THE CYCLIDE. j 53 



If the cyclide has conical points, and if one of them be made the point 

 of inversion, the cyclide becomes a right cone, whose semi-vertical angle is 



(5 2 rM /y j _c 2 \i 



-j J if r is less than b, or cos' 1 (;? n) if r is greater than 6. 



If the point of inversion be at any other point of the surface, the cyclide 

 becomes a parabolic cyclide. 



be 

 If the point of inversion be x = , y = Q, z= 0, the cyclide is inverse to 



itself. 



On the Conjugate Isothermal Functions on the Cyclide. 



DEFINITION. If on any surface two systems of curves be drawn, each 

 individual curve being defined by the value of a parameter corresponding to it, 

 and if the two systems of curves intersect everywhere at right angles, and if 

 the intercept of a curve of the second system between two consecutive curves 

 of the first system has the same ratio to the intercept of a curve of the first 

 system between two consecutive curves of the second, as the difference of the 

 parameters of the two curves of the first system has to the difference of the 

 parameters of the two curves of the second system, then the two systems of 

 curves are called conjugate isothermal lines, and the two parameters conjugate 

 isothermal functions. If the surface be now supposed to be a uniform con- 

 ducting lamina placed between non-conducting media, one set of these lines will 

 be isothermal for heat or equipotential for electricity, and the other set will 

 be lines of flow. (See Lame" on Isothermal Functions.) 



This property of lines on a surface is not changed by inversion. 



In the cyclide, we find the intercept ds of a line of curvature of the first 

 system is 



, r ccoshfi , ?ou , , . 



&- TO , (c--b-)*da ..................... (20), 



c cos A/3 o cos a v 



and the intercept ds t of a line of curvature of the second system is 



(21). 



c cos 



VOL. II. 20 



