62 The Rev. C. Graves on the Calculus of Operations. 



/"d'^v , 1 



M^ being used for brevity instead of / udjc^. 



The series of supplemental integrals here given differs from 

 that exhibited by Professor Young only as regards the signs 

 of its terms, which by an oversight he has made alternately 

 positive and negative. 



The reasoning remaining the same, it is enough merely to 

 indicate the corresponding mode of obtaining the general 

 formula for the finite integration of a product. Sir John 

 Herschel has given this formula in his excellent article on 

 differences and series, appended to the Cambridge translation 

 of Lacroix's treatise on the Differential and Integral Calculus : 

 and it is to be observed that he has taken care to supply those 

 supplementary integrals which are necessary to its correctness. 



Since 



^("A) = K^^. + ^.+ 1^^ = %^^. + ^''^.' ^«.' 

 we have the symbolical equation 



it being understood that A" and e^ operate only on t?^, and A' 

 only upon u^. And if we put A" and e^A' in place of D" and 

 D' in the formula given above for (D' + D")~", we shall at once 

 obtain Sir John Herschel's formula. 



The examples here discussed by the Calculus of Operations 

 are certainly instructive. Whilst they manifest the danger, 

 noticed by Professor Young, of substituting symbols of opera- 

 tion for those of quantity in divergent infinite series, they in- 

 dicate that, wherever we know how to express in a finite form 

 the value of the remainder after any given number of terms 

 of an infinite series, there is a safe way of effecting such a sub- 

 stitution. It must be made in the expression for the remai7i~ 

 derj as well as in the terms of the series. 



Dublin, November 30, 1848. 



