Section III., 1915 [199] Trans. R.S.C. 



Certain sets of orders of coincidence associated with an algebraic equation. 

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



(Read May Meeting, 1915.) 



Consider an equation 



1. /(c,M) = «''+/n-lW«-l+...+/o = 



where the coefficients fs are rational functions of s. In the neigh- 

 borhood of any value s = a we have a factorization of the form 



2. f{z,u) = {u-Pi)...{u-Pn) 



where Pi,. . ., Pn are series in powers of 2, — a. In the neighborhood 

 of 2= °° the element z~a is replaced by 1/z. Any rational function of 

 (s, u) can be written in the form 



3. H{k,îi)=h„^iu''-'^-\-. ..+ho 



where h„-i, . . ., ho are rational functions of z. In the neighborhood 

 of 2 = o it can also be represented in the form 



4. H(Z, U) = di Qi(z, ll)-\-...-]-dnQn (S, u) 



where du . . ., dn are series in powers of & — a and where the functions 

 Qs {z,u) are defined by the identities 



5. f{z,u)^{u-Ps)Qs {z,u), s=l, . . .,n. 

 On dividing f(2,iO by u—Ps we evidently have 



The representation (4) of H{z, u) will then take the form 



7. H{z,u) ='s' S ds {fnPV~\...+fa + ,Ps +/(r + i)w^. 



ff=0 S=l 



We know that the set of orders of coincidence which so condition 

 the general rational function of (2, u) that its principal coefficient 

 must be integral relatively to the element 2 — a is the set of adjoint 

 orders of coincidence corresponding to the value z = a. We here 

 propose to determine the set of orders of coincidence which so con- 

 dition the general rational function H(z, u) that the coefficient of 

 u<^ is integral with regard to the element z — a. The coefficients 

 hn-i, hn-2, • • • , ho in the expression for H{z, u) in (3) we shall call the 

 first, second, . . . wth coefficients of the function. The residue relative 

 to the value 2 = a in the coefficient hn-s we shall call the sth residue 

 of the rational function H{z, u) relative to the value z = a. The 

 expressions first residue and principal residue then have the same 

 signification. 



