246 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



similarly '^^ ^-l'fx(l + Bq+ &c.) ; 



and from the expression in Art 2. 



dar af- 



-(-1)''^'^(l + B7 + &c.)(l + ?logx&c.)| 



3. The expression given above for the differential coefficient of a logarithm is, 

 therefore, perfectly general, and is applicable to all cases. It is essentially ana- 

 lytical in its natiu-e, and does not appear to be reducible to a more arithmetical 

 form so as to retain its general character. The expression which I previously 

 gave exhibits very simply the mth differential coefficient of a logarithm as well as 

 its Hth integral, when n is a whole number, and may be, consequently, regarded as 

 the most comprehensive arithmetical form of this function which we can at pre- 

 sent obtain. 



It may not be considered out of place here to introduce the deduction of the 



value of „ , when m is a positive or a negative whole number, fi-om this form 

 also. 



The equation is 



ii^='-i^ [log— («+i)»((,rTi)»+«(«^i) + 

 1 1 1 1 „ \ -1 



2 (« - 1) (« - 2) + 3 («.- 2) {n - 3) + V ] 



(Part I, Art. 21.) 



(1.) If rt be a positive whole nimiber, the only terms in this expression which 

 are not indefinitely small, are, 



H--i);^2 / 1 1 \ 



/"ZTa." ^""^ ''"'\n{n-n + l){n-n)'^{n + l){n-n){n-n-V)) 



=^i=K^ (.+1) . (-1--^ ^1 \ 



/-la'" \n{n-n) (n+l){n — n)) 



^ /n(-l)" + ^ ^ /^(-ir + ^/i^^ 

 f—la^(n—n) j?"/» — »— l/re— n + 1 



_ /nC-lf+V-n-l) ^ ; ^(-l)"+^ ^ (-1)" + ^1- 2... (/»-!) 



the well known form. 



