320 WILSON AND MOORE. 



tangent to the curve X, since a depends only on the direction of the 

 tangent X^""^ as shown by (65). 



// a curve is projected on a ylane {or any plane space) passing through 

 a tangent line to the curve, the curvature of the projection at the point of 

 tangency is equal to the projection of the curvature of the given curve at 

 that point. To see this note first that the elements of arc on the given 

 curve {ds) and the projected curve {ds') differ at the point of contact 

 by infinitesimals higher than the second because their ratios involve 

 the cosine of a small angle. The elements ds and ds' are therefore 

 equivalent for first and second derivatives. The projection of a 

 vector r on a space Sk represented by a unit vector St is 



Then, 



r'= (- l)^-Ur-S,).S,. 



''=%=^-^^'~'^'^) .S.= (-l)-(c.S.).S.. 



We could in like manner show that if we project a curve on a plane 

 space through the osculating plane of the curve, the torsion of the 

 projection is equal to projection of the torsion at that point: and so on. 

 We have c = a + t^I. If we project the curve X on the tangent 

 plane to the surface, we have for the curvature of the projection, 



c' = - (c-M).]vi = - [(a + 7ii)-(|x'n)]-(|x'n) 

 = - Tl-d^'n) = 7^1. 



Hence the curvature of the projection upon the tangent plane is- 7 

 in magnitude. The invariant 7 is therefore the curvature of the 

 projection of the curve upon the tangent plane, — this is called the 

 geodesic curvature (which must be clearly distinguished from the 

 curvature of the geodesic tangent to the curve). 



If we project on a normal plane determined b}- 1 and any normal n 

 we have 



c' = - [(a + 7ii)-(|xn)]-(|xn) = (a.n)-n. 



Hence the curvature of the projection is the component of a along n. 

 If n had coincided with a in direction, the curvature of the projection 

 would have been a. 



Consider now a section of the surface by a normal space S„_i of 

 n — 1 dimensions containing the tangent line | and the normal space 



