relating to the Moon^s Orbit. ^ 101 



said, moreover, that while my solution leads necessarily to the 

 same numerical results as former methods, it adds something to 

 our knowledge of the moon's orbit. 



It will be proper to state here, also, my ideas respecting the 

 constants introduced by integration, when the solution of the 

 differential equations is only approximate. When an exact solu- 

 tion can be effected, I quite agree with Mr. Adams that, provided 

 the proper number of constants be obtained, it matters not what 

 is the process of integration and how it is suggested, simply for 

 the reason that there is but one such solution, and its form is 

 unique. Also the constants are arbitrary, and therefore entirely 

 independent of each other. From an exact integral obtained in 

 any manner, integrals applying to particular cases may be de- 

 rived by assigning particular values to the arbitrary constants, 

 or subjecting them to certain conditions. But when, as in the 

 instance of the lunar theory, an exact solution is not attainable, 

 that process is no longer possible, and recourse must be had to 

 approximate sohitions based on certain hypotheses. Now the 

 form of an approximate solution entirely depends on the hypo- 

 thesis on which the approximation proceeds, and may be different 

 for every different hypothesis. Also each hypothesis requires a 

 particular process of approximation which it is very important 

 to discover. The constants introduced by the integration, if 

 they should be the same in number as those of the exact solu- 

 tion, are not generally in the same degree arbitrary and inde- 

 pendent of each other, because an integration performed on a 

 certain hypothesis is equivalent to a particular case of the general 

 integral. These constants may therefore have certain relations 

 to each other, for the same reason that, in the common elliptic 

 theory, the arbitrary constants of the general integral satisfy cer- 

 tain conditions in the particular cases of motion in a circle and 

 a parabola. The process of approximation, if conducted strictly 

 according to rule, will itself determine those relations between 

 the constants which are appropriate to the circumstances defined 

 by the hypothesis of the approximation. I am persuaded, from 

 long consideration of the subject, that these principles are true, 

 although they do not coincide with those laid down by Mr. Adams. 



I proceed now to a particular discussion of Mr. Adams's 

 objections. The first in order is that which begins at the top of 

 page 30, to which an immediate answer may be given on the 

 principles I have just explained. The fallacy of the argument 

 consists in assuming that the constants a, e, e, and -cr are neces- 

 sarily arbitrary and independent of each other, because the values 

 of r and 6 containing them satisfy the di^erential equations (1) 

 and (2) at the bottom of page 29. Such reasoning is valid only 

 in the case of the complete solution of exact equations. Also 



