NUMBER OF CONDITIONS IMPOSED BY A SET OF ORDER 



NUMBERS 



By J. C. Fields, Ph.D., F.R.S. 



§1, Let 



1. f(z,u)=0 



be an equation in which f(z, u) is a polynomial in u of degree n with co- 

 efficients which are rational functions of z — or more generally series in 

 powers of z, involving only a finite number of negative exponents. The 

 branches of the function u corresponding to the value z = group them- 

 selves into a number r of cycles of orders fi, v^, . . . , v^ respectively. 

 Adjointness relative to this value of the variable z is defined by a set of 

 order numbers Mi-l + lAi, • • • , Hr—l-\-l/vr where in, . . ., n^ are 

 the order numbers of the function /'„(z, u) for z = 0. Two sets of order 

 numbers n, . . ., r^. and ti, . . ., T;. are said to be complementary 

 adjoint when we have 



2. T5+7^ = Ms-l + lAs; ^=1,2, . . ., r. 



On designating by z~\{z, u))^ the general function of (z, u) of rational 

 character for the value z = 0, which is conditioned by the order numbers 

 Ti, . . . , Tr, the vanishing of the principal residue relative to this value 

 of z in the product 



3. z-\{z, u))A{z, u) 



is, as we know,* the necessary and sufficient condition in order that 

 \p{z, u) may have the order numbers n, . . ., r^.. 

 We shall now consider a product 



4. S 2 ,8_,,„_,z-V-'S S a,_i,,_iz'-V-' 



<=1 r=-i+l /=! r=-«+l 



in which the two factors are of identically the same form so long as the 

 coefficients P_r.n-t and a^_i, <_i are all arbitrary. Holding the 2ni 

 coefficients ^^^.n-t all arbitrary and equating to the principal residue 

 in the product, it is readily seen that we make the 2ni coefficients ay_ i_ ;_i 

 all vanish. For if the second factor in (4.) is not identically we shall 



*" On the foundations of the theory of algebraic functions of one variable," 

 Phir. Trans. Roy. Soc. Series A, Vol. 212, p. 345. 



133 



