SEQUEL TO THE GENERALIZATION. 373 



to me that Professor Airy's treatise entitled Gravitation, published at 

 Cambridge in 1834, is of great value in supplying similar modes of 

 conception with regard to the mechanical origin of many of the prin- 

 cipal inequalities of the solar system. 



Bessel in 1824, and Hansen in 1828, published works which are 

 considered as belonging, along with those of Gauss, to a new era in 

 physical astronomy. 9 Gauss's Theoria Motuum Corporum Celesthm, 

 which had Lalande's medal assigned to it by the French Institute, had 

 already (1810) resolved all problems concerning the determination of 

 the place of a planet or comet in its orbit in function of the elements. 

 The value of Hansen's labors respecting the Perturbations of the Plan- 

 ets was recognized by the Astronomical Society of London, "which 

 awarded to them its gold medal. 



The investigations of M. Damoiseau, and of MM. Plana and Carlini, 

 on the Problem of the Lunar Theory, followed nearly the same course 

 as those of their predecessors. In these, as in the Mecanique Celeste, 

 and in preceding works on the same subject, the Moon's co-ordinates 

 (time, radius vector, and latitude) were expressed in function of her true 

 longitude. The integrations were effected in series, and then by re- 

 version of the series, the longitude was expressed in function of the 

 time; and then in the same manner the other two co-ordinates. But 

 Sir John Lubbock and M. Pontecoulant have made the mean lono-itude 

 of the moon, that is, the time, the independent variable, and have ex- 

 pressed the moon's co-ordinates in terms of sines and cosines of angles 

 increasing proportionally to the time. And this method has been 

 adopted by M. Poisson [Mem. Inst. xiii. 1835, p. 212). M. Damoiseau, 

 like Laplace and Clairaut, had deduced the successive coefficients of 

 the lunar inequalities by numerical equations. But M. Plana expresses 

 explicitly each coefficient in general terms of the letters expressing the 

 constants of the problem, arranging them according to the order of the 

 quantities, and substituting numbers at the end of the operation only. 

 By attending to this arrangement, MM. Lubbock and Pontecoulant 

 have verified or corrected a large portion of the terms contained in 

 the investigations of MM. Damoiseau and Plana. Sir John Lubbock 

 has calculated the polar co-ordinates of the Moon directly ; M. Poisson, 

 on the other hand, has obtained the variable elliptical elements; M. 

 Pontecoulant conceives that the method of variation or arbitrary con- 



9 Abhand. der Akad. d. Wissensch. zu Berlin. 1824 ; and Disquisitiones circa Thto- 

 riam Perturlationum. See Jahn. Gesch. der Astron. p. 84. 



