322 WILSON AND MOORE. 



by which we denote the rate of change of |, f\ with respect to the arc 

 upon the orthogonal trajectories Xr of Xr. 



Now the derived system (fr for Xr is related to the derived system (pr 

 for Xr by the relation 



'Pr = — (fr, 



as may be seen from (63) and (64). Let 



7 = 2.^.X('-) = - ^rvM\ (83) 



Then 7 is the geodesic curvature of the normal trajectories X. 



f.-V'-y^, '^=P + 7l. (84) 



as a"s 



By reasoning like that previously used we note that : The vector p 

 is the norvial curvature of the orthogonal trajectories of X. Moreover, 

 as the relation of t| to | is the same as that of — | to "H we may interpret 

 }!• as the rate of change of the surface-normal to the geodesic tangent 

 to X changed in sign, that is, the rate of change of the surface-normals 

 tangent to normal geodesies are equal and opposite (vectors). This 

 corresponds to the theorem in three dimensions that the geodesic tor- 

 sions in perpendicular directions are equal and opposite. In three 

 dimensions, where fx is scalar the inference is immediate that there 

 are a pair of orthogonal directions for which the geodesic torsion 

 is zero — the lines of curvature. But in the general case ji is a vector 

 and may change sign without passing through zero, and we cannot 

 affirm the existence of directions for which the rate of change of the 

 surface-normal vanishes. 



34. The mean curvature. From (68) we get, 



2rsa("Vr. = a2,X,.X'^^ + |a2,(XA<'-> + XA^")) + P2:XX(^\ 

 or 



2„a(") y.. = a + p. . , (85) 



This equation from its form on the right appears to depend on X, 

 but from the form on the left is seen to be independent of X. Hence: 

 The vector a + P is an invariant normal vector associated with a point of 

 the surface — it is a special and particularly important normal selected 



