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vector, and which is called the parameter ; from which it fol- 

 lows, that the semiparameter of the orbit is equal to the con- 

 stant area of the parallelogram under distance and velocity, 

 divided by the radius of the hodograph, and, consequently, that 

 it is equal to the square of the constant double areal velocity, 

 divided by the attracting 7nass. 



It is evident that these results agree with and illustrate 

 those by which Newton shewed that Kepler's laws were ma- 

 thematical consequences of his own great law of attraction. 

 In applying them to the undisturbed motion of any binary 

 system of bodies, attracting each other according to that law, 

 we have only to substitute the sum of the two masses for the 

 single attracting mass already considered, and to treat one of 

 the two bodies as if it were the fixed origin of the vectors of a 

 relative hodograph, which will still be circular as before. And 

 even if we consider a ternary, or a multiple system, we may 

 still regard each body as tending, by its attraction, to cause 

 every other to describe an orbit of which the hodographic 

 representative would be a perfect circle. 



When there is one predominant mass, as in the case of the 

 solar system, we may in general regard each other body of the 

 system as moving in an orbit about it, which is, on the same plan, 

 represented by a varying circular hodograph. For if, at any 

 one moment, we know the two heliocentric vectors of position 

 and velocity of a planet, we know the plane and area of the 

 parallelogram under those two vectors, and can conceive a 

 parallelepiped constructed, of which this momentary paral- 

 lelogram shall be the base, while the volume of the solid 

 shall represent the sum of the masses of the sun and planet ; 

 and then the height of the same solid will be equal to the 

 radius of the momentary hodograph ; so that, in order to con- 

 struct this hodograph, we shall only have to describe, in the 

 plane, and with the radius determined as above, a circle which 

 shall touch the side parallel to the heliocentric vector of posi- 

 tion, at the extremity of the vector of velocity, and shall have 



