344 
DR. J. C. FIELDS ON THE FOUNDATIONS OF THE 
assigned set of orders of coincidence , ..., for the branches of the corresponding 
cycles. Here tj, ..., may be any integral multiples—positive, negative, or zero—• 
of the numbers ijv^, ..., ijvj. respectively. We can write 
T, = s=l, 2 , .( 14 ) 
where the numbers are integral. In the form ( 8 ) each of the n elements on the 
right-hand side corresponds to a different one of the n branches. In the elements 
corresponding to the conjugate branches of the cycle of order v, substitute for the 
coefficients 0 corresponding conjugate series beginning with a term in or 
Do this for each of the r cycles and the resulting function H {z, ii) will have precisely 
the set of orders of coincidence tj , ..., r,, here in question, and will at the same time 
evidently be of rational character for the value z = a (or 2 = co). 
Not only can we construct a function H (2, n) of rational character for the value 
z = a (or 2 = 00), which possesses the arbitrary set of orders of coincidence 
but we can in particular construct a rational function of (2, u) which possesses 
precisely this set of orders of coincidence for the value of the variable in question. 
To obtain such a rational function, in fact, it evidently suffices in the function H (2, u), 
already constructed, to cut off in the series in powers of z — a (or I/2), which constitute 
the coefficients of the powers of u, terms of order sufficiently high to Ije unaffected by 
the orders of coincidence tj, t,., required of the function. 
Let us now suppose for the moment that the equation (l) has reference to the 
value 2 = oOj so that the coefficients ...j/o are series in powers of I/2 involving, 
it may be, a finite number of positive powers of 2. The aggregate degree of the 
equation in (z,u) we shall indicate by the letter N. Referring to the identities ( 7 ), 
then, we see that the degrees of the functions Q, (2, w) can in no case exceed N — 1 . 
If now the function Yi{z^ti) in ( 8 ) be adjoint for the value 2=00, the lowest 
exponents in the series @,(1/2) must, as we have already noted, be > — 1 , and the 
degrees in 2 of these series must therefore all be < 1 . The degrees in (2, u) of the 
elements 0 iQs( 2 , w) in ( 8 ) will consequently all be < N, and the same will be true of 
the degree of the function H (2, u). It follows that the degree of the function 
H (2, w) must be <N — 1 , because of the rational character of the function for the 
value 2 = CO. We have just proved then that a function H {z,u) which is of rational 
character for the value 2 = co , and which is also adjoint for this value of the variable 
2, must be of degree ^N — 1 , and we had already proved that the degree of the 
principal term in such a function must be 1 . 
§ 2 . If a function H {z,u), of rational character for the value 2 = a (or 2 = co)^ is 
also adjoint for this value of the variable 2, we have seen that its principal coefficient 
must be integral with regard to the element z — a (or I/2). We have also seen, in the 
case of a set of orders of coincidence corresponding to the value z = a (or 2 = 00), 
of which some one at least falls short of what is requisite to adjointness, that a 
