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LXXVI. Advance of Perihelion of a Planet. 



To the Editors of the Philosophical Magazine. 

 Gentlemen, — 



I AM much obliged to Mr. Pearson for pointing out an 

 error in my paper in the May number of the Philo- 

 sophical Magazine. 



If we take the expression 



ydt 2 -r 2 d6 2 - 1 dr 2 for ds 2 



and use the transformation 



H 1+ £)- 



we get for the new ds 2 



These gravitation potentials satisfy Einstein's equations, 

 and there is no restriction as to the magnitude of m. 

 It is therefore quite legitimate to use this expression for 

 ds 2 , and it makes the velocity of light at any point in the 

 sun's gravitational field independent of direction. 



Using the method of the Calculus of Variations, I find 

 that the differential equation of the path of a planet without 

 any restriction as to the magnitude of m is 



/ mu\ d mu 



2 4 



d 2 u in \ 2 / / , fd 



d6 2 It 2 mu 



where h is a constant. 



If now we make m small, we have as an approximate 

 solution 



u=(l-\-e cos 0)/L, where L = A 2 /m. 



Hence the equation for a second approximation, if small 

 terms are neglected, is 



d q u 1 2m ,_, 9N 6me cos 6 



which gives the correct amount for the advance of the 

 perihelion of Mercury. 



Recess, Yours faithfully, 



4th Sept., 1920. Alex. ANDERSON. 



