350 
DR. J. C. FIELDS ON THE FOUNDATIONS OF THE 
spending conjugate series 0 will not be divisible by z — a and will involve an arbitrary 
constant term which is independent of the coefficients in the remaining n — v^ series 6 . 
In this case, then, the coefficient of the principal term in the general function H (z, u) 
contains an arbitrary constant. The principal term in H (z, u) is, therefore, not 
divisible by z — a and the same is consequently true of H (z, u) itself. 
We see, then, that the general rational function conditioned by a certain set of 
orders of coincidence for the value z = a is or is not divisible by the element z — a 
according as its principal term is or is not divisible by this element, and we further¬ 
more see that the principal term is or is not divisible by z — a according as the orders 
of coincidence in question severally exceed the corresponding numbers or not. 
From this it follows in particular that the general rational function conditioned by a 
certain set of adjoint orders of coincidence for the value z = a is divisible by precisely 
the same power of the element z — a as the coefficient of its principal term. 
We sliall employ the letter A to designate the number of the independent 
conditions which must be imposed on the coefficients of the general integral rational 
function of (z, u) in order that it may be adjoint for the finite value z — a. Every 
extra coincidence over and above adjointness required from the function will impose 
an extra condition on its coefficients, for we liave seen that we can construct a 
rational function which actually possesses an arl)ltrarily assigned set of orders of 
coincidence corresponding to a value z = «, and we have also seen that an adjoint set 
of orders of coincidence already requires that the function be integral with regard to 
the element z — a. The number of the independent conditions then which are imposed 
on the coefficients of the general integral rational function of (z, u) by a set of orders 
of coincidence /x'l, adjoint for the value z = a, is given by the sum 
s = 1 
( 18 ) 
where we still have to determine the value of A. 
We have seen in § 2 that the necessary and sufficient conditions, in order that a 
function (z, n) may have a set of orders of coincidence complementary adjoint to a 
given set of orders of coincidence ti , ..., , corresponding to a finite value z = a, are 
obtained on equating to 0 the residue relative to this value of z in the coefficient of the 
principal term in the product ffi (z, u) 'F (z, m), where ffi (z, u) is the general rational func¬ 
tion of (z,w) conditioned by the orders of coincidence ti, for the value z = a. 
If in this theorem we give to each of the numbers ti, ..., t,. the value 0 and take for 
■F (z, u) the general integral rational function of (z, u), we evidently obtain the 
necessary and sufficient conditions which must be imposed on the coefficients of the 
general integral rational function in order that it may be adjoint for the value z — a, 
on equating to 0 the residue relative to this value of z in the coefficient of the 
principal term m the product <I> (z, u) "F (z, u), where (z, u) is the general rational 
function of (z, u), which is algebraically integral in character for the value z — a. 
