39 



It is entirely by accident that their results are exact in the case 

 of the Moon. Had the periodic time of the moon been different 

 from what it is, observations would have pointed out the error of 

 their conclusions. 



The differential equations and the expressions for the forces'acting 

 on the moon might have been |taken from Laplace or other authors, 

 but it is hoped that the manner in which the differential equations 

 are deduced, may afford an excuse for the introduction of the in- 

 vestigation. The method of limits, the undisputed invention of 

 Newton, seems deserving of more attention than is nowpaid to it. 



The differential equations are at once deduced without the inter- 

 vention of rectangular co-ordinates which in the present case would 

 uimecessarily have added to the length of the process, but in other 

 respects advantage has been taken of the facilities offered by La- 

 place's investigation of the Lunar motions which may be consi- 

 dered as the most satisfactory and as the most accurate that has 

 appeared. 



