I 



PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



287 



tional values of the index of difference will, in most cases, differ not at all from 

 the results obtained in the ordinary calculus of differences. We offer them only 

 for the purpose of exhibiting those formulse which possess all the generality which 

 can be desired, at a single glance. 



Suppose, then, ^u^-f K) = ""x 





and, according to the axiom of the calculus of operations that the repetitions of 



.; 

 equivalent operations are equivalent, we shall have generally A" n^ = ('"' '^-l)""a; : 

 whatever n may be. This, then, may be said to be the definition of A" «j;. 



d d 



Also, since "a;+K=e"''"M,i. by Taylor's Theorem, and A t(^=(e<*^-l)H,, ; it fol- 

 lows that M^ + n= (1 + a)" IIx. 



We proceed now to apply it to the demonstration of the theorems which con- 

 nect together A"" Ux+s, and M^+p, &c. 



(!)• 



.".. = (.^-1)" ..= (.-- ./""^^^+ '^4^/""^<^-^-&c.), 



n (re— 1) 

 " 172" 



— »x+n — n^x + n-\ + \ !)-' "aj + n-"? — <^C- 



CoR. 1. If «=-l; a-1m^ = «^_i + «^_2 + m.^_3 + &c. 

 or 2mj. = jf , _ 1 + M,, _ 2 + M.r _ 3 + &c. together with an arbitrary constant ; 



or Ux=^^x-\ + '^ «,;_2+ ^M.r-3 + &c. 



Cor. 2. If m=-2; 22m^=m.^_2 + 2«<^_3 + 3m^_4 +&c. together with A + B.r. 



(2). A»«,=(-i)"(i-.<'-)'v,=(-ir(i 



J. ■ ^ 



-&c.)m,. 



= (-!)"(«. -''".+i + '^^^«.+o-&c.) 



CoE. 1. Ifw=-1, A %.=2mj.= -(wj; + Mj,+i + %+2 + &c.); to which we may 

 add an arbitrary constant. 



CoK. 2. If n—-2, 22m„=««^ + 2«(j. + i + 3m,,+2 + &c., together with A + Bar. 



(3). 



n d 



-T- / ^dx . -« 



^X-^) ••.='"■(4^) 



gdx e-*^-! 



d _d_ \ —H 



