CHAP, in] PARTITIONS. 123 



Sylvester 54 cited Eider's 14 transformation of the problem of the Virgins 

 and noted that the general form of the problem is to find the number* of 

 ways in which a given set of numbers h, , l r [an r-partite number] can 

 be made up simultaneously of the compound elements ai, -, a r ; 61, , b r ; 

 etc. This problem of compound partition can be made to depend on simple 

 partition. Omitting details, he stated the following theorem: Given r 

 linear equations in n variables with integral coefficients such that the r 

 coefficients of each variable have no common factor, and such that not more 

 than r 1 variables can be simultaneously eliminated from the r equations, 

 then the determination of the number of sets of positive integral solutions 

 may be made to depend on like determinations for each of n derived inde- 

 pendent systems each in n 1 variables. The conditions are satisfied by 

 Euler's equations 



ax + + Iw = m, x + + w = p, 



if a, , I are distinct. Sylvester never published an explicit statement of 

 the theorem just sketched, nor of his obscure generalization. See the 

 following paper. 



Cayley 55 called (a, a) + (6, /?) + a double partition of (m, p) if 



a + 6 + = ra, a + & + = p. 



If a/a, 6/|8, are distinct irreducible fractions and if a, 13, - - are each 

 < p + 2, the number of such partitions is 



D(am dp; ab a/?, ac ay, ) 



+ D((3m bp', /3a ba, (3c 67, ) + , 



where the denumerant 107 D(m; a, b, ) is the coefficient of x m in 



(1 - z'O-Ki - z 6 )- 1 ---. 



He noted that Sylvester apparently eliminated each of the r variables in 

 turn from ax + by + = m, ax + fly + = M> obtaining r equations 

 of the form 



(ab a(3}y + (ac ay}z -{-= am ap, 



from which the above formula follows. 



* E. Mortara 56 treated partitions into three distinct elements. 

 Sylvester 57 delivered seven lectures on partitions in 1859. 



G. Bellavitis 58 proved that the number Qt, n, p~] of sets of integral solu- 

 tions ^ of a + i + + cL n = p, ai + 2 2 + + na n = p, equals 

 the number Q*, p, n} of sets of solutions ^ of + Pi + + &P n > 

 Pi + 2j3 2 + + pfi p = iJ~ For, every set of solutions of the first pair 



* Number of sets of integral solutions i of (nx + bty + = k (i = I, , r). 

 54 Phil. Mag., (4), 16, 1858, 371-6; Coll. Math. Papers, II, 113-7. 



66 Phil. Mag., (4), 20, 1860, 337-341; Coll. Math. Papers, IV, 166-170. 

 68 Le partizioni di un numero in 3 parti different!, Parma, 1858. 



67 Outlines of the lectures were printed privately in 1859 and republished in Proc. London 



Math. Soc., 28, 1897, 33-96; Coll. Math. Papers, II, 119-175. 



68 Annali di mat., 2, 1859, 137-147. 



