2 Sir J. Lubbock on the Perturbations of Planets 



fesses to give a general solution of the problem otherwise than 

 by mechanical quadratures, is clue to M. Hansen. This im- 

 portant work is translated in the Co7in. des Temps for 1847. 

 That great mathematician has considered the case when r< r', 

 that is, when the disturbed body is inferior; and has illus- 

 trated the question by the numerical calculation of the per- 

 turbations of the comet of Encke by Saturn. M. Hansen 

 develops the disturbing function according to multiple angles 

 of the eccentric anomaly of the disturbed planet literally; and 

 first, according to multiple angles of the true anomaly of the 

 disturbing planet; M. Hansen next converts the cosines and 

 sines of the multiple angles of the true anomaly of the disturb- 

 ing planet into sines and cosines of multiple angles of the mean 

 anomaly of that planet; so that finally the disturbing function 

 is exhibited in terms of the eccentricanomaly of the disturbed 

 planet and the mean anomaly of the disturbing planet ; but 

 those series which serve to give the sines and cosines of the 

 multiples of the true anomaly, in terms of sines and cosines 

 of the mean anomaly, are not very convergent ; and the pro- 

 cess becomes extremely laborious, even in the case which M. 

 Hansen has considered, in which, in consequence of the great 

 distance of Saturn, the approximation does not require to be 

 carried nearly so far as in the case of the perturbations of the 

 same comet by Jupiter, and in many others which may require 

 consideration. Moreover, in this as in every other mode 

 which can be devised of developing the disturbing function 

 literally, all quantities must be retained of a given order ; 

 although when they are of a different sign, in many instances 

 they destroy each other ; but such reductions cannot be fore- 

 seen. The numerical substitutions are also extremely labo- 

 rious, in consequence of the multitude of terms which have to 

 be considered. 



As the disturbing function, and others which require to be 

 integrated, are finally exhibited by M. Hansen in terms of 

 two variables, such that direct integration is impossible, M. 

 Hansen has recourse to the integration ^ar joar/zW, in which 

 each term by integration gives rise to a series of other terms, 

 the nature of which is complicated. 



The method which I propose differs from that suggested 

 by M. Hansen in every particular. Instead of attempting a 

 literal development, I insert the numerical values of the ellip- 

 tic constants in the earliest possible stage : by this means the 

 radical, which expresses the mutual distance of the planets, is 

 explicitly a function of sines and cosines of various angles 

 with numerical coefficients. When r < r', I develop in terms 

 of the eccentric anomaly of m, after having obtained expres- 



