Derivation of Symmetrical Gravitational Fields. . 19 

 the corresponding Fourier series being 



Q-2n (COS 6) 



= 2-«^{cos(2n + l)^ + |.g±|oos(2n + 3)^ 



1.3 An + 2 .4,11 + 4, ,_ cx/1 "1 • ' J/%N 



+ ^4- 4n + 3.4n + 5 COS(2 " + 5 ^ + -j- ' <«> 



Both series can be combined into the single formula, 

 whether n be odd or even, 



Q» (cos 6) 



= 22 " +1 ran { cos ( " + X) " + "2 • £r! cos (« + 3 > * 



2 . 4 2n + 3 . 2?2 + 5 x ' J v y 



The limitation of 6 between and tt is clearly not 

 necessary. 



II. The Derivation of Symmetrical Gravitational Fields. By 

 F. D. Mur'naghan, Ph.D., Assoc. Prof. Applied Math., 

 Johns Hopkins University *. 



IN a recent issue of the Philosophical Magazine, Hill 

 and Jeffery f call attention to a symmetrical gravita- 

 tional field which differs somewhat from the classical one 

 due to Schwarzschild and Einstein. In the usual treatment 

 of this problem a field is said to be symmetrical about a 

 point if the form for (ds) 2 is invariant under linear ortho- 

 gonal transformations of the " Cartesian coordinates " 

 (# l5 x 2i %z) ; then a transformation is made to " polar co- 

 ordinates " r, 6, cf>, where r = \/ x 2 -\- x 2 2 -r- x z 2 etc. and an 

 appeal is made, in choosing a form for (ds) 2 , to the 

 corresponding form in Euclidean space. Now the essential 

 assumption in the relativity theory of a permanent gravita- 

 tional field is that the physical space-time continuum of 

 four dimensions is no^-Euclidean ; the term " Cartesian 

 coordinates " for a non-Euclidean space requires definition, 

 and the equation for r given above assumes an underlying 



* Communicated by Prof. J. S. Ames. 



t Phil. Mag. May 1921. The result of this paper has already 

 appeared in a paper by Wevl, Ann. der Physik, liv. p. 132 (1917). 



2 



