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liodograph. The straight lines drawn to the moving body 

 from the centre offeree 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) is to the radius vec- 

 tor of the orbit as the element of the hodograph is to the ele- 



