24 Geometrical Propositions applied 



locus of all the points T and F; then if OS be perpendicular to 

 the plane which touches the surface in T, and OP to the plane 

 which touches the ellipsoid in Q, the lines OP and 08 will be 

 equal and perpendicular to each other, and the four straight 

 lines OP, OQ, OS, OT, will lie in the same plane at right 

 angles to Oq 



10. This theorem is taken from a former communication to 

 the Academy.* The surface to which it relates, being the wave 

 surface of FRESNEL, is one of frequent occurrence in optical in- 

 quiries, and it is therefore desirable to give it a distinctive name 

 not derived from any physical hypothesis. I shall call it a 

 biaxal surface, from the circumstance implied in its construction, 

 and adopted as the definition on which the preceding theorem is 

 founded namely, that any pair of its coincident diameters are 

 equal to the two axes of a central section made in the generating 

 ellipsoid abc, by a plane perpendicular to the common direction 

 of the two diameters. The name, perhaps, may appear the more 

 appropriate, as it reminds us of the place which the surface holds 

 in the optical theory of biaxal crystals. 



11. THEOREM IY. The biaxal surfaces generated by two 

 reciprocal ellipsoids are themselves reciprocal. 



For if Q and R (Fig. 11) be reciprocal points on the two 

 ellipsoids, abc and a'b'c, a tangent plane at Q will cut OR pen- 

 pendicularly in P ; a tangent plane at R 

 will cut OQ, perpendicularly in N', and 

 the rectangles ROP and NOQ will be 

 equal to each other and to k z (Art. 4). 

 Also if the straight line Oqr, at right angles 

 to the plane of the figure, cut the first ellip- 

 soid in q and the second in r, then (5) the 

 elliptic section QOq will have OQ and Oq 

 for its semiaxes, and the lines OR and Or Fig. 11. 



will be the semiaxes of the other section ROr. Draw, therefore, 

 in the plane of the figure, the right lines OTL and OSM pen- 

 pendicular to the right lines OQN and OPR, making OT, OL, 



* Tratwactionsofihe Royal Irish Academy, Vol. xvi.. pt. ii., pp. 67, 68. Supra, p. 4. 



