20] RELATING TO THE MOON'S ORBIT. 137 



But even if we were to suppose, with Professor Challis, that the equation 

 (C) contains the disturbing force as a factor (of which, as already remarked, 

 no proof whatever is given), it would not follow, as is inferred by him, 

 that I? C must be equal to /u. 2 . On the contrary, it is evident that the 

 required condition would be satisfied if A 2 C differed from /u. 2 by any quantity 

 involving the disturbing force as a factor ; whence it would follow that e 

 must be some function, indeed, of the disturbing force, but it could not be 

 decided what function. 



Professor Challis attempts to find the relation between r and t by 

 direct integration of the equation 



dr 



dt = 



mV 



fdr\ 2 

 Now it may be remarked that (->-) is a small quantity of the second order 



which vanishes twice in each revolution, and that the difference between the 



fdr\* 

 complete value of ( -7- ) and the approximate value 



_ S~1 . ^f** 



r 2 r ua" 



which is used instead of it in the above equation, is a periodic quantity 

 of the third order. 



Hence it follows that the quantity 



~ If 2/x, mV 2 



r 2 T 2a' 3 



may vanish for values of r different from those which make (-37) vanish, 



\MJ 



and that it may even become negative for actual values of r, which (-j-j 

 itself can never do. 



Therefore the coefficient of dr in the above differential equation may 

 become infinite, or even imaginary, within the limits of integration, so that 

 it is not surprising that Professor Challis should have met with such 

 difficulties in performing the integration. 



The relations between r, 6, and t, given in page 281 (which profess to 

 include all small quantities of the second order), are said to be derived from 

 the equations (B) and (C). It is easy to see, however, that they do not 



A. 18 



