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hodograph. The straight ae drawn to the moving body 
from the centre of force being called, in like manner, the vectors 
of the orbit, or the vectors of position of the body, we see that 
each such vector of position is now parallel to the tangent of 
the hodograph drawn at the extremity of the vector of velocity, 
as the latter vector was seen to be parallel to the tangent of 
the orbit, drawn at the extremity of the vector of position ; so 
that the two vectors, and the two tangents drawn at their ex- 
tremities, enclose at each moment a parallelogram, of which it 
is easily seen that the plane and area are constant, although 
its position and its shape are generally variable from one 
moment to another, in the motion thus performed under the 
influence of a central force. If, therefore, this constant area 
be given, and if either the hodograph or the orbit be known, 
the other of these two curves can be deduced, by a simple and 
“uniform process, on which account the two curves themselves 
may be called reciprocal hodographs. 
The opposite angles of a parallelogram being equal, it is 
evident, that if the central force be attractive, any one vector 
of position is inclined to the next following element of the 
orbit, at the same angle as that at which the corresponding 
vector of velocity is inclined to the next preceding element of 
the hodograph. Also, if from either extremity of any small 
element of any curve, a chord of the circle which osculates to 
that curve along that element be drawn and bisected, the ele- 
ment subtends, at the middle point of this chord, an angle 
equal to the angle between the two tangents drawn at the two 
extremities of the element ; that is, here, if the curve be the 
hodograph, to the angle between the two near vectors of posi- 
tion, which are parallel to the two extreme tangents of its 
element. We have, therefore, two small and similar triangles, 
from which results the following proportion, that the half 
chord of curvature of the hodograph (passing through, or 
tending towards the fixed centre of force) ts to the radius vec- 
tor of the orbit as the element of the hodograph is to the ele- 
