xlv 
the differential equation (21) of the locus (15) may be put 
under the form 
(re — a) da + da (re — a) = 0; (52) 
thus shewing that the vector re — a is perpendicular to the 
differential d of the focal vector a, or that it is parallel to the 
normal to the locus, at the extremity of that focal vector. 
That normal, therefore, intersects the axis of revolution in a 
point, of which the focal vector is r<; the position of the nor- 
mal is, therefore, entirely known, and every thing that depends 
‘upon it may be found, for the particular surfaces of revolution | 
which have been here considered. For example, in the ellip- 
soid, the vector of the second focus, drawn from the first, has 
been seen to be 2ae; if, then, we make 
z 2a —r=r’, (53) 
, 
so that 7’ denotes the length of the second focal vector, drawn 
to the same pointas the first focal vector, of which the length is 
r, we have — 7’< for the second focal vector of the intersection 
of the normal with the axis; the normal, therefore, cuts (in- 
_ternally) the interval between the two foci, into segments 
proportional to the two conterminous focal distances of the 
point upon the ellipsoid, and consequently bisects the angle 
between those focal distances. Again, if we divide the ex- 
_ pression re —a by the scalar quantity 7, and multiply the 
_ quotient by a, we find that « — « and ae — a are also ex- 
_ pressions for vectors in the normal direction; and because 
_ ae is the focal vector of the centre, while — a is a radius of 
the circumscribed sphere, opposite in direction to the focal 
_ vector of the point upon the ellipsoid, we see that if the focal 
vector of the extremity of this radius of the sphere be pro- 
| onged through the focus, it will cut perpendicularly the 
_ tangent plane to the ellipsoid. Again, the expression 
rTroa(e+) —a=(€A—rit+as, (54) 
VOL. III. d 
