SURFACJES IN HYPERSPACE. 321 



N„_2 . The geodesic tangent to X has for curvature a, as has been 

 seen, and hence its osculating pLane |xa hes in S„_i . The geodesic 

 has therefore three consecutive points in S„_i, i. e., to infinitesimals 

 of the third order the geodesic coincides with the normal section and 

 hence the curvature of the geodesic and of the normal section are equal. 

 Conseciuently : we may interpret a as the curvature of the normal section 

 of the surface. So far as curvature is concerned we may replace the 

 normal section by the geodesic. 



Now c = a + 7'n is the curvature of any section (for the curve on the 

 surface and the section of the surface by a space S„_i containing the 

 osculating plane of the curve are exchangeable as far as curvature is 

 concerned) and the projection of C = a + 7TI on the normal is a itself. 

 Hence we have Meusnicr's theorem that: The projection of the curvature 

 of any section on the normal section is the curvature of the normal section. 

 (Meusnier's theorem may be found in various degrees of generalization 

 in the literature, e. g., in Levi's long article cited in note 2). 



As c = a + 7'n, C- = ar-\- 7^ and hence: The magnitude of the curva- 

 ture of a section is the square root of the sum of the squares of the normal 

 and geodesic curvatures. 



33. Interpretation of P and |x. If we treat d"^ as we treated (/| 

 we find 



^ = 1^ - 7|. (82) 



as 



Now T| is a normal to the curve \ lying in the surface and dT]/ds is 

 the rate of change of this surface normal. If we consider the geo- 

 desic tangent to X we have 7 = 0, and hence : The vector [>• {which is 

 perpendicular to the surface) may be interpreted as the rate of change of 

 the surface-normal to a geodesic. In three dimensions the surface 

 normal is the binormal of the geodesic (with the proper convention as 

 to sign) and hence in three dimensions ix is the torsion of the geodesic 

 tangent to X. In higher dimensions this interpretation is no longer 

 valid because the osculating three space of the geodesic need not 

 contain the tangent plane M. 

 We may next form 



ds ds 



^ = S.il.^^ = 2,il,.X(^^ = P - IS.^.XC-', 

 ds d.s 



