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also with the angle $\scriptstyle {\theta ;}$ 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 order in the small fraction $\scriptstyle {{\frac {v}{\text{V}}},}$^[1] instead of the first order as has often been assumed in discussing the possible speed of propagation of gravitational force.^[2]

In the special case before us, the principle of relativity relieves us entirely from the difficulty of even these small variations 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 $\scriptstyle {f_{1}}$ and $\scriptstyle {f_{2}}$ be the components of the acceleration parallel and perpendicular to the common motion of the two bodies, we shall have

${\begin{array}{c}\scriptstyle {f_{1}={\frac {(\mathbf {E} )_{1}}{m_{1}}}={\frac {(1-\beta ^{2})^{\frac {3}{2}}}{m_{0}}}\mathbf {E} \cos \theta }\\\scriptstyle {f_{1}={\frac {(\mathbf {E} )_{2}}{m_{2}}}={\frac {(1-\beta ^{2})^{\frac {1}{2}}}{m_{0}}}\mathbf {E} \sin \theta (1-\beta ^{2})}\end{array}}$

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and the resultant of these is along $\scriptstyle {r,}$ so that there is no aberration 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, $\scriptstyle {(x)}$, are greater than their true values in the ratio $\scriptstyle {{\frac {1}{\sqrt {1-\beta ^{2}}}}.}$ Thus if the "true" coordinates of $\scriptstyle {P}$ (fig. 2) are $\scriptstyle {x,~y,}$ the "apparent" coordinates will be $\scriptstyle {x^{\prime },~y,}$ where $\scriptstyle {x^{\prime }={\frac {x}{\sqrt {1-\beta ^{2}}}}.}$ The "true" distance, $\scriptstyle {r,}$ will be the radius vector of an ellipse whose major axis is the "apparent" distance, $\scriptstyle {r^{\prime },}$ and whose minor axis is $\scriptstyle {{\sqrt {1-\beta ^{2}}}r^{\prime };}$ the polar equation of the ellipse ($\scriptstyle {\theta }$ being measured from the minor axis) gives

${\begin{array}{c}\scriptstyle {r^{2}={\frac {r^{\prime 2}(1-\beta ^{2})}{1-\beta ^{2}\sin ^{2}\theta }};}\\\scriptstyle {x^{\prime 2}+y^{2}=r^{\prime 2}.}\end{array}}$

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1. 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.
2. The general reason for this has been put very clearly by Lorentz, Amsterdam Proceedings, II, p. 573, 1900.