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column. Continuing the process we finally get nf square 

 lattices of n rows and n columns each containing a different 

 arrangement of n unities, but in each arrangement one 1 occurs 

 in each row and one in each column. 



This simple example depends upon the selection of a func- 

 tion and an operator. A variety of cases of greater com- 

 plexity is given and the relation of the whole to the differ- 

 ential operators of Section II with symmetric functions on 

 operands is discussed. This leads to the discussion of the 

 Latin Square. The author reminds us of the cognate problem 

 (which Euler propounded in 1782) of the arrangement of 36 

 officers arranged in six sets of six, six being in each of six 

 regiments and also arranged in sets of six according to rank ; 

 the disposition is required into a square in which no row 

 or column should contain two officers of the same rank and of 

 the same regiment. Though Euler was unable to solve this 

 problem, it suggested to him the problem of the Latin Square, 

 that is, the arrangement of n letters on a board of « 2 compart- 

 ments so that no letter occurs twice in the same row or 

 column. Euler recognised the difficulty of the problem in 

 its general form, and in 1890 Cayley, after enumerating the 

 number of possible second lines of the square when the first 

 is given, remarked that it was not easy to see how the whole 

 number of arrangements for the third line is to be calculated, 

 (adding) and the difficulty of course increases for the next 

 following lines. Now it is remarkable that Major MacMahon 

 has solved the general case and shown how to calculate the 

 number of sth lines that can be written down when (s — 1) 

 lines are known ; he has also solved similar problems of a more 

 difficult character. The last section of the book is devoted 

 to the enumeration of the partitions of a multipartite number. 



It is impossible to give anything but a most inadequate 

 idea in a review of the complicated and powerful analysis by 

 which the author attains his objects. The elaboration of their 

 methods has occupied thirty years of work ; most of the results 

 in the book are original, some have appeared in papers pub- 

 lished by the author in the Phil. Trans. R.S., the Proceedings 

 of the L.M.S., the Quarterly Journal, but most of them appear 

 here for the first time. Any one who wishes to measure the 

 magnitude of the task achieved in the book is recommended 

 to take the article on this subject published in the Encyclopedie 



