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LVI. On the Involution and Evolution of Quaternions. 
By J. J. Sylvester, F.R.S* 
THE subject-matter of quaternions is really nothing more 
nor less than that of substitutions of the second order, 
such as occur in the familiar theory of quadratic forms. A 
linear substitution of the second order is in essence identical 
with a square matrix of the second order, the law of multipli- 
cation between one such matrix and another being understood 
to be the same as that of the composition of one substitution 
with another, and therefore depending on the order of the 
factors ; but as regards the multiplication of three or more 
matrices, subject to the same associative law as in ordinary 
algebraical multiplication. 
Every matrix of the second order may be regarded as repre- 
senting a quaternion, and vice versa : in fact if, using i to 
denote v— 1, we write a matrix m of the second order under 
the form 
a + hi, c + di, 
— c + di, a — hi, 
we have by definition, 
i m = aa. + b/3 + cy + d8, 
1 x i 
where 
1 

= 
i 

a = 

1' 
- 
-i' 
Now 
« = 
■ a. 
/3 2 
= 7 2 
= o 2 
= - 
/3ry = — 7/3 = a, 78=— §7=/3, 8(3= — (38 = <y; 
so that we may for a, {3, y, 8, substitute 1, h, k, I, four sym- 
bols subject to the same laws of self-operation and mutual 
interaction as unity and the three Hamiltonian symbols. Now 
I have given the universal formula for expressing any given 
function of a matrix of any order as a rational function of 
that matrix and its latent roots; and consequently the ^th 
power or root of any quadratic matrix, and therefore of any 
quaternion, is known. As far as I am informed, only the 
square root of a quaternion has been given in the text-books 
* Communicated by the Author. 
