474 
Proceedings of Royal Society of Edinburgh, [july 16 , 
\ a A 
. . . . =^{m-n—p + q-r-\-2s-t)^^. 
For alternants of the fifth order the terms involving a are the same 
as those of the fourth order multiplied throughout by a, and the 
additional terms are the simple symmetric functions of c, c?, e of 
the fourth degree, namely, 
%bS 'W^C^ w-cd, bcde. 
For this order, then, must be of the form 
ma^ -{■ na^%b + p)a‘^%b^ + . . . Ata%hcd 
+ u-^fA + u^fdAc + u^dAc^ + ufihhd + ufcde ; 
and, denoting this expression by A, we have 
\ a a? A 
^{m-n- . . . - t 2 il^ + . 
In like manner the perfectly general expression for alternants of 
a higher order maybe found; and so the complete generalization 
referred to at the outset effected, by simply imposing the restriction 
that the coefficient of shall in each case be equal to ±1. 
6. The foregoing investigation obtains the most general expres- 
sions admissible in the elements of the last columns of the several 
orders of alternants. The preliminary proposition laid down in § 1 
is, however, true in the case of alternants of a more general 
character than those considered. For, not only are alternants of 
the form 
1 a . . . cf)(ahc . . .) 
equal to multiplied by a constant, but also alternants of the more 
general form 
1 cf)fa he . . .), (f>fa 6c...), . . . , he .. .) 
in which the functions are (1) all symmetric with respect to 
