50 



1. To shew by a compendious process that the niraii motion of 

 the Lunar perigee may be satisfactorily accounted for by tiie New- 

 tonian Theory of Gravitation. 



2. To shew that the term e cos {v—w) occurring in the first ap- 

 proximation for the value of u becomes by succeeding approxima- 

 tions of the form e cos (cv — n) and therefore if we use this form in 

 the first approximation and then add other terms of the proper 

 combination of angles with indeterminate coefficients, as Laplace 

 has done, considerable advantage will be thus afforded of obtaining a 

 sufficiently exact value of*/, for the purpose of an exact determination 

 of the Lunar orbit. No farther difficulty afterwards occurs ex- 

 cept that arising from the great length of the computations. 



It is remarkable that in the above expressions for the mean 

 motion of the perigee, the part \- j/i* arising from the first 

 approximation is nearly equal to ?- m" (5 i-5m ) A ; because 

 m = ,0748 and A =, 1814 and therefore {5 + 5m) A = 0,97 or 

 unity nearly. But this coincidence is merely accidental, had the 

 periodic time of the moon been double what it is, the first approxi- 

 mation would not have given ; of the whole motion, but had the 

 Moon revolved about the Earth in one day, the Newtonian re- 

 sult, according to the principles of the 9th sect, of the first book 

 of the Principia, would have been within -^-^ part of the whole, 

 instead of being within one half only. 



Hence we see the results of the investigations of Machin, 

 Walmsly, Mathew Stewart and Frisi are all quite erroneous, as 



their methods give the mean motion of the perigee :r -5- nearly, 



which only affords an exact result when the periodic time of the 

 Moon = _'_ that of the Earth nearly. 



