1905-6.] Dr Muir on the Theory of Alternants. 
375 
or T say, the sign of summation being meant to indicate that of 
the two series of elements the one is to remain unaltered and the 
other is to be permitted in every possible way. The development 
of this function according to descending powers of t , t 1 , t 2 , . . . , t n 
leads to those simplest types of integral symmetric functions of 
a , a x , a 2 , . . . , a n which originate by permutation from a single 
product of integral powers of the said variables. The determina- 
tion of such functions is thus reduced to the problem of trans- 
forming T so as to have no longer occurring therein the single 
elements a , a x , a 2 , . . . , a n , but instead those combinatory 
sums of them which are the coefficients of the powers of 2 in the 
development of (z - a)(z - a-^)(z - a 2 ) . . . (z- a n ) or f(z) say. 
Without further preparatory statement the announcement is 
made that the solution is readily reached when the relation of T 
to the determinants 
y ±_L • — •••• 
t (X Ctj 
y + J__ . 1 
1 
t n - a, 
or A , 
1 
(*n-0 2 
or 
is known, namely, the relation 
D = T-A. 
D, 
In proof of this relation it is pointed out that 
{M ■ fih) • f(h) • • • /( < »)} 2 - D 
being an integral alternating function both with respect to the 
elements t , t Y , t 2 , . . . , t n and with respect to the elements 
a , a x , a 2 , . . . , a n is exactly divisible by the two difference- 
products 
5 t^ 1 t 2 , . . . , t n ) , II(a , a ] , a 2 , . . . , a , 
and that although we cannot with equal promptness tell the 
remaining factor, we are able to determine it from knowing a 
sufficient number of its special values, namely, those values got 
by putting each t equal to one of the a’s. Since the number 
of ways in which the n + 1 a’s can be taken when repetitions 
are allowed is (n + l) n+1 , this gives us (w+l ) n+1 values, of 
which, however, only two are different, namely, the value 
(- l) in{n+1) • f\a) • /'(af - f'(a 2 ) . . . /'( a n ) obtained in the n !. 
cases where all the a’s used are different, and the value 0 obtained 
