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 a + P tvill be xcritten as 2h, where h is called the 

 mean (vector) curvaturc.^^ 



Since a + P is constant and thr \ector 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 = XrC0s9 + Xrsin^, X'r = X^cos^ — Xrsin0, 

 whence cos9 = SX^^^X',, sin0 = SX(^)X'„ 



X, = X'rCOSd - X'rSin^, X, = X'rSind + X'rCOS^, (86) 



X('-) = X'^^)cos0 - X'^'hrnd, X(^^ = y^^Vnid + X'^cos^. 



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



a = ZrsX^r'X^'^yrs', P = S..X('->X(^)y,, , (87) 



\i = 2„X''-)X(^^V„ = S.,X(''^X(^)y., . 



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

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

 Thus 



a = a'cos^e - 2|i'sin^ cos0 + p' sin-0 



P = p'cos^e + 2|x'sin0cos0 + a'sin-0 (88) 



|x = |i'(cos20 - sin'd) + (a' + P')sin0cos9, 



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 generally stands for the sum of the curvatures (See Eisenhart, 

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

 quote as Levi does a theorem of lulling : the sum of the squares of the mean 

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

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



