473 
1888.] Professor Anglin on Alternants. 
and the sign is positive or negative according as N is even or 
odd. 
5. We can now effect the generalization referred to at the outset 
of the paper. 
Taking the case of alternants of the third order, we have seen, in 
order that 
1 a (f)(a h c) 
==K^\abc), 
it is necessary and sufficient that cfj should be (1) symmetric with 
respect to b and c, and (2) of the second degree. IS'ow <f> being 
integral, and the simplest symmetric functions of c which are of a 
lower degree than the third being b + c,b^ + and be, it follows that 
the only terms admissible in are a?, a{b + c), b‘^ + and be ; that is 
to say, ^ must be of the form 
7110? + na{b + c) + + c^) + qbe . 
Denoting this general expression by A, we have by the law 
established for simple symmetric functions 
\ a A 
1 b B 
1 c C 
= . 
= (in - n -p + q )^ . 
To obtain the corresponding general expression in the case of 
alternants of the fourth order, we have 
1 a c? <ji{a bed) 
= bed), 
where is (1) symmetric with respect to b, c, d ; and (2) of the third 
degree. The simplest symmetric functions of b, c, d which are of a 
lower degree than the fourth being 
%b ; ^b\ %be ; tb% %b\ bed ] 
it follows that ^ must be of the form 
7110? + na?%b -\-]pa%b^ + qa%be + r%b^ + s%b\ + tbed . 
Denoting this general expression by A we have, in accordance 
with the law for simple symmetric functions, 
