Conditions Imposed by a Set of Order Numbers 135 



see that the difference in the numbers of the conditions imposed on the 

 general function of the form z~*{{z, u)) by the sets n, . . ., r^ and 



r r 



h, . . ., /y is 2 (tj — Ov^ while 2 irs — ts)vs is the difference in the numbers 



S=l 5=1 



of the conditions imposed on the general function in question by the 

 sets ri, . . ., T, and ^i, . . ., /;.. It follows that the difference in the 

 numbers of the conditions imposed on the general function of the form 



r 



s~'((2, u)) by the sets n, . . ., r^. and ti, . . ., r^ is 2 (r^— rjv,. We 



r 



consequently have 2X — 2w' = S (r^ — T J i^^, whence 



5=1 



r 



7. X = m"+i2 (r,-T,)i/,. 



5=1 



Combining (7) with (2) we evidently have 



r r 



8. X = m+S r,v,-iS U-l + lAJj',. 



5=1 5=1 



In deriving the last two formulae we have assumed the equation (1) 

 to be integral. Supposing this not to be the case we can choose an integer 

 j such that on writing z^u = v equation (1) transforms into an integral 

 equation F{z, v)—0. Here F(z, v)=z^^f{z, u) and we have F[{z, v) = 

 z^"~^^Yu(Zj m). On designating the order numbers of F'^{z, v) for the r 

 cycles corresponding to the value 2 = by the notation ju'i, . . ., ^t'^ we 

 evidently have /i' J =(w — l) J +M5; ^ = 1, • • •, ^- For the number X' of 

 the conditions imposed on the general function of the form s~*((s, v)) 

 by the set of order numbers n, . . ., t^. we can, in analogy with formula 

 (8), then write 



5=1 5=1 



and from this we derive 



9. x'+Xw-i)j=w+s T,v,-\i: U-i + i/v,)v,. 



5=1 5=1 



The general form z'\{z, v)), expressed in terms of z and u, may be 

 written 



10. 2-'((2, Z;))=2-'+^«-')^P„_lW«-^ + . . . +S-'+'Pl«^ + 2"'>0. 



where P„_i, . . . , Pq are series in power of z not involving negative ex- 

 ponents. This form, however, is included under the general form z~\{z,u)) 

 and lacks just (w — 1) j + (w — 2)j+ • • • +2j+i of the terms included 

 under the latter form. The general form (10) is then derived from the 



