528 



REPORT — 1901. 



write down .r" as the product of ?« letters, viz., xi'.c.i' . . ., it is obvious that if we 

 substitute unity in place of a single .v in all possible ways, and add together the 

 results, we shall obtain n.i"'^. We have, therefore, 7i different elementary opera- 

 tions, each of which consists in substituting unity for .v. We may denote these 

 diagrammatically by 



the 



and from this point of view — is a combinatorial symbol, and denotes by 



coefficient n the number of ways of selecting one out of n different things. 



Similarly, the higher differentiations give rise to diagrams of two or more 

 rows, the numbers of which are given by the coefficients which result from such 

 diflerentiations. Following up this clue much progress has been made. For a 

 particular problem success depends upon the design, on the one hand, of a func- 

 tion, on the other hand, of an operation such that diagrams make their appearance 

 which have a one-to-one correspondence with the entities whose enumeration is 

 sought. For a general iuA-estigatiou, however, it is more scientific to start by 

 designing functions and operations, and then to ascertain the problems of which 

 the solution is furnished. The difficulties connected with the Latin Square and 

 with other more general questions have in this way been completely overcome. 



The second new method in analysis that I desire to bring before the Section 

 had its origin in the theory of partition. Diophantus was accustomed to consider 

 algebraical questions in which the symbols of quantity were subject to certain con- 

 ditions, such, for instance, that they must denote positive numbers or integer 

 numbers. A usual condition with him was that the quantities must denote posi- 

 tive integers. All such problems and particularly those last specified are qualified 

 by the adjective Diophantine. The partition of numbers is then on all fours with 

 the Diophantine equation 



a + ^ + y+ ... +v = n, 



a further condition being that one solution only is given by a group of numbers 

 a, P, y . . . satisfying the equation ; that in fact permutations amongst the quanti- 

 ties a, /3, y . . . are not to be taken into account. This further condition is bi-ought 

 in analytically by adding the Diophantine inequalities 



a ^/3^7^. . .^v^O 



V in number. The importation of this idea leads to valuable results in the theory 

 of the subject which suggested it. A generating function can be formed which 

 involves in its construction the Diophantine equation and inequalities, and leads 

 after treatment to a representative, as well as enumerative, solution of the problem. 

 It enables further the establishment of a group of fundamental parts of the parti- 

 tions from which all possible partitions of numbers can be formed by addition with 

 repetition. In the case of simple unrestricted partition it gives directly the com- 

 position by rows of units which is in fact carried out by the Ferrers-Sylvester 

 graphical representation, and led in the hands of the latter to important results 

 connection with algebraical series which present themselves in elliptic functions 

 and in other departments of mathematics. Other branches of analysis and geometry 

 supply instances of the value of extreme specialisation. 



What we require is not the disparagement of the .specialist, but the stamping out 

 of narrow-mindedness and of ignorance of the nature of the scientific spirit and of 

 the life-work of those who devote their lives to scientific research. The specialist 

 who wishes to accomplish work of the highest excellence must be learned in the 

 resources of science and have constantly in mind its imity and its grandeur. 



