138 HISTORY OF THE THEORY OF NUMBERS. [CHAP, in 



at least two of the parts are > j, say j + k, j + k f , equals the number of 

 partitions of w (j + k) (j + k'} into i 2 parts; etc. Hence 



F-^- 1 



-a;*) 



(1 -)(! -re*- 2 ) 



|_ . . .) 4. 



(I -X)'"(l- Z*'- 3 ) 



_ (1 . (1 . <r*\2.(<rJ+l yi+1 



' /I \ /i ,-\ I A * J^l^* } J J 



+ ^1 ^,t IN/'I /yAy /'W+i ^ . .1 



v 1 *' A- 1 Jt j*ti\*> ) j j> 



where S m (o: J '' +1 , ) is the sum of the ?w-ary combinations of x j+l , x j+z , 

 By induction, 



Hence 



Euler's theorem that a number can be partitioned into odd parts as 

 often as into any distinct parts is proved constructively and extended. 

 The number of ways of forming w additively with an indefinite number of 

 parts not divisible by k and with m distinct parts (each repeated indefinitely) 

 divisible by k is equal to the number of ways of forming w with an indefinite 

 number of parts each occurring fewer than k times and with m distinct 

 parts each occurring k or more times. The proof is made for k = 10, 

 though the argument is general. First, let m = 0. Consider any partition 

 consisting only of parts not divisible by 10 and let the number of times any 

 such part X occurs be written in the decimal notation, say -c6a; then if 

 in place of - -cba times X we write a times \ b times 10X, c times 100X, -, 

 we get a partition in which no part occurs as many as 10 times, and the 

 correspondence is 1 to 1, so that the theorem is proved if m = 0. Next, if 

 along with the non-tenfold parts we introduce m distinct parts each divisible 

 by 10 and at the same time introduce in the corresponding partition of the 

 other set 10 times these same parts, each divided by 10, the partitions of 

 the second set will contain m parts occurring 10 or more times, while the 

 1 to 1 correspondence will not be disturbed. 



A. Cayley 116 remarked that Franklin's 105 theory does more than group 

 the partitions into pairs. In addition to the existing division E + of the 

 partitions into even and odd, it establishes a new division / + D of the 

 same partitions into increasible and decreasible. There is thus a fourfold 

 division El, 01, ED, OD. For instance, if N = 10, the arrangement is 



El : 8 + 2, 7 + 3, 6 + 4 



01 : 10, 5 + 3 + 2 



ED : 9 + 1, 4 + 3 + 2 + 1 OD : 7 + 2 + 1, 6 + 3 + 1, 5 + 4 + 1 



116 Johns Hopkins Univ. Circ., 86 (in full). 



