xliv 



though it touches this last surface at the two points where it 

 meets the axis of revolution. On the other hand, when 

 e^ <^ I, the locus is entirely contained within the spheric 

 surface (45), touching it, however, in like manner in two 

 points upon the axis of revolution. K finite surface of revo- 

 lution (the ellipsoid) might thus have been discovered, of 

 which each point has a constant sum of distances from two 

 fixed foci ; and an infinite surface (the hyperboloid), with 

 two separate sheets, of which each point has a constant dif- 

 ference of distances from two such foci : and all the other 

 properties of these two surfaces of revolution might have 

 been found, and may be proved anew, by pursuing this sort 

 of analysis. A third distinct surface of the same class, but 

 infinite in one direction only (the paraboloid), might have 

 been suggested by the observation that the reduction to a 

 centre fails in the case e^ = 1,£^ zi — 1. Its equation may 

 be put under the form 



{ea" - a"ef = ^p{m" + a^'e), (46) 



by making 



a = a"-P-^, (47) 



so that a" is the vector from the vertex : and it lies entirely on 

 one side of the plane which touches it at the vertex, namely, 

 the plane 



ta" + a"£ = 0. (48) 



In general whatever e or t may be, and therefore for all the 

 three surfaces, the length of the focal vector perpendicular 

 to the axis is p ; for, by (33), if we make 



m + a£ = 0, (49) 



we get 



a^ + p^- = 0. (50) 



Indeed (15) then gives r — p. 

 Since 



a^ 4- r^ = 0, a . da + da . a + 2r Ar = 0, (51) 



