It will be useful to give the values of $G_{\mu\nu} - \frac{1}{2} g_{\mu\nu} G$ corresponding to the symmetrical formula for the interval~\Eq{(38.2)}. By varying $\lambda$~and $\nu$ this can represent any distribution of continuous matter with spherical symmetry. We have \[ G = -e^{-\lambda} \bigl(\nu'' - \tfrac{1}{2}\lambda'\nu' + \tfrac{1}{2}\nu'^{2} + 2(\nu' - \lambda')/r + 2(1 - e^{\lambda})/r^{2}\bigr)\Add{,} \] \[\left. \begin{aligned} &G_{11} - \tfrac{1}{2} g_{11} G = -\nu'/r - (1 - e^{\lambda})/r^{2}\Add{,} \\ &G_{22} - \tfrac{1}{2} g_{22} G = -r^{2} e^{-\lambda} \bigl(\tfrac{1}{2}\nu'' - \tfrac{1}{4}\nu'\lambda' + \tfrac{1}{4}\nu'^{2} + \tfrac{1}{2}(\nu' - \lambda')/r\bigr)\Add{,} \\ &G_{33} - \tfrac{1}{2} g_{33} G = -r^{2}\sin^{2}\theta\, e^{-\lambda} \bigl(\tfrac{1}{2}\nu'' - \tfrac{1}{4}\nu'\lambda' + \tfrac{1}{4}\nu'^{2} + \tfrac{1}{2}(\nu' - \lambda')/r\bigr)\Add{,} \\ &G_{44} - \tfrac{1}{2} g_{44} G = \Neg e^{\nu-\lambda} \bigl(-\lambda'/r + (1 - e^{\lambda})/r^{2}\bigr)\Add{.} \end{aligned} \right\} \Tag{(46.9)} \] \Section{47.}{Experiment and deductive theory} \index{Postulates, list of}% So far as I am aware, the following is a complete list of the postulates which have been introduced into our mathematical theory up to the present stage: 1. The fundamental hypothesis of \SecRef{1}. 2. The interval depends on a quadratic function of four coordinate-differences (\SecRef{2}). 3. The path of a freely moving particle is in all circumstances a geodesic (\SecRef{15}). 4. The track of a light-wave is a geodesic with $ds = 0$ (\SecRef{15}). 5. The law of gravitation for empty space is $G_{\mu\nu} = 0$, or more probably $G_{\mu\nu} = \lambda g_{\mu\nu}$, where $\lambda$~is a very small constant (\SecRef{37}). \PageSep{105} No.~4 includes the identification of the velocity of light with the fundamental velocity, which was originally introduced as a separate postulate in \SecRef{6}. In the mathematical theory we have two objects before us---to examine how we may test the truth of these postulates, and to discover how the laws which they express originate in the structure of the world. We cannot neglect either of these aims; and perhaps an ideal logical discussion would be divided into two parts, the one showing the gradual ascent from experimental evidence to the finally adopted specification of the structure of the world, the other starting with this specification and deducing all observational phenomena. The latter part is specially attractive to the mathematician for the proof may be made rigorous; whereas at each stage in the ascent some new inference or generalisation is introduced which, however plausible, can scarcely be considered incontrovertible. We can show that a certain structure will explain all the phenomena; we cannot show that nothing else will.