60 
Proceedings of the Royal Society 
appear to be new, and our methods indicate how to obtain an 
unlimited number of new ones; but all are proved much more 
simply than by the usual processes. 
(1.) We have at once, by quaternions, 
S . (a + P) (J3 + 7 ) (7 -f a) = 2 S . afiy . 
In its algebraic form this is 
X X 1 
y +Vi 
z + h 
= 2 
X 
y z 
X 1 + a? 2 
Vi + 2/2 
Z 1 + * 2 
2/i z i 
+ x 
Vi + y 
z 2 ~h z 
X 2 
2/2 *2 
(2.) Again 
S . Y a/3 Y/3y Y ya = — (S . a/3y) 2 , 
y*i 
“ Z V\ 
ZX Y 
— xz x 
x y\ 
- y x 1 
= 
X 
y 
2 
Vl Z 2 
~ Z $2 
Z 1 X 2 
- X 1 Z 2 
x iy- 2 
- y^2 
x l 
2/1 
V2 Z 
- z 2 y 
z 2 x 
~ ^2 Z 
x 2 y 
- y 2 x 
x 2 
2/2 
*2 
This is the well-known property of the Reciprocal , or determinant 
formed from the minors of the original determinant. 
(3.) Since we have 
V a/3 — * 
y z 
+ / 
2 X 
+ h 
x y 
2/1 
Z L X i 
X x 2/, 
(whose application to (2) is obvious), we may also write 
Y a/3 = 
i k 
x y z 
Vi 
(4.) But, if we put 
x y z 
*1 Vi z i 
x, y 2 z. 
we have 
= ( i 4+>s+ t i> 
