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The tangential equations of these sections are 



. . (13) 



Now the former of these is the tangential equation of the evolute of 

 an ellipse, while the other is that of an ellipse whose semiaxes are 

 the radii of curvature at the extremities of a and c in the planes of 

 xy and zy diminished by a and c. 



It is easy to show, that if through the four umbilici of the ellipsoid 

 normals to the surface be drawn, they will lie in the plane of xz y they 

 will touch the evolute internally and the ellipse externally in the same 

 points, so that the lozenge formed by the four normals will be in- 

 scribed in the evolute and circumscribed to the ellipse, and the distance 



of the point of contact to the umbilicus will be equal to . 



The respective areas of the lozenge, of the inscribed ellipse, and of 

 the circumscribed evolute, are connected by relations independent of 

 the axes of the ellipsoid. 



It is in these four points, and in these four points only, that the 

 two sheets of the surface of centres touch each other. We should find 

 on investigation, that the points of intersection of the sections of the 



