Theorem relating to Parallel Surfaces. 43 



coordinates, 



p + fI/ + v=C+ Vp* + fi* + v*-b*-<*, 



or 



pp + pv + fiv-C(p + p + v) + i(C* + b* + c*)=:0. . (IV.) 



The equation (III.) of the cyclide, transformed into an equation 

 between x, y, z, becomes 



|^ + y « + ^HA 2 + c s -C 2 } 2 ==4{(J 2 + ^)a7 8 + cy + AV + 2icCjF + i 2 c B } 



This equation representing a system of parallel surfaces, if we 

 regard C as a parameter, we at once deduce the first negative of 

 any one of them corresponding to a particular value of C by ap- 

 plying our theorem. Accordingly let us write \x, \y, ±z for x,y,z, 

 and C+ \ Vx 2 + y 2 + z 2 for C, and we shall find for the equation 

 of the first negative in reference to the centre of the elliptic 

 system as origin, 



{(C 2 -4 2 -c 2 )^ 2 +(C 2 -c 2 )y 2 + (C 2 -&V-- 4 *cC^+(c 8 --* 2 ) a 

 -2C*(b* + c*)+C 4 }*=4>{C 3 -b 2 C-c' 2 C--bcx}*{x* + y* + z*} } 



as we might also have found by transforming (IV.) out of the 

 elliptic system. I have already published this result in a note 

 presented to the French Academy of Sciences (Comptes Rendus, 

 December 16, 1861). 



We now proceed to show how, from the equation of a surface 

 parallel to a given one, we can deduce the equation of a parallel 

 to the surface derived from the given by the method of reciprocal 

 radii vectores. Let x', y [ f , z ] be the coordinates of a point on the 

 given surface, and m 2 the constant rectangle formed by the coin- 

 cident radii vectores of this surface and its inverse. The parallel 



surface to the inverse is the envelope of the spheres represented 



by the equation 



{m 2 #' \ 2 j f m<2 y' \ ~ 



X ~ x n + y'* + z ,2 J + V" x l2 + y' 2 + z , *J 



r _™v_V 2 



But this equation may be written 



(dx-x ! )*+ (dy-y'f+ (0z-J)* = 0*k*, 



where 



6= 



m 2 



ffa + ya + sS-.* 2 ' 



whence it appears that if, in the equation of the parallel to a 

 given surface, we replace x, y, z, and k respectively by 0x, 6y, 

 6z, 6k y we shall have the equation of the parallel to the inverse of 



