284 Lanavicensis on the Partition of Numbers. 



of equations as for the system derived from it by interchanging 

 p with q. 



This, in fact, is a generalization of Euler's theorem, with which 

 it becomes identical on making one of the quantities p or q (say 

 q) infinite ; for the theorem of Bellavitis then assumes the fol- 

 lowing form : " The number of ways of making up 7n with the 

 numbers 1, 2, . . .p, is the same as the number of ways of break- 

 ing up m into p or fewer parts." 



It is capable of an intuitive proof in precisely the same way as 

 Euler's particular case by means of Mr. Ferrer's method of dis- 

 integration given in this Magazine. For it can be shown that 

 every group of integers x-^, x^, . . . Xp which satisfies one of the 

 two systems, can be transformed into a group x■^^, x^,... Xq satis- 

 fying the other system, and vice versa. 



Bellavitis's theorem may be stated otherwise as f ows: — ''The 

 number of distinct combinations of letters a^, a^, . . . Up, figuring 

 in the coefficient of .r'" in {aQ + ayX + a^x'^+ . . .-\-apxP)i, is the 

 same as that of the distinct combinations of b^, b^,. . . bq in the 

 coefficient of x'>^ in {b^ -j- b-^x + b^x"^ + . . . bqX^y." 



The theorem in this form is an intensification of the obvious 

 theorem, that the total number of combinations of letters in the 

 expansion of 



{a^ + a.+ .-.+CpY 

 IS the same as that in 



{b^ + b, + ...+bqY, 



each being equal to 



1.2. ..{p + q) _ 

 {l.2...p){1.2...q)' 



that is to say, we see now (thanks to M. Bellavitis) not only that 

 these two totals are equal, but that they are respectively made up 

 of a like number (jiq + l) of equal parts. 

 September 3, 1859. 



P.S. Bellavitis's theorem may be best stated in general terms as 

 follows : — " The number of modes of composing any integer with 

 other integers subjected to two given limits, viz. one of number and 

 another of magnitude, remains unaltered when those two limits are 

 interchanged." The ground of its truth, as I have on another 

 occasion remarked upon Euler's, is identical with that upon which 

 the law of reciprocity in the Multiplication Table, known to every 

 schoolchild, reposes. Bellavitis's theorem, as an idea, is little 

 more than a slight extension of the aperqu that four sevens make 

 up seven fours : but it is none the less a most valuable intuition 

 gained to the arithmetician's stock in trade. 



