ESSAY-REVIEWS 603 



in which the denominator is in symbolic form in such wise 

 that on multiplication the factors a n a 2 2, «n«22<233> • • • ar e to 

 be placed in determinant brackets (a n a 22 ,), («n«22«33) • • • 

 and denote coaxial minors of the determinant of transforma- 

 tion." Now by selecting different types of the guiding 

 determinant various interesting algebraic forms of the generating 

 function are written down ; the application of these enables 

 the author to solve problems of great interest : amongst these 

 we may cite the Probleme des Rencontres, the number of 

 circular substitution of n letters and certain cases of lattice 

 permutations. A by-product of the theory expounded occurs 

 in the interesting case of dealing out the cards of m packs 

 of playing cards ; it is proved that the average number of 

 pairs met with in dealing is equal to 4m — 1 . The Master 

 Theorem is a remarkable instance of the insight of the author 

 into the inner nature of those questions which are so well 

 known as recreations and which have been effectively dis- 

 played by Mr. Rouse Ball and M. Edouard Lucas. The par- 

 ticular problems do not interest our author : he goes for the 

 general algebraical theorem upon which they depend, and when 

 that has been solved the particular problem falls into its 

 proper relation. It is the same philosophical spirit at work 

 in the sphere of number as that which Newton showed in his 

 great dynamical discoveries ; when one studies Newton one 

 is apt to forget the magnitude and diversity of the problem 

 which suggested his immortal speculations — so it is with the 

 Combinatory Analysis, the infinite perplexities of the problems 

 are reduced by Major MacMahon to order and simplicity, not 

 by solving particular problems, but by revealing the laws 

 which these particular cases, so puzzling in themselves and 

 so apparently unrelated to each other, must obey. 



In Section IV the theory of the compositions of numbers 

 is treated. The simplest case is the unipartite composition of 

 n, and by composition the author means partitions in which 

 order is regarded ; thus 4 has eight compositions — 



( 4 ) (30 (13) (2 2 ) (I 4 ) (l 2 2) (21*) (121) 



Various graphical representations of unipartite numbers are 

 explained and then the general problem of multipartite 

 numbers is entered upon, that is, the number of distributions 

 of objects of the type (pqr . . .) into n parcels of the type 



