602 SCIENCE PROGRESS 



before us these problems are considered and in certain cases 

 solved ; the theory of symmetric functions, which is intimately 

 bound up with the theory of distribution, is developed. The 

 section also gives an account of the laws of symmetry and of 

 Hammond's operators which are to play an important part 

 in the development of the subject. 



Major MacMahon in his preface notes the position of the 

 subject of Partitions in the international organisation of 

 the subject-matter of mathematics ; in this division partitions 

 are placed in Theory of Numbers : our author pleads for it as 

 a subdivision of Combinatory Analysis. His plea is supported 

 by the treatment of the theory of the separations of a partition 

 which is given in Section II, and its intimate relation to the 

 expression of symmetric functions as well as to the part which 

 this theory plays in other portions of the subject developed 

 in this book. In this section the theory of distribution and 

 the theory of symmetric function are both advanced by 

 the notion of the unitary partitions of a number ; as the 

 author says, at one time the theory of distributions is urged 

 on by the algebra and at another pulls the algebra after it. 

 In Section III permutations in their most general form are 

 discussed. This section contains a theorem which is called 

 by the author the Master Theorem on account of the success 

 with which he has applied it to the solution of many problems 

 of great variety and difficulty. It is of great algebraical interest 

 and is quoted here because it gives, as well as an extract can, 

 some notion of the problems solved in the book. The theorem 

 is : "Given 



X r = a rl x r + a r2 x 2 + . . . + a m e n (r = i , . . . n) 

 then the coefficient of 



x^ l x 2 h . . . xj n . 

 in the development of the product 



XfXJ* . . . X* 

 is equal to the coefficient of the term 



X\ X% • • • Xfi 



in the expansion of the function 



( i - a n x x ) ( i - a n x 2 ) . . . ( i - a% n x n ) 



