64 
ON THE THEORY OF DETERMINANTS. 
[12 
In the first section I have denoted a determinant, by simply writing down in the 
form of a square the different quantities of which it is made up. This is not concise, 
but it is clearer than any abridged notation. The ordinary properties of determinants, 
I have throughout taken for granted; these may easily be learnt by referring to the 
Memoirs of Cauchy and Jacobi, quoted above. It may however be convenient to write 
down the following fundamental property, demonstrated by these authors, and by Binet. 
.. (©), 
a, fi,... 
P , o-, 
= 
p CL + (T fi ... 
, p a! + o- fi'..., ... 
... 
to 
/ / 
P, a , 
p'a + a fi ... 
, poi + p'a ... , 
an equation, particular cases of which are of very frequent occurrence, e.g. in the 
investigations on the forms of numbers in Gauss’ Disquisitiones Arithmetica [1801], in 
Lagrange’s Determination of the Elements of a Comet's Orbit [1780], &c. I have applied 
it in the Cambridge Mathematical Journal [1] to Carnot’s problem, of finding the relation 
between the distances of five points in space, and to another geometrical problem. With 
respect to the notation of the second section, this is so fully explained there, as to 
render it unnecessary to say anything further about it at present. 
§ 1. On the properties of certain determinants, considered as Derivational Functions. 
Consider the function 
U— x(a£ + firj + ...) + (1), 
x ' + fi'v + ...) + 
(n lines, and n terms in each line) ; 
and suppose 
KU = 
a, fi, 
a', ¡3', 
.(2). 
(The single letter k being employed instead of K U, in cases where the quantity KU, 
rather than the functional symbol K, is being considered.) And write 
FU = — 
JU = - 
A.X “I* A'x -f- ..., BiC B X -|- .. 
r£+St;+..., a , fi 
R £+ S77 + ..., a , fi 
R&’ -f- R X -f- . . . , S# -f- S X -f- .. 
a£ + Bì7 + ..., a , fi 
a'£ + b't7+ ..., a! , fi’ 
(3). 
(4). 
The symbols K, F, J possess properties which it is the object of this section to 
investigate.