336 On the Extension of the Theotem of Leibnitz to Integration* 

 that is, still omitting the supplementary integrals, 

 P* 7 2 o dv o&v A d3v , « 



And by repeating this process for a third integration, a fourth, 

 and so on, it is obvious, from the law of their formation, that 

 the numerical coefficients will be those furnished by the cor- 

 responding negative powers of a binomial ; so that, by restoring 

 the original symbols for u v w 2 , u 3 , &c, we shall have 



I vudx n — v I ndx n —n-r- I udx n+1 +-^— — --p^l 



which is the form in which the theorem is usually given. 



From the suppression of the supplementary integrals, it is 

 plain that this theorem cannot be generally true. In fact, it 

 is strictly accurate only when v is such a function that one of 

 its differential coefficients, and therefore every one that follows, 

 becomes zero. 



If we supply the omissions which have been made, and em- 

 ploy the former notation for brevity, we shall have 



/ 2 7 a _ dv n d 2 v , d s v d m ~ l v . 



vudx^vu,-2^u 3+ 3^u 4 -^ ^« 5 + • a pB=r«fc + . 



fd m v , _ f 2 d m v . 9 



±m Jd^ Um ^ dx+ J d^ Umdx 



/ 3 j ^ „dv ■ tPv 

 vuaar — vii 3 — 3 -r- u 4 + 6 -r^u?, to m terms 



. m(m+l)f*d m v . __ s*H m v »'»' p 3 d m v 



± ~-2—Jd^ Um ^ dx + m J d^ Um ^ d ' V± J d^ Umd ^ 

 and so on; the leading sign in the lower series being the con- 

 trary sign to that with which the upper terminates, that is, 

 contrary to the sign of the mth term, which may be any term 

 arbitrarily chosen. In both series the signs are alternately 

 plus and minus; and it is to be observed, that the coefficient 

 with which the second or supplementary series commences is 

 the same as that with which the first series terminates ; for 

 m(m+\) _n(n+l) (n + 2)....(n + m— ( 2) 



~%T 1.2.3 (m— 1) 



If we call this ?«th coefficient M, the supplementary series for 

 the nth integral will be 



/ n d m v . p n - l d m v . ,_ml?n + l) 



> . . (Jj. ) 



^— u m+2 dx»->± M/ —u m+n - x dx 



