applied to the Wave Theory of Light. 243 
the ellipsoids, it is evident that two planes intersecting in this semi-axis and passing 
through the reciprocal radii, are reciprocal planes. 
7. Tueoren II. If three right lines at right angles to each other pass through a 
: fixed point O, so that two of them are confined to given planes; the plane of these 
two, in all its positions, touches the surface of a cone whose sections, parallel to the 
: given planes, are parabolas ; while the third right line describes another cone, whose 
sections parallel to the given planes are circles. 
Let the plane of the figure, (/¥%g. 3) supposed parallel to one of the given planes, 
be intersected by the other given plane in the right line J/N; and let OQ be per- 
pendicular to the latter plane, while OP is perpendicular to the former and to the 
plane of the figure, so that PQ being joined will meet WN at right angles in R.. Let 
OA, OB, OC, be the three perpendicular lines, of which OA is parallel to the plane 
of the figure ; this plane will be intersected by the plane of OA and OB in a right 
line BT parallel to O_4, and therefore perpendicular to both OL and OP, and to 
the plane BOP, and to theline BP. ‘Thus the angle PBT is always a right angle, 
and therefore 6T' always touches the parabola whose focus is P and vertex R; or, 
which comes to the same thing, the plane 4OBT always touches the cone which has 
0 for its vertex, and the parabola for its section. 
Again,,since OB, OP, OC, are all at right angles to O.4, 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# is also a right angle, and therefore QP x PR=OP’; 
whence BP x PC=QP x PR, and therefore the points B, R, C,Q, are in the cir- 
cumference of a circle, so that the angle at Cis a right angle, being in the same seg- 
| _ ment with the angle at R. Thus the point C describes the circle whose = 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 
isthe 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. Turorem III. In a straight line at right angles to any diametral section QOq 
z of an ellipsoid abc whose centre is O, let OT and OV be taken respectively equal to 
OQ and O¢ the semi-axes of the section, and imagine the double surface which is 
the locus of all the points T and V; then if OS be perpendicular to the plane which 
Bit ouches the surface in 7',and OP to the plane which touches the ellipsoid in Q, the 
f lines OP and OS will be equal and perpendicular to each other, and the four straight 
. lines OP, OQ, OS, OT, will lie in the same plane at right angles to Og. 
Pee vOL,. XVII, : 3E 
