For a circular orbit the Newtonian theory gives \[ m = \omega^{2} r^{3} = v^{2} r, \] the constant of gravitation being unity. Applying this to the earth, $v = 30$ km.\ per~sec.\ $= 10^{-4}$ in terms of the velocity of light, and $r = 1.5 · 10^{8}$~km. Hence the mass~$m$ of the sun is approximately $1.5$~kilometres. The mass of the earth is $1/300,000$th of this, or about $5$~millimetres\footnotemark.\footnotetext {Objection is sometimes taken to the use of a centimetre as a unit of gravitational (i.e.\ gravitation-exerting) mass; but the same objection would apply to the use of a gram, since the gram is properly a measure of a different property of the particle, viz.\ its \emph{inertia}. Our constant of integration~$m$ is clearly a length and the reader may, if he wishes to make this clear, call it the gravitational radius instead of the gravitational mass. But when it is realised that the gravitational radius in centimetres, the inertia in grams, and the energy in ergs, are merely measure-numbers in different codes of the \emph{same} intrinsic quality of the particle, it seems unduly pedantic to insist on the older discrimination of these units which grew up on the assumption that they measured qualities which were radically different.} More accurately, the mass of the sun, $1.99 · 10^{33}$ grams, becomes in gravitational \index{Gravitational mass of sun}% \index{Sun, gravitational mass of}% units $1.47$~kilometres; and other masses are converted in a like proportion. \PageSep{88} \Section{40.}{The advance of perihelion} \index{Perihelion!advance of}% The equation \Eq{(39.5)} for the orbit of a planet can be integrated in terms of elliptic functions; but we obtain the astronomical results more directly by a method of successive approximation. We proceed from equation~\Eq{(39.61)} \[ \frac{d^{2}u}{d\phi^{2}} + u = \frac{m}{h^{2}} + 3mu^{2}. \Tag{(40.1)} \] Neglecting the small term~$3mu^{2}$, the solution is \[ u = \frac{m}{h^{2}} \bigl(1 + e\cos(\phi - \varpi)\bigr), \Tag{(40.2)} \] as in Newtonian dynamics. The constants of integration, $e$~and $\varpi$, are the eccentricity and longitude of perihelion.