TRANSACTIONS OF SECTION A. 527 



systematic advance in certain questions which depend upon the partitions of 

 numbers was only possible when Euler showed that the identity .r" .r'' = x^*'' 

 reduced arithmetical addition to algebraical multiplication and vice verbid. Starting 

 with this notion, Euler developed a theory of generating functions on the expan- 

 sion of which depended the formal solutions of many problems. The subsequent 

 work of Cayley and Sylvester rested on the same idea, and gave rise to many im- 

 provements. The combinations under enumeration had all to do with what may 

 be termed arrangements on a line subject to certain laws. The results were im- 

 portant algebraically as throwing light on the theory of Algebraic series, but another 

 large class of problems remained untouched, and was considered as being both 

 outside the scope and beyond the power of the method. I propose to give some 

 account of these problems, and to add a short history of the way in which a 

 method of solution has been reached. It will be gathered from remarks made 

 above that I regard any department of scientific work, which seems to be narrow 

 or isolated, as a proper subject for research. I do not believe in any branch 

 of science, or subject of scientific work, being destitute of connection with other 

 branches. If it appears to be so, it is especially marked out for investigation by 

 the very unity of science. There is no necessarily pathless desert separating 

 different regions. Now a department of pure mathematics which appeared to be 

 somewhat in this forlorn condition a few years ago, was that which included prob- 

 lems of the nature of the magic square of the ancients. Conceive a rectangular 

 lattice or generalised chess board (cf. ' Gitter,' Klein), whose compartments are 

 situations for given numbers or quantities, so that there is a rectangular array of 

 certain entities. The general problem is the enumeration of the arrays when both 

 the rows and the columns of the lattice satisfy certain conditions. With the 

 simplest of such problems certain progress had undoubtedly been made. The 

 article on Magic Squares in the 'Encyclopaedia Britannica,' and others on the 

 same subject in various scientific publications, are examples of such progress, but 

 the position of isolation was not sensibly ameliorated. Again the well-known 

 ' probleme des rencontres ' is an instance in point. Here the problem is to place 

 a number of different entities in an assigned order in a line and beneath them the 

 same entities in a different order subject to the condition that the entities in the 

 same vertical line are to be different. This easy question has been solved by 

 generating functions, finite differences, and in many other ways. In fact when the 

 number of rows is restricted to two, the difliculties inherent in the problem when 

 more than two rows are in question do not present themselves. The problem of 

 the Latin Square is concerned with a square of order n and « different quantities 

 which have to be placed one in each of the >r compartments in such wise that 

 each row and each column contains each of the quantities. The enumeration of 

 such arrangements was studied by mathematicians from Euler to Cayley without 

 any real progress being made. In reply to the remark ' Cui bono ? ' I should say 

 that such arrangements have presented themselves for investigation in other 

 branches of mathematics. Symbolical algebras, and in particular the theory of 

 discontinuous groups of operations, have their laws defined by what Cayley has 

 termed a multiplication table. Such multiplication tables are necessarily Latin 

 Squares, though it is not conversely true that every Latin Square corresponds to a 

 multiplication table. One of the most important questions awaiting solution in 

 connection with the theory of finite discontinuous groups is the enumeration of 

 the types of groups of given order, or of Latin Squares which satisfy additional 

 conditions. It thus comes about that the subject of Latin Squares is important in 

 mathematics, and some new method of dealing with them seems imperative. 



A fundamental idea was that it might be possible to find some mathematical 

 operation of which a particular Latin Square might be the diagrammatic repre- 

 sentative. If, then, a one-to-one correspondence could be established between such 

 mathematical operations and the Latin Squares, the enumeration might conceivably 

 follow. Bearing this notion in mind, consider the differentiation of .r" with 

 regard to x. Noticing that the result is ??.i"-i (n an integer), let us inquire 

 whether we can break up the operation of differentiation into n elementary por- 

 tions, each of which will contribute a unit to the resulting coefficient ??. If we 



