349 



curve is its own osculating circle, and the rectangle under the 

 segments of the chord is, therefore, constant, by an elemen- 

 tary theorem of geometry contained in the third book of Eu- 

 clid ; if, then, the square of the velocity be subtracted from 

 the double of the attracting mass, divided by the distance of 

 the body from the centre of force, at which that mass is con- 

 ceived to be placed, the remainder is a constant quantity, 

 which is positive if the orbit be a closed curve, that is, if the 

 centre of force be situated in the interior of the circular hodo- 

 graph. 



In this case of a closed orbit, the positive constant, which 

 is thus equal to the product of the segments of a hodographic 

 chord, or the constant product of any two opposite velocities 

 of the body, is easily seen, by the foregoing principles, to be 

 equal to the attracting mass divided by the semisum of the two 

 corresponding distances of the body, which semisum is, there- 

 fore, seen to be cotistant, and may be called (as in fact it is) 

 the mean distance. The law of living force, involving this 

 mean distance, may, therefore, be deduced as an elementary 

 consequence of this mode of hodographic representation, for 

 the case of a closed orbit ; together with the corresponding 

 forms of the law, involving a null or a negative constant, in- 

 stead of the reciprocal of the mean distance, for the two cases 

 of an orbit which is not closed, namely, when the centre of 

 force is on, or is outside the circumference of the hodographic 

 circle. 



Whichever of these situations the centre of force may have, 

 we may call the straight line drawn from it to the centre of 

 the hodograph, the hodographic vector of eccentricity ; and 

 the nu?nber which expresses the ratio of the length of this 

 vector to the radius of the hodograph will represent, if the 

 orbit be closed, the ratio of the semidifFerence to the semisum 

 of the two extreme distances of the body from the centre of 

 force, and may be called generally the numerical eccentricity 



