relating to the Moon's Orbit. 107 



and be solved as a quadratic, the resulting value of a is 



which is not true. If, without altering the form of (F), a be 

 put for r in the small term, and the equation be solved as a 



quadratic, the value of a is found to be '^. These contradictory 



and false results show that the apsidal distances can be obtained 

 only by solving (F) as a biquadratic, and consequently that the 

 equation contains the disturbing force as a factor. Mr. Adams 

 is not happy in the inference he draws (in the first paragraph of 

 page 34) respecting the different values of a given by his two 

 processes. Plainly he had not bestowed on the subject all the 

 consideration it demands. 



In reply to the argument in the next paragraph, that " the 

 required condition would be satisfied if h^C differed from fi^ by 

 any quantity involving the disturbing force as a factor,^^ it is 

 enough to say that the equation h^C = fju^ is sufficient , and that 

 it would be a violation of the principles of analytical reasoning 

 to introduce gratuitously a quantity to determine which there 

 are no conditions. It is true that on carrying the approximation 

 to small quantities of a higher order,- it will be found that 

 h^C~-fju^= a quantity containing the disturbing force as a factor, 

 but at the same time the approximation itself determines the 

 form and value of this quantity. 



The next attack on the equation (C) is made on the principle, 

 that an approximate equation, formed so as to include all quan- 

 tities of the second order, may be proved to be false by reference 

 to quantities of the third order which have been neglected in 

 deducing it. No one, I think, ever heard of such a principle 

 before. To state this argument is to refute it. 



In the first paragraph of page 35 it is contended that the 

 relations between r, 6 and t, in page 281 of my article in the 

 April Number, do not satisfy the equations (B) and (C) from 

 which they are derived. They satisfy these equations by taking 

 account of a term involving the sun^s longitude, which by the 

 integration rises to the third order, and which on that account 

 was omitted. Surely there was no necessity to bring forward 

 such an argument as this. 



The three differential equations which are the basis of the whole 

 of the analytical reasoning, were formed on the supposition that 

 the ordinate z is small compared to the radius-vector r. In 

 Titan^s orbit, z may be very nearly the half of r, for which 

 reason the theory does not apply to that body. In fact, it can- 

 »ot be tested by reference to a system of satellites, the mutual 



