Direct Derivation of the Complementary Theorem 245 



DIRECT DERIVATION OF THE COMPLEMENTARY 



THEOREM.* 



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



The statement of the Complementary Theorem involves only 

 differences between complementary orders of coincidence and the 

 difference between the number of the conditions imposed on a sufficiently 

 general rational function by a given basis and the number of the 

 conditions imposed on the same function by the complementary basis. 

 It would appear therefore that it should not be a necessary preliminary 

 to the proof of the theorem to determine a formula for the actual 

 number of the conditions imposed on a rational function of given form 

 by a given set of orders of coincidence corresponding to a specific value 

 of the variable z. The proof developed in what follows does as a matter 

 of fact dispense with the aid of formulae of the type just referred to. 

 Let 



1. /(z,«) = W«+/n-iW»-l4- +/o= 



be an algebraic equation in which /„- 1, , /„ are rational functions 



of s. A rational function of (z, u) can be written in the form 



where N (s, u) and P (z, u) are polynomials in (z, ii!) of degree n—\ in m, 

 the degree in z of N{z, u) being less than the degree of Q (z) . We shall 

 suppose (r) to be a basis of coincidences relative to the equation (1), the 

 partial bases corresponding to the finite values of 2 and to the value 

 z= 00 being indicated by (t)' and (t)^°°^ respectively. The number of 

 cycles corresponding to a value z = ajfe we indicate by r* . The elements 

 of the basis (t) are connected with the elements of the complementary 

 basis (t ) by the relations 



3. rf+7f=^f)_l + l/.f ;, = !,...,,,, 



where the orders of the cycles are represented by the symbols vf^ and 

 where the symbols iJf> designate the orders of coincidence of the function 

 /«' (z, u) . 



*This paper was read at the Harvard Meeting of the American Mathematical Society 

 in September, 1916. The writer has however neglected to publish it in the interval. 



