ANNALS 



OF 



PHILOSOPHY, 



APRIL, 18 



Article I. 



On Solutions of the Function \J/' x, and their Limitations. 

 By Mr. W. G. Homer. 



{Concluded from p. 173.) 



9. From the preceding train of argument, in which I hope I 

 have erred, if at all, only in preferring minuteness to obscurity, 

 it will be evident, among other things, that I was correct in 



excluding k = ^ from the solution 



^ X = (p~' J ft-^ - 2 COS. i^bc + c' i (28)* 



^"^ (2 + 2 C0S.2 3) a '^ * J 



and that two ferfectly distinct genera must be recognised, in the 

 solution of periodic equations. This is a circumstance which 

 has not yet, as it appears to me, attracted the attention of 

 mathematicians, in any degree proportioned to its importc.nce ; 

 and, nevertheless, there are few instances of the practical appli- 

 cation of functional principles which do not exhibit strong 

 indications of its influence. In the majority of examples, the 

 solution of the ultimate differential equation has been only 

 effected by making i]/ x constant ; and 1 am not aware that any 

 exists which has admitted of solution on the principles -^"^ x = x, 

 and ^|/* ""' a; = x, at the same time ; or that a solution by means 

 of a function of any one order above the second exists, which 

 does not hold true for any other of the superior orders. 



For two very instructive examples in illustration of these 



Soints, I refer the reader to Questions 408 and 409 of the 

 lathematical Repository, proposed and solved by Mr. Herschel, 

 In the former of those questions, however, Mr. H. seems to 



express a belief that the solution " — r-. as well as all others, is 



'^ c + dx' ' 



contained in the formula 



^x = <?-■{(!)"•. <px] 



• In all tlic fonnula distinguished by an asterisk, S is to be understood as 

 kn 

 = — restricted as in Art. 8. 

 n 



Nexti Series, vol. xi. r 



