Classes of Congruent Integers. 1 9 



This shews that all the elements of 21 are represented by B, each 

 being repeated ?' = ~ times. 



Now, it is known that in any Abelian group, if G u C 2 , ■•■, C t 

 be a system of elements of orders c u c 2 , • ■-, c t respectively, such that 

 all the elements of the group are represented in the form 



T=C? l c/----C? t , c'^O, 1,2, —,e t -l, 



each being repeated U times, then we can, by rejecting some C's or 

 by replacing some C's by other elements of lower orders, succes- 

 sively diminish h, until finally we obtain a system of bases, which 

 consequently consists of not more than t bases. In virtue of this 

 theorem we infer that in the present case the raiik of 31 is 

 réfd+ï. 



§• 6. 

 Case IV {Continued) 



Let us now proceed a step further and shew thatr=/<i+l 

 exactly. 



To begin with, suppose e kl <ze k and consequently li=- JL <:—. 



Then, evidently e ai me d+k >h for all values of a and i, except for 

 a=h, i = l ; and e kl >h. Thus, since all e's are greater than h, if we 

 reject any one of the fd+1 numbers in R, then the number of 

 elements of 2t represented by means of the remaining fd numbers 



is certainly less than "\ * , =p Kn "°. This shews that the re- 

 jection can never take place, therefore r=fd + l. 



Next suppose e kl =e k . Such a case is not necessarily im- 

 possible for some values of n and x u even if (L + fX : ) p =l + {f +d+x i}, 



1) Weber : Mathematische Annalen, Bd. 20, or Elliptische Punktionen und Algebraische 

 Zahlen, §. 54. 



