xlix 



hyperbola, and infinite for the parabola, and the harmonic 

 mean determined accordingly. We have also, for every conic 

 section (if r'a~^ be suitably interpreted), 



a/« = - = - = -= Vl — K (64) 



c s p \paJ ^ 



so that the semiparameter, the normal, the focal half chord of 

 curvature, and the radius of curvature, are in continued geo- 

 metrical progression : and the analysis may be verified, by 

 calculating directly, on the same principles, the length of the 

 normal, as follows : 



ii=^ {-(ra-ay\ = V \rXe'+l)+2r(p-r)}=v{^).(65) 



The general relation of a conic section to a directrix is 

 an immediate geometrical consequence of the equation (15), 

 which has been here (in part) discussed, and may be regarded 

 as its simplest interpretation. Some of the foregoing sym- 

 bolical results respecting such a section admit of dynamical 

 interpretations also ; and, in particular, the expression at — at, 

 which has been seen to represent, both in length and in direc- 

 tion, the perpendicular let fall from the second focus on the 

 tangent, may suggest, by its composition, what is, however, 

 a more immediate consequence of the equation (12), that in 

 the undisturbed motion of a planet or comet about the sun, 

 the whole varying tangential velocity may be decomposed into 

 two partial velocities, of which both are co?istant in magni- 

 tude, while one of them is constant in direction also. The 

 component velocity, which is constant in magnitude, but not 

 in direction, is always in the plane of the orbit, and is 

 perpendicular to the heliocentric radius vector of the body ; 

 the other component, which is constant in both magnitude 

 and direction is parallel to the velocity at perihelion ; and the 

 magnitude of this fixed component is to the magnitude of the 

 revolving one in the ratio of the excentricity e to unity. The 

 author supposes that this theorem respecting a decomposition 



