INTEGRATION OF DIFFERENTIAL EQUATIONS. 107 



and from (a'), which is true for n 1, we get 



Now _! is obviously = 0, 



_2 



and these two quantities are independent of a lt 2 , &c., 

 is a particular solution, and 



is the complete solution of the proposed equation. 



The method of proceeding suggested by this example is to 

 obtain a solution, neglecting all factors analogous to (n 1), and 

 then to complete it by reference to the assumptions of trans- 

 formation, such as (a), which have been made use of. 



The equations which we have solved are not a very numerous 

 nor perhaps an important class. But one of them, at least, is 

 susceptible of a physical application of great interest ; and so 

 few equations of the higher orders are integrable in finite terms, 

 that the discussion of those which are, has always some degree 

 of value. 



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