686 
Proceedings of the Royal Society of Edinburgh. [Sess. 
£ = a Y x + a 2 y + a 3 z 
r i = b Y x + b 2 y + b 3 z - 
£ = cpr + c 2 y + c 2 z 
may, he says, be written in the form 
ft Vi £ 
( «l « 2 
! \ \ 
l C. Cr. 
a s $x, y,z), 
and consequently the solution of the set in the form 
x,y,z = ( Aj Ci $£,y,0- 
AAA 
^2 ft ft 
AAA 
The latter matrix he calls the inverse of the former, and is naturally led to 
propose that it be denoted by 
( a x a 2 ct 3 )~ 1 
h i \ h 
C \ C 2 C 3 
Next, supposing that along with the original set there exists the set 
x j y » 2 ( a l a 2 a 3 $ X , Y , Z) , 
ft $2 ft 
7l 12 73 
so that by substitution ft y, £ are expressible in terms of X, Y, Z, Cayley is 
led by comparison of the old and the new notations to the conception of 
the product of two matrices, and to the use of 
( «1 
a 2 
a 3 l 
) cq 
a 2 
a 3 ) 
\ 
b 2 
h 
ft 
ft 
ft 
C 1 
C 2 
C 3 
7i 
72 
73 
for 
( K a 2 a A a \ ft 7l) K % ft 72 ) ( a 3 «2 « 3 iS a 3 ft 7s) ) 
ft b 2 frgft ft y^) ft & 2 ^ 3 $ a 2 ft 72) ft ^2 ^ 3 $ a 3 ft 73) 
ft c 2 Cgft ft yd ft c 2 c 3 $a 2 ft y 2 ) ft c 2 c 3 $a 3 ft y 3 ) 
Lastly, he explains his related notations for lineo-linear functions and 
quantics.* These we need only exemplify by saying that 
* Cayley’s first memoir on quantics was presented to the Royal Society of London on 
20th April, and this paper on the notation of matrices is the first of five which appeared 
together in Crelle’s Journal with the date 24th May affixed by the author. 
