84 Proceedings of Royal Society of Edinburgh. [sess. 
[Memoire sur les hyperdeterminants. Crelle’s Journ ., xxx. 
pp. 1-37.] * 
[On linear transformations. Camb. and Dub. Math. Journ., 
i. pp. 104-122; or Collected Math. Papers, i. pp. 95- 
112 .] 
These memoirs, afterwards so famous in the history of what is 
now known as the algebra of quantics, contain exceedingly little 
on determinants. It is important, however, to direct attention to 
them, because the basis of them is a generalisation of determinants. 
Using language which came into vogue two or three years later, 
we may say that just as the idea and notation of determinants 
provided the means of expressing one of the invariants (viz., the 
discriminant) of a function, the idea and notation of hyper- 
determinants were brought forward for the purpose of expressing 
all the invariants.! The generalisation is of great width, hyper- 
determinants including as a very special case the generalisation 
previously made, viz., commutants. 
The first memoir gives incidentally a more general mode of 
using what we may call the notation of multiple determinants than 
that specified in his paper of 1843. The first usage, it will be 
remembered, is exemplified by 
a 1 
a 2 
<*. 3 
a 4 
\ 
h 
h 
which is meant to signify that 
“i 
a 2 
a 1 
a s 
“i 
a i 
II 
5 
CO 
a 2 
a i 
CO 
B 
a t 
W 
b 2 
h 
h 
h 
A 
V 
*2 
h 
h 
A corresponding example of the new usage is 
a l 
a 
« 4 
1 x i 
x 2 
X B 
x i 
h 
b 2 
h 
*4 1 
r 
i Vi 
V2 
Vz 
Vi 1 
* This is stated to be a translation of the preceding paper, with certain 
additions by the author ; and as such it is not reprinted in Collected Math. 
Papers. It also contains the substance of the paper which follows, the latter 
having been delayed in publication, 
f And indeed the covariants also. 
