710 Mr. W. Wilson on Space- Time Manifolds 



Integrating these last equations, we obtain 



/jl = — Am log r + A, ^ 

 Am 



v = 



l_2m' 0gr + B '| (20) 



* = T^» lo g'- + c >J 



where A, B, and C are constants of integration. 



We shall see, when the equations of motion are written 

 down, that m can be identified with the mass per unit length 

 of the z axis. When m is zero the square of the element of 

 length will take the form 



d 5 2 = „ dr 2 - dz 2 - r 2 d$ 2 + dt 2 • 



i.e., when w = 0, 



\ = 0, fi = 0, v = 0. 

 Therefore 



A = 0, B = 0, 0= 0, 



and the values of \, p, and v are 



/x = — Am log r, 



Am . 



log r, 



l-2m 



8771 2 



X = -z ^— logr. 



1 — 2m ° 



The square of the element of length (1) is therefore 



The Christoffel expressions which do not vanish are : — 

 _ 4<m 2 1 , _ -2m 



r / _ 2m JrSi) r 3 - x 



J- 22 — * ? ' l 13 — » 



1- 



1 33 — 1 ' 



1 44 = 



2m 1 



14- 1 _ 2m '/ 



2?7l (4m -1) 



1-2, 



