xlvi 



is easily seen to denote here a vector perpendicular to a — r£, 

 and therefore to the normal, because 



ar + Ta =: r (et + te) = 2 {ap — rr') ; (55) 



but T is also in the same plane with a and e, 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 ; 

 at 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 «£, we have 

 ai — «E 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 t — e 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 — r 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 = rr' — ap, and therefore that it is less 

 than the product rr' of the lengths of those two vectors by the 

 constant quantity ap, or 6^, 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 



pp' = ap = b% (56) 



if p and p' denote the lengths of the perpendiculars let fall from 

 the two foci on the tangent, while b 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 



