Prof. Sylvester on the Problem of the Virgins. 371 



the N^ values of y corresponding to x,^. To effect this, we have 

 only to write down the values of '%xy, %x-y, . . . '^x'^ . y. In like 

 manner we may find the sum of the Nj values of ?/^ correspond- 

 ing to x^, the N2 values of y'^ corresponding to X:^) ^^v ^^^ ^^ 

 in general for y" . Thus, then, we may obtain the requisite num- 

 ber of sets of equations for determining independently by means 

 of equations of the degrees N„ No, . . • N^ respectively the values 

 of y corresponding to each of the distinct values of x ; and in 

 like manner for all the other variables. The principal interest 

 of this note consists, however, in the appreciation of the fact that 

 we can represent algebraically, as has been shown above, the 

 value of 2.r«.7/P. £•■''. . . , where the sign of summation extends 

 over all the simultaneous solutions of 



ax + by + cz + &c. = n. 

 This is a considerable advance upon the conception (itself before 

 my discovery entirely unrecognized*) of the explicit represent- 

 ability of the mere number of the solving systems x, y, z . . . by 

 general algebraical formulae. By this new theorem we pass, as 

 it were, from the shadow to the substance. 



XLII. On the Problem of the Virgins, and the general Theory of 

 Compound Partition. By J. J. Sylvester, M.A., F.R.S., 



Professor of Mathematics in the Royal Military Academyf. 



IN the Opera Minora of the great Euler, in the last page of 

 his seco7id memoii- on the partition of numbers (vol. i. p. 400), 

 occur these words : — " Ex hoc principio definiri potest quot 

 solutiones problemata qu^ ab arithmeticis ad regulam virginum 

 referri soleut, admittunt ; hujusmodi problemata hue redeunt ut 

 invcniri debeant numeri p, q, r, s, &c., ita ut his duabus condi- 

 tionibus satisfiat, 



ap + bq + cr + dsSic. =n, et up + ^q -{- <^r -\- hs kc. =vj 

 et jam quajstio est quot solutiones in numeris integris positivis 

 locum sint habiturse ubi quidem tenendum est numeros a,b,c,d...n 

 et x,/3,j,B...v esse integros ;" and he then proceeds to observe 

 that the number in question is the coefficient of x"Ky" in the 

 expansion of the expression 



(i -x^y') (1 -x^y^) (1 -x" .y^) ... 

 in terms of ascending positive powers of x and y. 



* As witness the comparatively unfructuous labours of Paoli, Ilcrschel, 

 Kirkman, and even of Cayley. But as honest labour is seldom entirely 

 wasted, so in the ))iesent case it was my valued friend Mr. Kirkman's Man- 

 chester memoir on partitions which first drew and fixed my attention on 

 the subject. 



t Communicated by the Author. 



2B» 



