124 Proceedings of the Royal Society 
Now, using capitals for minors we have, by the first property, 
X 2 + X y 2 + y z 2 + z 
2 
X Y Z 
X + Y + Y 1 Z + Z l 
= 
X 2 y, z 2 
Xj + X 2 y 2 + y 2 z +z 2 
x, Y, Z, 
- 2 
X 
Y 
Z 
- 2 
X 
y 
z 
= 
X, 
Y. 
z, 
= 
X x 
2/1 
x 2 
Y a 
Z 2 
X '2 
y -2 
*2 
Q. E. D. 
This, of course, may be generalised like the first formula ; and 
the result is 
1 , n 
A l,l + A l,2 + 
A„ „ -4“ A, „ -4“ 
+ A. . , &c., &c. 
* 1, n — 2 J 5 
-O-i) 
A i, 
3 + 
+ 
1 , n — 
1 > 
&c., 
K 
4 + 
+ 
A l,l 
? 
&C., 
K 
1 + 
+ 
a, 
1 , n — 
i 5 
&C., 
a , 
a , 
n 
- 1 
1 
1 1 
1 , n 
• • 
a 
1 
, n 
n , n 
The sixth formula is got by a reduplication of the proof of the 
fifth , whether the method used be Quaternion or Determinant. 
As to the reciprocal, the following seems quite elementary : — 
Let 
y* 1 
- *y 1 
ZX x 
- *iy* 
zx. 
yz 
- w 
xz x 
x y 1 
- y x 1 
X '2 Z 1 
x iy -2 
- 
X Z. 2 
x « y 
- yp 
Multiply the first row by x 2 , the second by x, the third by x x ; add 
all the rows for a new first row ; divide the second row by and 
the third by x r 
x y z 
X x Vi \ 
X< 2 V'2 Z 2 
X 
X, 
0 0 
Y Z 
Yi Z A 
= V 
