28 Geometrical Propositions applied 



ellipse QOq is that semidiameter of the principal ellipse be which 

 lies in the intersection of the plane be with the plane QOq ; and 

 as V is equal and perpendicular to this semidiameter, the point 

 V describes an ellipse equal to be, but turned round through a 

 right angle, so that the greater axis of the ellipse described by V 

 coincides in' direction with the less axis of the ellipse be. As the 

 radius a of the circle is greater (4) than both the semiaxes b, c, 

 of the ellipse, the circle will lie wholly without the ellipse. 



In like manner, the section made in the biaxal surface by 

 the plane ab consists of a circle with the radius c, and an ellipse 

 with the semiaxes a, b ; and as the radius of the circle is less than 

 both the semiaxes of the ellipse, the circle lies wholly within the 

 ellipse. 



18. But when the section lies in the plane of the greatest 

 and least semiaxes a, c, the circle and ellipse, of which it is com- 

 posed, intersect each other. For the radius b of the circle is less 

 than one semiaxis of the ellipse ac and greater than the other. 

 Leaving the ellipse ac in the position which it has as a section of 

 the ellipsoid abc, if we describe the circle b with the centre and 

 radius b, 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 b of the ellip- 

 soid, 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 con- 

 stitute the section of the biaxal sur- 

 face, and will cut each other (Fig. 12) 

 in four points n at the extremities of 

 two diameters nOn, nOn, which are 

 perpendicular to the two former dia- 

 meters, and therefore perpendicular 

 to the planes of the two circular 

 sections. Consequently, the biaxal 

 surface has four nodes at the four Flg - 12> 



points n. These nodes, it is manifest, are alike in all their 1 pro- 

 perties ; and they are the only points common to the two biaxal 



