1898-99.] Dr Thomas Muir on Multiplication of an Alternant. 541 
Now, if we look carefully at the four partial products obtained 
during this process, we see that one alternant occurs in every 
one of them, viz., the alternant | alWcPd^ | , and further, that its 
coefficient in each is simply the coefficient of the corresponding 
partial multiplier ; so that its coefficient in the complete product is 
the aggregate of the coefficients in the expression for 2a 3 Z> in terms 
of 2a, 2afr, 2a&c, . . . viz., 1 - 2 - 1 + 4. 
(5) The general theorem, of which the foregoing is a particular 
case, may be formulated thus : — 
If the alternant [affi’u 2 . . . k n “ 2 l n_1 | be multiplied by any sym- 
metric function of a, h, c, d, . . . of the t th degree , t being not 
greater than n, one term of the product is got from the multiplicand 
by increasing each of its last t indices by 1, and the coefficient of 
this term is the same symmetric function of the roots of the eguation 
x n_ x n-l + x n-2_ . _ _ +(-^ 1 = 0 . 
By way of proof we may reason as follows : — 
The symmetric function may he expressed in terms of 2a, 2a5, 
'Zabc, . , the expression being of the form 
C 1 (2a) a i(2a6)^i(2a5c) 7 i . . . . + C 2 (2a) a 2(2a6)02(2a&c) Y 2 
where cq + 2(d^ + 3y^ + ... = a 2 4- 2/? 2 + 3y 2 + ... = .... = t. 
Now the multiplication of | a% x <? . . . k n ~H n ~ 1 \ by 2a raises the index 
of l by 1, and in the multiplication of this result by 2a one term 
of the product will be got by raising the index of k by 1, and so 
on : consequently, the multiplication by (2a) a i will give rise to one 
term having each of its last cq indices increased by 1. Similarly 
in the multiplication by 2 ab, which then follows, there must arise 
a term got from the multiplicand by increasing the (oq + l) th and 
(cq + 2) th indices from the end by 1 each, and in multiplying this 
result by 2 ab a term must arise which is got from the multiplicand 
by increasing the (cq + 3) th and (cq + 4=) th indices from the end by 1 
each, and so forth through the remaining multiplications, the final 
result necessarily containing a term having the last cq + 2/^ + 3y x + . . 
(that is, t) of its indices increased by 1, and having further Cq for 
its coefficient. For the same reason a like term must occur with 
the coefficient C 2 , and so on : so that the aggregate of its co- 
