Since the sun far exceeds all other bodies of the solar system in mass, the motion of each particular planet is primarily conditioned by the gravitational field of the sun. Under its action the planet describes, according to Newton's theory, an ellipse (Kepler's law), the major axis of which—defined by connecting the point of its path nearest the sun (perihelion) with the farthest point (aphelion)—is at rest, relative to the stellar system. Upon this elliptic motion of a planet there are superimposed more or less considerable influences (disturbances) due to the remaining planets, which do not, however, appreciably alter the elliptic form; these influences partly only call forth periodical fluctuations in the defining elements of the initial ellipse (i.e. major axis, eccentricity, etc.), partly cause a continual increase or decrease of the latter. In this second kind of disturbance are also to be classed the slow rotation of the major axis, and consequently also of the corresponding perihelion, relative to the stellar system; which has been observed in the case of all planets. For all the larger planets, the observed motions of the perihelion agree with those calculated from the disturbing effects (except for small deviations which have not been definitely established, as in the case of Mars); on the other hand, in the case of Mercury the calculations give a value which is too small by 43" per 100 years. Hypotheses of the most diverse description have been evolved to explain this difference; but all of them are unsatisfactory. They oblige one to resort to still unknown masses in the solar system: and, as all the searches for masses large enough to explain this anomalous behaviour of Mercury prove fruitless, one is compelled to make assumptions about the distribution of these hypothetical masses, in order to excuse their invisibility. In view of these circumstances, there is no shade of probability in these hypotheses.
According to Einstein's theory, a planet, at the distance of Mercury for instance, moves, under the action of the sun's attraction, along the "straightest path," according to the equation:
The
's can be derived from the differential equations, which were given for them above, and which result from the assumed sole presence of the sun and the planet being regarded as a mass concentrated at a point. Einstein's developments give the ellipse of Kepler too as a first approximation for the path of the planet: at a higher degree of approximation, however, it is found that the radius vector from the sun to the planet, between two consecutive passages through perihelion and aphelion, sweeps out an angle, which is about 0.05" greater than 180°; so that, for each complete revolution of the planet in its path, the major axis of the path—i.e. the straight line connecting perihelion with aphelion—turns through about 0.1" in the sense in which the path is described. Therefore, in 100 years—Mercury completes a revolution in 88 days—the major axis will have turned through 43". The new theory, therefore, actually explains the hitherto inexplicable amount, 43 seconds per 100 years, in the motion of Mercury's perihelion, merely from the effect of the sun's gravitation. (The deviations due to such disturbances would only differ very inappreciably from the values obtained by Newton's theory in the case of the remaining planets.) The only arbitrary constant which enters into these calculations is the gravitational constant which figures in the differential equations for the gravitational potentials
as has already been mentioned on [page 50]. This achievement of the new theory can scarcely be estimated too highly.
The reason that a measurable deviation from the results according to Newton's theory occurs in the case of Mercury, the planet nearest to the sun, but not in the case of the planets more distant from the sun, is that this deviation decreases rapidly with increasing distance of the planet from the sun, so that it already becomes imperceptible at the distance of the earth. In the case of Venus, the eccentricity of the path is, unfortunately, so small, that it scarcely differs from a circle: and the position of the perihelion can, therefore, only be determined with great uncertainty.
Of the other two possibilities of verifying the theory, one arises from the influence of gravitation upon the time an event takes to pass. How such an influence can come about, will be evident from the following example: According to the new theory, an observer cannot immediately distinguish whether a change, which he observes during the passage of a certain event, is due to a gravitational field or to a corresponding acceleration of his place of observation (his system of reference). Let us assume ah invariable gravitational field, denoted by parallel lines of force in the negative direction of the