ME. A. CAYLEY’S SIXTH MEMOIE UPON QUANTICS. 
69 
and the equation is 
00 , 
01 , 
02 
10 , 
11 , 
12 
20 , 
21 , 
22 
an equation which may also be written in the form 
cos' 
01 
\^00 Vll 
cos- 
12 
\/ll -^22 
COS' 
02 
VOO \/22 
as it is easy to verify by reducing this equation to an algebraical form. The various 
formulae have been given in relation to the establishment of the notion of distance in 
the geometry of one dimension, but it will be convenient to defer the consideration of 
this theory so as to discuss it in connexion with geometry of two dimensions, 
On Geometry of Two Dimensions, Nos. 169 to 208. 
169. In geometry of two dimensions we have the plane as a space or locus in quo, 
which is considered under two distinct aspects, viz. as made up of points, and as made up 
of lines. The several points of the plane are determined by means of the point-coordinates 
{x,y, z), viz. attributing to these any specific values, or writing x, y,z=a, h, c, we have a 
particular point of the plane ; and in like manner the several lines of the plane are 
determined by the line-coordinates (|, t], t), viz. attributmg to these any specific values, 
or writing yi,t=a, (3, y, we have a particular line of the plane. And we may say that 
the plane is the locus in quo of the point-coordinates {x, y, z), and of the line-coordinates 
(I, ri, 1). It is not necessary to consider separately the analytical theories of point-coor- 
dinates and of line-coordinates ; for the theory of the former in relation to points and 
lines respectively is identical with the theory of the latter in relation to lines and points 
respectively; but it is necessary to show how either system of coordinates, say the system 
of point-coordinates, is applicable to both points and lines, or in fact all loci whatever, 
and to explain the mutual relation of the two systems of coordinates, 
170. Considering then point-coordinates, the equations 
X, y,z^a, h, c, 
determine, as already mentioned, a point. 
A linear equation, 
2)'=o, 
determines a line, viz. the line which is the locus of all the points, the coordinates of 
which satisfy this equation. And in like manner an equation 
{*\x, y, zy=^ 
determines a curve of the wth order, viz. the curve which is the locus of all the points, 
the coordinates of which satisfy this equation. In particular, an equation of the second 
degree, 
{*Jx,y,zf=i), 
determines a conic. 
MDCCCLIX. L 
