QWs Geometrical Propositions 
10. This theorem is taken from a former communication to the Academy*. The 
surface to which it relates, being the wave surface of FResNEt, is one of frequent occur- 
rence in optical inquiries, and it is therefore desirable to give it a distinctive name not 
derived from any physical hypothesis. I shall call it a biaral surface, from the cir- 
cumstance implied in its construction, and adopted as the definition on which the pre- 
ceding 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, per- 
haps, may appear the more appropriate, as it reminds us of the place which the surface 
holds in the optical theory of biaxal crystals. 
11. Turorem IV. The biaxal surfaces generated by two reciprocal ellipsoids are 
themselves reciprocal. 
For if Q and R (Fig. 4.) be reciprocal points on the two ellipsoids, abe and aé’c’, 
a tangent plane at Q will cut OR perpendicularly in P ; a tangent plane at R will 
cut OQ perpendicularly in V7; and the rectangles ROP and NOQ will be equal to 
each other and to A? (4rt. 4). Also if the straight line Ogr, at right angles to the 
plane of the figure, cut the first ellipsoid in g and the second in 7, then (5) the elliptic 
section QOq will have OQ and Og for its semi-axes, and the lines O# and Or will be 
the semi-axes of the other section ROr. Draw therefore, in the plane of the figure, 
the right lines OTL and OSM perpendicular to the right lines OQN and OPR, 
making OT, OL, OS, OM, equal to OQ, ON, OP, OR, respectively ; the angles at 
Sand L being of course right angles. Then it is evident that the point Z’is on the 
biaxal surface generated by the ellipsoid abc, because OT" is perpendicular to the 
plane of the ellipse QOg and equal to the semi-axis OQ ; and by Theorem III. it ap- 
pears that OS is perpendicular to the tangent plane at 7. In like manner, the point 
Mis on the biaxal surface generated by the other ellipsoid a’b’c’, and OL is perpen- 
dicular to the tangent plane at M. Moreover, the rectangles MOS and LOT, being 
equal to the rectangles ROP and NOQ, are each equal to k?, Hence the proposi- 
tion is manifest. 
12. As the ellipsoid whose semi-axes are a,5,c, may be called the ellipsoid abc, so 
the biaxal surface generated by this ellipsoid may be called the biaxal abc; and that 
which is generated by the ellipsoid a’b‘c’ may be called the biaxal a’b’c’. 
13. Prorostrion V. To find what properties of biaxal surfaces are indicated by 
the cases wherein one of the two sections QOq, ROr, in the preceding theorem, 
is a circle. 
Case 1. When QO is a circular section of the ellipsoid abc, the points T' and V, 
(9) in the description of the biaxal surface abc, coincide in a single point m. At this 
* Transactions of the Royal Irish Academy, Vol. XVI. Part II. pp. 67, 68. 
