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



MacMahon 195 applied his 174 second memoir to find the probability that 

 in the election of P by m votes to Q's n votes (m > ri) the order of the ballots 

 is such that P has at all times more votes than Q, and similarly for n candi- 

 dates. 



Start with any Ferrers' graph of an ordinary partition and place the 



parts of the partition at the nodes so that the numbers in a row, read 

 from west to east, and in columns, read from north to south, are in descend- 

 ing order. We obtain a two-dimensional partition of 19 : 



3222 

 2111 

 2 1 



2 



E. Landau 196 considered the maximum value f(ri) of the l.c.m. of a\, 

 - , a p in all the partitions of n into positive parts, n = a\ + + a p 

 (P ^ n). Thus, forn = 5 = 4 + l=2 + 3, /(5) = 6. He proved that 



*=* ^x log x 

 R. W. D. Christie 197 noted that, if 1 =i M ^ 5, QN + M has 



v = (3N + M)(N + 1) 



partitions into parts ^ 3, and v + 1 partitions if M = 0. 



J. W. L. Glaisher 198 treated various questions of partitions by solving 

 equations in finite differences which were constructed by means of L. F. 

 A. Arbogast's 850 rule of derivations. The capital letters A, B, C, 

 signify any distinct numbers in ascending order of magnitude, while Greek 

 letters denote any distinct numbers. The only partition of 8 of the form 

 A*BC is 1, 1, 2, 4; the only one of the form AB 2 C is 1, 2, 2, 3; while either 

 partition is of the form cPfiy. Denote by P n (i, j, k, ; A p B q - - -}x the 

 number of partitions of x into the elements i, j, k, , each partition con- 

 sisting of n parts and being of the form A p B q - . When the elements are 

 0, 1, 2, -, this number P is the number G n (x, A p B q - ) of terms of that 

 form in the xth derivation of a n ; its values for n = 2, 3, 4 and all possible 

 forms are tabulated (p. 67), and by simple additions, we deduce 



P n (0, 1, ; a *p. "}x = G n (x, a p p q - - ) 



The latter are computed for n ^ 7; likewise G n (x) = P(l, 2, , n)x 

 and P n (l, 2, ) for n ^ 9, and P n (l, 2, ; a p /3- -}x for n ^ 7. It is 



196 Phil. Trans. Roy. Soc. London, 209, A, 1909, 153-175. Memoir IV on Partitions. 



196 Handbuch . . . Vcrtcilung der Primzahlen, 1, 1909, 222-9. Cf. Landau. 184 



197 Math. Quest. Educ. Times, (2), 16, 1909, 104. 



198 Quar. Jour. Math., 40, 1909, 57-143. 



