140 Transactions of the Royal Canadian Institute 



the maximum order numbers consistent with the limitations imposed 

 on their forms. In (14) we shall assume that the functions R„^s and 

 Ps-i = Rs-i have maximum order numbers. These order numbers we 

 shall designate as a„_j and a^.i respectively. Since the product 

 i?''~Vs-i is of degree n — 1 in w with 1 for the coefficient of w"~^ it is 

 evident that an^s-^o-s-i cannot be greater than fx. We shall here 

 assume a„_s+a,_i =fx and shall justify our assumption later on in a 

 supplemental lemma. 



From the nature of the basis functions in (11) it is plain that we 

 must here have /i — 1 + !/»'< f^n-s +a„ -5 < M+l/f whence a^.i —1 + 1/;; 

 < Kn-5 < O's-i + 1/'' and therefore a^-i — 1 < k„_j < a^^i . From the 

 nature of the basis functions in (12) we derive immediately 

 a^_i — 1< X^-i < Us-i- From the foregoing inequalities we conclude 

 K„^s = K-i ', s = 1, 2, . . . n, and consequently k = \. From /c+X = 

 (fi — l-\-l/v)v then follows 



16. X = /c = Km-1 + 1A)^ 



the special case of formula (15) when equation (1) consists of a single 

 cycle of order n = v for the value z = 0. For the number of the conditions 

 imposed by an adjoint order number r on the general polynomial of 

 degree n — l = v—l in u, with coefficients of integral rational character 

 relative to the value 2 = 0, we immediately derive the expression 



17. t^-Hm-I + IA)''- 



To complete the above proof we have to establish the supplemental 

 lemma a„-s +<^s-i =M- With this end in view we shall proceed to con- 

 struct a polynomial of degree n — 1 in u with coefficients of integral 

 rational character in z, the coefficient of u"~^ being 1, the order number 

 of the polynomial for s = being /x. 



Let us consider the successive terms in one of the branches u—P = 

 of the cycle of order v with which we are here concerned. The full order 

 V of the cycle to which the branch belongs may not immediately declare 

 itself. The cyclical characters successively presenting themselves we 

 shall designate by 1, v', v", . . . , v^'\ v where evidently each element in 

 the succession is a factor of the one which follows. We shall write 



Suppose P , P , . . . ,Pj,(t) to be portions of ^(0 series P^, P^, . . . , Pj,(^) 

 these part series characterizing a cycle of order v \ Furthermore 

 suppose these partial series to contain every term of the respective series 



P, P, P^(;) up to and including the last terms in these series 



antecedent to the first terms which determine the series as belonging to 



