relating to the Moon^s Orbit. 105 



give accurate information. I should have thought that he would 

 have concluded the difficulty to be a confirmation of that opinion^ 

 and not regarded it as ^' purely imaginary/' 1, who have no 

 doubt of the truth of my equation, may well believe that the 

 difficulty is only apparent, and admits of explanation. Mr. Adams 

 even volunteers to give an explanation, but, as might be ex- 

 pected, is not successful. He is equally unsuccessful in stating 

 the difficulty. In the quotation from my article, inserted towards 

 the bottom of page 32, the reference to the mean distance might 

 have been omitted without detriment to the conclusion, as may 

 be thus shown. Put a for r in the small term of equation (C), 

 and we have 



dr^ h^ 2fA> 7n'a'^ .n-n 



Now this equation informs us that the orbit is a fixed ellipse, 



and the force is wholly central and equal to -^^ The difficulty 



consists in its giving this information, although the effect of the 

 disturbing force to small quantities of the second order has been 

 taken into account. Also there are other ways in which the 

 difficulty presents itself. By putting the equation under the form 



,, —rdr 



at= 



— Fa' 

 m'r* 



\/-Cr^-h'-^2f.r + '^, 



2ft'3 



substituting a for r in the small term, and integrating, I find a 

 relation between r and t, which, combined with the differential 

 equation between r and 6, leads to apsidal motion, the amount 

 of which is entirely false. Again, if in the small term a-^r — a 

 be put for r and be expanded to the first power of r—a, the 

 result is an apsidal motion equal in amount to the know^n first 

 approximation, but contrary in sign. These absurd and con- 

 tradictory results plainly indicate that the equation (C), though 

 it contains the disturbing force explicitly when a is put for r in 

 the small term, contains it so as to give no information respect- 

 ing the orbit. The only inference from this fact is, that the 

 equation contains the disturbing force as a factor. Analysis 

 supplies no other language by which the fact may be expressed. 

 Mr. Adams asserts that this inference is "unwarranted," and 

 "wholly unsupported by any proof." But here, as in other 

 parts of the discussion, his assertions are stronger than his argu- 

 ments. For if we now turn to the explanation he offers of the 

 " supposed difficulty," we shall find that he leads us unwittingly 

 to the very same conclusion respecting the existence of a factor. 



