Geometrical Propositions applied 



will be (1) perpendicular to OR and to the plane 0,0 R. Let a 

 plane also perpendicular to the plane QOR pass through OQ, 

 cutting the surface to which the point Q belongs in a certain 

 curve, and the tangent plane at Q in a tangent to this curve. 

 The tangent is evidently perpendicular to 0Q, and therefore the 

 point Q is an apsis of the curve. 



In like manner, the point R is an apsis of the section made 

 in the other surface by a plane passing through OR and perpen- 

 dicular to the plane QOR. 



26. From these observations, and from Prop. VI., it appears 

 that if the points Q, It, in the figure of Theorem IV., be reciprocal 

 points on any two reciprocal surfaces, and if the same construc- 

 tion be supposed to remain, the points T and M will be points 

 on the apsidal surfaces generated by these reciprocal surfaces, and 

 the tangent planes at T and M will be perpendicular to the lines 

 OM and OT respectively. Also the rectangles ZOTand MOS 

 will be equal to A' 2 . Hence we have another general theorem : 



PROP. VII. THEOREM. The apsidal surfaces generated by 

 two reciprocal surfaces are themselves reciprocal. 



27. A very simple example of apsidal surfaces, with nodes 

 and circles of contact, may be had by supposing the generatrix 

 G to be a sphere, and the pole to be within the sphere, between 

 the surface and the centre C. 



It is evident that the apsidal surface in this case will be one 

 of revolution round the right line OC 

 as an axis. Therefore taking for the 

 plane of the figure (Fig. 13) a plane 

 passing through OC and cutting the 

 sphere in a great circle of which the 

 radius is CS, let a plane at right 

 angles to the figure revolve about 0, 

 cutting the circle 08 in the points 

 A, A'. The section of the sphere 

 made by the revolving plane will have 

 only two apsides A, A', with respect Fig. 13. 



to the point 0, except when the plane is perpendicular to OC. 



