Einstein's correction to the perihelion of Mercury has removed the principal discordance in the table, which on the Newtonian theory was nearly $30$~times the probable error. Of the $15$~residuals $8$~exceed the probable error, and $3$~exceed twice the probable error---as nearly as possible the proper proportion. But whereas we should expect the greatest residual to be about $3$~times the probable error, the residual of the node of Venus is rather excessive at $4\frac{1}{2}$~times the probable error, and may perhaps be a genuine discordance. Einstein's theory throws no light on the cause of this discordance. \Section{41.}{The deflection of light} \index{Deflection of light}% \index{Light!deflection in gravitational field}% For motion with the speed of light $ds = 0$, so that by~\Eq{(39.62)} $h = \infty$, and the orbit~\Eq{(39.61)} reduces to \[ \frac{d^{2}u}{d\phi^{2}} + u = 3mu^{2}. \Tag{(41.1)} \] The track of a light-pulse is also given by a geodesic with $ds = 0$ according to~\Eq{(15.8)}. Accordingly the orbit~\Eq{(41.1)} gives the path of a ray of light. We integrate by successive approximation. Neglecting $3mu^{2}$ the solution of the approximate equation \[ \frac{d^{2}u}{d\phi^{2}} + u = 0 \] is the straight line \[ u = \frac{\cos\phi}{R}. \Tag{(41.2)} \] Substituting this in the small term~$3mu^{2}$, we have \[ \frac{d^{2}u}{d\phi^{2}} + u = \frac{3m}{R^{2}} \cos^{2}\phi. \] A particular integral of this equation is \[ u_{1} = \frac{m}{R^{2}} (\cos^{2}\phi + 2\sin^{2}\phi), \] so that the complete second approximation is \[ u = \frac{\cos\phi}{R} + \frac{m}{R^{2}} (\cos^{2}\phi + 2\sin^{2}\phi). \Tag{(41.3)} \] Multiply through by~$rR$, \[ R = r\cos\phi + \frac{m}{R} (r\cos^{2}\phi + 2r\sin^{2}\phi), \] or in rectangular coordinates, $x = r\cos\phi$, $y = r\sin\phi$, \[ x = R - \frac{m}{R}\, \frac{x^{2} + 2y^{2}}{\Chg{\surd(x^{2} + y^{2})}{\sqrt{x^{2} + y^{2}}}}. \Tag{(41.4)} \] \PageSep{91} The second term measures the very slight deviation from the straight line $x = R$. The asymptotes are found by taking $y$ very large compared with~$x$. The equation then becomes \[ x = R - \frac{m}{R}(±2y), \] and the small angle between the asymptotes is (in circular measure) \[ \frac{4m}{R}. \] For a ray grazing the sun's limb, $m = 1.47$ km., $R = 697,000$ km., so that the \index{Eclipse results}% deflection should be~$1''.75$. The observed values obtained by the British eclipse expeditions in 1919 were \[ \begin{array}{l@{\qquad}c} \text{Sobral expedition} & 1''.98 ± 0''.12 \\ \text{Principe expedition} & 1''.61 ± 0''.30 \\ \end{array} \]