Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., late sadlerian professor of pure mathematics in the University of Cambridge (Vol. 11)

790] GEOMETRY. 567 
of a point in the plane may be regarded as the coordinates of the point; or, if 
instead of a single point they determine a system of two or more points, then as the 
coordinates of the system of points. But, as noticed under Curve, [785], there are also 
line-coordinates serving to determine the position of a line; the ordinary case is when 
the line is determined by means of the ratios of three quantities f, y, £ (correlative to 
the trilinear coordinates x, y, z). A linear equation ci^ + by + c^ — 0 represents then the 
system of lines such that the coordinates of each of them satisfy this relation, in fact, 
all the lines which pass through a given point; and it is thus regarded as the line- 
equation of this point; and generally a homogeneous equation (*]££, y, £) n = 0 represents 
the curve which is the envelope of all the lines the coordinates of which satisfy this 
equation, and it is thus regarded as the line-equation of this curve. 
II. Solid Analytical Geometry (§§ 26—40). 
26. We are here concerned with points in space,—the position of a point being 
determined by its three coordinates x, y, z. We consider three coordinate planes, at 
right angles to each other, dividing the whole of space into eight portions called 
octants, the coordinates of a point being the perpendicular distances of the point from 
the three planes respectively, each distance being considered as positive or negative 
according as it lies on the one or the other side of the plane. Thus the coordinates 
in the eight octants have respectively the signs 
X, y, z 
+ + + 
+ - + 
+ + 
- - + 
+ + - 
+ 
+ 
Fig. 16. 
z 
The positive parts of the axes are usually drawn as in fig. 16, which represents 
a point P, the coordinates of which have the positive values OM, MN, NP.
	        
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