to the Wave Theory of Light. 23 



plane, while OP is perpendicular to the former and to the plane 

 of the figure, so that PQ being joined will meet MN at right 

 angles in P. Let OA, OB, OC, be the three perpendicular lines, 

 of which OA is parallel to the plane 

 of the figure ; this plane will be inter- 

 sected by the plane of OA and OB in 

 a right line BT parallel to OA, and 

 therefore perpendicular to both OB 

 and OP, and to the plane BOP, and 

 to the line BP. Thus the angle PBT 

 is always a right angle, and therefore 

 BT always touches the parabola whose 

 focus is P and vertex R ; or, which 



comes to the same thing, the plane AOBT always touches the 

 cone which has for its vertex, and the parabola for its section. 

 Again, since OB, OP, OC, are all at right angles to OA, 

 they are in the same plane, and therefore the points B, P, C, 

 are in the same straight line ; and as BOC is a right angle, the 

 rectangle under BP and PC is equal to the square of the per- 

 pendicular OP ; but QO R is also a right angle, and therefore 

 QP x PR = OP 2 ; whence BP x PC = QP x PR, and therefore 

 the points B, R, C, Q, are in the circumference of a circle, so 

 that the angle at C is a right angle, being in the same segment 

 with the angle at R. Thus the point C describes the circle whose 

 diameter is PQ, and OC describes the cone of which this circle 

 is the section. 



8. Of the two right lines OP and OQ perpendicular to the 

 given planes, one is also perpendicular to the plane of the section. 

 That one is OP. Its extremity P is the focus of the parabola. 

 The extremities of both are the extremities of the diameter PQ 

 of the circle. The vertex of the parabola is the point R, where 

 the diameter of the circle intersects that given plane to which 

 the plane of section is not parallel. 



9. THEOREM III. In a straight line at right angles to any 

 diametral section QOq of an ellipsoid abc whose centre is 0, let 

 OT and OF" be taken respectively equal to OQ and Oq, the semi- 

 axes of the section, and imagine the double surface which is the 



