230 Proceedings of Boyal Society of Edinburgh. [sess. 
obtained from the original three by multiplying by y, 2 , x respec- 
tively, and then by z, x, y respectively. The seventh is got by 
writing the original equations in the form 
(Ax 4- cy)x — - (bx + ay)z 
(My + lz)y = - (ny + mz)x 
(Hz + qx)z = - (pz + rx)y 
and multiplying, the immediate result clearly being 
(A# + cy)(My + lz)(fiz + qx) + (bx + ay)(ny + mz)(pz + rx) = 0 , 
whence we have 
(AMg + bnr)x 2 y + (MRc + anp)y 2 z + . 
The new form is thus seen to be 
A a 
M ml. 
= 0 . 
c b 
n 
r 
R . q 
b a A 
P 
c 
n . . . m M l 
. p . R . r q 
AMg + bnr MRc + anp AR£ + bmp R le + amp A ql + mrb Alcq + anr cr + o-' 
where, for shortness’ sake, a- + a-' is written for AMR + amr -f bnp 
+ clq. 
(15) To show that this is symmetrical with respect to the inter- 
change of § 8, we have only got to interchange three of the columns 
with other three, viz., 1, 2, 3 with 4, 5, 6 respectively, and then 
three of the rows with other three, viz., 1, 2, 3 with 4, 5, 6 respec- 
tively, when it is found that the interchange of § 8 has been 
effected. 
The symmetry with respect to the cyclical substitutions is 
equally easily made manifest. Taking, for example, the sub- 
stitution 
/A MR am rbnpclq 
U 
\M RA mranpblqc 
we find that the effect of it upon the determinant is exactly the 
same as the effect of passing the 1st column over into the 3rdplace^ 
