THE CYCLIDE. 



The two curves are therefore plane oonics, such that the cone whose base 

 is one of the conies, and its vertex any point of the other, is a right cone. 

 The conies are therefore in planes at right angles to each other, and the foci 

 of one are the vertices of the transverse axis of the other. We shall call tin > 

 curves the focal conies of the cyclide. 



Let the equations to a point on an ellipse be 



a; = ccosa, y = (c 1 - 6')* sin a, 2 = (1), 



where a is the eccentric angle, and let the equations to a point on tin- 

 hyperbola be 



a: = 6sec-B, y = 0, z = (c 1 - &')' tan B (2), 



where B is an angle, then these two conies fulfil the required conditions. For 

 uniformity, we shall sometimes make use of the hyperbolic functions 



cosfy8 = |(e /l + e-), and sin/j = (<f-e~ f ) (3), 



and we shall suppose /? and B so related, that 



cos hft = sec B, whence sin hft = tan B } , . 



sec hfi = cos B, tan lift = sin B \ 



The equations to a point in the hyperbola may then be written 



x = bcoahp, y = 0, z = (c 1 - ft 1 )* sin lift (5). 



Construction of the Cyclide by Points. First Method. 



Let P be a point on the ellipse, Q a point on the hyperbola, then 



PQ = csecB-bcosa (6 ). 



\.i\v take on PQ a point R, so that 



/'/,' = ? -b cos a (7). 



or QR = r c&ecB ( 8 ), 



where r is a constant, then if a and B vary, R will give a system of points 

 on the cyclide (bcr). 



For if P be fixed while Q varies, R will describe a circle, and if Q 1> 

 fixed while P varies, R will describe another circle, and these circles cut at 

 right angles, and are both at right angles to PQR at their intersection, and 



