applied to the Wave Theory of Light. Q47 
Leaving the ellipse ac in the position which it has as a section of the ellipsoid abe, if 
we describe the circle b with the centre O and radius 4, the ellipse and the circle will 
cut each other in four points at the extremities of two diameters ; and planes, passing 
through these diameters and through the semiaxis 4 of the ellipsoid, will evidently be 
the planes of the two circular sections of the ellipsoid. Now turning the ellipse ac 
round through a right angle (17), the circle and the ellipse in its new position will 
constitute the section of the biaxal surface, and will cut each other (F% 1g. 5.) in four 
points » at the extremities of two diameters n On, nOn, which are perpendicular to 
the two former diameters, and therefore perpendicular to the planes of the two cir- 
cular sections. Consequently, the biaxal surface has four nodes at the four points 7. 
These nodes, it is manifest, are alike in all their properties ; and they are the only 
points common to the two biaxal sheets, since the points Z’and V (9), in the deserip- 
tion of the biaxal surface, cannot coincide unless the section Q Og, perpendicular to 
OTF, 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 n On, n On, passing 
through the nodes, may be called the nodal diameters. 
At one of the nodes n (Fig. 5) draw tangents nf, nk, to the ellipse and the circle 
that compose the biaxal section ; and through O draw Op perpendicular to On, cutting 
the circle inp. 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 perpendicular to the plane 
which is reciprocal to the plane of the circular section. But the tangent at p is per- 
pendicular to the tangent mf, 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 thereforegwo planes perpendicular to the plane of the figure and passing through 
_ these tangents, are the planes that we have called (14) the principal tangent planes 
at 2. 
20. Produce Op to meet vf in v, aad conceive a parabola having its focus at Q, its 
vertex at v (8), and its plane perpendicular to the plane of the figure. A cone, with 
its yertex at ” and this parabola for its section, is (14) the nodal tangent cone. 
Draw Of perpendicular to nf at f, and meeting nk ink. ‘The perpendiculars let 
fall from O upon the nodal tangent planes form a cone, of which the circles described 
in planes perpendicular 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 perpendi- 
cular to On. This circle of contact is (16) another section of the cone last mentioned. 
21. To the circle b and to the principal section ae of the ellipsoid abc conceive a 
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