d' (z, u) gives, in the case of a finite value 
z = a, the necessary and sufficient condition that the orders of coincidence of the 
function d'(z, m) should be complementary adjoint to the numliers ti, while, 
if the functions and numbers here in question have reference to the value z =oo, 
the vanishing of the corresponding residue in the product (z, v) d^ (z, gives the 
necessary and sufficient condition that the orders of coincidence of the function d^ (z, \C) 
should be complementary adjoint to the order 2 to the numbers ti, ..., t,.. 
In the foregoing statement it would evidently suffice to let (z, u) represent the 
general rational function of (z, u) whose orders of coincidence for the value of z in 
question do not fall short of the number ti, respectively—-at least so long as 
these numbers are all finite. Where we are concerned with the finite value z = a we 
might, without detriment to the truth of our statement, further impose on the 
rational function (z, u) the condition that its coefficients should be integral with 
regard to all finite values of z save only the value z = a, with regard to which value 
the coefficients will oi' will not be integral according as this is or is not required by 
the set of orders of coincidence tj, In the statement here in question the 
function d' (z, n) was simply assumed to Ije a function of (z, u) of rational character 
for the value z = a (or z = co), and the statement therefore holds good in particular 
when "F (z, u) is a rational function of (z, u). 
Tlie product of any two functions <I>(z, u) and "T (z, ?t) can be written in the form 
(z, u) d' (z, w) - ^ (z, u) f{z, w) + X (2, w),.( 17 ) 
