374 Pi'or. Sylvesteron the Prohlem of the Virgins, 



cients have the same ratio ; and for this problem the process is 

 always limited to ouly two modes of application. The method, 

 however, in a very important class of cases admits of being ap- 

 plied in one, and only one mode when these conditions are not 

 strictly fulfilled. 



Thus the virgins who appeared to Euler, but with their forms 

 muffled and their faces veiled, have not disdained to reveal 

 themselves to me under their natural aspect. Wonderful indeed 

 has been the history of this theory of partitions. Notwithstand- 

 ing that the immortal Euler had written two elaborate memoirs 

 on the subject, that Paoli, and I believe other Italian mathema- 

 ticians, had taken it up from another but less advantageous point 

 of view, so completely had it fallen into oblivion, as far as the 

 mathematicians of this country are concerned, that Sir John 

 Herschel has written a memoir upon it, inserted iu the Philo- 

 sophical Transactions, without any reference to, and evidently in 

 complete unconsciousness of, the labours of his predecessors, 

 and subsequently Professor De Morgan, so justly celebrated for 

 his mathematical erudition, in a paper in the Cambridge and 

 Dublin Mathematical Journal, refers to the doctrine of partitions 

 as being of quite recent creation. The importance of the sub- 

 ject iu these later times has been vastly augmented by the^ mag- 

 nificent applications which our great mathematical luminary has 

 made of it to the doctrine of invariants. 



Postsci'ipt. — In the first instance I discovered the theorem 

 above given by a method of induction, aided by an effort of 

 imagination, and confirmed by numerous trials; but I have 

 since obtained a very simple, although somewhat subtle general 

 proof of it. Mr. Cayley on his part, and independently, has also 

 laid the foundation of a most ingenious and instructive method 

 of demonstration entirely distinct from my own. I reason upon 

 the equations, Mr. Cayley upon the Eulerian generating func- 

 tion ; but it was by operations performed upon this function 

 that I was myself originally led to a perception of the transcen- 

 dental analogies out of which I was enabled to evolve the law. 



The very interesting case of the composition of a proposed 

 integer out of elements given both in number and specie, to 

 which Euler has called particular attention, falls without pre- 

 paration under the standard form ; for this question is in fact 

 merely that of determining the number of solutions of the 

 binary system of equations, 



ax + by + c~ + . . . +lw = m, 



a:+ y + z + . . .+ w = iJ., 



a, b, c, . . . I being supposed to be all different. 



