7 4 Proceedings of Royal Society of Edinburgh. [sess. 
It is thus seen that the vanishing of |1234] and |1235| entails the 
vanishing of |1245|, |1345|, [2345|. Similarly from the vanishing of 
|1234| and 1 1 236 1 we infer the vanishing of |1246, |1346|, |2346| ; 
and from the vanishing of jl235| and |1236| we infer the vanishing 
of |1256|, |1356|, |2356|. 
In the next place all the three original determinants are con- 
nected with one of the twelve by a linear relation, and from this 
like consequences ensue. The relation is 
|1234| |1567| - |1235| |1467| + |1236||1457| - |1237| |1456| = 0, (B) 
from which by interchange as before we have also 
|2134||2567i - |2135||2467| + |2136||2457| - |2137| |2456| = 0 , (B') 
|3214| |3567| - |3215||3467| + |3216||3457| - |3217| |3456| = 0 . (B") 
It is thus seen that the vanishing of |1234|, |1235|, |1236| entails the 
vanishing of |1456|, |2456|, |3456|, which are the last three deter- 
minants of the twelve. 
The identities (A) and (B) have long been known ; the one is an 
extensional of 
|34||57| - |35| |47| + |37||45| = 0, 
and the other an extensional of 
|234||567| - |235| |467| + |236| |457| - |237||456| = 0. 
These are the first two cases of a general theorem discovered and 
brought into notice by Sylvester, hut included in a wider generalisa- 
tion of earlier date. Had the determinants with which we started 
been of a higher order than the 4th, we might have required to use 
the next case, viz., the extensional of 
|2345]|6789| - |2346||5789| + |2347j |5689| - |2348||5679| + |2349| |5678 | = 0. 
Cayley’s mode of enunciation is : — The 15 { C 6 , 4 ) equations 
a l 
a 2 
CO 
e 
a 5 
a 6 
h 
h 
h 
h 
h 
C 1 
C 2 
C 3 
C 4 
C 5 
*1 
d 2 
d% 
^5 
d§ 
are not independent, hut are reducible to 3 ; and if these he 
( 1 ) = 0 , ( 2 ) = 0 , ( 3 ) = 0 , 
