304 WILSON AND MOORE. 



and hence 



a„ = 7r-Ys. (44') 



Differentiate covariantly by the rule for a composed system (§19) 



arst = = yrt'Vs-h yr'Yst. 



As this relation holds for any r, s, t, we have also 



= Ytr-Ys + yrYrs, = Yrs'Jt + Yr'^ts. 



As y is a function of xi, x^, the second covariant derivative is commu- 

 tative like an ordinary derivative (§ 19), and by addition and sub- 

 traction among the three equations we have 



TfrYr, = or Yr-y.t = 0, ' (45) 



for all values of r, s, t. 



22. Normal vectors. Equations (45) mean that the second 

 covariant derivatives Yst are perpendicular to the first derivatives yr. 

 As yr lies in the tangent plane and as y,t is perpendicular to yr for 

 r = 1, 2, we infer that the vectors Yst lie in the normal plane ^* to 

 the surface.^^ If z and w are any two unit vectors in the normal 

 plane we may write 



Yrs = brsZ + CraW (46) 



with Z'Z=ww=l, z«w = 0, (47) 



z-yr= wyr= 0, (47') 



brs = Z'Yrs, Crs = W'Yrs- (47") 



Here z, w are particular unit vectors in the normal plane and con- 

 sequently are invariant of the coordinate system, .ti, x^; they are, 



24 By the normal plane we mean the plane which is completely perpendicular 

 to the tangent plane, that is, such that any line in one is perpendicular to every 

 line in the other. These planes intersect in only one point. 



25 One great advantage of the covariant derivative is therefore brought to 

 light; for the ordinary second derivative of y would not lie in the normal 

 plane. 



