766 Proceedings of Royal Society of Edinburgh. [sess. 
The identity is readily recognisable as Bezout’s (1779). The mode 
of arriving at it, however, is fresh, and worthy of every attention. 
The determinant of the sixth order on the left is shown to he equal 
to zero ; and it is implied that the identity is got by transforming 
the said vanishing determinant into an aggregate of products of 
pairs of determinants by means of Laplace’s expansion-theorem. 
The method is far-reaching in its application, and manifestly Cayley 
could have used it to produce a host of identities of similar 
kind. 
The equatement of the two determinants of the sixth order 
deserves also to be noted, and may be taken as evidence that Cayley 
was familiar with the theorem that a determinant is not altered if 
each element of one row be diminished by the corresponding 
element of another row. No such theorem had been formulated or 
used before his time. (lix.) 
Lastly, it may be pointed out that we have here the first 
instance of a practice which afterwards became very general, 
viz., putting a dot instead of a zero element when writing a deter- 
minant. 
The other lemma and the main body of the paper are geometrical ; 
but as an important determinant identity is implicitly established 
in the course of the investigation, and as it is of the greatest histori- 
cal importance to make evident the wonderful command which 
Cayley with his new notation had suddenly obtained over deter- 
minants, we shall give the full text of these portions also, at least 
up to a certain point. 
“ Lemma 1. Let U = kx + ¥>y + Cz = 0 be the equation of a 
plane passing through a given point taken for the origin, and 
consider the planes 
U^O, U 2 = 0, U 3 = 0, IX 4 = 0, U 5 = 0, U 6 = 0; 
the condition which expresses that the intersections of the 
planes (1) and (2), (3) and (4), (5) and (6), lie in the same 
plane, may be written down under the form * 
* The commas which Cayley prints after the elements in a determinant we 
omit here and henceforth. 
