156 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. Ill 



which reduces by cancellation to (1 + P x z)(l + P x+l z) (! + 



if n is a positive integer. The simplest of the general formulas proved is 



(pnl _ lVp<n-l)l _ 1) . . . (p(n-H-l)l _ 1)' 



i -- L 



n _ 1 _|_ 



( p l - 1)(P.. 1) ... (Prl.. 1) 



which includes as special cases formulas of Euler 3 - 9 and Cauchy. 29 



A. S. Werebrusow 188 gave a recursion formula for the number of sets of 

 positive integral solutions of a\x-i+ + a n x n = A, where the positive 

 integers a have no common factor. Then he considered the number of 

 sets when at least one x is ^ 0. 



P. A. MacMahon 189 treated a " general magic square," consisting of n 2 

 integers (zeros and repetitions permitted) arranged in a square such that 

 the rows, columns and diagonals contain partitions of the same number 

 (whereas in an ordinary magic square the n 2 integers are 1, 2, , n 2 }. 

 The treatment applies to all arrangements of integers which are denned by 

 linear homogeneous Diophantine equations or inequalities such that the 

 sums of corresponding elements of two solutions give a solution [cf. Mac- 

 Mahon 174 ]. 



0. Meissner 190 noted that to decompose n into positive integral sum- 

 mands whose product is a maximum, the summands must be equal or differ 

 at most by unity, and must include as many threes as possible. 



G. Mignosi 191 wrote c n for the number of sets of integral solutions ^ 

 of 0,1X1 + + a m x m = n, and a(j) for the sum of those of ai, , a m 

 which are divisors of j, and proved the recursion formula 



<r(2)Ci_ 2 + ' ' + ff(i)Co = Wij C = 1. 



Taking i = 1, , n, we obtain n\c n as a determinant of order n. If each 

 a,- = 1, then <r(i) = m and c n is the number of combinations of m + n 1 

 things taken n at a time. 



S. Minetola 192 wrote R m , n for the number of different ways m distinct 

 objects can be separated into n groups, where n ^ m. For example, 

 #4,2 = 7, the separations being a\ a 2 asa 4 , , a 4 aia 2 s, 



We have 



R m , n = nR m ^, n + # m _ lt n _ l} R m , 2 = 1 + 2 + 2 2 



_ _ j ) Rm-k-l, n -kRk+l, k 



k, 



188 Matem. Sbornik (Math. Soc. Moscow), 24, 1904, 662-688. 



189 Phil. Trans. Roy. Soc. London, 205, A, 1906, 37-59. Memoir III on Partitions. Abstract 



in Proc. Roy. Soc., 74, 1905, 318. 



190 Math. Naturw. Blatter, 4, 1907, 85. 



191 Periodico di Mat., 23, 1908, 173-6. 



192 Giornale di Mat., 45, 1907, 333-366; 47, 1909, 173-200. Corrections, generalizations and 



simplifications in II Boll, di Matematica Gior. Sc.-Didat., Rome, 11, 1912, 34-50, with 

 errata corrected pp. 121-2. 



