tr 



V ' 



VII, 



') 



r 



S^marlcs on tJie usual Bemonstratim of the permanency of the 



I— 



solar system^ tenth respect to the- McceTttricities and luclina- 

 ' timx s of th^ rb its of the Plane ts^- 



BY NATHANIEL BOWDITCH, iiL. D. 



object of this paper is to show 



noted erqwfctlbir 



of Gondiiion betvveen the eccentricities of the orbits of the planets, 

 and from which it is generally inferred ttiat those orbits will foiv 

 ever remain nearly circular, is not sufficient fbr a complete demon- 

 stration of this permanency, though it may render it highly prob- 

 able by considerations of analogy. 



The equation referred to is contained in Book I, § 57 of the 

 ^'Mecanique Celeste,^' where it is shewn that if m, m', m\ &c. 

 represent the masses of the planets ; a, a', a", &c. their mean dis- 

 tauces from the sun ; e, e', e% &c. the eccentricities of their or- 

 bits expressed in parts of o, a', a\ &c. respectively ; we shall have 



by neglecting quantities of the order of the fourth power of the 



\ 



eccentricities 



w e^ va + to' e'^ v^ + m" e^, v^' + & 



constant. 



The signs of the radicals being all positive ; and we may 



lect the terms depend 



a- 



the newly discovered pi 



ellites and the comets, on account of 



tion must be satisfied, whatever changes may be 



1 bis equa- 

 in the val- 



of e, e% e^, &c. by the secular variations of those qw 



Xow in the present state of the solar 



sy 



ery term of the 



• 



