572 
GEOMETRY. 
[790 
Thirdly, supposing Q to be a fixed point, coordinates (a, b, c) and the distance 
QP, = p, to be constant, say this is = d, then, as before, the values of f, ri, £ are 
x — a, y — b, z — c, and the equation f 2 4- rj 2 + £ 2 = p 2 becomes 
(x — a) 2 + (y — b) 2 + (z — c) 2 = d 2 , 
which is the equation of the sphere, coordinates of the centre = (a, b, c) and radius = d. 
A quadric equation wherein the terms of the second order are x 2 + y 2 4- z 2 , viz. 
an equation 
x 2 4- y 2 4- z 2 4 Ax + By 4- Cz 4- D — 0, 
can always, it is clear, be brought into the foregoing form; and it thus appears that 
this is the equation of a sphere, coordinates of the centre — \A, — ^B, — ^C, and 
squared radius = ^ (A 2 + B 2 + C 2 ) — D. 
Cylinders, Cones, Bided Surfaces. 
33. A singly infinite system of lines, or a system of lines depending upon one 
variable parameter, forms a surface; and the equation of the surface is obtained by 
eliminating the parameter between the two equations of the line. 
If the lines all pass through a given point, then the surface is a cone; and, in 
particular, if the lines are all parallel to a given line, then the surface is a cylinder. 
Beginning with this last case, suppose the lines are parallel to the line x — mz, 
y — nz, the equations of a line of the system are x = mz + a, y = nz + b,—where a, b 
are supposed to be functions of the variable parameter, or, what is the same thing, 
there is between them a relation f(a, b)= 0: we have a = x — mz, b = y — nz, and the 
result of the elimination of the parameter therefore is f(x — mz, y — nz) = 0, which is 
thus the general equation of the cylinder the generating lines whereof are parallel to 
the line x = mz, y = nz. The equation of the section by the plane z = 0 is f(x, y) — 0, 
and conversely if the cylinder be determined by means of its curve of intersection 
with the plane z = 0, then, taking the equation of this curve to be f(x, y) = 0, the 
equation of the cylinder is f(x — mz, y — nz) = 0. Thus, if the curve of intersection 
be the circle {x — a) 2 + {y — ¡3) 2 = y 2 , we have {x — mz — a) 2 + {y — nz — /3) 2 = y 2 as the 
equation of an oblique cylinder on this base, and thus also (x — a) 2 + (y — /3) 2 = y 2 as 
the equation of the right cylinder. 
If the lines all pass through a given point (a, b, c), then the equations of a line 
are x — a = a (z — c), y — b = /3(z — c), where a, (3 are functions of the variable parameter, 
or, what is the same thing, there exists between them an equation /(a, /3)=0; the 
elimination of the parameter gives, therefore, / ( X ^ , -—-]=0; and this equation, or, 
what is the same thing, any homogeneous equation fix — a, y — b, z — c) = 0, or, taking 
f to be a rational and integral function of the order n, say (*) (x — a, y — b, z — c) n = 0, 
is the general equation of the cone having the point (a, b, c) for its vertex. Taking the 
vertex to be at the origin, the equation is (*)(#, y, z) n = 0; and, in particular, 
(■*)(x, y, z) 2 = 0 is the equation of a cone of the second order, or quadricone, having the 
origin for its vertex.