

23] SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. 161 



In order to give a more complete view of the subject, however, and 

 to obviate the necessity of my returning to it in a controversial manner, 

 I shall not confine myself to the observations of M. de Ponte"coulant, but 

 shall likewise say a few words in reply to the objections of M. Plana and 

 those of M. Hansen. I shall also take the opportunity of making some 

 preliminary remarks which may tend to remove certain misapprehensions, 

 which I have reason to believe exist in some minds with respect to the 

 real nature of the matter in dispute. 



First, then, I would call attention to the fact that the question is a 

 purely mathematical one, with the decision of which observation has nothing 

 whatever to do. It may be simply stated thus : if the eccentricity of the 

 Earth's orbit be supposed to change at a given uniform rate and very 

 slowly, what will be the corresponding rate of change, according to the 

 theory of gravitation, in the mean motion of the Moon ? Now the solution 

 of this question is effected by means of a purely algebraical process, the 

 validity of each step of which admits of being placed beyond all possible 

 doubt. 



What conclusion must be drawn, then, supposing that ancient obser- 

 vations should shew that the secular variation of the Moon's mean motion 

 is different from that which, according to theory, is due to the known 

 change of the eccentricity of the Earth's orbit ? 



Why, simply this ; that the mean motion of the Moon is affected by 

 some other cause or causes, besides the variation of eccentricity which has 

 been taken into account. This fact, if established, would be a most inte- 

 resting one, and might put us on the traces of an important physical 

 discovery. It is not difficult to imagine the existence of causes which 

 may affect the mean motion of the Moon, but whether it were so or not, 

 any question respecting the validity of a mathematical process must be 

 decided on mathematical grounds alone, quite independently of the agree- 

 ment or disagreement of theory and observation. 



In the case before us the mathematical question as stated above may 

 be greatly simplified, without its ceasing to involve the point which is in 

 dispute. The values of the secular acceleration given by M. Plana's theory 

 and mine, differ in terms which are independent of the eccentricity and 

 inclination of the Moon's orbit ; consequently in deciding which of the 

 theories is right, we may suppose the eccentricity and inclination to vanish. 



A. 21 



