Conditions Imposed by a Set of Order Numbers 143 



Let us now increase the orders of coincidence of the function ((s, u)) 

 with the branches of the equation /^(z,?^) =0 to what is requisite for 

 adjointness relative to the equation f{z, it) = on the part of these 

 branches. This means imposing on the function gt{z, u) the order of 

 coincidence ni—\-\-l/vt— m relative to a branch of the cyclical equation 

 ft{z, u) =0 where, for the moment, we employ the notation nt to designate 

 the order of coincidence of the product /i(z, ii) . . . ft-\ (s, w) with a 

 branch of the equation /^(z, m)=0. The number of the conditions so 

 imposed on the coefficients of the function gt{z, u), and therewith the 

 number of the additional conditions imposed on the coefficients of the 

 function ((s, u)) is then, in accord with formula (17), given by the 

 expression 



■ {nt-l-\-l/vt- ixt)vt-hiiJ-t -l + '^/v^Vf 



The total number of the conditions imposed on the general function 

 ((2, 11)) by adjointness relative to the value 2 = is then given by the sum 



20. 2 {{^,,-lJrl/v,-~y.t)v,-Wt-l + '^/vt)vt}. 



t=\ 



r 



Now 2 jxt vt is evidently equal to half the aggregate of all those numbers 

 each of which is the sum of the orders of coincidence of a branch with 



r 



the branches of all the cycles other than its own. The sum S m^i-^ 



<=i 



r 



\ S lit' Vt is then equal to half the aggregate of all those numbers each 



of which is the sum of the orders of coincidence of a branch with the 

 remaining n — 1 branches of the equation / (2, m) =0- The sum in 



r 



question is then equal to ^ 2 ntVf The expression given in (20) for 



t=\ 



the number of the conditions imposed on the coefficients of the general 

 function ((z, u)) by adjointness relative to the equation /(z, w)=0 for 

 the value z = can therefore be written in the form 



i i {nt-i + i/vtW 



§ 5. We shall now develop another method for reaching our results 

 in the general case, on starting out from the formulae (16) and (17) 

 obtained for the case of a single cycle. In connection with the proof of 

 these formulae we had occasion to construct a function R{z, u) syntheti- 

 cally as a product of a number of cyclical factors. The extension of 

 these formulae to the general case we shall here effect through an analysis 

 of the conditions under which factorization of the general function 



