316 A Short History of Astronomy [Cn. xi. 



The second is precisely similar, save that eccentricity is 

 replaced by inclination.* 



The first of these propositions establishes the existence 

 of what may be called a stock or fund of eccentricity shared 

 by the planets of the solar system. If the eccentricity cf 

 any one orbit increases, that of some other orbit must 

 undergo a corresponding decrease. Also the fund can 

 never be overdrawn. Moreover observation shews that the 

 eccentricities of all the planetary orbits are small ; conse- 

 quently the whole fund is small, and the share owned at 

 any time by any one planet must be small. t Consequently 

 the eccentricity of the orbit of a planet of which the mass 

 and distance from the sun are considerable can never 

 increase much, and a similar conclusion holds for the 

 inclinations of the various orbits. 



One remarkable characteristic of the solar system is 

 presupposed in these two propositions; namely, that all the 

 planets revolve round the sun in the same direction, which 

 to an observer supposed to be on the north side of the 

 orbits appears to be contrary to that in which the hands 

 of a clock move. If any planet moved in the opposite 

 direction, the corresponding parts of the eccentricity and 

 inclination funds would have to be subtracted instead of 

 being added ; and there would be nothing to prevent the 

 fund from being overdrawn. 



A somewhat similar restriction is involved in Laplace's 

 earlier results as to the impossibility of permanent changes 

 in the eccentricities, though a system might exist in which 

 his result would still be true if one or more of its members 

 revolved in a different direction from the rest, but in this 

 case there would have to be certain restrictions on the 

 proportions of the orbits not required in the other case. 



* S e*m */a = c, S tan~im \/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. 



f 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. 



