468 
Proceedings of Boyal Soeiety of Edinburgh, [july 16, 
9. Alternants which are constant Multiples of the 
Difference-Product of the Variables. By Professor 
Anglin, M.A., LL.D., &c. 
1. In a recent paper on “ Certain Theorems mainly connected 
with Alternants (II.),”* the possible generalization of the series of 
theorems (1.)^, (II.)d (Ill-)n referred to in § 6. The object 
of the present communication is to endeavour to effect this generali- 
zation, considering for the present Integral functions only. 
First in order we notice the following general proposition : — In 
order that the alternant 
\ a .. . cj>(a h c , . . 1) 
which involves n letters, may be equal to i^idbc . ... T) multi- 
plied by a constant, it is necessary and sufficient that be (1) 
symmetric with respect to ... I, and (2) of the {n - \)th 
degree. For, the alternant vanishes if any two of the variables are 
equal, and is therefore divisible by K^\abe . . . l)\ also the degree 
of the alternant, namely \n{n - 1), is the same as that of (which 
is equal to the number of combinations of n things taken two at a 
time), so that the co-factor is a constant. 
2. We now propose to investigate the Law by which the value 
of the numerical coefficient of ^[abe . . . ) is obtained in the case 
of the simplest symmetric functions of &, c, c?, . . . ; and thence for 
any symmetric function whatever. 
For convenience we may denote y>(abc . . . ) by A, and the 
numerical coefficient of I"- hj K; and as it will be obvious from the 
mode of investigation that the appearance in A of a factor of the 
form as a^cj){bcd . . .), does not affect the value of K corre- 
sponding to (fi(bGd . . . ), it will only be necessary to consider 
functions of the latter form. 
[For greater clearness, we may illustrate this statement by an 
easy example — the value of K is the same for the functions 
%b^c and a^%b‘^c. For, in the former case. 
Proc. Eoy. Soc. Edin.^ vol. xv. p. 381. 
