MATHEMATICAL NOTES. 131 



Hence p is generally the perpendicular on a tangent of an ellipse 

 of eccentricity e cos a. Hence the chord touches such an ellipse. 

 The latus rectum is diminished in the same ratio as the ec- 

 centricity. 



2. The relation between the long inequalities of two mu- 

 tually disturbing planets, may be easily found without having 

 recourse to the development of the disturbing function. 



Let w, m\ be the masses of the planets, a, a, the major axes 

 of their orbits, n, ri, their mean motions, A, A', twice the areas 

 described in 1" ; then we have 



a* 



fju being the mass of the Sun, in comparison with which the 

 masses of the planets are neglected, so that it is the same for 

 both. Taking the logarithmic differentials of these equations, 

 and replacing the differentials by differences, we find 



A^__3Aa Ari _ 3 Aa' 

 n ~ 2~cT' ~nT~ "~2~oT* 



But by the principle of the conservation of areas, 



mh + m'ti = const. 



so that wAA + m'AA' = 0. 



Now the orbits being supposed circular, we have 



AA 1 Aa AA' 1 Aa 

 hence T" = o ~ -rr = r'- 



h 2 a h 2 a 



Therefore we have 



Aw t An' _ Aa a _ AA h' _ m f a'* t 

 n ' n' Aa' " a AA' * A ~ m #4 ' 



and - - and r are the inequalities due to the disturbances, so 

 n n 



that their ratio is thus given. 



92 



