relating to the Moon's Orbit. 281 



the moon's orbit are the same as those in an elUptic orbit de- 

 scribed by the action of the central force ^. Now as the first 



power of the disturbing force has been fully taken into account, 

 it is impossible that this result can be true. Some step in the 

 reasoning inconsistent with the hypothesis of the approximation 

 must have been taken, which it is necessary to correct before we 

 advance further. 



After various fruitless attempts I discovered at length the 

 following solution of this difficulty. Let it be supposed that the 

 equation (C) contains tlic disturbing force as a factor. In that 

 case, putting the equation under the form 



— rdr 



dt= 



^ -Cr^^^f,r-h^+'^'''^ 



and substituting a + r—a for r, it is necessary to expand to the 

 second power of r—a in the small term, in order to embrace 

 small quantities of the second order. This proceeding is legiti- 

 mate, because it is only required that the value of r should be 

 consistent with the hypothesis that the true longitude of the 

 moon difi"ers by a small quantity from a mean longitude. The 

 approximation being conducted in this manner, the two equations 

 (B) and (C) readily give by integration the following results, 

 which include all small quantities of the second order : — 



r=a + ae cos -v/r 





/m'f/ir \i_ the moon's periodic time (p) 

 xa'^C'^ ) ~" the earth's periodic time (P) 



We have now to satisfy, by ineans of the arbitrary constants, 

 the condition on which alone the above results can be obtained, 

 viz. that the disturbing force is a factor of the equation (C). 



By substituting in that equation the values of r and -r given by 



the first and second of the above equations, it will appear that 

 the condition is satisfied if the arbitrary constants //. and C are 

 Phil. May. S. 4. Vol. 7. No. 45. April 1854. U 



