THE CYCLIDE. 



155 



Having found these conjugate isothermal -functions we may deduce from 

 them any number of other pairs, as 1 and < where 



( 26 )- 



On Confocal Cyclides. 



A system of cyclides in which the focal ellipse and hyperbola remain the 

 same, while r has various values, may be called a confocal system. This system 

 of cyclides and the two systems of right cones which have their vertices in 

 one conic and pass through the other, form three systems of orthogonal surfaces, 

 and therefore intersect along their lines of curvature. By inversion we may get 

 three systems of cyclides intersecting orthogonally. 



A system of confocal cyclides may also be considered as a system of wave 

 surfaces in an isotropic medium, corresponding to a pencil of rays, each ray of 

 which intersects the two focal conies. Each cyclide corresponds to a certain 

 value of r, which we may call the length of the ray of that cyclide. 



Now let us consider the system of confocal conicoids, whose equation is of 

 the form 



By putting p = c, we get the ellipse 



I " _1 n O ff)Q\ 



<?*<? ty~ * '' 



By putting p = b, we get the hyperbola 



I-^P" 1 ^- ( 29 )' 



These two conies therefore belong to the system and may be called its focal 

 conies. If, with any point R for vertex, we draw cones through the ellipse and 

 hyperbola, these will be confocal cones, whose three axes are normal to the 

 three conicoids of the system which pass through R. The two cones will inter- 

 sect at right angles along four generating lines r 1} r 2 , r t , r t> which are normal 

 to four cyclides passing through the point R. 



20 '2 



