438 
[357 
357. 
A SUPPLEMENTARY MEMOIR ON THE THEORY OF MATRICES. 
[From the Philosophical Transactions of the Royal Society of London, vol. clvi. (for 
the year 1866), pp. 25—35. Received October 24,—Read December 7, 1865.] 
M. Hermite, in a paper “ Sur la theorie de la transformation des fonctions 
Abeliennes,” Gomptes Rendus, t. XL. (1855), pp. 249, &c., establishes incidentally the 
properties of the matrix for the automorphic linear transformation of the bipartite 
quadric function xw' + yz' — zy — wx, or transformation of this function into one of 
the like form, X W' + YZ' — ZY' — WX'. These properties are (as will be shown) 
deducible from a general formula in my “Memoir on the Automorphic Linear Trans 
formation of a Bipartite Quadric Function,” Phil. Trans, vol. cxlviii. (1858), 
pp. 39—46, [153]; but the particular case in question is an extremely interesting one, 
the theory whereof is worthy of an independent investigation. For convenience the 
number of variables is taken to be four; but it will be at once seen that as well 
the demonstrations as the results are in fact applicable to any even number whatever 
of variables. 
Article Nos. 1 and 2. Notation and Remarks. 
1. I use throughout the notation and formulae contained in my “ Memoir on the 
Theory of Matrices,” Phil. Trans, vol. cxlviii. (1858), pp. 17—37, [152], and in the 
above-mentioned memoir on the Automorphic Transformation. With respect to the com 
position of matrices, the rule of composition is as follows, viz., any line of the compound 
matrix is obtained by combining the corresponding line of the first or further com 
ponent matrix with the several columns of the second or nearer component matrix ; it 
is very convenient to indicate this by the algorithm, 
(a, a', a"), (/3, /3', /3"), (?', y', y") 
a , h , 
{ « , $ , 
7 
) = (a , 
b , c) 
» 
33 
a', h', 
d 
/3', 
7 
(a', 
b', d) 
„ 
33 
a" V, 
d' 
«" /3", 
y" 
(a", 
b", c") 
??