824 Mr. Hill and Dr. Jeffery on the Gravitational 



minor of g p<T divided by g, and we introduce the Christoffel 

 three index symbols: 



Einstein's equations of gravitation in free space are 

 then 



G aT = -^ {o-t, a} + {o-«, /3}\t/3, a} 



+ *£h log ' - g - { ° T ' a>< lv log v/ ~ g = °- (4) 



In onr case, we have from (1) 



g n = -e\ g 22 = —r 2 e"; g 3B = -r 2 sin 2 <9^, # 44 = g", 



# p<r = ,'V = it p ^ <r, 

 and <7 PP = l/g pp . 



It is then found that of the 40 Christoffel symbols only 

 9 survive, and their values are easily calculated. Sub- 

 stituting these into (4), it is found that there are only three 

 equations between X, /x, v. As given by de Sitter *, these 

 are : 



2 1 



fjL +iv + - fjb — - X -t 7j/x ^ — ^X /x — \ X v + \v - — U. 



- 1 + **-*[l + 2r/*' + i^V' + M"' -■>•') 

 K+iiZ+ivV + iv'-iV) = 0. 



Putting X=//,, we have 



V' + i^ + ix 7 -iW + i^ 2 = 0, (5) 



X" +?V + ^ r +iV» + iXV =0, (6) 



-K -i*' -iXV-i^ 2 = 0. (7) 



* ' Monthly Notices, Royal Astronomical Society,' lxxvi. p. 712 (1916). 



