xlv 



the differential equation (21) of the locus (15) may be put 

 under the form 



(n — a) da + da (/•£ — a) = ; (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 rs ; 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 2a£ ; if, then, we make 



2a - r = r', (53) 



so that r' denotes the length of the second focal vector, drawn 

 to the same point as the first focal vector, of which the length is 

 r, we have — r's 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 r, and multiply the 

 quotient by a, we find that e — t and ue — ai are also ex- 

 pressions for vectors in the normal direction ; and because 

 as is the focal vector of the centre, while — at 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- 

 longed through the focus, it will cut perpendicularly the 

 tangent plane to the ellipsoid. Again, the expression 



rzza{£ + i) — u= (a — r) I + Of, (54) 



VOL. III. d 



