340 
DE. J. C. FIELDS ON THE FOUNDATIONS OF THE 
be an equation in which we shall ultimately suppose the coefficients fn-i, ••• 5/0 to be 
rational functions of 0 . For the moment, however, it will suffice to assume that these 
coefficients have the character of rational functions for the value z — a (or 0 = co), 
that is, that they are developable in series of integral powers of z — a (or ijz) in 
which, at most, a finite number of terms have negative exponents. We say that a 
function has the character of a rational function of {z, u) for the value z = a (or 0 = c») 
if it is built up by rational operations out of u and functions of 2 which have the 
character of rational functions for the value z = a (or 2 = 00). Here it is to be 
understood that the function is to have a meaning for each of the branches of the 
equation (1), corresponding to the value of the varialde 2 in question—otherwise said, 
that the rational operations do not involve division by a factor of f ( 2 , u). The 
equation (l) may or may not be reducible in the domain of functions of rational 
character for the value z = a (or 2 = go). In any case, however, without detriment to 
the generality of our theory, we may assume that the equation does not involve a 
repeated factor. 
Any function possessing the character of a rational function of ( 2 , u) for the value 
2 = a (or 2 = CO) can evidently be written in one, and oidy one, way in the form 
H ( 2 , ?f) =+ ...+/?-o.( 2 ) 
where the coefficients h, possess the character of rational functions for the value of 
the variable 2 in question. This form we call the reduced form of the function. The 
coefficient of in the reduced form of a function of ( 2 , u) we call the prmcipal 
coefficie7it of the function. The term itself we call the principal term. In 
what follows we shall take for granted that a function of ( 2 , u) is expressed in its 
reduced form where nothing in the context implies the contrary. 
Corresponding to the value z = a (or 2 = 00 ) we have a representation of the 
equation (l) in the form 
f{z,u) = {u-V,){u-V^)...{u-V,) = Q, .(3) 
where Pi, ...,P„ are series in powers of z — a (or I/ 2 ) with exponents, integral or 
fractional, of which, it may be, a finite number are negative. These power-series 
group themselves into a number, r, of cycles of orders z/j, ..., v^. respectively where the 
series of a cycle of order proceed according to ascending integral powers of the 
element (2 —or 2 “^^’'“, as the case may be. As a general rule we have = 1. 
We shall speak of the order of coincidence of a function H ( 2 , u) with a branch 
—Pj = 0, or of the order of coincidence of the branch with the function, meaning 
thereby the lowest exponent in H ( 2 , PJ arranged according to ascending powers of 
z — a (or 1 / 2 ). The order of coincidence of the branch u — Y, = 0 with the product 
Q,(2 ,w) = (?t-Pi) ... (w-P,_i) (t^-P,^i)... (m-P„) .... (4) 
we shall indicate by the symbol This is plainly also the order of coincidence of 
