604 
NOTES AND REFERENCES. 
but what is done is only to find two or more equations satisfied in virtue of the 
system of the two equalities between the roots. And similarly in the case of a 
system of more than two equalities. See my paper 77, where this notion of the order 
of a system of equations was established. 
152. The next later memoir on the theory of Matrices, so far as I am aware 
is that by Laguerre, “ Sur le Cacul des Systemes Lineaircs,” Jour. Ec. Polyt. t. xxv. 
(1867), pp. 215—264. A “système lineaire ” is what I called a matrix, and the mode 
of treatment is throughout very similar to that of my memoir; in particular we 
have in it my theorem of the equation satisfied by a matrix of any order. The 
memoir contains a theorem relating to the integral functions of two matrices A, B 
of the same order, viz. this is expressible in the form m + pA + qB + rAB. For 
later developments see the papers by Sylvester in the American Mathematical 
Journal. 
158. The notion of the “Absolute” was I believe first introduced in the present 
memoir. In reference to the theory of distance founded upon it and here developed, 
I refer to the papers 
Klein, Ueber die sogenannte Nicht-Euklidische Geometrie, Math. Ann. t. iv. (1871), 
pp. 573—625. 
Cayley, On the Non-Euclidian Geometry, Math. Ann. t. v. (1872), pp. 630—634. 
Klein, Ueber die sogenannte Nicht-Euklidische Geometrie, Math. Ann. t. vi. (1873), 
pp. 112—145. 
In his first paper Klein substitutes, for my cos -1 expression for the distance 
between two points, a logarithmic one; viz. in linear geometry if the two fixed points 
are A, B then the assumed definition for the distance of any two points P, Q is 
dist. (PQ) = c log ; 
this is a great improvement, for we at once see that the fundamental relation, 
dist. (PQ) + dist. (QB) = dist. (PB), is satisfied : in fact we have 
dist. (QB) = c log 
and thence 
AQ.BB 
ABPBQ’ 
dist. (PQ) + dist. (QB) = c log , = dist. PB. 
But in my Sixth Memoir, the question arises, what is meant by “coordinates”: 
if in linear geometry (x, y) are the coordinates of a point P, does this mean that 
x : y is the ratio of the distances in the ordinary sense of the word of the point 
P from two fixed points A, B: and if so, does the notion of distance in the new 
sense ultimately depend on that of distance in the ordinary sense ? And similarly 
in Klein’s definition, do AP, BQ, AQ, BP denote distances in the ordinary sense