partictilar Differential Equaiion. 41 S 



Secondly, it is quite within the province of theory to determine 

 how the superfluous constants of the solution of a differential 

 equation shall be disposed of, — to decide which of the opera- 

 ting factors 7niist be retained, and which of them jnai/ be re- 

 jected. It was upon a theoretical consideration that I decided 

 upon placing certain elements of the solution at which I arrived, 

 in the latter class; but instead of enforcing that consideration 

 here, I prefer to prove the correctness of the conclusion to 

 which it led me, by exhibiting the result in a purely quanti- 

 tative form. It will be interesting to show how, in a case of 

 more than ordinary difficulty, the conclusions of theory may 

 be practically verified. For the laws which govern the appli- 

 cation of the inverse factors, in connexion with the more 

 general method already adverted to, I must refer, in illustra- 

 tion of the above remarks, to the original memoir in the Phi- 

 losophical Transactions, p. 249. 



Let us then first consider the solution 



in which all the factors are retained. 



Since 7rop=^7r„ we have 7r,=p-'7r„p, and by induction, 

 7r,„=p~"'7rop'", from which we readily obtain 



Again, since tt,, =p -"Trap's we have on inversion 



"^n —? To P • 



Lastly, reducing in this way the remaining inverse factors, we 

 have 



TT-i rr-^ . .7r-»=p-("' + '•)^•7r~V)'■+'p•"~'• 

 and substituting these forms in the general solution, there 

 results 



« = p-™(p-',ro)'-p™^''*"'^o"' p-(-+'-"'(7r-V)'-+'p'"-'. . (5.) 



Now the forms of tt^ and p employed in my paper were 



i. 



dt 



'^o='PW:77-P = 'PW' 



in which 

 Hence we have 





d\-' 





