Theorem relating to Parallel Surfaces. 41 



point of symmetry to the solution since given by Mr. Cayley, I 

 think it worth mentioning, and therefore subjoin it. 



Consider any one of the circular sections of the surface, and 

 suppose the annular surface described which is the envelope of 

 a sphere with constant radius, whose centre moves along the cir- 

 cumference of the section : it is evident that the parallel surface 

 will be the envelope of all the annular surfaces belonging to the 

 circular sections of one series. Let the axis of x be the trace of 

 one of the central circular sections on the plane of the greatest 

 and least axes, that of z the diameter in this plane perpendicular 

 to the axis of x, and that of y the mean axis of the surface. 

 Transfer the origin, for the moment, to the middle point (<zV) of 

 a chord parallel to the axis of x. If 28 be the length of this 

 chord, the equation of a sphere with radius k, the centre of 

 which lies on the circumference of the circular section whose 

 trace is the chord, is 



{x — 8 cos c/>) 2 + (y — 8 sin cf)) 2 -\- z 2 ■= k 2 ; 



and the equation of the envelope of these spheres, (j> being a 

 parameter, or of the annular surface round this circular section, 

 will be 



^ 2 (x 2 + y 2 ) = (x 2 + y 2 + z 2 -k 2 + h 2 ) 2 ; 



or, when we transfer the origin back again to the centre, 



48 2 {{x—x ! ) 2 + y 2 \ = {{x-x , ) 2 + y 2 +{z-z') 2 -k 2 + S 2 } 2 . 



But it may be easily shown that 



s' = Ix'j 8 2 = 7W + nx n , 



I, m, n being absolute constants depending only on the axes of 

 the ellipsoid. Hence the equation of the annular surface, invol- 

 ving a single parameter x 1 , may be written 



4|m+ nx n \ $x 2 -\-y 2 —2xx' + x' 2 l 



== ^ + yZ + z <2-k 2 + m-2(x + lz)x'+{l+n + l 2 )x ,2 \*. 



Since the parameter x 1 enters in the fourth degree, the equation 

 of the envelope or of the parallel surface may be written in the 

 form S 3 = T 2 , according to the well-known type of the discrimi- 

 nant of a quartic function, due to Messrs. Boole and Cayley. 

 The solution of the problem, to find the equation of the surface 

 parallel to an ellipsoid, involves, we repeat, that of the two 

 following problems : — 



1. To find the equation of the surface which is the envelope 

 of planes passing through the points of an ellipsoid, and perpen- 

 dicular at each point to the radius vector drawn from any fixed 

 point whatever. 



2. To find the equation of the surface which is the envelope 



