of Edinburgh, Session 1871 - 72 . 
677 
We, in fact, have 
L = £(M + q); 
Also this gives (M + q) 2 - p 2 M = 0, that is 
M 2 - (p 2 - 2q) M + q 2 = 0 , 
which must agree with 
consequently, 
that is, 
M 2 - pM + q = 0; 
P 2 - 2q = p, q 2 = q , 
q = Vq, p = V p + 2fq_ , 
and then, 
L = t (M + q), 
which is the required solution; viz., this signifies 
L = ( a + q b 
P ’ P 
c d + q 
p ’ P 
where p, q have the above-mentioned values—a result which can 
be at once verified. Observe that there are in all 4 solutions, but 
these correspond in pairs of solutions, differing only in their sign ; 
the number of distinct solutions is taken to be = 2. 
Passing now to the case of a matrix of the third order, 
M = ( a, b, c ), 
d, e, f 
9, h , i 
let the expanded value of the determinant 
a - M, 
d , 
9 , 
b, 
e - M, / 
h 
i - M 
be = - (M 3 - p M 2 + - r ); 
and let the required square root be 
L = ( a, b, c ) 
d, e, f 
h > 1 
4 x 
vol. vi r. 
