xliv 
though it touches this last surface at the two points where it 
meets the axis of revolution. On the other hand, when 
e? <1, the locus is entirely contained within the spheric 
surface (45), touching it, however, in like manner in two 
points upon the axis of revolution. A finite surface of revo- 
lution (the ellipsoid) might thus have been discovered, of 
which each point has a constant swm 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? = 1,2 = — 1. Its equation may 
be put under the form 
(ea — a’’c)? = 4p(ca” + a”), (46) 
by making 
a se a”’ = a (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 
ca’ ta’e = 0. (48) 
In general whatever e or ¢ 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 
ca + ac = 0, (49) 
we get 
a+p-od. (50) 
Indeed (15) then gives r = p. 
Since 
er = 9, a.datda.a+2rdr=0, (51) 
