= eS Ce 
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 énterior 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 constant, 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 hodographie 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 dine drawn from it to the centre of 
the hodograph, the hodographic vector of eccentricity ; and 
the number 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 
