10 THE ORBIT OF NEPTUNE. 



S log rrzS log a 

 -\-mv{2 i'h -{-leE—\e'E}cosN — is ehnv {eL — e W— 3^ } cos {N—f) 

 + 1 mv {eL — e W— E } cos {N— f) + i~, e'mv { eZ — e Pr+ 3^ } cos (V+/) 

 — lmv{eL — eW+E}cos{N-\- /) — etc. 



+ 1 emv {eL — e W— E } cos {N— 2/) (5) 



— lemv{eL — eW + E}cos{N-\-2f) 

 + a ernv {eL — e W— 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 



8v = e'{f+f+f")smN 



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 fs, it should include all. This end may be attained 

 by omitting from h, its differential coefficients, and 7i cot 4-, all terms of a higher 

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

 the form 



h = e" (^1 + e'- JC2 -f «'"' ^4 + • • • } 



if we limit ourselves to the power s + 1? we should put 



, dh dxi 



h := e'xi; a -j- == e a -y— ; 

 da da 



~ = .se'^-i;ci+(5 + 2)e'^ + i;£2 



s7iCot4' = se"~^Xi +se'* + ^ ( — Isci + Xa). 



§ 7. Perturbations of latitude. 



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



dQ' a'ii' dR 



dt sin ^' cos 4'' d^' 



dip' _ a'n' dR (6) 



dt sin ^' cos '^' ' dd' 



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

 of the plane of the orbit, we have 



dR_dR d^ dR do dR d?,' dR d(J clR dy^ 

 d^' ~ d?i d^' ~*~ dco dip' d?i' d^' da' dip' "*" dy d<p' 

 dR _ dR cU dR dco dR dW dR dco' dR dy^ 

 d¥~dX dd'^ d^' W^dX' M''^ iU (W ^ d^ dW 



