924 Proceedings of Royal Society of Edinburgh. [sess. 
determinants belonging to a given system of minors, I call 
the Homaloidal law, because it is a corollary to a proposition 
which represents analytically the indefinite extension of a 
property, common to lines and surfaces, to all loci (whether 
in ordinary or transcendental space) of the first order, all of 
which loci may, by an abstraction derived from the idea of 
levelness common to straight lines and planes, be called 
Homaloids.” 
A further advance is made just before the close of his paper. 
Leaving the square array and taking m lines and n columns, he 
says 
“ This will not represent a determinant, but is, as it were, a 
Matrix out of which we may form various systems of 
determinants by fixing upon a number p and selecting at 
will p lines and p columns, the squares corresponding to 
which may be termed determinants of the p ih order.” 
Here there is to be noticed the first use of the word matrix in 
connection with determinants, as well as the change back to 
Cauchy’s mode of particularising the minors. The corresponding 
more general ‘law’ is said to be — The number of uncoevanescent 
determinants constituting a system of the p th order derived from a 
given matrix , n terms broad and m terms deep , may equal but can 
never exceed the number 
(n-p + 1 )(m-p + 1) . 
Ho proof of this is given. 
Spottiswoode, W. (1851). 
[Elementary theorems relating to determinants, viii + 63 pp. 
London.] 
This is noteworthy as being the first separately-published 
elementary work on the subject, the author explaining that he had 
been led to write it because determinants had come to be in 
frequent use, and there was no accessible text-book to which 
students could be referred. It consists of a preface and ten short 
chapters or sections, the mode of partitioning and arranging the 
matter being such as has often subsequently been followed. 
