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Whatever may be the complication of the accelerating forces 

 which act on any moving body, regarded as a moving point, and, 

 therefore, however complex may be its orbit, we may always 

 imagine a succession of straight lines, or vectors, to be drawn 

 from some one point, as from a common origin, in such a man- 

 ner as to represent, by their directions and lengths, the varying 

 directions and degrees(or quantities) of the velocity of the mov- 

 ing point : and the curve which is the locus of the ends of the 

 straight lines so drawn may be called the hodograph of the 

 body, or of its motion, by a combination of the two Greek 

 words, oSoC) a way, and jpacpu), to write or describe; because the 

 vector of this hodograph, which may also be said to be the vector 

 of velocity of the body, and which is always parallel to the tan- 

 gent at the corresponding point of the orbit, marks out or in- 

 dicates at once the direction of the momentary path or way in 

 which the body is moving, and the rapidity with which the 

 body, at that moment, is moving in that path or way. This 

 hodographic curve is even more immediately connected than 

 the orbit, with the forces which act upon the body, or with 

 the varying resultant of those forces, for the tangent to the 

 hodograph is always parallel to the direction of this resultant; 

 and if the element of the hodograph be divided by the element 

 of the time, the quotient of this division represents (to the 

 usual units) the intensity of the same resultant force ; so that 

 the whole accelerating force which acts on the body at any 

 one instant is represented, both in direction and in magnitude, 

 by the element of the hodograph, divided by the element of the 

 time. We have also the general proportion, that i\\e force is 

 to the velocity, in any varied motion of a point, as the eletnent 

 of the hodograph is to the corresponding element of the orbit. 



These general remarks respecting varied motion, under 

 the influence of any accelerating forces whatever, having been 

 premised, let it be now supposed that the force is constantly 

 directed towards some one fixed point or centre, which it will 

 then be natural to choose for the origin of the vectors of the 



