566 
GEOMETRY. 
[790 
Polar Coordinates. 
23. The position of a point may be determined by means of its distance from a 
fixed point and the inclination of this distance to a fixed line through the fixed 
point. Say we have r the distance from the origin, and 6 the inclination of r to the 
axis of x; r and 6 are then the polar coordinates of the point, r the radius vector, 
and 6 the inclination. These are immediately connected with the Cartesian coordinates 
x, y by the formulae x = r cos 0, y — r sin 6; and the transition from either set of 
coordinates to the other can thus be made without difficulty. But the use of polar 
coordinates is very convenient, as well in reference to certain classes of questions 
relating to curves of any kind—for instance, in the dynamics of central forces—as in 
relation to curves having in regard to the origin the symmetry of the regular polygon 
(curves such as that represented by the equation r = cos m6), and also in regard to 
the class of curves called spirals, where the radius vector r is given as an algebraical 
or exponential function of the inclination 6. 
Trilinear Coordinates. 
24. Consider a fixed triangle ABC, and (regarding the sides as indefinite lines) 
suppose for a moment that p, q, r denote the distances of a point P from the sides 
BC, CA, AB respectively,—these distances being measured either perpendicularly to the 
several sides, or each of them in a given direction. To fix the ideas each distance 
may be considered as positive for a point inside the triangle, and the sign is thus 
fixed for any point whatever. There is then an identical relation between p, q, r: if 
a, b, c are the lengths of the sides, and the distances are measured perpendicularly 
thereto, the relation is ap + bq + cr = twice the area of triangle. But taking x, y, z 
proportional to p, q, r, or if we please proportional to given multiples of p, q, r, then 
only the ratios of x, y, z are determined; their absolute values remain arbitrary. But 
the ratios of p, q, r, and consequently also the ratios of x, y, z determine, and that, 
uniquely, the point; and it being understood that only the ratios are attended to, we 
say that (x, y, z) are the coordinates of the point. The equation of a line has thus 
the form ax + by + cz = 0, and generally that of a curve of the wth order is a homo 
geneous equation of this order between the coordinates, (*$&•, y, z) n = 0. The advantage 
over Cartesian coordinates is in the greater symmetry of the analytical forms, and in 
the more convenient treatment of the line infinity and of points at infinity. The method 
includes that of Cartesian coordinates, the homogeneous equation in x, y, z is, in fact, an 
equation in -, - , which two quantities may be regarded as denoting Cartesian coordi- 
z z 
nates; or, what is the same thing, we may in the equation write z=l. It may be 
added that, if the trilinear coordinates (x, y, z) are regarded as the Cartesian coordi 
nates of a point of space, then the equation is that of a cone having the origin for 
its vertex ; and conversely that such equation of a cone may be regarded as the equation 
in trilinear coordinates of a plane curve. 
General Point-Coordinates.—Line-Coordinates. 
25. All the coordinates considered thus far are point-coordinates. More generally, 
any two quantities (or the ratios of three quantities) serving to determine the position