On the Involution and Evolution of Quaternions. 395 
on quaternions, notably by Hamilton in his Lectures on Qua- 
ternions. 
The latent roots of m are the roots of the quadratic equation 
X 2 -2aX + a 2 + b 2 + c 2 + d 2 = 0. 
The general formula 
m _v^ (m—\ 2 )(m—\ 3 )...(m—\i) 
* m - 2 ^(x 1 -x 2) (x 1 -x 3 )...(x 1 -x i ) ; 
where i is the order of the matrix to, when i=2 and (pm=mk, 
becomes 
II I - 
\ _ Xv i — X? 2 X 3 X q x — X x X q 2 
A x — A-2 X x — A-2 
where X l3 X 2 are the roots of the above equation. If, then, fi 
is the modulus of the quaternion, viz. is \/a 2 + b 2 + 6 l + d~\ and 
/a cos #=a, the latent roots X x , X 2 assume the form 
[x (cos 0±zsin 6). 
When the modulus is zero the two latent roots are equal to 
one another, and to a the scalar of the quaternion; so that in 
this case the ordinary theory of vanishing fractions shows that 
I -/to q — 1\ 
\,q =a« I h ). 
\ a q J 
In the general case there are q 2 roots of the </th order to a 
quaternion. Calling — = a>, and writing m q = Ato + B, 
-iCOS 
H> q V£ 
s (- + 2Jca^ + ism (- + 2kco)- cos(- + 2k' <o\ + zsin(- + WaS 
2i sin 6 
i 
B = fJbq 
_ C os (L 1 ^ + 2F W ) + t sin (?— - 6 + 2&'<A 
- cos(^— - 6> + 2Jfa>) + * sin (^-^ + 2#») 
2i sin <9 
