we obtain 



PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 

 5_=Wa: 1- -5^ — ■ — L . 



589 



rf«»+i 



<?«" + ! 



dx" 



_{n + V)n 1 rf'-ia: (w + l)w(w -l) 2 (^"-^^ 

 1.2" «2 (^^,^1 + 1.2.3 Ifi rf :»■«-» 



- &c 



= iogx~ (-1)"+' Al + ^±1 . _i_/_i-)«/«-i 



x" 



/-I 



("- + !)'» J_ _JL_/ iV'-i /"^-s 



1.2 • a:^ • /3j (. a; ~^S=2- 



(-l)»+i r ,_ 



^=5 ;;- { logx /n — /n — 1 . (« + 1) 



/ 



-i!L±lL^» /^:r2 _ 1.2 . ^-J-^l--') ^3 - &c. } 



= t;E:lL hog x/^-/n-in + l)/i^i-ttM^/„'-2 

 l~lx" 1 * / V ^/ 2 ' 



_ in + l)n(.-l) /^-3- (. + l)W(n-l)(>.-2) /^4 _ ^^ _ J 

 flf'loga; , . (-1)"** fn 



= ^J^T^J {aog^-l) /» - (r^ + l) /^^ 



Now the series may be put under forms as follows : 



l^-l = {n-2) l^i^2 

 &c. = &c. 



., (, + i)/^+-^±i>/ir:2 + ... 



-In (''-±^ + (" + 1)^ 1 , {n + \)n{n-l) J^ \ 



''* \n-\ {n-\){n-2) 2 (»-l)(w-2)(«-3) ' 3 7 



This series is divergent, except when n is negative or fractional. 



d~''(uv) ud-''v du d-'' + ^v 



Now, -d^^ = -d^-'-^^^^^ + --- 



if, therefore, u=tf+\ v=x-^"' 



d-'juv) _ a;" + ' ( - 1)*- a;- "-^' 

 dx-^ ~ (n-l) (n-2) . . . (n-r) 



(»-l)T.. (»J-»— 1) 



?-(r + l)a:'-' (-1)'-+^ a;-"-'-^' (w + 1) w _ „ 

 "^ 1.2 (n-1) ... (»-r-2) "" 



