of Edinburgh, Session 1871-72. 
677 
We, in fact, have 
L = J(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 
M 2 - pM + q — 0 ; 
consequently, 
that is, 
p 2 - 2q = p, q2 = q , 
and then, 
q = A /q, p = V p + 2 A? , 
L = J (M + q) , 
which is the required solution ; viz., this signifies 
L = ^ a + q b ), 
P ’ P 
c f i + <4 
p ’ P 
where p, q have the above-mentioned values— a result which can 
be at once verified. Observe that there are in all 1 solutions, but 
these correspond in pairs of solutions, differing only in their sign ; 
the number of distinct solutions is taken to be = 2. 
Passing jiow to the case of a matrix of the third order, 
M = ( a, b, c ) , 
| d , «, / I 
I 9, h i\ 
let the expanded value of the determinant 
a — M, b, c 
d , e - M, / 
9 , h , i - M 
be 
(M 3 
pM. 2 + qK - r ) ; 
and let the required square root be 
L = ( a, b, c ) 
I d, e, f I 
I g. h > i I 
VOL. VII. 
