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COMBINATORY ANALYSIS— THE INTERPRETATION 

 THEREOF, by C. : on Combinatory Analysis, by Major Percy 

 A. MacMahon, F.R.S., D.Sc, LL.D. Volume I. [Pp. xix + 300.] 

 (Cambridge: at the University Press, 1915. Price 15.J. net) 



Combinatory Analysis is well known even to students of 

 elementary algebra, where it figures under the title of Permu- 

 tations and Combinations ; while in theory of equation the 

 subject is intimately connected with symmetric functions. 

 The formal statement of the objects of this branch of mathe- 

 matics is that it includes the formation, enumeration, and other 

 properties of the different groups of a finite number of elements 

 which are arranged according to prescribed laws. By its 

 subject-matter combinatory analysis is related to some of the 

 most ancient problems which have exercised human ingenuity. 

 It would be difficult to assign a date to such a question as the 

 Ferry-boat problem ; its historian describes it under the 

 extensive title of mediaeval, but it would surprise no one to 

 find it enclosed in a still-unwrapped papyrus roll, or to hear 

 that a version of it had been discovered in some ancient manu- 

 script which described the pastimes of King Solomon's ample 

 families. The scientific subject commences with the works of 

 Pascal, Leibniz, Wallis, and John Bernoulli. In the hands of 

 these mathematicians the problems do not always retain the 

 charms of their earlier romantic setting ; jealous husbands 

 and frail wives no longer meet at ferries, and letters are 

 no longer placed at random in their envelopes. The problems 

 are enunciated in set and formal terms ; now the chief aim is 

 to determine the number of distributions of n objects of which 

 p are of one kind, q of another kind, r of a third and so on 

 ( n= p-\-q-\-r + ...) into parcels, that is unarranged classes, 

 of which p x are alike and of one kind, q x of a second kind, t\ 

 of a third kind, and so on {n x = p\ -f- q\ -f- r x + . . .), while a 

 similar question arises in connection with groups in which 



the objects are arranged. In the first sections of the book 



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