[150] ∑e2m√a = c, ∑tan2im√a = c′, where m is the mass of any planet, a, e, i are the semi-major axis, eccentricity, and inclination of the orbit. The equation is true as far as squares of small quantities, and therefore it is indifferent whether or not tan i is replaced as in the text by i.
[151] Nearly the whole of the “eccentricity fund” and of the “inclination fund” of the solar system is shared between Jupiter and Saturn. If Jupiter were to absorb the whole of each fund, the eccentricity of its orbit would only be increased by about 25 per cent., and the inclination to the ecliptic would not be doubled.
[152] Of tables based on Laplace’s work and published up to the time of his death, the chief solar ones were those of von Zach (1804) and Delambre (1806); and the chief planetary ones were those of Lalande (1771), of Lindenau for Venus, Mars, and Mercury (1810-13), and of Bouvard for Jupiter, Saturn, and Uranus (1808 and 1821).
[153], The motion of the satellites of Uranus (chapter XII., [§ 253] 255) is in the opposite direction. When Laplace first published his theory their motion was doubtful, and he does not appear to have thought it worth while to notice the exception in later editions of his book.
[154] This statement again has to be modified in consequence of the discoveries, beginning on January 1st, 1801, of the minor planets (chapter XIII., [§ 294]), many of which have orbits that are far more eccentric than those of the other planets and are inclined to the ecliptic at considerable angles.
[155] Système du Monde, Book V., chapter VI.
[156] In his paper of 1817 Herschel gives the number as 863, but a reference to the original paper of 1785 shews that this must be a printer’s error.
[157] The motion of Castor has become slower since Herschel’s time, and the present estimate of the period is about 1,000 years, but it is by no means certain.
[158] More precisely, counting motions in right ascension and in declination separately, he had 27 observed motions to deal with (one of the stars having no motion in declination); 22 agreed in sign with those which would result from the assumed motion of the sun.
[159] The method was published by Legendre in 1806 and by Gauss in 1809, but it was invented and used by the latter more than 20 years earlier.