162 
Proceedings of Boyal Society of Edinburgh. [sess. 
which involve complete generality since i,j , Jc are undefined, we 
have for the cubic (1) in g 
A-pg 
c 
b 
c 
B-qg 
a 
b 
= 0 . 
a 
C-rg 
The transformation of (1) given above is equivalent to dividing 
the successive rows, and also the columns, of this determinant by 
Jp, Jq, Jr respectively. It thus becomes 
A !p - g cjjpq 
c/Jpq B/q-g 
b/Jpr a/Jqr 
b/Jpr 
a/Jqr 
G/r-g 
from the form of which the reality of the roots is obvious. 
A somewhat similar process shows that the roots of 
A- x b 
c 
= 0 
d B — x f 
g h I- x 
are always all real, provided the single condition, 
cdh — bfg , 
be satisfied. 
It is easy to see that this statement may be put in the form : — - 
The roots of M 3 = 0 are real, provided a rectangular system can be 
found such that 
The quaternion form, of which this is an exceedingly particular 
case, expresses simply that the roots of the cubic in cj> are all real, 
if a self-conjugate function w can be found, such that o o<f> is self- 
conjugate. This is merely another way of stating the chief result 
of § (1) above. But it may be interesting to illustrate it from this 
point of view. We may write, in consequence of what has just 
been said, 
S.P 1 P 2 P 3 $P=9l Y P2PzSplP + 92 V P3PlSP2P + 93 V PlP-2®P3P > 
(00- =P 1 Pi&p 1 (T + P 2 pSp-2 (T +l ? 3P3 S P3 (r • 
and 
