404 Proceedings of Royal Society of Edinburgh. [sess. 
a r = a rJrn = a r _ n 
a n — a 0 — a_ n 
b r b r _^ n b r _ n 
K=h = b_„ 
c r = c r+n = c r _ n 
C n = C 0 ~ C -n 
lr I'r+n I'r-n 
£ 
• 
II 
• 
. II 
r being any number whatever. Then 
£_iPD(0 abc . . . 1) = i{B x (dbc . . . l)f¥V(0abc . . . 1)) 
£-2PD(0 abc . . . I) = {(S 2 (abc . . . Z).£PD(0a6c . . . 1)) 
£_ r PD(0 abc . . . 1) = £(S r (abc . . . l).£PD(0abc . . . Z)).” 
The limitation made upon the arguments of the base would seem 
to imply that the theorem only concerned determinants of a very 
special kind. Such, however, is not the case. A special example 
in more modern notation will bring out its true character. Let the 
determinant chosen be 
| af 2 cfl^ I » 
and let the symmetric function be 
ab + ac + ad + bc + bd + cd . 
Multiplying the two together “ zeta-ically, ” that is to say, in 
accordance with the law 
a r x a s = a r ^. s , 
we find that 120 of the total 144 terms of the product cancel each 
other, and that the remaining 24 terms constitute the determinant 
| af) 2 c^d b | , 
the identity thus reached being 
£(l ®1^2^3^4 I * = I ^1^2^4^5 I * 
4 
Now Sylvester’s £PD notation being unequal to the representation 
of the determinant | a l b 2 c 4 d b | in which the index-numbers do not 
proceed by the common difference 1, he would seem to have been 
compelled to give a periodic character to the arguments of the 
