PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 589 



we obtain -nrr?- = '^g ^ - , „,-, + - ^ • -7-^ 



(?J + 1)« 1 (/"-I a; (w + l)w(w — 1) 2 fl?"-2a; 

 O ^~^«^i "*" 17273 1^ c?««-i ■ 



- &c. 



/—I «" a; /—I 



_ (w + ]> 1 1 / -.Nn-l /^7I2 ^ 



1.2 • a:2 • /3i - ^ 



"T^TT^ yogx/n-/n-l.(n + l) 



= (y_}y:l" /log ;. ^- /n-(«+i) /^i - MIK^ /,r-2 



3 ^ 4 ' J 



rfMogx , . ( -1)"+^ /n 



— r-^— = above series — i— ^ — • —ji 



= lzi}!77/ (log ^ _ 1) /% - (,^+1) /^^ 



- (*^±^- /5r=2.-^?^^:tl)|K-l) /.^ _ &e. } 

 Now the series may be put under forms as follows : 



/»-r=(«-2)/^^ 



&c. = &c. 



,. (.+1)/^ + -^^/^ + ... 



_ (n + l {n + l)n 1 (n + l)n{n-l) J^ \ ^^ 



"/'' \n-l (n-V){n-2) 2 («-l)(?J-2)(«-3) '3 ■■/ 



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



d~'' (u v) _ u d-'' V du rf~'" + i tj 



^0^' «?«-'• Jx^' ^~d^ dx-^ ^ "" 



if, therefore, m=«»+i, ?j=a;-(") 



rfa;-^ ~'(w-l)(^j-2)... («-r) 



/•(/^ + l)a: » . (-l y + i a:-^"-'-- ^' 

 (» — i) . . . (n—r — V) 



r(/- + l)a:"-i (-!)'•+ 2 a;-(«-'-2' (« + l) « _ ^^ 

 "^ 1.2 (»-l) TT («-r-2) 



