124 HISTORY OF THE THEORY OF NUMBERS. [CHAP, in 



of equations consists of a partition of n into a n numbers n, , i numbers 

 1, where p is the total number of parts. To such a partition corresponds 

 as conjugate 

 (a n + <x n -i + + i) + (a n + + a 2 ) + + (a n + n-i) + a = M, 



which gives a partition of ju into n parts si p. These parts occur in the 

 second pair of equations as j3 p numbers p, , ft numbers 1. Again, 



O, w, P] = O, n - I, p~\ + [_n - n, n, p - 1], 

 - M, n, p]. 



There are Qi, w, p] partitions of N into p parts from c, c + d, - -, c + nd, 

 it At = (N cp)/d, since if each part be diminished by c and the remainder 

 be divided by d, we get the parts 0,1, , n whose sum is p. Application 

 is made to semin variants. 



L. Oettinger 59 stated and J. Derbes 59 proved that (k iyk r ~" is the 

 maximum of the products of the r equal or distinct integers into which the 

 positive integer N = rk v can be partitioned, where v is the least positive 

 integer such that k is integral. 



Sylvester 60 noted that Bellavifcis' 58 first theorem reduces for p infinite 

 to Euler's theorem that the number of partitions of /j, into parts ^ n equals 

 the number of partitions of /JL into n or fewer parts. Bellavitis' theorem, 

 which is capable of intuitive proof by Ferrer's 35 method, may be stated as 

 follows : The number of distinct combinations of a , , a n figuring in the 

 coefficient of af in (a + a\x + + a n x n ) p is the same as the number of 

 distinct combinations of 6 , , b p in the coefficient of x" in (6 + 



S. Roberts 61 proved Sylvester's 43 formula for waves. 

 Sylvester 62 noted that, if ILn = nl, 



V _ 1 _ = 



where the summation extends over all ways of expressing n as a sum of a 

 parts each a, p parts each 6, etc. 



E. Fergola 63 proved the analogous result: 



Tin A 



TLa ' 

 summed for all positive integers satisfying 



i + + d n = a, ai + 2 2 + + na n = n, 

 where A0 n denotes the ath order of difference of x n for x = 0. He evaluated 



69 Nouv. Ann. Math., 18, 1859, 442; 19, 1860, 117-8. 



60 Phil. Mag., (4), 18, 1859, 283-4, under pseud. Lanavicensis. 



61 Quar. Jour. Math., 4, 1861, 155-8. 



62 Comptes Rendua Paris, 53, 1861, 644; Phil. Mag., 22, 1861, 378; Coll. Math. Papers, II, 



245, 290. 

 88 Rendiconto dell'Accad. Sc. Fis. e Mat., Napoli, 2, 1863, 262-8. 



