xlvi 
is easily seen to denote here a vector perpendicular to a — re, 
and therefore to the normal, because 
ar + ra =r (er + re) = 2 (ap — rr’); (55) 
but 7 is also in the same plane with a and ¢, and therefore is a 
vector parallel to the tangent to the elliptic section of the 
locus made by a plane passing through the axis of revolution ; 
a is therefore the central vector of a point upon this tangent, 
because a — ae is the central vector of the point of contact ; 
and the central vector of the second focus being az, we have 
a — de as an expression for the second focal vector of the 
same point upon the tangent; this second focal vector is 
therefore parallel to the normal, because « — « is parallel 
thereto, and, consequently, it is the perpendicular let fall 
from the second focus on the tangent line or plane: and the 
foot of this perpendicular is thus seen to be at the extremity - 
of that radius of the circumscribed circle or sphere, which is 
drawn in a direction similar (and not, as lately, opposite) to 
the direction of the first focal vector of the point on the ellipse 
or ellipsoid. We see, at the same time, that — 7 is a symbol for 
the projection of the second focal vector upon the tangent line 
or plane ; from which we may infer, by (55), that the product 
of the lengths of the two projections of the two focal vectors 
on the tangent is = 77” — ap, and therefore that it is less 
than the product 77” of the lengths of those two vectors by the 
constant quantity ap, or b*, which constant must thus be equal 
to the product of the lengths of the projections of the same two 
vectors on the normal, so that we may write the equation 
BP! = api= 5b’, (56) 
if p and p’ denote the lengths of the perpendiculars let fall from 
the two foci on the tangent, while J is the axis minor of the 
ellipse. Analogous reasoning may be applied to the hyper- 
bola, or to the surface formed by its revolution round its 
transverse axis. Most of the foregoing geometrical results are 
well known, and probably all of them are so: but it may be 
