Sir J, Lubbock on the Perturbations of Planets. 87 



tities P;^, ; into explicit functions of sines and cosines of mul- 

 tiples of / is an arithmetical process ; while the calculation of 

 the quantities A„,,- in p. 29 of M. Hansen's paper, is literal 

 or algebraical, 



I regard as very difficult any development of the disturbing 

 function, either literal or quasi-literal^ when the eccentricity 

 of the disturbed body is considerable and the perturbations 

 are large, as in the case of Encke's comet disturbed by Jupiter; 

 and if such were possible, the replacement of the numerous 

 symbols by numbers at the conclusion, would be an operation 

 of almost insurmountable difficulty. On the other hand, in 

 performing an arithmetical development according to the rules 

 which I have invented, not only no quantity can be introduced 

 which has a numerical value beneath any given limit (say 

 beneath unity in a given decimal place), but it is equally im- 

 possible, except by a numerical mistake, that any quantity 

 which is above that limit can be omitted. The developments 

 may also be effected by mechanical quadratures, as explained 

 by M. dePontecoulant(T//eor.y47m/.,vol.iii.), or by the method 

 given by M. Le Verrier in the first number of the Developpe- 

 ments sur plusieurs points de la Theorie des Perturbations des 

 Planetes. 



If ar, J/, z are rectangular co-ordinates of a comet or planet 

 7W, and/' the true anomaly, 



x-r\v cos/+a- sin/}, 



^=;'{^cos/+(7sin/}, 



2= rig- cos/+^ sin/ }. 



*Fj 8^5 O'i (7' Sr> i ^''s constants which depend only on the 

 elliptic constants of the planet tw, and such that 



^ = cos TT + 2 sin^ — sin (tt — v) sin v, 



3"= — sin 7r+2 sin^— cos (tt — v) cos V, 



^ = sin TT — 2 sin^ — - sin (w — v) cos V, 



(J = cos TT— 2 sin^ — cos (tt — v) cos v, 

 3"= sin(7r— v) sin i, 



^= cos (tt— v)sin I. 



* I have had occasion to use so many alphabets in the course of the 

 work from which this is extracted, that I have had recourse to this artifice 

 of reversing the letters in order not to use the same symbol in two different 

 significations. 



