582 Prof. F. D. Murnaghan on the Deflexion of a 



difficulty may be overcome * by considering, instead of 

 the arc integral, the integral 



No matter what the parameter t is, this has an extreme 

 value (zero) when extended over a light-ray, since the 

 quadratic form (ds) 2 is supposed one-signed. (In the 

 discussion of the new-minimal geodesies which give 

 the path of a material particle it was convenient f to 

 take for t the arc length along the extremal ; but this 

 cannot be done here, since this length is no longer a 

 variable but a constant — zero.) Writing for convenience 



gmn%m%n—^> We find 



\O«/a=0 tj Q \OXr OOL cW ^CL^T J j a=0 



the limits of the integral being independent of a J. On 

 integrating the second term of the integrand by parts, after 

 interchanging the order of differentiation with respect to a. 

 and t, we find 



SI = -r — j 8x r 

 qx t 



i JcVcUv dr"dxr'J 



(r umbral), 



where the symbols 1, 2 refer to the ends of the curve 

 and ^ = (^1 dot. If the ends are first kept fixed, 



\O*/a=0 



we get the familiar Euler-Lagrangian equations 



h%r dr ~dx r ' 



* Cf. Weil, H.. Ranm. Zeit, Materie, Dritte Auflage (Berlin, 1919), 

 p. 210. 



Attention is directed to the fact that the integral l(x) is not, properly 

 speaking, a line integral attached to the curve C(a). Its value depends 

 not mevely on the curve C(«) alone, but on the particular parameter r 

 chosen to specify points on it. 



t Eddington's Report, p. 48. 



% This does not imply that the end-points of all the curves Ca coincide. 

 Although r takes the same values at these end-points, the functions 

 x r =x r (T,a) do not necessarily do this, since there is a second variable a 

 on which they depend. 



