26 Geometrical Propositions applied 



words, the feet of the perpendiculars OS, let fall from on the 

 nodal tangent planes, occupy the circumference of a circle pass- 

 ing (8) through the nodal point. 



14. Parallel to the plane of the circle and to its reciprocal 

 plane, conceive two planes passing through the node, and call 

 them the principal tangent planes at n. The plane of the circle 

 and its reciprocal plane are intersected in the right lines Oq, OR, 

 by the plane qOR, which is parallel to a tangent plane at n. 

 Consequently this tangent plane at n intersects the two principal 

 tangent planes in lines that are parallel to Oq, OR ; and as Oq, 

 OR are perpendicular to each other, it follows that every nodal 

 tangent plane intersects the two principal tangent planes in lines 

 that are at right angles. 



Hence again, the nodal tangent planes touch (7) the surface 

 of a cone whose sections, parallel to the principal tangent planes, 

 are parabolas. As this cone touches the biaxal surface all round 

 the point n, it may be called the nodal tangent cone. 



15. Case 2. When ROr is a circular section of the ellipsoid 

 a'6V, any two perpendicular radii of the circle may be taken for 

 OR, Or : and because OR = b', and OR x OP = k z = bb', we have 

 OP or 08 equal to b, the mean semiaxis of the ellipsoid abc. 

 Hence OS is given both in position and length ; for it is perpen- 

 dicular to the fixed plane R Or, and it is equal to b. Now, a 

 plane cutting OS perpendicularly at S is a tangent plane to the 

 biaxal abc ; and we have just seen that this tangent plane re- 

 mains the same, whatever pair of rectangular radii are taken for 

 OR, Or. But the point of contact T is variable, for the plane 

 ROS in which it lies changes with OR. Therefore as OR re- 

 volves, the point T describes a curve of contact on the tangent 

 plane of the biaxal abc. 



The lines OR, Or, are in the fixed plane ROr; and as OQ 

 is reciprocal to OR, it lies in a fixed plane reciprocal to the plane 

 Ror (6). Therefore the first two of the three perpendicular right 

 lines Or, OQ, OT, are confined to fixed planes. Hencl the third 

 line OT describes a cone, whose sections parallel to these planes 

 are circles. But the tangent plane is parallel to the fixed plane 



