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LXXYI. The Gravitational Field of a Particle on Einstein's 

 Iheory. By F. W. Hill, M.A., late Fellow of St. John's 

 College, Cambridge, and G. B. Jeffery, M.A., I).Sc, 

 Fellow of University College, London *. 



rpHE solution of Einstein's contracted tensor equation 

 JL G Ml/ =0 for a single attracting point mass may be 

 expressed by means of the line element 



ds 2 = -G^dr 2 -e"-(r 2 d0 2 + r 2 sin 2 0d<j> 2 )+e v dt 2 , . (1) 



where r, 0, d> are polar co-ordinates and \, /jl. v are functions 

 of r only. On substitution into the equations G jav = 0, it 

 is found that the resulting equations are insufficient to 

 determine \, fi, v. One relation between them must be 

 laid down, and this corresponds to the way in which the 

 radius vector r is measured. Perhaps the best known 

 form of the solution is that for which /x = 0, in which case 

 we have f 



d s 2 = - 7 - l dr 2 -r 2 d6 2 -r 2 sm 2 '0d<l> 2 + vdt 2 , . (2) 



where *y=l — 2m/r and m is the mass of the particle. 



De Sitter % gives approximate solutions for which X=/x 

 and \ + 2u + v=0. 



Difficulty is sometimes felt in applications of the solutions 

 for which \ and /i are different, and this is often met by 

 writing r-fra for r in (2) by which, neglecting squares of 

 m/r, we have 



ds 2 = - y -i(dr 2 + r 2 d0 2 + r 2 sin 2 0dcf> 2 ) +ydt 2 , . (3) 



y having the same meaning as before. 



The purpose of this paper is to show that there is an exact 

 solution for which \ = /jl. 



In the general theory with co-ordinates ajj, x 2 , x s , # 4 , 

 we have 



ds 2 = g^dxpdx^, 



where the occurrence of the same suffix twice in any term 

 indicates that that term is to be summed for values 1, 2, 3, 4 

 of that suffix. The sixteen quantities g pcT form a sym- 

 metrical covariant tensor whose determinant is denoted 

 by g. The contravariant tensor g^ is defined to be the 



* Communicated by the Authors. 



t Cf. Eddington, < Report on the Relativity Theory of Gravitation,' 

 p. 46. 



t ' Monthly Notices, Royal Astronomical Society,' Ixxvi. p. 699 

 (1916). 



