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double of the elementary area divided by the square of the 

 distance (of the body from the centre of force), while the ele- 

 ment of the hodograph has been seen to be equal to the force 

 multiplied by the element of time, or multiplied by the same 

 double element of orbital area, and divided by the constant of 

 double areal velocity, therefore this radius of curvature of the 

 hodograph must, for any central force, be equal to the force 

 multiplied by the square of the distance, and divided by the 

 double areal velocity. 



The point on the hodograph which is the termination of 

 any one vector of velocity may be called the hodographic re- 

 presentative of the moving body, and the foregoing principles 

 show, that with a central force varying as the inverse square of 

 the distance, this representative point describes, in any proposed 

 interval of time, a circular arc, which contains the same num- 

 ber of degrees, minutes, and seconds, as the angle contempo- 

 raneously described round the centre of force by the body itself 

 in its orbit, or by the revolving vector of position ; because, 

 whatever that angle may be, an equal angle is described in 

 the same time by the revolving tangent to the hodograph. 

 Thus, with the law of Newton, the angular motion of a body 

 in its orbit is exactly represented, with all its variations, by 

 the circular motion on the hodograph ; and this remarkable re- 

 sult may be accepted, perhaps, as an additional motive for the 

 use of the new term which it is here proposed to introduce. 



Whatever the law of central force may be, if the square of 

 the velocity in the orbit be subtracted from the double rectan- 

 gle under the force and distance, which has been seen to be 

 equal to the rectangle under the same velocity and the chord 

 of curvature of the hodograph, the remainder is the rectangle 

 under the segments into which that chord is cut by the centre 

 of force, being positive when this section takes place inter- 

 nally, but negative when the section is external, that is, when 

 the centre of force is outside the osculating circle of the hodo- 

 graph. In the case of the law of the inverse square, this latter 



