xl 



the constant/) being here not only a scalar but an essentially 

 positive quantity, because theforce is supposed to be attractive, 

 or M > 0, while ]3^ < 0. The equation (15) thus obtained, 

 contains the law of elliptic, parabolic, or hyperbolic motion. 

 For if we make (by way of comparison with known results), 



V{ - e') =€, (17) and (a, - ,) = v, (18) 



(a, — a) denoting here the angle between the directions of 

 a and £, we have (by the formula (a) of the abstract of last 

 November), 



at + ta = 2€rcosv; (19; 



and therefore, by (15), 



r = j-—^ , (20) 



I + e cos V ^ 



which is the known equation of a conic section, referred to a 

 focus. The Greek letters, throughout, represent vectors : and 

 the Italics, scalar quantities. 



Supposing that we had no previous knowledge of the pro- 

 perties of cosines or of conies, we might have proceeded thus to 

 investigate the nature of the locus represented by the equation 

 (15). This locus is a surface of revolution round the line t ; 

 because the differential of its equation being 



da.E + E.da 4- 2dr = (21) 



if we cut it by a series of concentric spheres round the origin 

 of vectors, the sections are contained in a series of planes per- 

 pendicular to £ ; since 



dr = 0, (22) 



which is the differential equation of the first series, gives, 

 by (21), 



da£ +£da = 0, (23) 



which is the differential equation of the second series. To study 

 more closely this surface of revolution (15), make 



a = 7 + «', (24) 



