SURFACES IN HYPERSPACE. 323 



from all possible normals. As a is the curvature of one section and P 

 of the orthogonal section, we have the result that: The sum of the 

 normal curvatures in two orthogonal directions is independent of the 

 directions. The sum ci + P will be written as 2h, where h is called the 

 mean (vector) curvature}^ 



Since a + P is constant and the vector a — P is the other diagonal of 

 the parallelogram on a and P, the vector a — P must pass through a 

 fixed point on the mean curvature vector (namely, the extremity of 

 that vector) and the termini of a and P must describe a central curve 

 about that point. 



If we introduce a new pair of orthogonal directions X' making an 

 angle 6 with Xr we have 



X'r = X,cos9 + Xrsin^, X'r = XrCOsS — Xrsin^, 



whence cos0 = ^X^'^W, sin0 = Z:X<^)X'„ 



X, = X',cos9 - \'r^md, \r = \'rs\\\d + X'^cos^, (86) 



X(^) = X'^^^cos0 - X'('-)sin0, X^ = X'^^^sin^ + V^'^co^d. 



Now from (65) we have, in vector form, 



a = S^XC-^X^^-^y^ , p = S.,X(^'X(^)y,, , (87) 



If we substitute for the X's in terms of the X"s we get the relations 

 between a, P, |x and a', p', |J.' for different directions in the surface. 

 Thus 



a = d'cos-0 - 2|i'sin0 cos0 + P' sin^^ 



P = P'cos-0 + 2|i'sin^cos0 + a'sin^^ (88) 



|i = jx'(cos20 - sin20) + (a' + P')sin0cos0, 



33 By mean curvature we designate the half sum of the curvatures a and p. 

 This is a true mean. In three dimensional surface theory the mean curvature 

 often if not generalh' stands for the sum of the curvatures (See Eisenhart, 

 Differential Geometry, page 123;. E. E. Levi, loc. cit., page 69). We may 

 cjuote as Levi does a theorem of Killing: the sum of the squares of the mean 

 curvatures of the n — 2 three dimensional surfaces obtained by projecting an 

 » -dimensional surface on n — 2 mutually perpendicular three spaces passing 

 through the tangent plane, is constant. That is, is independent of the n — 2 

 normals selected to determine the three -spaces. The value of this invariant 

 is (2hj-. The theorem is of course merely the scalar form of our relation (67')- 



