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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 mass. 
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 
i, 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 
