78 
Proceedings of Royal Society of Edinburgh. [sess. 
will be true ; that is to say, we shall have a new set of 
C w>g+ i equations, which follows logically from the original set. 
Further, and conversely, if the set (6) hold, we can deduce the 
set (3) provided that the determinant (5) be not zero. The other 
theorem is quite similar, being to the effect that the equations 
(3) may be replaced by the set 
*1 
x 2 
AiAi + • 
• ■ + OqKj 
^1^2 4 b W 1^2 ' 
A. n + 
• f- oqK,, 
+ * 
• * + (OgKj 
A. 2 A 2 + • • • + WgK 2 • 
\A n + 
• • • + COgK.,, 
and that conversely from the set (8) the set (3) is deducible 
provided the determinant 
fh * * 
* w i 
x 2 
H * ' 
* w 2 
A 
Tq ‘ • 
• Wg 
be not zero. 
As the “ derivation of coexistence ” came prominently before us 
in examining Sylvester’s early work, it may be noted here in 
passing that Cayley’s second chapter, extending to about a page, 
consists of the enunciation of a theorem on this subject. 
Cayley (1843). 
[On the theory of determinants. Trans. Cambridge Philosph. 
Soc., viii. pp. 1-16 ; or Collected Math. Papers , i. pp. 
63-79.] 
Up to this point Cayley had dealt with determinants, only, as it 
were, incidentally. Now, however, he devotes a memoir of sixteen 
quarto pages to the study of them. 
The introductory page shows a pretty wide acquaintance with 
previous writings on the subject, the authors mentioned being 
Cramer, Bezout (1764), Laplace, Vandermonde, Lagrange,* Bezout 
* As the memoir of Lagrange which Cayley refers to is not one of those 
brought into notice in the early part of our history, but is one bearing the 
title “ Sur le probleme de la determination des orbites des cometes d’apres trois 
