438 Professor Booth on the Focal Properties 



revolution, its axis being the line joining the conjugate foci 

 of the surface. 



When (2) is a surface of revolution round the transverse 

 axe, the conjugate directrix planes coincide, therefore the 

 planes through the vertex of the cone and the intersections of 

 the directrix planes by its base, coincide, hence the cone is 

 a right cone, and we obtain the known theorem, that " the 

 cone whose vertex is the focus, and base any plane section 

 of a surface of revolution, round the transverse axe, is a 

 right cone." 



Hence also the cone whose vertex is the centre of an ob- 

 late spheroid and base any plane section of ihis surface, has 

 its circular sections parallel to the diametral planes passing 

 through the right lines in which the base of the cone inter- 

 sects the parallel directrix planes. 



PROP. III. The preceding theorem may be generalized 

 thus : let a cone enveloping (S), having its vertex anywhere 

 on the line QQ' joining the conjugate foci, be cut by any 

 plane passing through the intersection of the conjugate di- 

 rectrix planes in a conic section, the cone whose base is this 

 section and vertex the focal centre of the surface, is a surface 

 of revolution. 



PROP. IV. Let a right line be drawn meeting a surface of 

 the second order, in the points T and T', and the conjugate 

 directrix planes in the points m and m 1 , the segments of this 

 right line, m T and m 1 T', subtend equal angles at the focal 

 centre. 



When the line is parallel to one of the directrix planes, 



it may be easily shown that the rectangle m T x m T' = m O 2 > 

 O being the focal centre, and m the point in which the right 

 line meets the directrix plane to which it is not parallel. 



Hence, if a tangent plane is drawn to a surface of the se- 

 cond order at the umbilicus, meeting the other conjugate 

 directrix plane in a right line, the distances of any point in 

 this right line from the umbilicus and focal centre are equal. 



When the surface is one of revolution round the transverse 

 axe the points m and m 1 coincide, or the line O m bisects the 

 supplement of the angle T O T', which is a known property of 

 such surfaces. 



Let a series of surfaces of the second order, having the 

 same focal centre, and the same pair of conjugate directrix 

 planes, be cut by any transversal; the segments of this right 

 line, between each pair of surfaces, subtend equal angles at 

 the common focal centre. 



When (S) is a cone, if from m any point in one of the 

 directrix planes a right line is drawn parallel to the other, 



