Intelligence and Miscellaneous Articles. 161 



motion are 



df I dx* 



dt 2 I y dy ' 



and the variations of the four elements are given by the formulae 



d ± i a -^ +b VTT« 



dt g a 2 —b 2 



dR ~' fg 



9 dR 



db i b ^^ a V im 



dt 



or 



dR , dR 



, a— — — o — 



act, / 1 db da 



VI 



dt V g a 2 -b 2 



,dR dR 



o — — a — ■ 



dr _ I db da 



dt" g TiTl? 



As in the Mecanique Celeste, the perturbations are of two kinds, — 

 the one periodic and producing no appreciable effect ; and the other 

 the so-called secular, and which, accumulating themselves, end, in 

 virtue of their action repeated during a sufficiently long period, in 

 producing an appreciable effect, however feeble the disturbing force. 

 These latter it is important to calculate. In order to limit the ques- 

 tion, we supposed that the perturbing force emanated from a fixed 

 point A situated in the horizontal plane at a distance h from the 

 centre O of the ellipse, and varied inversely as the square of the 

 distance, which would be the case if there were an attractive or re- 

 pulsive spherical mass fixed in this point. We supposed, moreover, 

 that the initial value of the minor axis was null — that is, that the 

 oscillation of the pendulum was originally plane. Under these con- 



dR 

 ditions, it is easily seen that the secular part of -rr contains b as a 

 jp mo 



factor; further, ~- only contains periodic terms. It follows that 



-j- and -T- are small quantities of the second order (the perturbing 



force being taken as a small quantity of the first order), and therefore 

 may be neglected. Hence the direction of the major axis and its 

 magnitude are virtually invariable. But the two other elements 

 undergo secular perturbations, and we get approximately 



db_ 1 fl_ dR dr__l!_dR t 

 dt ~ a\/ g \da! dt ~ a g da 



