6 Sir J. Lubbock on the Perturbations of Planets. 



And then in each of these examples Q will contain no term of 

 which the coefficient exceeds -1 in numerical amount. 



If ^ - 



1+P + Q 



{\+P]-^ = A + B-\-C+D, 



^=(l + P+Q)-t, 5=|(a^, C=~(^B, 



o o 



The calculation of {1 +P+ Q}~^ is much facilitated by the 

 use of a table which gives the values of the coefficients of 



{1— ^cosa}"^ 

 -J and 1 +Pcontam terms multiplied by ~j-cosj', -jsmj', -j, 



and -j2, and none others require consideration. If the eccen- 

 tricity of m' is small, they may be developed in terms of ^', the 

 mean anomaly of m'; and it will be sufficient to consider the 

 terms depending on 



cos ^'5 cos 2^', cos 3^', sin g', sin 2^', sin 3f'. 



If/ is the time reckoned from the time of the perihelion 

 passage of w, r<r', 



w/ = y — esin w ; 



and if 360° — n'c' is the mean anomaly of m' at the time of the 

 perihelion passage of w, 



^' = n't—n'c'; 

 and if 



— v — n'c'=n, 

 n 



. r n! . \ 



cos?fc' = cos?-< It ^sino >-, 



L w J 



sin ?^'=sin I < U' e sin > ; 



and as — is a fraction, cos /^'and sin i^' can be developed in 



a series rapidly converging, and containing explicitly only the 

 variable quantity u. 



[To be continued.] 



