xlix 
hyperbola, and infinite for the parabola, and the harmonic 
mean determined accordingly. We have also, for every conic 
section (if 7’a— be suitably interpreted), 
BR Geo N rr’ 
ee sistcia than tat v(2), (64) 
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: 
prr’ 
pr’) .(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 a: — ae, 
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 constant 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 
N= ¥ {—(re—a)"}=V $r(C+1)+:2r(p—r)} =v ( 
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 
