10 THE ORBIT OF NEPTUNE. 



5 log r =. & log 



+ mv { 2 i'h + J ?# i e 3 ^ } cos JV A rm v { c c TF 3# } cos (N/) 

 + lmv{eL eWE}c,os(N- /) + A^mvj eZ eTF+ 3# } cos (V-|-/) 

 \mv{eL eTF+ J}cos(JV + /) etc. 



+ i em* { eL e W E } cos (N 2/) (5) 



\emv\eL eW + E] cos(N + 2/) 

 + H e 2 mi/ { eL eW E } cos (N 3/) 

 etc. 



By these formulae all the perturbations of the longitude and radius vector have 

 been computed, except that the computation was so conducted as to reject all 

 terms above a certain order with respect to the eccentricities. The sum of all 

 the factors (functions of the ratio of the mean distance) of any power of the 

 eccentricity in any coefficient in the perturbations of the co-ordinates will generally 

 be much smaller than each individual factor, as we shall presently show. If, for 

 example, we have 



to = e* (/+/+/") sin N 



the sum /+/+/" will, in general, nearly destroy itself, being much smaller than 

 the individual components,/,/', and/". Hence, if the computation is arranged so 

 as to include any one of the/'s, it should include all. This end may be attained 

 by omitting from /*, its differential coefficients, and 7<coti]/, all terms of a higher 

 order with respect to the eccentricities than the assigned limit. Thus, h being of 

 the form 



if we limit ourselves to the power + 1> we should put 



7 dh (7x, 



n d 'x^; a - 7 - = da =-= ', 

 da da 



s7t cot ^ = set*- 1 Xl + se' s + ' ( J Xl + x a ) . 



* 



7. Perturbations of latitude. 



The equations which determine the change in the plane of a planet's orbit are 



dtt a'ri dR 



_ 



dt ' ~ sin q>' cos fy dq>' 



d<p' _ a'ri dR (6) 



dt ' sin <p' cos fy ' dO 1 



R being a function of /I, ^,', w, o', and y, each of which depends on the position 

 of the plane of the orbit, we have 



d R _ dR d^ dR <k>_ , dR dX dR d^ dR 

 d$ ~ <fa d$ ~*~ fa d$ + dX d<j>' ~T~ dti d$ "^ dy 

 dRdR fa dR rfw dR dX dR <k>'dR 



