CHAP, ill] PARTITIONS. 163 



and to still more exact results in which the sum of a number of approximating 

 functions appears on the right hand side. Six terms of the series thus 

 obtained give p(200) = 3972999029388, with an error of .004, a result 

 confirmed by MacMahon by direct calculation. Here 0(g(i)) denotes 

 a function whose quotient by g(t} remains numerically under a fixed finite 

 value for all sufficiently large values of t. At the end of the paper occurs 

 a table, calculated by MacMahon, of the number of partitions of n for 

 n ^ 200. 



P. A. MacMahon 222 proved that, if pi, -, p t are integers in descending 

 order of magnitude and (mi raj is the partition conjugate to (pi - p t ), 

 the number of ways of distributing n objects of specification (n) into boxes 

 of specification (mi raj is the coefficient of x n in the expansion of 



1 -^ {(1 - x) Pl (l - z 2 )" ---(I- x^}. 



MacMahon 223 established a (1, 1) correspondence between combinations 

 derived from m identical sets of n distinct letters and general magic squares 

 of order n in which the numbers in any row or column have the sum m 

 [MacMahon 189 ]. 



S. Ramanujan 224 proved that, if p(ri) is the number of partitions of n, 



p(5m + 4) = (mod 5), p(7m + 5) = (mod 7), 



p(35m + 19) = (mod 35), p(25ra + 24) = (mod 25), 



p(49n + 47) = (mod 49); 



, 

 p(4) + p(9)s + p(14)* 2 + . . . - 6 



^. * j_ MG^-i. ^ 



p(5) + p(12)* + p(19)x- + .-.- 7 



{(I - 



{(1- 



which imply the first two congruence theorems. 



H. B. C. Darling 225 gave elementary proofs of the first two of Ramanu- 

 jan's 224 congruence theorems. 



L. J. Rogers 226 gave a new proof of his 156c two identities. J. Schur 227 

 gave two proofs. Finally, Rogers 228 and S. Ramanujan 228 each gave a proof 

 which is much simpler than all earlier proofs. 



P. A. MacMahon 229 solved the problem of multipartite partition. 



222 Proc. London Math. Soc., (2), 16, 1918, 352-4. 

 Ibid., (2), 17, 1918,25-41 

 224 Proc. Cambridge Phil. Soc., 19, 1919, 207-210. 

 226 Ibid., pp. 217-8. 



226 Proc. London Math. Soc., (2), 16, 1917, 315-7. 



227 Situngsber. Akad. Wiss. Berlin (Math.), 1917, 302-321. 



228 Proc. Cambridge Phil. Soc., 19, 1919, 211-6. 



229 Phil. Trans. Roy. Soc. London, 217, A, 1916-7, 81-113. Memoir VII on Partitions. 



