MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
603 
Chapter V. —General Equation of the Second Degree in Power-Coordinates. 
Nature of the Surface. —§§ 241, 242. 
241. The most general equation of the second degree in power-coordinates may he 
written 
(f)(x, y, z, w, v)==a lt 1 ad+ a 2t 3 y *+.* 3t3 z 2 + a 4 4 iv 2 -j-a 5}5 v* 
+ 2a h2 xy-j- 2 a 1)3 xz-{-2a. 1A xiv-\-2a lt5 xv 
+ 2a 2,3 V Z + 2a 2,4 V W + 2a 2,5 V V 
-\-2o. 3i4; zw+2u 3 ' 5 zv+2oi 4:i5 vu; =0 ; 
and since the coordinates of any point must satisfy the equation of the absolute, 
which is also of the second degree, we see that this equation contains only 14 
constants. 
Now (X, Y, Z) denoting Cartesian coordinates of a point, we can express the power- 
coordinates ( xyzwv) of the point as linear functions of X 3 -f- Y~+Z' 3 , X, Y, Z, 1 ; and 
if we substitute for [xyzwv), the equation becomes of the form 
(X s +Y 2 +Z 2 ) s +Ui(X s +Y s +Z?)+U 8 =Q, 
where Uj is of the first degree, U 3 of the second degree in (X, Y, Z); this equation 
has 14 constants, and represents a surface having the circle at infinity as a nodal 
curve, and is usually called a cyelide. It thus appears that <£= 0 , is a form to which 
the equation of every cyclide can be reduced. 
242. It is also evident that, since the equation of every plane is satisfied by the 
coordinates of the plane at infinity, the surface (f)= 0 will represent a cubic surface 
and the plane at infinity, if the equation </>=0 be satisfied by the coordinates of the 
plane at infinity. Such surfaces have been called cubic cyclides ; they intersect the 
plane at infinity in the imaginary circle, and also a straight line. 
Equation of Tangent at any Point. —§§ 243-247. 
243. If 17 £ w or) be the coordinates 
point (x'y'z'w'v), we must have 
of any sphere touching the cyclide <f> a, t ^ ie 
4 h 2 
