Lanavicensis on the Partition of Numbers. 283 



respectively «, /3, y . . . \, and also not to be simultaneously equal 

 to a, /3, 7, . . . A,. 



Hence excluding from v, v" &c. the value corresponding to 

 «'=0, /S' = O...X' = 0, 



and 





But by supposition^ since a' + /3'+ . . . A.' <s, ^i'(l) = l, except 



when all but one of the quantities a', fi', y',. • . V is zero, in 



which cases <^^(1)= the prime number, whose index in v is that 



one of these quantities which is not zero. 



Hence « of the factors in the denominator will be a, /S of 



them b, y of them c, . . .\ of them I, and all the rest unity. 



n 

 Hence 6(1) — ^ ,„ jr = 1. 



So that if the theorem is true for the superior limit s, it is true 

 for s + 1 ; and since it is true for s = \, it is true universally. It 

 must be well observed that this induction is only legitimate upon 

 the basis of the theorem being stated as relating at one and the 

 same time to both forms of the value of <^;j(l). 



The theorem in question, or (as I should rather say) the less 

 obvious part of it, may also be stated as follows : If m be a 

 number containing more than one distinct prime factor, and if 

 Vj, Vg, . . . Vi arc the numbers (including unity) less than ^n and 

 prime to it, 



sm 



I - TT Isml — TT I . . . SmI — TT J = vp-. 



\n J \n / \n J 2* 



XLV. Note on a Theorem ofM.. Bellavitis on the Partition of 

 Numbers. By Lanavicensis*. 



M BELLAVITIS has given, in Tortolini's 'Annals,' the 

 • annexed beautiful theorem of reciprocity, as of import- 

 ance in the theory of invariants ; viz. if 



'^^O "r" *'l ' ^"2 ' • • • "I */) ~" ?/i 



and 



a;, + 2a^2+ . . . +])Xp=m, 



tlie number of holutions in integers is the same for this system 

 * Communicated by the Author. 



