538 Chief Justice Cockle on the Conversion of Integrals, 



or, if the integration with respect to v in (2) be definite and be- 

 tween limits m and n, to 



£, +(?-e)y = X-ey{0(ar,n)-.0(a?,»i)}— /(a?,n)+/(a?,m). (6) 



4. Either of the last two equations is a linear differential in 

 y, whence y can be expressed by indefinite integrations with re- 

 spect to x only. The function X is to be determined by the 

 conditions of the question*. 



5. This process may be extended. Let <f> satisfy the relation 



then, integrating with respect to v, we have 



$*+&+<*-*$*} ; ... (8) 



which is equivalent to 



and, subject to the determination of X, the expression of ?/ by a 

 formula free from integration with respect to v depends upon 

 the solution of a linear differential equation of the second order. 

 In the last three equations /of course represents /(#, v), and 

 throughout this paper it is considered that <£ and /are free from 

 integration with respect to v ; so that our ultimate results in- 

 volve, in the case of equations (5) and (6), no other integrations 

 than those with respect to x. 



6. The process maybe further extended and conditions ascer- 

 tained which, when satisfied, enable us to make the expression 

 of the y of (2) depend upon the solution of linear differential 

 equations of an order higher than the second. When (1) is a 

 linear differential equation of the order n } and the y of (2) can 

 be made to depend on a linear differential equation of the order 

 m by the foregoing process, the determination of the function X 

 in general depends upon a linear differential equation of the 

 order n — m. 



7. Starting with the equations 



S= p > §= Q ' < 10 ' n > 



* When (1) is a linear differential equation of the nth. order, X is deter- 

 mined by a linear differential equation of the (n — l)th order, and (5) or (6 ) 

 gives an internal factor of (1). 



