344 Mr. Stubbs on a new Method in Geometry. 



of the corresponding surface in the first, and similarly the sur- 

 face of normals by inverse of that in original surface ; hence 

 it may be seen that the normals to the inverse surface along 

 the inverse points of a line of curvature meet consecutively, 

 or the inverse of a line of curvature on a surface is the line of 

 curvature of the inverse surface; or if the line of curvature of 

 a surface be known, that of its inverse surface is had by de- 

 scribing a cone with the pole as vertex and passing through 

 the line of curvature on direct surface, the line in which it 

 pierces the inverse surface is a line of curvature. 



Hence the umbilici of one surface correspond to the um- 

 bilici of the second ; and in general to a tangent plane cor- 

 responds a polar sphere, and to all the singular points of one 

 surface correspond singular points of the second. 



From the known theorems of surfaces of the second order 

 may be deduced numberless theorems of surfaces of the fourth 

 order by inversion, similarly as in plane curves. I shall con- 

 fine myself to the inverse of the central ellipsoid, which is 

 Fresnel's surface of elasticity in the wave theory of light. 



1. By the construction for the tangent plane to an inverse 

 surface, the tangent plane to the surface of elasticity may be 

 found from knowing the tangent plane of the ellipsoid. 



2. The lines of curvature on that surface may be found by 

 producing the cone passing through the lines of curvature on 

 the ellipsoid to meet it, but 



3. The intersection of a confocal ellipsoid and hyperboloid 

 determines the line of curvature on either, as they cut at right 

 angles; hence as the equation of the inverse ellipsoid is 



Ou u & 



— 5- + tts- -\ s- = (# 2 + y 2, -+ 2 2 ) 2 > if two inverse central sur- 

 er b l (r 



faces of the second order have their constants connected by 

 the condition a 2 — a' 2 = b 2 — b n = c 2 — c' 2 , and they intersect, 

 they cut at right angles, and their intersection is the line of 

 curvature on either. 



4. By subtracting the equations 



-^+^+-^=(* 2 + y + * 2 ) 2 > ™d 



we get in the case above mentioned, 



or a'" w 0"- c 2 c 12 ~ ' 

 the equation to a cone of the second order: the intersection 



