of the different Equations of Motion. 97 



^ + £(*)+4 l +^R=o, ■ (6.) 



neglecting the powers of the disturbing force above the first, 

 8r and Iv being the variations in r and r; produced by that 

 force ; but instead of (5.) another is sometimes employed. 



To compare these with the equations before found, we may 

 observe that (3.) is more simple than (5.), but (4.) is consider- 

 ably more complex than (6). The higher powers of the dis- 

 turbing force are more easy to develope (by the method 

 suggested) in the former set, and are attended with some dif- 

 ficulty in the latter, on account of the dB, in 2/"e?R relating 



only to the co-ordinates of the disturbed body. Upon the 

 whole, therefore, perhaps (3.) and (4.) may be quite as easily 

 solved as (5.) and (6). 



I shall not enter upon the determination of the latitude and 

 reduction, as these must be found by some of the ordinary 

 methods. 



Reply to Mr. Boole's paper in the last Number of this Journal. 



Mr. Boole says that the example which I adduced by way 

 of illustration is not analogous to his. I know it. It was 

 given as one instance in which the supernumerary arbitraries 

 must be determined by substitution. It appeared to me that 

 he did not consider it necessary to retain them in any case. 

 I might form this opinion on insufficient grounds, but I do 

 not yet see how I could come to any other conclusion. In 

 the Number of this Journal for January 1847, and in the 

 seventh Number of the Cambridge Mathematical Journal, he 

 made assertions which implied that the supernumerary arbi- 

 traries might be rejected. In neither case did he allege the 

 peculiar nature of the example or process, or say that his as- 

 sertions might be proved. And moreover these examples are 

 not analogous. If in certain cases it can be proved that the 

 arbitraries in question will go out, of course in those cases 

 they may be at once rejected. And now he has only given 

 the proof in one particular case or interpretation of his equa- 

 tion ; and the completeness of the solution is only proved in 

 this particular case. I pass over the various other points 

 contained in his paper, believing that I may safely leave them 

 to the consideration of the candid reader. 

 Gunthwaite Hall, June 15, 1848. 



Phil. Mag. S. 3 . Vol. 33. No. 220. Aug. 1 848 . H 



