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IX. Combinatorial Analysis. The Foundations of a New Theory, 

 tly Majvr P. A. MACMAHON, D.Sc., F.R.S. 



Received March 19, Read April 5, 1900. 



INTRODUCTION. 



IN the 'Transactions of the Cambridge Philosophical Society' (vol. 16, Part IV., 

 p. 262), I brought forward a new instrument of research in Combinatorial Analysis, 

 and applied it to the complete solution of the great problem of the " Latin Square," 

 which had proved a stumbling block to mathematicians since the time of Euler. The 

 method was equally successful in dealing with a general problem of which the Latin 

 Square was but a particular case, and also with many other questions of a similar 

 character. I propose now to submit the method to a close examination, to attempt 

 to establish it firmly, and to ascertain the nature of the questions to which it may be 

 successfully applied. We shall find that it is not merely an enumerating instrument 

 but a powerful reciprocating instrument, from which a host of theorems of algebraical 

 reciprocity can be obtained with facility. 



We will suppose that combinations defined by certain laws of combination have to 

 be enumerated ; the method consists in designing, on the one hand, an operation 

 and, on the other hand, a function in such manner that when the operation is 

 performed upon the function a number results which enumerates the combinations. 

 If this can be carried out we, in general, obtain far more than a single enumeration ; 

 we arrive at the point of actually representing graphically all the combinations under 

 enumeration, and solve by the way many other problems which may be regarded as 

 leading up to the problem under consideration. In the case of the Latin Square it 

 was necessary to design the operation and the function the combination of which was 

 competent to yield the solution of the problem. It is a much easier process, and 

 from my present standpoint more scientific, to start by designing the operation and 

 the function, and then to ascertain the questions which the combination is able to 

 deal with. 



1- 



Art. 1. I will commence by taking the simplest possible question to which the 

 method is applicable. Let us inquire into the number of permutations of different 

 letters. A knowledge of the result would at once lead us to design 



An operation. A function. 



(d/dx)' X" 



VOL. CXCIV. A 260. 3 A 30.7.1900 



