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



in the x r \ we have (by E alerts Theorem on homogeneous 

 functions, or directly) 



x r '^ = 2F (V umbral). 



On multiplying then, the equations ^ r ( ^ — 7) = 



r ; 6 ' H ^dx r dr\dx r 'J 



by x r ', and using r as an umbral symbol, we find 



O^r «T V d^r dr 



verifying the fact that F is constant along the extremals — 

 the constant in the case of the minimal geodesies being zero. 

 The differential equations are 



O X m 



(m, n umbral), 

 or gmOOn + \vnn, r\ x m * x n ' = 0, 



Since I'dffm d,</r»A , , / _ p dffr 



On multiplying by ^ rp and using r as an umbral symbol, 



this gives 



x p " + \mn, p}x m 'xn = 0. 



Writing, for convenience, 



we have, on account of the absence of t and <£ from F, 

 the integrals 



BF , BF 



— , = const. ; 4— p = const. 



ck 09 



or ^' = const. ; g d (p' = const., 



which give, on substituting the values of g± and g 3 , 

 (1 - 2m\r)t' = C 1 r 2 sin 2 . </>' = h, 



where C and h are constants. The equation for is found, 

 by writing p = 2 in 



a?/' -f {mw, p}x m r ,x n ' — 0, 



to be 



A {fO') -r 2 sin (9 cos 6 \$) 2 = * ; 

 * Of. Eddington's Report, Equation (30.12), p. 49. 



