[ 378 ] 

 LIX. Proceedings of Learned Societies. 



ROYAL ASTRONOMICAL SOCIETY. 



[Continued from vol. v. p. 147.] 



May 13, /^|N a new Solution of the Problem of Planetary Pertur- 

 1853. " bation. By Professor Encke. (Extract of a Letter 

 to the Astronomer Royal.) 



" The subject of the small planets has continued to occupy my 

 attention so much, that I could not communicate with you until I 

 had relieved my mind of this load of anxiety. I consider myself 

 very fortunate on this account in being enabled, through the meri- 

 torious services of my excellent friend Dr. Brunnow, to transmit to 

 you in the annexed paper the mean elements of Flora, with the per- 

 turbations of the same planet as produced by Jupiter and Saturn. 

 From the mode in which the calculation of these elements has been 

 worked out, I am induced to entertain a confident hope that we 

 have it now in our power to compute the perturbations of the first 

 power of the masses for all the small planets, not even excepting 

 Pallas, and that the time bestowed on the calculations is not immo- 

 derately long. 



" The perturbations produced by Jupiter upon Flora, which pos- 

 sesses the advantage of being pretty distant from the disturbing 

 planet, but which, on the other hand, has an excentricity of 9° and 

 an inclination of 6°, will admit of being computed in four weeks 

 without any immoderate haste ; and even with respect to those of 

 Pallas, I would pledge myself, without hesitation, to compute them 

 for Jupiter and Saturn in the course of a winter, if nothing unusual 

 interfered with the discharge of my ordinary duties. 



" On a former occasion I had the honour of transmitting to you 

 a copy of a paper which I communicated to the Academy, and in 

 which I proposed an indirect method. This was not suitable to the 

 object in view, as any one may easily convince himself, since the 

 approximations conducted to the required result only in a few rare 

 cases. Still the proposed method has not been useless, since it has 

 given me the form which is most convenient for calculation, of the 

 integral of the assigned differential equation which always reappears 

 in the problem. By a mere simple consideration of the combinations 

 which present themselves in this inquiry, the following solution has 

 been obtained. 



" If the differential equation has the form 



*+5 i=:s ( o '' oos(, ' M - , " M ' ) )' 



in which the cosine may be exchanged for the sine, and M, M' 

 denote the mean anomalies of the disturbed and disturbing planets, 

 the mean motions of which are represented by jx and // ; then, in 

 every case, will the coefficient of cos (t'M— i'M') or of sin(iM— i'M') 

 be thus determined by integration :— 



