MATHEMATICAL NOTE*. 



Stability of Eccentricities and Inclinations. The equation 

 for proving the stability of the eccentricities and inclinations 

 of the planetary orbits may, as has been shown by Laplace, 

 be deduced from the principle of the conservation of areas, joined 

 to the fact of the invariability of the major axes. 



Let m be the mass of a planet, h be twice the area de- 

 scribed by the radius vector about the sun in an unit of time 

 projected on the ecliptic, and i the inclination of its orbit to 

 the ecliptic ; then, by the principle of the conservation of 

 areas, 



2 (mh) = const. 



Every term under the sign of Summation is positive, be- 

 cause all the planets move round the sun in the same direction. 

 But 



h = {a (I - e 2 )}* cos t = a* (1 - e 8 ) 4 (1 + tan 2 *)"*. 



Now 6 and i are small at the present time; hence, if we 

 neglect their fourth power and the products of their squares, 

 we have 



= <^{l-Je 2 -itan 2 ;}. 



Hence 2 {wa* (I - Je 2 - \ tan 2 *)} = const. 



But since the major axes have no secular inequalities 



2 (mat) = const.; 

 hence the preceding equation is equivalent to 



2 (ma e* + ma? tan 2 *) = const. 



Now the left-hand side of the equation being small at the 

 present time, the second side is also small, and therefore the 

 first side is always small, and therefore 



2 (wa^e 2 ) and 2, (ma* too? f) 

 are both always small. 



* Cambridge Mathematical Journal, No. XVIII. Vol. m. p. 290, May, 1843. 



