Geometrical Propositions. 2 i 



plane ; for from any one of them R let Rn be drawn perpendi- 

 cular to the right line Oq and meeting Ogmn; then, on account 

 of the similar right-angled triangles OP'q and Ontf, the rect- 

 angle nOq will be equal to the rectangle R'OP, or to the 

 constant quantity k\ so that the point n, or the foot of the per- 

 pendicular let faU upon Oq 9 will be the same for aU the points 

 R, R', R", R m , &c., and consequently all these points will lie in 

 a plane cutting the right line Oqn perpendicularly in , so as to 

 make the rectangle nOq equal to k\ Now, while the point Q 

 remains fixed, let the point q approach to it without limit in the 

 tangent plane at Q ; and the points R, R", R m , &e., will in like 

 manner approach without limit to the fixed point R ; the plane 

 which contains all those neighbouring points having for its limit- 

 ing position the tangent plane at R. Also the point n will 

 ultimately coincide with N. It follows, therefore, that the tan- 

 gent plane at R cuts the right line OQ perpendicularly in N, so 

 as to make the rectangle NOQ, equal to k*. 



3. Corollary. If any point Q upon the surface A should be 

 a point of intersection, where the surface admits an infinite 

 number of tangent planes, the perpendiculars from upon these 

 planes will form a conical surface having for its vertex. In 

 OQ take, as before, a point N, so that ON x OQ = k\ and let a 

 plane passing through N at right angles to OQ cut the conical 

 surface. The intersection will be a certain curve. From the 

 preceding demonstration it is evident that every point of this 

 curve belongs to the surface B, and that the plane which touches 

 this surface at any point of the curve cuts OQ perpendicularly 

 in J^; or, in other words, that the same plane touches the surface 

 B through the whole extent of the curve. 



4. Two surfaces related to each other like A and B in the 

 preceding theorem may be called reciprocal surfaces, and points 

 like Q and R reciprocal points ; the radii OQ and OR may like- 

 wise be termed reciprocal. A familiar example of such surfaces 

 is afforded, as I have shown on a former occasion,* by two ellip- 



* Transactions of the Royal Irish Academy, Vol. xvi., pt. ii., pp. 67, 68. Supra, p. 3. 



