106 Prof. Challis on certain Questions 



Irinnp 



dt 



dv 

 After deducing from the equation (C), by making -t; =0, the 



equation for finding the apsidal distances, viz. 



A«-2Mr + Cr»- ^^/'=0, . . . . (P) 



Mr. -Adams goes on to obtain values of the apsidal distances by 

 an approximate solution of this equation . The process he adopts, 

 though conducted in an unusual manner, is nothing more than 

 the ordinary Newtonian method, applied to approximate to two 

 roots of an equation of /owr dimensions. On the reason for this 

 process he is quite silent. The roots being put under the form 

 a(l ± e) the resulting value of a is 



C "^ '^^' 

 which is known from independent considerations to be true. 

 Mr. Adams might have added the resulting^ value of e^, which is 

 equally true, viz. 



■ .--? -^; 



This very remarkable result was first obtained in my paper, and 

 ought alone to have saved it from unqualified condemnation. 



Now, as to the rationale of the above process, it is known from 

 the theory of equations, that since the term of highest dimen- 

 sions contains a very small coefficient, the process would be quite 

 illusory unless that coefficient be a factor of the equation. But 

 it is a factor of the equation in case ^^— 2ftr + Cr^ be a com- 

 plete square, that is, if h^C=fi^. For then the equation takes 

 the form 



7^ = 0, 



i'-ff- 



2a'^ 

 and, putting —■ +/8 for r, may be transformed into 



Hence the required condition is satisfied if the unknown quan- 



m 

 tity / be such that /* = ^-^. 



All that remains to complete this reasoning is to show that 

 no result applicable to the moon^s orbit can be derived from the 

 equation (F), unless it be treated as an equation of four dimensions. 

 Here, again, Mr. Adams comes to my assistance. For he has 

 shown, that if the equation be put under the form 



A«-Vr+(c-^)r'=0, 



