350 



of the hodograph, or of the orbit (without violating the re- 

 ceived meaning of the term). 



Whatever the value of this numerical eccentricity may be, 

 the constant area of the parallelogram under the vectors of 

 position and velocity may always be treated as the sum or 

 difference of two other parallelograms, of which one is equal 

 to the rectangle under the constant radius of the hodographic 

 circle and the varying radius vector of the orbit, while the 

 other is equal to the parallelogram under the vectors of posi- 

 tion and eccentricity ; and hence it is not difficult to infer, 

 that the length of the vector of position, or of the radius vec- 

 tor of the orbit, varies in a constant ratio, expressed by the 

 numerical eccentricity, to the perpendicular let fall from its 

 extremity, that is, from the position of the body, on a constant 

 straight line or directrix, which is situated in the plane of the 

 orbit, and is parallel to the hodographic vector of eccentricity. 

 The orbit, therefore, whether it be closed or not, is always 

 (with the law of the inverse square) a conic section, having 

 the centre of force for a focus — a theorem which has, indeed, 

 been known since the time of Newton, but has not, perhaps, 

 been proved before from principles so very elementary.* 



Conceive a diameter of the hodograph to be drawn in a 

 direction perpendicular to the vector of excentricity ; the ex- 

 tremities of this diameter correspond to the extremities of that 

 chord of the orbit which is perpendicular to the shortest radius 



* The hodograph of the earth's annual motion may be considered to be ex- 

 hibited to observation in astronomy, as the curve of aberration of a star; and 

 it is known that this aberratic curve is a circle, notwithstanding the eccen- 

 tricity of the earth's orbit ; but the author is not aware that this circularity 

 of the aberratic curve (for a star near the pole of the ecliptic) has ever been 

 shewn before to be a consequence of the law of the inverse square, except by the 

 help of the properties of the elliptic orbit; whereas the spirit of the present 

 communication is to derive that orbit from the circle, and to regard that circle 

 itself as a sort of geometrical picture of Newton's law, instead of being only 

 one of many corollaries from the laws of Kepler. 



