to the Wave Theory of Light. 29 



sheets, since the points Tand V (9), in the description of the 

 biaxal surface, cannot coincide unless the section QOq, perpen- 

 dicular to OTV, be a circle. 



19. The plane of the greatest and least semiaxes, a, c, of the 

 generating ellipsoid, may be called the plane of the nodes ; and 

 the two diameters nOn, nOn, passing through the nodes, may 

 be called the nodal diameters. 



At one of the nodes n (Fig. 12) draw tangents nf, nk, to the 

 ellipse and the circle that compose the biaxal section ; and 

 through draw Op perpendicular to On, cutting the circle in p. 

 Then as On is perpendicular to the plane of a circular section of 

 the ellipsoid abc, this circular section will have Op for its radius, 

 and its circumference will cross that of the ellipse ac (belonging 

 to the ellipsoid) in the point p. A line touching the ellipse ac 

 at p will be parallel to every plane that touches the ellipsoid in a 

 point of the circular section, and will therefore (6) be perpendi- 

 cular to the plane which is reciprocal to the plane of the circular 

 section. But the tangent at p is perpendicular to the tangent nf, 

 since the two tangents would coincide if the ellipse ac were turned 

 round (18) through a right angle, the point p then falling upon 

 n. Hence the circular section and its reciprocal plane are 

 parallel to the tangents nk, nf; and therefore two planes perpen- 

 dicular to the plane of the figure, and passing through these tan- 

 gents, are the planes that we have called (14) the principal 

 tangent planes at n. 



20. Produce Op to meet nf in v, and conceive a parabola 

 having its focus at 0, its vertex at v (8), and its plane perpendi- 

 cular to the plane of the figure. A cone, with its vertex at n 

 and this parabola for its section, is (14) the nodal tangent cone. 



Draw Of perpendicular to nf&tf, and meeting nk in k. The 

 perpendiculars let fall from upon the nodal tangent planes 

 form a cone, of which the circles described in planes perpendi- 

 cular to the figure upon the diameters nf, nk, are sections (8). 

 On the other biaxal surface a'b'c there is (16) a circle of contact 

 whose plane is perpendicular to On. This circle of contact is 

 (16) another section of the cone last mentioned. 



