448 A SUPPLEMENTARY MEMOIR ON THE THEORY OF MATRICES. [357 
we obtain with the s coefficients the equivalent result, 
XW + YZ'-ZY'- WX' = xw' + yz' — zy' — wx. 
We thus see conversely that the Hermitian matrix is in fact the matrix for the auto- 
morphic transformation of the function xw' + yz' — zy' — wx'. 
22. Considering any two or more matrices for the automorphic transformation of 
such a function, the matrix compounded of these is a matrix for the automorphic 
transformation of the function—or, theorem, the matrix compounded of two or more 
Hermitian matrices is itself Hermitian. 
Article No. 23. Theorem on a Form of Matrices. 
23. I take the opportunity of mentioning a theorem relating to the matrices 
which present themselves in the arithmetical theory of the composition of quadratic 
forms. Writing 
a , a , 5 + /3) and .*. = 
7 , -o , b-/3) 
~7 > • , b + /3, -a 
c , ~(b + /3), . , a 
-(b-(3), a , -a, . 
. , y , -c',b'-/3') 
— 7' , . , b'+fi', — a' 
c' , -(b'+(3'), . , a' 
~(b'~/3'), a' , - a , . 
№ = ( • , 
— cl , . ,5-/3, c 
-a , -(b-/3), . , 7 
- (b + /3), - c , - 7 , . 
where D = ac — b 2 , A = ay — /3 2 ; and similarly, 
X'=( . , «' , a' ,b’+!S')aad.-.(X0-= I) .^7( 
— a' , . , b'—/3', c' 
-a' , -(b'-P), . , y' 
-(b'+/3'), -c' , -y , 
where D' = a'c' — b' 2 , A' = a'y' — /S' 2 ; then 
(X'&X') + (D - A) (.D' - A') 
or, what is the same thing, 
(X$X') + (D — A) (D - A') ((X$X'))- 
is to a factor pres equal to the matrix unity; viz. writing 
A = aa + 2bf3 + cy + a' a! + 25'/3 / + c'y', 
the foregoing expression is 
= A( 1 . . . ). 
1 . . 
. 1 . 
. . 1 
The theorem is verified without difficulty by merely forming the expressions of the 
compound matrices (XQX') and (X /-1 ][X -1 ).