T 



Some Applications of a Theorem relating to Parallel Surfaces. 39 



Expressing A, B, C, and D in terms of a, b, c, and d, I find 

 A=-a 4 d 3 + 9a 3 bcd 2 -4 : a 3 c 3 d-6a 2 b 3 d 2 -lDa 2 b 2 c 2 d+l2a 2 bc i 



+ 2±ab 4 cd- 1 TaPc 3 + 6b'°c 2 - Sb 6 d, 

 B=-2a 4 cd+2a 3 b 2 d+6a s bc 2 -~10a' 2 b 3 c + 4:ab 5 , 

 C = a*d 2 -\0a 3 bcd + 8a 2 b 3 d+2la 2 b 2 c 2 -32ab 4 c + 12b 6 , 

 D = oPbd i —±cPc*d+ \%a%& + %ab A d- \7ab 3 c 2 + 6b b c. 

 Every root of the cubic (1) satisfies the linear differential equa- 

 tion (2), which is of the second order. Therefore among the in- 

 tegrals of (2) the roots of (1) are included. 



4 Pump Court, Temple, London, 

 June 14, 1862. 



VII. On some Applications of a Theorem relating to Parallel 

 Surfaces. By the Rev. William Roberts*. 



HEOREM. — In the equation of a surface parallel to a given 



one, let x, j, z be replaced by \x, \\, Jz, and let ^ ^x 2 + y 2 + z 2 

 be written instead of ' k, the constant length taken on the normals of 

 the given surface ; the surface represented by the equation which 

 results from these substitutions will be the envelope of planes passing 

 through the several points of the given surface, and perpendicular 

 at each point to the radius vector drawn to the point from the 

 origin of coordinates. 



This theorem, which I have already published under a slightly 

 different enunciation {Comptes Rendus, November 14, 1859), 

 may be demonstrated as follows : — 



Let cc 1 , y f , z' be the coordinates of a point on the given sur- 

 face : the parallel surface will be the envelope of spheres repre- 

 sented by the equation 



( X -Jf + (y-y<f+(z-z'f=k\ 



or 



2xx J + 2yy' + 2zz'-x' 2 -y' 2 -z' 2 = x 2 + y* + z*-k 2 . 



Now suppose the eliminations requisite for finding this envelope 

 performed, it is plain that the result, when transformed by the 

 substitutions mentioned in our theorem, will be identical with 

 the equation representing the eDvelope of the planes, 



xx J + yy' + zz ] = x n + y' 2 + z n , 



which pass through the extremities of the radii vectores (from 

 the origin) of the given surface, and are perpendicular to them. 

 The surface envelope of planes perpendicular to the radii 

 vectores of a surface and passing through their extremities has 

 been treated of by Mr. T. A. Hirst in a very valuable memoir, 

 * Communicated by T. A. Hirst, Esq. 



