NOTES TO BOOK VI. 131 



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. Ponte'coulant have made 

 the mean longitude of the moon, that is, the time, the 

 independent variable, and have expressed the moon's 

 co-ordinates in terms of sines and cosines of angles in- 

 creasing proportionally to the time. And this method 

 has been adopted by M. Poisson (Mem. Inst. xm. 1835, 

 p. 212). M. Damoiseau, like Laplace and Clairaut, has 

 deduced the successive coefficients of the lunar inequalities 

 by numerical equations. But M. Plana expresses ex- 

 plicitly each coefficient in general terms of the letters 

 expressing the constants of the problem, arranging them 

 according to the order of the quantities, and substi- 

 tuting numbers at the end of the operation only. By 

 attending to this arragement, MM. Lubbock and Pon- 

 tecoulant have verified or corrected a large portion of 

 the terms contained in the investigations of MM. Da- 

 moiseau 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 of arbitrary constants may most conveniently 

 be reserved for secular inequalities and inequalities of long 

 periods. 



MM. Lubbock and Pontecoulant have made the 

 mode of treating the Lunar Theory and the Planetary 

 Theory agree with each other, instead of following two 

 different paths in the calculation of the two problems, 

 which had previously been done. 



Prof. Hansen, also, in his Fundamenta Nova Investi- 

 gationis Orbitce verce quam Luna perlustrat (Gothce, 1838) 



K2 



