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



of planes passing through the points of a surface parallel to an 

 ellipsoid, and perpendicular at each point to the radius vector 

 drawn to the point from any fixed origin. 



We now proceed to another application of the theorem. M. 

 Liouville has shown (Journal de Mathematiques, January 185]) 

 that the system of parallel surfaces, which have for loci of their 

 centres of curvature a determinate pair of confocal surfaces of 

 the second order (represented by the constants a and /3), is 

 given by the equation in elliptic coordinates 



j*P^ + JM^ + jN^v=C, .... (II.) 

 where 



- (p*-b*)(p*-c*)' ~~ (c 2 -/* 2 )^ 2 -^) ' 



(aW )(/3 2 - y 2 ) 

 1 ~~ (6 2 -v 2 )(c 2 -v 2 ) ' 



If C be another value of C, C — C will be the portion of any 

 common normal to the two parallel surfaces which correspond 

 to these two values of the constant intercepted between them. 

 Hence the first negative of (II.), in reference to the centre, or, 

 more correctly, the surface similar to it by doubling its radii 

 vectores, will have for equation 



as appears at once if we remember that 



p 2 + ^ 2 + v 2 = # 2 + 2/ 2 + * 2 + 6 2 + <? 2 . 



If the loci of the centres of curvature be supposed to degenerate 

 into the focal conies, which is expressed by making a = c, /3 = #, 

 the parallel surfaces become algebraic, and their equation in 

 elliptic coordinates takes the very simple form 



p + ^ + v=C (III.) 



This surface was discovered long ago by M. Charles Dupin, 

 and he gave it the name of cyclide. It forms the subject of a 

 variety of interesting articles in the Correspondance de I'Ecole 

 Poly technique. M. Mannheim has lately published a complete 

 and elegant discussion of its properties in the nineteenth volume 

 of the Nouvelles Annates de Mathematiques*. 



The surface similar, by doubling its radii vectores, to the first 

 negative of the cyclide (III.), is given by the equation in elliptic 



* In mentioning this periodical, I cannot refrain from joining in the 

 general expression of regret which the recent death of its editor, M. Ter- 

 quem, has called forth. Even those who knew him only by corresponding 

 with him feel as if they had lost an esteemed personal friend. 



