TACTIC 141 



Invariant-theory, in many ways, brought into a new light the old theory 

 of symmetric functions. A semin variant of a form is a symmetric function 

 (S.F.) of the differences of the roots, and as such has to satisfy an equation 

 i20 = o, where 12 is a differential operator, called the annihilator of 0. 

 One of MacMahon's discoveries may be explained as follows. Taking forms 

 of indefinitely high degree («), let 



/= I — nuj^x + n^a^x^ — n^a^x^ + . . . 



(^ = I - a^sr + ^ - 



3 



-! 3! 



then every seminvariant of / is, up to a numerical factor, equal to a non- 

 unitary S.F. of the roots of <p, and conversely. For instance, the semi- 

 variant 2(a - /3)^ for / is n^{n — i) (0^ — 02). where {a^^ - a^) is the value 

 of 2n'" for (f). In this connection MacMahon greatly simplified the theory 

 of perpetuants, and published a conjectural enumerating function for 

 them, afterwards verified by Stroh. His researches on differential operators 

 bore fruit in various directions, notably in the theory of reciprocants and 

 other differential invariants. In the theory of symmetric functions they 

 are of primary importance, and have illuminated the subject to a surprising 

 degree. 



In his two volumes Combinatory Analysis, Major MacMahon has codified, 

 with additions, his previous work on tactical problems, properly so called. 

 The results are so numerous, and many of them so technical, that it must 

 suffice to give a general outline, and a few particular examples. Vol. I 

 deals with symmetric functions, combinations, and partitions of numbers. 

 The principal method is that of constructing and transforming enumerative 

 (or generating) functions ; and the theorem which the author calls the 

 " Master Theorem " is probably one of the most general results in rational 

 algebra that have ever been proved. Graphical methods, such as those of 

 Durfee, Franklin, etc., are occasionally used ; among particular problems 

 considered are S. Newcomb's problem, and various chess-board arrange- 

 ments. In Vol. II we have a great extension of graphical methods, and 

 a most original discussion of partitions of numbers from various points of 

 view. A sub-section deals with Diophantine inequalities ; here it is possible 

 to explain the nature of the problem in simple terms. Suppose that it 

 is required to enumerate all pairs of positive integers (a, y3) such that 



779a \207i3 ; the answer is given by 



a = a + b + c + d + 2e + 3f+4g + 2ih + 38^ + 207; 



/3 = b + 2c + sd + je + iif + i5g + jgh + 143? + 779; 



where (a, b, . . . j) are arbitrary positive integers (some of which may be 

 zeros). More complicated problems of a similar kind will be found in the 

 book. 



In Chapter III mention is made of two theorems stated, without proof, 

 by Raman u Jan. One of these may be put into the following form. Firstly, 

 let any given integer (n) be expressed in all possible ways as the sum of 

 integers, every pair of which differ by 2 at least ; secondly, let the same 

 integer («) be expressed in all possible ways as the sum of integers (with 

 or without repetitions) selected from 



I. 4. 6. 9, II. 14 . . . 



i.e. all integers of the form 5m ± i. Then the number of representations 

 is the same in each case for all values of n. For instance, if n = 11, the 

 partitions are 



(i) II, 10. 1, 9.2, 8.3, 7.4, 7.3.1, 6.4.1 



(ii) II. 9.12. 6.4.1, 6.1^, 42.13. 4.17, 111 



