242 Geometrical Propositions 
plane at Q ; and the points R’, R”, R”’, &c. will in like manner approach without 
limit to the fixed point R; the plane which contains all those neighbouring points 
having for its limiting position the tangent plane at #. Also the point m will ulti- 
mately coincide with V. It follows therefore that the tangent plane at # cuts the 
right line OQ perpendicularly in N, so as to make the rectangle VOQ equal to k’. 
3. Corollary. If any point Q upon the surface 4 should bea point of intersection, 
where the surface admits an infinite number of tangent planes, the perpendiculars 
from O upon these planes will form a conical surface having O for its vertex. In 
O@ take, as before, a point N, so that ON x OQ =k*, and let a plane passing through 
NV 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 thé plane which touches this surface at any point of 
the curve cuts OQ perpendicularly in VV; 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 4 and B in the preceding theorem may 
be called reciprocal surfaces, and points like Q and -# reciprocal points ; the radiz 
OQ and OR may likewise be termed reciprocal. A familiar example of such sur- 
faces is afforded, as I have shown on a former occasion*, by two ellipsoids having a 
common centre at the point O, and their semi-axes coincident in direction, and con- 
nected by the relation aa’ =bb'=cc’'=k’ ; where a, b,c, are the semi-axes of one el- 
lipsoid in the order of their magnitude, a being the greatest ; and a’, 6’, c’, those of 
the other ellipsoid, a’ being the least. The mean semi-axes 6 and b’ coincide, and the 
circular sections of both ellipsoids pass through the common direction of 6 and 6’. 
5. It has also been shown with regard to those ellipsoids, that if Q and # be reci- 
procal points on the surfaces of abc and a’b’c’ respectively, and if a right line Ogr, 
perpendicular to the plane QO, cut the first ellipsoid in g and the second in 7, the 
lines OQ and Oq will be the semi4fxes of the section made in the ellipsoid abc by a 
plane passing through them; and the lines O# and Qr, in like manner, will be the 
semi-axes of the section made in the other ellipsoid a@’b’c’ by the plane in which 
they lie. 
6. It may further be remarked, that if the radius OQ in one of the reciprocal el- 
lipsoids describe a plane, the corresponding radius OF will describe another plane. 
For the planes touching the ellipsoid abc in the points Q will all be parallel to a cer- 
tain right line, and therefore the perpendiculars O# to these tangent planes will all 
lie in a plane perpendicular to that right line. “These two planes, containing the re- 
ciprocal radii, may, for brevity, be called reciprocal planes. 
When two reciprocal radii lie in a principal plane, at right angles to a semi-axis of 
-* Transactions of the Royal Irish Academy, Vol. XVI. Part II. pp. 67, 68. 
