44 The Rev. W. Roberts on some Applications of a 



the given surface. It is evident that the direct application of 

 our theorem in order to obtain from the equation of the parallel 

 to a given surface that of the first negative derived from its in- 

 verse, would introduce infinite terms ; it will be necessary there- 

 fore to take the expression x 2 + y 2 + z 2 — k 2 out of all the deno- 

 minators before applying our theorem for this purpose. 



To verify our last result in conjunction with the first theorem, 

 let us seek to deduce the first negative of a sphere from the equa- 

 tion of a plane parallel to a given one. The parallel to the plane 

 z = a is z = <*+■%. Hence the parallel to the sphere which is the 

 inverse of z = u has for equation 



and to find the first negative from this sphere, we have only to put 

 \x, \y, \z for x, y, z, and k + ± Vx 2 + y 2 + z 2 for k. This gives 



the equation 



(m 2 + cik) V ' x 2 + y 2 -j-z 2 = m 2 z + uk 2 — 2m 2 k, 



which represents a surface of revolution of the second order, the 

 origin being one of the foci, as it ought to do. 



We are now in a position to solve the problem of finding the 

 equation of the parallel to the surface of the fourth order which 

 is derived from an ellipsoid by the method of reciprocal radii 

 vectores drawn from any point whatever. It will only be neces- 

 sary to place the origin at any point we choose, and to make in 

 the equation of the parallel to the ellipsoid, referred to this origin, 

 the substitutions indicated above. We are also led to the solu- 

 tion of the two following problems : — 



1. To find the equation of the surface parallel to the first po- 

 sitive derived (in reference to any origin) of an ellipsoid. 



2. Being given an ellipsoid, and two fixed points situated any- 

 where : let P be the foot of a perpendicular dropped from either 

 of the fixed points on any tangent plane to the ellipsoid; it is 

 required to determine the surface which is the envelope of the 

 plane passing through P, and perpendicular to the line joining 

 P with the other of the fixed points. 



For the inverse of the locus of P is an ellipsoid; hence the 

 equation of the parallel to this locus can be found, and by our 

 first theorem its first negative in reference to any origin may be 

 deduced from this equation. 



All the foregoing results apply, mutatis mutandis, to the case 

 of curves. As an application of them, I subjoin the following 

 transformation of the equation of the curve parallel to the ellipse, 

 into that of the parallel to the first positive derived from the 

 ellipse, referred to the centre, and whose equation is 



a 2 x 2 + b 2 y 2 ={x 2 + y 2 ) 2 . 



