relating to the Moon's Orbit, 108 



this untruly, why does not Mr. Adams contest it, or even allude 

 to it ? One sound argument against the reasoning employed to 

 deduce the equation would suffice to settle the whole question. 

 I can assure Mr. Adams, that while he has been unable to con- 

 test that reasoning, I have had no difficult task in discovering 

 the fallacy or irrelevancy of every one of his attempts to nullify 

 the equation. 



The argument in the first paragraph of page 31 shows that 

 the integral of the equation (C) gives the value of r only to the 

 first order of small quantities. As this result perfectly accords 

 with my views, I have no remark to make upon it. Neither in 

 my paper nor in the Philosophical Magazine had I occasion to 

 take into consideration the value of r to the second order of small 

 quantities. 



I come now to an argument which, among others, Mr. Adams 

 communicated to me to justify his advising the Council of the 

 Cambridge Philosophical Society to reject my paper. The argu- 

 ment then appeared to him to prove beyond a doubt that the 

 equation (C) was erroneous : now he proposes it as a " test of 

 the degree of accuracy to be attained by the use of that equa- 



dr 

 Hon J' By differentiating the equation (C) and dividing by -j. 



there results 



dt^ ^3 -+-^2 2«'3 ^^ 



But the equations (1) and (2) already referred to, give 

 d'^r W- fjb mJr 3m'a 



^-^3 + ^-3^3- ^co8(3n< + 6-2»'< + ^)=0, (E) 



which equation is certainly true to small quantities of the second 

 order. Since, therefore, the equation (D) is not true, it was 

 argued that the equation (C) from which it was deduced could 

 not be true. If Mr. Adams had known the days of the Cam- 

 bridge school exercises for mathematical honours, he would have 

 got credit by propounding such an argument as this. As oppo- 

 nent he might have put the syllogism thus : if (C) be true, (D) 

 is true, being '^ a strict deduction from (C).^^ But (D) is not 

 true, because (E) is true. Therefore (C) is not true. Ergo, 

 cadit qucestio. A clever respondent would, however, have imme- 

 diately answered, nego major em; that is, he would not have 

 admitted (D) to be a strict deduction from (C) by a retrograde 

 step, unless (C) is deducible from the premises of the question 

 through (D). But (D) is the antecedent of (C) only in case the 

 force is wholly central and th§ integration is emctf which are not 



