1898-99.] Dr Thomas Muir on Multiplication of an Alternant. 539 
The Multiplication of an Alternant by a Symmetric 
Function of the Variables. By Thomas Muir, LL.D. 
(Read February 6, 1899.) 
(1) As is well known, the simplest form of alternant is 
| a°5 W .... | 
and the problem of multiplying it by any symmetric function of 
a, b, c, d, . . . has been in a manner fully solved. 
(2) When the symmetric function is linear in each of the 
variables — that is to say, when it takes any of the forms 2a, 2a&, 
labc, . . . . — the result is an alternant got from the multipli- 
cand by increasing the last index, the last two indices, the last 
three indices, .... respectively by 1. Thus, writing for short- 
ness’ sake five variables only, we have 
| a°& 1 c 2 ^ 3 e 4 | . 2a = | a°5 1 c 2 ^ 3 e 5 1 , 
| a°& 1 c 2 6Z 3 e 4 | . 2 ab = | aWc^e 5 1 , 
| a 0 /fic W | . 2 abc = j aWcW j , 
| a°& 1 c 2 <i 3 e 4 j . 2 abed = | a°5 2 c 3 <i 4 e 5 1 , 
| aWc W | . 2 abode = | aWsW* | . 
This was first established in 1825 by Schweins in his Theorie der 
Differenzen und Differentiate , p. 378 ; but it is also barely possible 
that it was known to Prony in 1795 (see Journ. de VEc. Polyt ., 
i. pp. 264, 265), and Cauchy in 1812 (see Journ. de VEc. Polyt., 
x. pp. 49, 50). 
(3) When the symmetric function is non-linear, the result takes 
the form not of one alternant, but of an aggregate of alternants. 
These cannot be so readily specified, but the mode of obtaining 
them can be made clear without any difficulty. Let us take the 
case of the function 2a 3 5, the multiplicand being | aPDfid? 1. Since 
the term a z b = a 3 ^ 1 ^ 0 we may specify it by the four indices alone, 
viz., 3 10 0, and in this way shall have 
