3O Geometrical Propositions applied 



21. To the circle b and to the principal section ac of the 

 ellipsoid abc conceive a common tangent d'i' to be drawn, in a 

 quadrant adjacent to that which contains the node n, and let it 

 touch the circle in d' and the ellipse ac in i'. A radius Od' , 

 drawn through the point d' to meet the ellipsoid a'b'c in the 

 point d", will be reciprocal to the radius Oi', because it is perpen- 

 dicular to a tangent at i', and it will be equal in length to b', be- 

 cause Od" x Od' = k* = bb', and Od' = b ; whence Od" = b. There- 

 fore Od" is in a circular section of the ellipsoid a'b'c'. Two planes 

 perpendicular to the plane of the figure, and passing through 

 the reciprocal radii Od", Oi', are (6) reciprocal planes, and we 

 have seen that the first of them makes a circular section in the 

 ellipsoid a'b'c'. They are therefore (15) the fixed planes in the 

 second case of Prop. Y. 



22. Now draw di a common tangent to the circle b and 

 ellipse ac composing the biaxal section, and let it touch the circle 

 in d and the ellipse in i. The lines Od, Oi, are of course perpen- 

 dicular to the lines Od', Oi', and therefore perpendicular to the 

 fixed planes just mentioned. Hence the line Od and the point 

 d are the same as the fixed line 08 and the point S in the second 

 case of Prop. Y. The plane of the circle of contact is therefore 

 perpendicular to Od at the point d (15) ; and the points d and i, 

 where its plane intersects the right lines Od, Oi, perpendicular 

 to the fixed planes, are (8) the extremities of a diameter. 



These things agree with the obvious remark, that the points 

 of contact d and i must be points of the circle of contact ; and 

 that di must be a diameter, because the plane of the circle is per- 

 pendicular to the plane of the figure, and this latter plane divides 

 the biaxal surface symmetrically. 



As the circle and ellipse may have a common tangent oppo- 

 site to each node, there are four circles of contact in planes per- 

 pendicular to the plane of the nodes.* 



23. The biaxal surface belongs to a class that may be called 



* The curves of contact on biaxal surfaces, and the conical intersections or nodes, 

 were lately discovered by Professor Hamilton, who deduced from these properties 

 a theory of conical refraction, which has been confirmed by the experiments of 



