xl 
the constant p being here not only a scalar but an essentially 
positive quantity, because theforce is supposed to be attractive, 
or M > 0, while 8B’? <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=. y=, (7) and (a, — «) =v, + (18) 
(a, — «) denoting here the angle between the directions of 
a and «, we have (by the formula (a) of the abstract of last 
November), 
at + ca = 2er cosv; (19) 
and therefore, by (15), 
eu" + ae (20) 
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 conics, we might have proceeded thus to 
investigate the nature of the locus represented by the equation 
(15). This locus is a suxface of revolution round the line « ; 
because the differential of its equation being 
da.e + ¢.da + 2dr = 0 (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), 
daz + eda = 0, (23) 
which is the differential equation of the second series. ‘To study 
more closely this surface of revolution (15), make 
a=ytd, (24) 
