676 Proceedings of the Royal Society 
Consider the matrix M — 
( a , b ) ; write down the equation, 
\ c, d \ 
a — M, b 
c id — M 
0, 
where the function on the left hand is a determinant, M being 
therein regarded in the first instance as a quantity, viz., this equa¬ 
tion is 
M 2 - (a + d) M + {ad - be) M° = 0; 
and then substituting for M 2 , M, M°, their expressions as matrices, 
this equation is identically true, viz., it stands for the four iden¬ 
tities— 
a? + be - (a + d~) a + ad — be — 0, 
b(a + d) - (a + d) b = 0, 
c(a + d) - (a, + d) c = 0, 
d? + be - (a + d) d + ad - be = 0, 
and the like property holds for a matrix of any order. 
To extract the square root of the matrix M = ( a, b ) ; in 
\ c, d \ 
other words, to find a matrix L = ( a, b ) such that L 2 = M; 
| c, d I 
that is 
( a 2 + be, b(a + d) ) = ( a, b ) , 
c(a + d), d 2 + be | 
'c, d | 
(four equations for the determination of a, b, c, d) 
The solution is as follows : write 
a - M, 5 
c , d — M 
/ 
= M 2 - + q , 
(q is here written for ^M 0 , and so in other cases); and similarly 
= L 2 - pL -f- q, 
then we have 
a - L, b 
c , d - L 
M 2 - pM + q = 0, 
L 2 - pL + q = 0, 
L 2 = M; 
and from these equations we may express L as a linear function of 
M, M°, with coefficients depending on p, q; and also determine 
the unknown quantities p, q in terms of p, q. 
