140 SCIENCE PROGRESS 



thus if 5 = 6, the probability is 5/36 ; if s = 7, it is 6/36, or 1/6, and so on. 

 Since all problems of annuities, insurance, and the like, ultimately depend 

 upon calculation of this kind, it is clear that some parts of tactic, at any 

 rate, have important practical applications. Chess has provided a number 

 of tactical problems ; the best known are those of the knight's tour and 

 the eight queens. In the former, a knight is placed on any square, and 

 has to move so as to occupy successively each square once and only once ; 

 in the latter, eight queens have to be placed on the board in such a way 

 that no two of them attack each other. 



Since every combinatorial problem is ultimately one of enumeration, 

 the whole theory is, strictly speaking, a branch of arithmetic. Strange as 

 it may seem, some of the most important theorems in tactic have been 

 deduced from algebraical identities. As a first illustration, we may take 

 the binomial expansion {n a positive integer) 



(i + x)** = I + 71X + n2X^ + . , . + nrX^ + . . . + x» 



Since the coefficients are unaltered if written in the reverse order, it is clear 

 that if we square the expression on the right the coefficient of x** will be 



2 



i^ + n^ + n^+ ' • ■ +^l+ . . . + w2 + I = 2m', 



But the square being (i + ^r)^", the coefficient of x*^ must be (2m)„ ; so we 

 have the theorem that 5«^ = (2w)„. For instance, if w = 4, 



(2w)„ = 8i = 70 = i2 + 42 + 62 + 42 + i2 



Very many theorems relating to the binomial coefficients have been dis- 

 covered by algebraic methods more or less similar to the above. 



Euler was one of the first to apply algebraic analysis to tactical problems ; 

 and his work on infinite products acquired new and unexpected importance 

 after the discovery of the theta-f unctions. Both as infinite products and 

 as infinite series, theta-functions have been of great service in yielding 

 theorems about partitions of numbers. By their aid Jacobi was able to 

 prove the famous theorem of Fermat, that every number is expressible as 

 the sum of four or fewer squares ; not only so, but he also enumerated the 

 number of such partitions for a given integer. It is significant that Jacobi 

 thought it worth while to give subsequently a purely arithmetical proof 

 without any use of infinite series or products. 



With the advent of invariant-theory came a new development of tactic. 

 If a^, «!, ^2. etc., are the coefficients of a form, the product «o "^J ^2 • • • 

 is said to be of degree {p + q + r + . . . ) and of weight {q + 2r + . . . ) , 

 the last being obtained by adding the products of each sufifix by the corre- 

 sponding exponent. The problem of enumerating all products of given 

 degree and weight is a fundamental one in the theory. With coefficients 

 Oq, a^, a^, flg, for example, the products of degree 4 and weight 6 are 



«gOg, aoflifla^s. (^o^l> «i^3. «i«2 



Invariants and seminvariants may be defined as homogeneous isobaric 

 functions which satisfy certain linear partial differential equations. 



It is time to end this preamble, because the main object of this article 

 is to give an outline of the work of a single man. Captain MacMahon, as he 

 was in the old days, soon showed an extraordinary power of dealing with 

 problems of what we may call rational symmetrical algebra, and he has 

 devoted many years to the subject, with uninterrupted and brilliant success. 

 An attempt will now be made to classify some of his most conspicuous 

 achievements. 



