The Line, Plane, and Sphere. 
32. The foregoing formulas give at once the equations of these loci. 
For first, taking Q to be a fixed point, coordinates (a, h, c) and the cosine- 
inclinations (a, /3, y) to be constant, then P will be a point in the line through Q 
in the direction thus determined; or, taking (x, y, z) for its coordinates, these will be 
the current coordinates of a point in the line. The values of £, y, £ then are x — a, 
y — b, z — c, and we thus have 
which (omitting the last equation, = p) are the equations of the line through the 
point (a, b, c), the cosine-inclinations to the axes being a, /3, y, and these quantities 
being connected by the relation a? + /3 2 4- y 2 = 1. This equation may be omitted, and 
then a, /3, y, instead of being equal, will only be proportional to the cosine-inclinations. 
Using the last equation, and writing 
x, y, z= a+ cep, b + /3p, c + yp, 
these are expressions for the current coordinates in terms of a parameter p, which is in 
fact the distance from the fixed point (a, b, c). 
It is easy to see that, if the coordinates (x, y, z) are connected by any two 
linear equations, these equations can always be brought into the foregoing form, and 
hence that the two linear equations represent a line. 
Secondly, taking for greater simplicity the point Q to be coincident with the origin, 
and a, ¡3', y, p to be constant, then p is the perpendicular distance of a plane from 
the origin, and a, /3', y are the cosine-inclinations of this distance to the axes 
(a' 2 + (3' 2 + y' 2 = 1). Now P is any point in this plane; taking its coordinates to be 
(x, y, z), then (£, y, £) are = (x, y, z), and the foregoing equation p = a'f -f ¡3'y + y'£ 
becomes 
u'x + /3'y + y z =p, 
which is the equation of the plane in question. 
If, more generally, Q is not coincident with the origin, then, taking its coordinates 
to be (a, b, c), and writing p x instead of p, the equation is 
a' (#-«) + ^ (y~b) + y (z - c) =p x ; 
and we thence have p± =p - {aa! + bj3' + cy), which is an expression for the perpendicular 
distance of the point (a, b, c) from the plane in question. 
It is obvious that any linear equation Ax + By + Cz + D = 0 between the coordinates 
can always be brought into the foregoing form, and hence that such equation 
represents a plane. 
72—2