512 
the constant part of this expression (14) for the vector of the 
velocity, r, being the vector of the centre of the hodograph, 
drawn from that one of the two bodies which is regarded as 
the centre of force; while the variable part of the same ex- 
pression for 7 represents the variable radius of the same hodo- 
graphic circle, or the vector of a point on its circumference, 
drawn from its own centre of figure as the origin. 
Multiplying this integral equation (14) by §rd¢, taking the 
vector part of the product, dividing -by m, and multiplying 
both members of the result into the constant denominator of 
the second member of (13) or of (14), we find, by the rules of 
the present calculus, 
—(v.r(rdr)? 
eee = s.§rdé+7.(rde ; (15) 
where s and 1 are the characteristics of the operations of taking 
respectively the scalar and tensor of a quaternion, so’that, as 
applied to the present question, they give the results, 
TArdt=Ta= ¥(—a’)=r; (16) 
and 
S.e\rdt =3(ea+ ae) = ercosv; (17) 
where 
€=Te= ¥ (—«”)=const. ; (18) 
while v is the angle (of true anomaly) which the variable vec- 
tor a of the orbit makes with the fixed vector —< in the plane 
of that orbit ; and 7 denotes the length of a, or what is usually 
called (and may still in this theory be named) the radius vec- 
tor of the relative orbit. The first member of the equation 
(15) is a positive and constant number, representing the quo- 
tient which is obtained when the square of the double areal 
velocity in the relative orbit is divided by the sum of the two 
masses ; if then we denote, as usual, this constant quotient (or 
semiparameter) by p, and observe that the constant e is also 
numerical (expressing, as usual, the eccentricity of the orbit), 
we shall obtain again, by this process, as by that of July, 
