271 



of the plane of each orbit to the invariable plane is absolutely con- 

 stant ; and that the two nodes are always in opposition to each 

 other, and move with a uniform angular velocity round the inva- 

 riable plane. 



This theorem is then extended to a system of n bodies moving 

 about a central predominant body ; and it is shown that the ag- 

 gregate effect of the disturbing forces of such a system upon the 

 plane of any one of the bodies can always be represented by stating 

 that its node upon a certain fixed plane revolves with a uniform 

 angular velocity, the plane of the orbit always remaining at the 

 same inclination to the fixed plane. The rate of this angular move- 

 ment, and the coordinates of the fixed plane upon which the move- 

 ment takes place, are found by means of formulae of remarkable 

 simplicity. These three quantities may be ascertained once for all 

 for each planet (viz. the inclination of the fixed plane on which the 

 node moves to any coordinate plane, the longitude of the node of the 

 fixed plane in relation to any coordinate line, and the angular rate of 

 movement of the node of the orbit upon this fixed plane), and, when 

 once ascertained, may be regarded as fixed elements of the planet, 

 from which the position of the plane of its orbit can always be deter- 

 mined without the use of tables. 



It is then shown that a system of the form 



x" + n*x=P n &c., 



where n 2 and P#, P^, and P are any variables, may be solved by the 

 same set of final integrals, and the same values of x', y', and z' t 

 by supposing the elements a, , </>, \^ t z, I, and p to become variable. 

 These elements are those of an ellipse tangential to the actual 

 curve of motion ; and the following formulae are obtained for their 

 variation : 

 Let 



and let (putting T for nt+p) 



a cos cos T b sin tf> sin T = , 

 a sin cos T -f b cos sin T=ij ; 

 VOL. IX. U 





