373 Pi'of. Sylvester on the Pruhlem of the Virgins, 



Why tlie solution in integers of two simultaneous equations 

 with an indefinite number of variables should be referred to " the 

 rule of the Virgins '' I am at a loss to conjecture, unless indeed 

 it be supposed to have some mystical reference to the alligation 

 or coupling of the coefficients of the two equations*. The problem 

 in question may be otherwise stated as having for its object to 

 discover the number of modes in which the couple m, n may be 

 made up of the couples a, a.; b, ^ ; c, y &c. 



I need hardly remark that Euler's form of representation is 

 no solution, but merely a transformation of the question. The 

 problem in its most general form is to determine the number 

 of modes" in which a given set of conjoint partible numbers 

 l^, Zj, . . . Ir can be made up simultaneously of the compound 

 elements, 



«!, a^, ... a,. ; b^, bc^ . .- . b^ ; c„ Cg, . . . c,. ; &c. 



The problem of simple partition has been already completely 

 resolved by the author of this notice ; but the resolution of the 

 problem of double, and still more of multiple decomposition in 

 general, seemed to be fenced round with insurmountable diffi- 

 culties. 



Let the reader imagine then with what surprise and joyful 

 emotion, within a few days of despatching my previous paper 

 on Partitions to this present Number of the Magazine, follow- 

 ing out a train of thought suggested by the simple idea in that 

 paper ccntained, I found myself led, as by a higher hand, to 

 the marvellous discovery that the problem of compound parti- 

 tion in its utmost generality is capable of a complete solution — 

 in a word, that this problem may in all cases be made to depend 

 on that of simple partition. The theorem by which this is eflfccted 

 has been already confided to the great mathematical genius of 

 England, and will be shortly committed to the 'Transactions^ of 

 one of our learned societies ; for the present I shall confine myself 

 to a disclosure of the general character of the theorem without 



* Professor De Morgan lias kindly furnished me with the following in- 

 formation as to the use of this singular phrase : — 



" I have seen this process cited as the rule of — Ceres, Series, Verginum, 

 Virginum, Ceres and Virginum, Series and Virginum, Ceres and Verginum, 

 Series and Verginum. 1 do not think any one of the eight is missing. I 

 cannot find that Ceres is attended b}^ any maidens, and I cannot guess who 

 the ladies were. It is applied by the arithmeticians to the rule of alliga- 

 tion when of an indeterminate mmiber of solutions — ^just Euler's problem 

 which you quote." Mr. Ue Morgan subsequently writes, " I forget 

 whether they wrote Series or Ceries; I think the latter:" and adds a plea- 

 sant caution .igaiust indulging a passion for one of these algebraical virgins ; 

 "for that though Jupiter did once animate a statue maiden at the prayer of 

 an enamoured sculptor, yet even Jupiter himself could not impart a body 

 to au algebraical abstraction." 



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