CURVILINEAR CO-ORDINATES IN THREE DIMENSIONS. 299 



surface 17, drawn parallel to a. Hence the normals at A and B will intersect 

 at some point 0. Let us call OA, the radius of curvature of 17 in the plane 

 of a, R^, then 



AB : DC ':: R : R 



Hence R ^ = '' similarl J **** 



These values of the radii of curvature are true for any system of orthogonal 

 surfaces, but since in this case a=y, we find 



or the principal radii of curvature of the surface 77 are equal to each other at 

 every point of the surface. Hence the surfaces rj must be spherical. 



Since a = j3 = y, the other families of surfaces f and must also be spherical. 



Now take one of the points where three spherical surfaces meet at right 

 angles for the origin, and invert the system. These spheres when inverted 

 become three planes intersecting at right angles, and the other spheres become 

 spheres, each intersecting at right angles two of these planes. Hence their 

 centres lie in the lines of intersection of the planes. Let a, b, c be the distances, 

 from the point of intersection of the three planes, of the centres of spheres 

 belonging each to one of the three systems, and let their radii be p, q, r 

 respectively. Then the conditions that these spheres intersect at right angles 



are 



whence p' = a\ q s = b\ r 2 = c 2 ; 



or all the spheres pass through the origin, and each system has a common 

 tangent plane. Hence if we take these three planes as co-ordinate planes, and 

 write 



we have rp = R?, a constant quantity, 



x _y _ z 



?"c; 



382 



