216 Proceedings of the Eoyal Society of Edinburgh. [Sess. 
and the second, which is the complementary of the first, is 
A"' | ^ 3 ^ 4 ^] ^4^] ^2 1 — 
Ai ^3^4 
a 1 a 2 A 4 
a x a 2 K 3 
B 2 ^3&4 
B A & 4 
^A B 4 
& A B 3 
C 2 c 3 c 4 
Ci c 3 c 4 
C l C 2^4 
C]C 2 C 3 
D 
D 
d Y d 2 D 4 
d]d<^D 3 
A further result, however, is clearly necessary in order to establish 
identity analogous to (V.). * 
* Prof. Nanson in the Educ. Times for 1902, pp. 515-516, gives 
| A t A 2 B 2 B 3 C 3 C 1 = - a 2 . | a x a 2 b 2 b 3 | , 
where A is | a 1 b 2 c 3 | ; and it is not difficult to show also that 
| A x 2 B 2 2 C 3 2 | — | A x A 2 B 2 B 3 | — A 2 1 1 a T 2 b 2 c 3 2 j — | a 1 a 2 b 2 b 3 c 3 c± j |. 
(X.) 
an 
{Issued separately April 8, 1908.) 
