THE CYCLIDE. 145 



Of the two lines of curvature through R, the first touches the plane 

 RPT, and the second is perpendicular to it. Hence, if the plane RPT turn 

 about the tangent PT as an axis, it will always be normal to the second 

 line of curvature. The second line of curvature is therefore a circle, and PT 

 passes through its centre perpendicular to its plane. 



All the normals belonging to the second line of curvature are of equal 

 length, and equally inclined to PT, so that they may be considered either 

 as the generating lines of a right cone whose axis is the tangent to the 

 fixed curve, or as the radii of a sphere whose centre is at P, and which 

 touches the surface all along the line of curvature. 



The surface may therefore be defined as the envelope of a series of 

 spheres, whose centres lie on the fixed curve, and whose radii vary according 

 to any law. 



If the normal passes through two fixed curves, the surface must also be 

 the envelope of a second series of spheres whose centres lie on the second 

 fixed curve, and each of which touches all the spheres of the first series. 



If we take any three spheres of the first series, the surface may be 

 defined as the envelope of all the spheres which touch the three given spheres 

 in a continuous manner. 



This is the definition given by Dupin, in his Applications de Geometric 

 (p. 200), of the surface of the fourth order called the Cyclide, because both 

 series of its lines of curvature are circles. 



If the three fixed spheres be given, they may either be all on the same 

 side (inside or outside) of the touching sphere, or any one of the three may 

 be on the opposite side from the other two. There are thus four different 

 series of spheres which may be described touching the same three spheres, 

 but we cannot pass continuously from one series to another, and the normals 

 to the four corresponding cyclides pass through different fixed curves. 



Let us next consider the nature of the two fixed curves. Since all the 

 normals pass through both curves, and since all those which pass through a 

 point P are equally inclined to the tangent at P, the second curve must 

 lie on a right cone. If now the point P be taken so that its distance from 

 a point Q in the second curve is a minimum, then PQ will be perpendicular 

 to PT, and the right cone will become a plane, therefore the second curve is 

 a plane conic. In the same way we may shew, that the first curve is a plane 

 conic. 



VOL. n. 19 



