Sir John W. Lubbock on the Theory of the Moon. 83 



Or, 2. By finding values of a and b considered as variable, which, 

 substituted in the expression 



x Nfe a sin (t + b), 

 will satisfy the differential equation. 



Or, 3. By finding a quantity z, such that 

 x = a sin (t + z + b), 

 so that x is the same function of t + z in the disturbed, as of t in 

 the undisturbed motion. The two latter of these methods are more 

 fully developed. 



This case, illustrative of M . Hansen's method, is investigated at large ; 

 and the author, stating that in the more complicated question of the 

 determination of the longitude of the planets and their satellites, 

 the reasoning is precisely similar, proceeds to exemplify its applica- 

 tion to that problem, in accordance with the expressions previously 

 obtained by Laplace and Pontecoulant. 



The increased labour, trouble, and difficulty of the method pro- 

 posed by M. Hansen, are then pointed out, and the special discus- 

 sion of that method is closed by the following observations : — 



" It is impossible to estimate the policy of adopting M. Hansen's 

 methods sufficiently without considering at the same time the ac- 

 tual state of the lunar theory. M. Plana, after much labour, ex- 

 hibited results obtained with no less skill than honesty of purpose, 

 and containing few numerical mistakes. His results have been ex- 

 amined by me, and more extensively by M. de Pontecoulant ; and 

 as we pursued an independent process, it is probable that few if any 

 errors remain undetected in M. Plana's expressions for the longitude 

 and latitude of the moon. If M. Hansen contemplates the exhibi- 

 tion of the value of z to the same degree of approximation, and not 

 developed according to powers of m, such an expression will be in- 

 capable of verification, and useless. If the value of z be developed 

 according to powers of m, no opinion will be possible of its accuracy 

 until it has been verified independently by other mathematicians, 

 for only the terms which are independent of the eccentricities will 

 be found in M. Plana's expression for the longitude." 



So far as these observations relate to the great work of Plana on 

 the lunar theory, they may, we think, be instructively placed in ap- 

 position, by the student of analysis and of physical astronomy, with 

 the character of M. Plana's labotirs given by Sir John F. W. Her- 

 schel, in his address to the Royal Astronomical Society on the sub- 

 ject of the award of the medal to the Italian astronomer and analyst, 

 which has already been transferred to our pages. (Phil. Mag. Third 

 Series, vol. xviii. p. 153.) 



The most advantageous method of calculating the perturbations 

 of the small planets, is stated by Sir John Lubbock to be that de- 

 scribed by M. de Pontecoulant (Theor. Anal., vol. iii. p. 505), which 

 he briefly recapitulates, and next proceeds to compare with the 

 method of M. Hansen, pointing out the far greater facility of the 

 operations required in the former ; and terminating at once the con- 

 sideration of the subject and the Preface to the work, with the sub- 

 joined reflections : — 



G2 



