84 Mr. J. C. Adams on Professor Challis's new Theorems 



or 



fi m' fi? 



to the same degree of approximation as before. 



Hence we see that the values of a, in the two cases supposed, 

 differ by a quantity of the second order. Consequently the dif- 

 ficulty into which Professor Challis is led by the conclusion that 

 these values are the same, disappears, and the solution of the 

 difficulty with it. 



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 h^Q> must be equal to /x^. 

 On the contrary, it is evident that the required condition would 

 be satisfied if /t^C differed from yu,^ 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\^ 

 -jr) is a small quantity of the 



second order which vanishes twice in each revolution, and that 



/dr\^ 

 the difference between the complete value of i-rf and the ap- 

 proximate value 



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

 quantity of the third order. 



Hence it follows that the quantity 



_p_A2 2^ f^2 



may vanish for values of r different from those which make 

 \lf) ^^^^^^f ^^^ ^^^* ^^ '^^y ^v®^ become negative for actual 

 values of r, which f ^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 in- 



