Mr. J. W. L. Glaisher on Partitions. 309 



An application of the same principle affords another trans- 

 formation of some interest for P( L, 3, 5 . . . )x. It was proved by 

 Euler that P(l, 3, 5 . . ,)x is equal to the number of partitions of 

 x into the parts 1, 2, 3 . . . in which each part only appears once 

 in each partition, or, say, is equal to P(l, 2, 3 . . .)x without 

 repetitions. Now, applying Mr. Ferrers's method to the latter, 

 we see that a partition without repetitions corresponds to a par- 

 tition without omissions. For example, writing 1 + 2 + 3 + 5 as 



1 



1, 1 



1, ], 1 



1, 1, 1, 1, 1, 



and adding the columns, we obtain 1 + 1+2 + 3 + 4; and it is 

 clear that though any part may appear any number of times, on 

 part can appear unless all the lesser parts appear also. We thus 

 see that 



P(l, 3, 5 . . .)# = P(l, 2, 3 . . .)x without omissions ; 



and this latter quantity is equal to 



1+P(1, 2)(*-l-2) + P(l, 2, 3)(tf-l-2-3) 



+ P(l,2,3,4)(ar-l-2-3-4)+&c; . (4) 



for the second term represents the number of partitions of x into 

 parts in which both 1 and 2 must occur and no higher part can 

 occur; the third term represents the number of partitions in 

 which 1, 2, 3 must, and no higher part can, occur, &c. By use 

 of the well-known theorem 



P(l, 2, 3 ... n) (x-n) +P(1, 2, 3 . . .w-l)^ = P(l, 2, 3 . . . n)x 



(which can be readily established by general reasoning), the last 

 equality can be transformed into 



P(l,3,5...)* = P(l,2)(tf-l) 



+ P(l,2,3,4)(#-l-2-3) + &c, . . (5) 



which can be at once identified with the formula 



l+x.l+x*.l + x 3 ... = l + 



1—x . 1 — x- 



,1+2 + 3 



+ &C. 



' 1-^.1-^.1-^.1-. 



That Mr. Ferrers' s method should, as it were, afford a demon- 

 stration of this identity is what we should expect, as it was in 

 effect remarked by Mr. Sylvester (Brit. Assoc. Report, 1871, 

 p. 25, Sect. Proc.) that it established Euler's more general 



