504 H. A. Bwnstead — Lor entz-Fitz Gerald Hypothesis. 



also with the angle ; and there is also an aberration in the 

 direction of the force. It is important, however, to notice 

 that the variation and aberration of the force is of the second 



v 

 order in the small fraction ^,* instead of the first order as 



has often been assumed in discussing the possible speed of 

 propagation of gravitational force. f 



In the special case before us, the principle of relativity 

 relieves us entirely from the difficulty of even these small varia- 

 tions from the Newtonian law. This is apparent from the 

 general statement of the principle ; but it is of some interest 

 to see how the matter works out in detail. What is subject to 

 observation is not the force but the acceleration ; if we let f\ 

 and f 2 be the components of the acceleration parallel and 

 perpendicular to the common motion of the two bodies, we 

 shall have 



fi = (g) ! = (i-/n* Eoog , 



= fflj = (!_£)* E 



and the resultant of these is along r, so that there is no aberra- 

 tion of the acceleration. With regard to the variation of the 

 acceleration with the distance, it must be remembered that, to 

 an observer moving with the system, apparent distances in the 

 direction of motion, (a?), are greater than their true values in 



the ratio , Thus if the " true " coordinates of P 



(fig. 2) are x, y, the " apparent " coordinates will be x\ y, 



x 

 where x' = —= ==. The " true " distance, r. will be the 



V 1 — p 



radius vector of an ellipse whose major axis is the "apparent" 

 distance, r\ and whose minor axis is *J \ _ w r' ; the polar 

 equation of the ellipse [6 being measured from the minor axis) 

 gives 



" 1 - & sin 2 6 ' 



CO 



x' 2 + if = r'\ 



*This was pointed out by Heaviside, who was the first, so far as I know, 

 to apply the modern electrodynamics to gravitation. Electrician, 1893, July 

 14 and Aug. 4. Electromagnetic Theory, Vol. I, Appendix B. 



f The general reason for this has been put very clearly by Lorentz, 

 Amsterdam Proceedings, II, p. 573, 1900. 



