4 Sir J. Lubbock on the Perturbations of Vlanets 



and to integrate par parties. I have obtained the law of the 

 coefficients in the series which resuhs in this process, and they 

 are highly convergent. I am confident that, by the processes 

 which I have attempted thus so briefly to describe, the per- 

 turbations of planets moving in orbits, however eccentric and 

 inclined, may be calculated jvith nearly as great facility as 

 they are given by existing methods, in orbits nearly circular 

 and in the same plane, and may be exhibited in tables, giving 

 their values for an indefinite period, if required. If these me- 

 thods, which I have described in detail elsewhere, possess the 

 advantages which I ascribe to them, I hope the time is not 

 distant when the perturbations of Pallas and of some of the 

 comets may be reduced to a tabular form; but as the labour 

 will be very considerable, it will be necessary to limit the in- 

 quiry in the commencement to the cases of the greatest emer- 

 gency. 



Although my methods are specially adapted to the deter- 

 mination of the perturbations of bodies moving in eccentric 

 orbits which cannot be developed in terms of the mean mo- 

 tions, yet they embrace also the case of a planet moving in an 

 orbit nearly circular ; and it is easy to show in what manner 

 the labour is increased by the greater eccentricity. If the 

 reciprocal of the radical which expresses the mutual distance 

 of the planets be called 



the chief difficulty arises in developing {1+P}~5. If the 

 numerical values of the elliptic constants are introduced, 



1 +P=1— ^1 cos «!— ^2^08 «2 + &c.. 



Ay, A^ &c. are numerical coefficients, which I here suppose 

 ranged in the order of their numerical magnitude. I make 



{1— -4iCosa,}{l— ^gcosag} • • • • {1— ^;COSa,.} = H-P+Q, 



including a limited number of terms in 1-|-P+Q. 



{l-^jcosai}"^, {1— .^gcosag}"^ &c., 



can be obtained at once by means of a table. But as the 

 coefficients given by such a table do not readily furnish, by 

 interpolation, the values required unless it be considerably 

 extended, I take for A^, A^, &c. the nearest value given by 

 the table, and I leave the residue to form part of Q. In this 

 way it will generally be found sufficient to include not more 

 than six terms in 1+P+Q, so as to leave Q consisting of 



