PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 247 



(2.) If w be a negative integer =-m; 



,TO + 1 1 2™ + ^ 1 »"« + 3 



^^^y^n{n-l)^ 2(»-l)(»-2)"^3(n-2)(«-3)"^*'^- 



+ 5 7 Tow o\ + «C. 



'm{m+V) ^ 2(m + l)(w + 2) ^ 3(m + 2)(wj + 3) 

 as" <5 



= -«"'-- log (1-2) 



whence, by integration, 



y = 7 T\log(l-'S') + ^= iri — + — 1 +&C. +s[ 



1 , ,. s 1 f S'" Z™-^ „ ^' \ 



+ —, T\l0g(l~■^)^ T \ ^ T\-* TT7 ON + °^C. + i^^ \ 



" log(l-^)--^--— ilog(l-4 



»/j — 1 ^^ ' m — \ m — 1 



Consequently, the value of p between the limits and 1 is 



1 /I 1 . 1\ 



V ^ m(m — l)\l 2 m/ 



^J_(J_.J_ ^ 1 \ 1_ 



\i ^m-l\l-2'^2-3'^ ~^{m-r)m) m-1 



•: 1 /I 1 . 1 \ 1 1 



'- =—f T^ It + o + '^c. + — ) + ^_ 



m{m — \) \1 2 m) m m—1 



m(m—l) 12 3 m ) 



:* a:"'(-i)-'"+i r, A 1 „ i\ 1 



= ? v^^ -^r\ — ? — o^ 1 log*'- lT + ?f + &c- + — \ 



{ — m){ — m + l)...{-2){ ^ \1 2 mj J 



<■ »»(»!— 1)... 2 l ° \1 2 3 m/ } 



rfir^logx. 



4. In my previous memoirs, I have obtained the general differential coeffi- 

 cients of several functions, and have applied the results to the solution of analy- 

 tical and mechanical problems. It will be my object at present, to extend the 

 science itself by exhibiting the solution of differential equations, and by investi- 

 gating some of the properties of finite differences. In every instance I shall select 

 the most simple problems which will serve to illustrate the process employed. 

 Of the process itself, consisting entirely of the application of the calculus of opera- 



