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LVII. Note on the Conversion of Integrals, 

 By Chief Justice Cockle, F.R.S.* 



ON reconsidering investigations suspended for some months 

 and examining my paper " On the Conversion of Inte- 

 grals " printed in the last July Number (Supplement), I perceive 

 that I have improperly left v outside one of the brackets in equa- 

 tion (6) of page 538. The terms 



— ev j(/> (x } n) — (j) (x, m) j- 

 should be replaced by 



— e\ty{Qc,n)—ty(x,m)\, 



where 



yjr(x,v) = v<j>{x,v). ...... (23) 



A.nd I further perceive that throughout the numerators of (18) 

 the quantity a can only have the value -<-l ; so that (19) dege- 

 nerates into 



vto ±_p 



x dv oo av v f 



which is identical and useless for our purpose. It will be neces- 

 sary to substitute for V its value as given by 



v n-l 



V=T^ ....... (25) 



and to form the function 



« 



which when developed will be a rational function of x, v } e, and 

 f ; and the possibility of satisfying (3) will depend upon our being 

 able so to assign e and f as to render this function a perfect dif- 

 ferential coefficient with respect to v. In other words, we must 

 be able to integrate the resulting rational fraction without com- 

 pletely decomposing its denominator or having to solve a certain 

 trinomial equation. The question of the conversion of Boole's 

 integrals is an important one. We know that the definite inte- 

 grals of the quadratic cubic and quartic Boolian resolvents are 

 equivalent to indefinite integrals, which become in fact alge- 

 braical expressions; and any general process by which their con- 



* Communicated by the Rev. Professor Harley, F.R.S. 



