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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, 6ddc, a way, and ypagw, towrite 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 the force is 
to the velocity, in any varied motion of a point, as the element 
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 
