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sections of two confocal hyperbolas, a single- sheet one and a double- 

 sheet one. It is also known that these three surfaces are reciprocally 

 orthogonal, or that any two of them cut the third along its lines of 

 curvature where the three intersect in a point. If we fix on the 

 ellipsoid as the surface whose lines of curvature are in question, and 

 normals be drawn to the surface of the ellipsoid along any given line 

 of curvature, the radii of curvature will not only lie on these normals 

 at the successive points, but they will all, taken indefinitely near to 

 each other, constitute a developable surface, and the line of centres 

 of curvature will constitute its edge of regression. Hence if we draw 

 tangent planes to the two hyperboloids at this point, they will inter- 

 sect in the normal to the ellipsoid, and will also be tangent planes to 

 the above developable surface. Let the equation of the ellipsoid be 



or as the surfaces are confocal, we may put 2 - 6 2 =A 2 , a 2 - c 2 =A- 2 . 

 Hence this equation may be written 



Let a be the transverse axe of the hyperboloid pas.sing through 

 the point x'y'z', and we shall have 



Now the tangential equation of this hyperboloid is 



a ^ + ( a 2_AV+(a 2 -F) 2 =l. ... (3) 

 But the equation of the tangent plane to the hyperboloid at the 

 point (x'y'z 1 ) is 



and as the planes which touch the ellipsoid and hyperboloid at the 

 common point (#', y' } z') are at right angles to each other, we have, 

 moreover, 



Hence eliminating x\ y', z', and a, we shall have, finally, 



-l). . (6) 



