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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 
