302 Proceedings of Royal Society of Edinburgh, [april 16, 
The process is seen to be perfectly general, so that, for example, 
we can affirm, without any preparatory ciphering at all, that 
a* a^%h^cHh 
yr y: 
yr y y 
dg d" d* 
= A(r + x, s + y, t + z,u) + A(r + x,s + z,t + y,u) + A{r + y, s + z, t + x, u) 
+ A(r + y, s + X, t + z, u) + A{r + z, s + x, t + y, u) + A{r + z, s + y,t + x, u). 
(5) There is a quite different mode, however, of considering the 
whole matter, — a mode which, resting as it does merely on the 
definitions of a determinant and affialternating function, is simplicity 
itself. Taking the determinant of the example just given, Ave 
reason as follows. The principal term of the determinant is 
a^b^dd^%a^by& , 
or, at full length, 
a^h^dd'"[a^b^c^ + a^edf + + e^a^b^ + d^b^a?} . 
Since, however, the determinant is an alternating function, it cannot 
have the term 
gT^xy-ryy-\-zgu 
without having at the same the 23 other terms of 
I yr+xy+yy+zyu I ^ 
Similarly we conclude that it must have all the terms of 
I I j I a^+^b"+^c^^^d^ \ , &c. And as this accounts for the 
whole 144 terms which on starting Ave knew to exist, the desired 
identity has been established. 
(6) As soon as this new point of vieAv is taken it is seen that the 
symmetric functions need not be confined to one column of the 
given determinant. Let us consider, for example, the complicated 
case of the determinant 
1 be -{-bd^-cd b^c + h^d + c% + . . . b^e^d + b^cd^ + bc^d^ 
1 ed ^rca^- da cH e^a d^c . d^d^a + cHa^ + cd^a^ 
1 da + db->tab d^a + d% + a^d + . . . d^a^h + + da^b^ 
1 ab +ac be a% + a^c + h'^a + . . . a^b'^e + a%d^ + ab'^d 
