708 Mr. W. "Wilson on Space-Time Manifolds 



viewed from a system (r, z, <£) rotating in the inertia system 

 with angular velocity (l— VA) times that of the particle in 

 the inertia system. 



We now proceed to the investigation of the manifold (1), 

 where 



ds 2 — - f\ dr 2 - f 2 dz 2 —j 3 r' 2 d<j> 2 +/ 4 dt 2 , 



and the functions 



-Air), -Mr,, -r'h(r), AW 



must be solutions of the equations (5). 



For our purpose there is no loss in generality in using any 

 function of r in place of r. We therefore write 



-f/,(r) = -r" (10) 



and obtain for the square of the line element the form : 



ds 2 = -f^-fzdS-r'dtf+fidt 2 , . . (11)* 



where the accent has been dropped after making the sub- 

 stitution (10). 



It is easier to deal with the equations (5) if we write ds 2 

 in the form 



ds* = -e*dr 2 '-e*dz 2 -^d<l> 2 + e v dt 2 , . . (12) t 



where e K =f\(r) etc., and \, /x, and v are functions of r 

 which have to be determined. 



The Christoffel expressions F* y are then found to be 



( r'jj = \x' } ( r 2 12 = \\x ', 



i p » = ? 



I 

 iF M = iv'e»-\ {I n u=iv', 



V — _-l,,'/-A I 1 



1 22 — 2^ e 5 ' ™ X 



where 



or or or 



Substituting these expressions in the equations (5) the 



* I am indebted to Dr. Wilson, of King's College, for su<>geeling the 

 investigation of a line element of this type. 



f -Uddington, Report. 



