THE CYCLIDE. 



Since the quantities p> c, p, b, v, 



are in descending order of magnitude, it is evident that 



r i> P, ^, r 3 , -p, r t 

 are also in descending order of magnitude. 



The general equation to the cyclide in elliptic co-ordinates is 



(r-p-p. + v)(r-p + p.- v )(r + p-f J ,- v )(r + p + i Ji + v ) = () (36), 



which may be expressed in Cartesian co-ordinates thus : 



When b = c there are two focal points F and F' and the values of the 

 four rays are 



r l= RF+c 

 r,= RF-c\ 



r,= -RF+c\ < 38 )' 



r t =-RF-c. 

 The equation of the ellipsoid 



2p = r 1 + r 1 = RF+RF' (39) 



in this case expresses the property of the prolate spheroid, that the sum of the 

 distances of any point from the two foci is constant. 



In like manner, the equation 



r t + r, = 2v = RF' RF (40) 



expresses the property of the hyperboloid of revolution of two sheets, that the 

 difference of the focal distances is constant. 



In order to extend a property analogous to this to the other conicoids, let 

 us conceive the following mechanical construction : 



Suppose the focal ellipse and hyperbola represented by thin smooth wires, 

 and let an indefinite thin straight rod always rest against the two curves, and 

 let r be measured along the rod from a point fixed in the rod. Let a string 

 whose length is b + c be fastened at one end to the negative focus of the ellipse 

 and at the other to the point ( + b) of the rod, and let the string slide on the 

 ellipse at the same point as the rod rests on it. To keep the string always 

 tight let another equal string pass from the positive focus of the ellipse round 



