Manchester Memoirs, Vol. Ivi. (191 2), No. 4. 7 



an element of the line of curvature and PQ an element of 

 th.c j)erpenciicular line of curvature. Let S/\, SQ be 

 lines of curvature through S. Let PIIG, QH'G, RH and 

 SH' be the normals to the surface at /-*, Q, R and 5. 

 Then PG, PH, QH' are ultimately the radii of curvature 

 of the normal sections through PQ, PR and QS respec- 

 tivelv. "Lei PG^R', PH^R, QH' = R + dR. Now the 



Fii:. -x. 



plane PGQ is obviously the normal plane of the curve 

 PR at /-*, and since RH and SH' intersect the plane 

 RHH' is the normal plane at A*. These two planes 

 intersect in ////', which is therefore perpendicular to 

 the osculating plane POR. Draw H'N perpendicular 

 ioPG. 



