Conditions Imposed by a Set of Order Numbers 141 



a cycle of order v. The product {u—P^ (u—P^). . . (u—Pp(t)) we shall 

 call a truncated function of cyclical character v . 



We shall now build up a polynomial of degree w — 1 in w constructing 

 it as a product of cyclical truncated functions, as follows. We first 

 multiply together X^'^ — 1 of the truncated functions referred to above 

 as of cyclical character v^*\ This leaves a factor of degree ^^ — 1 still 

 to provide for. From one of our truncated functions of cyclical character 

 v^^^ truncate X^'~^^ identical funct'ons of cyclical character v^'~^'. By 

 X^'~^^ — 1 of these functions multiply the product of X^''^ — 1 functions of 

 cyclical character v^^'' already formed. This leaves a factor of degree 

 j*^^"^^ — 1 still to provide for. So proceeding we ultimately construct a 

 product R{z, u) of degree n — 1 in w, made up of X^'^^ — 1 identical factors 

 of degree v}^^ of X^^~^^ — 1 identica factors of degree v}^~^^ . . . , of X'— 1 

 identical factors of degree v' and of j^' — 1 identical factors of degree 1 in u. 

 The order number of this product is evidently p,. 



Out of the factors of R{z, u) we now construct as products two 

 functions R,t-s (z.w) and Rs-\{z,u) the degrees of which in u are indi- 

 cated by their suffixes. This can be done in one way only. For in fitting 

 into Rn-s and Rs-i the maximum numbers of factors of cyclical character 

 v^*^ permitted by the degrees of these functions in u we just dispose of 

 the X^*^^ — 1 factors of this cyclical character contained in R{z, u). There 

 remain to be fitted into Rn-s and Rs-i factors whose degrees in the 

 aggregate are j'^'^^ — 1. Just as we distributed between R„-s and Rg-i the 

 X^'^^ — 1 factors of cyclical character v^^^ contained in R{z, u) so now we 

 further distribute between i?„_jiand Rs-i the X^'~^^ — 1 factors of R(z,u) 

 of eye ical character v^^~^^ which evidently can also be done in one and 

 only one way. So proceeding it is evident that we can factor R(z, u) 

 uniquely in the form R{z,u)=R,i-s{z,u)Rs-\ {z,u), the degrees in u of 

 the factors being indicated by the suffixes. The fact that we can split 

 R{z, u) into two such factors suffices to establish the lemma 

 a„_^ -\-as-\ =iJL and the proof of formula (16) is therewith complete. 



§4. Having obtained the formula (16) with reference to a single cycle 

 we can deduce the more general formula (15) in various ways. 



Representing /(z, u) as the product of its cyclical factors we write 



f{z,u)=fi{z, U)f2{z,u) . . .fr(2,u) 



where vu V2, . ■ - , Vr are the degrees in u of the respective factors. The 

 sum of the orders of coincidence of one branch of the equation /^(s, u)=0 

 with the remaining Vs — 1 branches we shall indicate by /x,'. Adjointness 

 relative to this cyclical equation for the value 2 = will then be defined 

 by the order number /i,' — 1 + 1/j'j. 



