Section III, 1913. [249] Trans. R.S.C. 



Proofs of Certain Theorems Relating to Adjoint Orders of Coincidence. 

 By Prof. J. C. Fields, F.R.S. 



(Read May 28th, 1913.) 



In the present paper the method of the deformation of a product 

 is employed to prove certain elementary theorems relating to orders 

 of c incidence. When one considers the left-hand side of the funda- 

 mental equation as represented in formula (2) and when one remem- 

 bers that any rational function of (2, ti) can be written uniquely in 

 the form (4) it is not unnatural to attempt to co-ordinate factors in 

 the products (2) and (4) with reference to the orders of coincidence 

 of the latter product and this immediately suggests the method of 

 the deformation of a product. As here employed the method appears 

 to better advantage than in the writer's book ^ on the algebraic 

 functions where it was first made use of, so that on this account and 

 because of the possibility that the method may still have its uses in 

 other connections it is perhaps worth while to give the proofs which 

 follow, although other and probably preferable methods of proof of 

 the theorems here in question have been furnished elsewhere' by the 

 writer. 



Let 



(1) /(0, U)=U" +/n-l«''-l+ . . . +/o = 



be an algebraic equation, reducible or irreducible. The degree in 

 (2, ti) of /(s, u) we shall indicate by N. We can write 



(2) /(2, w) = (m-Pi) . . . {u-P„) 



where Pi, . . . , P„ are n distinct series in powers of z — a (or I/2) with 

 exponents integral or fractional. A finite number of these exponents 

 may happen to be negative. 



Any rational function of (2, u) can be written in the reduced form 



(3) H{z,u)=h„-ru"-'-^ . . . +h„ 



where the coefficients A„-i, . . . , Aq are rational functions of 2. Adding 

 f{z, u) to the reduced form here in question, we can write identically 



(4) /(z, u) +H{z, u) = (« - GO ...(«- (2» ) 



where Qi, . . . , Q,i are series in powers of z — a (or I/2) with exponents 

 integral or fractional. A given rational function of (z, u) will have 

 a certain set of orders of coincidence with the n branches 

 u — P\ — 0, . . . ,u—P„ =0. The orders of coincidence of the function 



1 Theory of the Alg-ebraic Functions of a Complex Variable. Mayer & Millier, 

 Berlin, 1906. 



2 A method of proving^ certain theorems relating' to adjointness. Proc. London 

 Math. Sac, Sen 2, Vol. 11, pp. 127-132. 



On the Foundations of the Theory of Algebraic Functions of one Variable 

 Phil. Trans. Roy. Soc, 1912. 



Sec. Ill, 1913—16 



