292 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



When n is less than 1 , therefoi'e, it is necessary to seek some other method 

 of obtaining the «th difference. The following method, analogous to that by 

 which we obtained the Mth differential coefficient of a logarithm in Art. 2 appears 

 to be the most simple. 



Let r be represented by ? — — — , where y is of a higher order than p, and 



both are 0. 



Then a" .r = (eP-ire>'^'-(e",-ire"-^ 



P 



{p +^ + &c.)"(l+iB2; + &c.)-(9 + y^ + &c.)"(l+?a; + &c.) 

 H Wjo"+l w(3»+l) „ + 2 , . n nq^ + '^ M(3«i-1) „ + .> . 



^ +"^ +~^d: — -p^ +^^--<i — ^ ~~^4 — -9 +&C. 



P 

 p+ 2 +&C.-? — ^ -&c. 



P 



2 



+ — 



P 



+ &C. 



If «::^0-^l, every part vanishes except the constant, which is infinite : if 

 «=l, Aa = l ; if nz^\, every term is zero. 



If n is neo^ative, there will still exist the infinite constant which may be re- 

 warded as part of the arbitrary constant ; there will also exist in some instances 

 infinite functions of x, which, as will easily be seen, may be considered in those 

 cases as part of the arbitrary functions. 



Let w=-l; then 



1 ^ ip + kiQ. , 2 



A~ :«■ = const. x + ^gr' 



P 



= const. H p 



Let n=-2; and 



A " "a; = const. + const. ''^ ~ "o" + ^ 



and so on. 



Ex. 4. a" r"' = (;i; + nV-»»(^ + «-l)'" + ^X^ (a; + M-2)"'-&c. 

 l)y the first formula. 



