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 T 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 ^rdt, 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, 



^^^■^^S.ESrd^ + T.Srd^; (15) 



where s and t are the characteristics of the operations of taking 

 respectively the scalar and tensor of a quaternion, sothat, as 

 applied to the present question, they give the results, 



T.^Tdt = Ta=V{-a'') = r; (16) 



and 



s.£§Td^ = ^(£a + a£) r crcosi;; (17) 



where 



e=:T£= V( — £^) = 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 r 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 sura 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, 



