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 formulze which possess all the generality which 
can be desired, at a single glance. 
Suppose, then, Aw,.=/(w,)=%, 
d ad 
Bo oa (et i) Gye (t= 1)?u,, &e. 
and, according to the axiom of the calculus of operations that the repetitions of 
d 
equivalent operations are equivalent, we shall have generally A“ u. = (e¢*—1)"u, : 
whatever » may be. This, then, may be said to be the definition of A” uz. 
d d 
Also, since ve+n=¢ ¢* uw by Taytor’s Theorem, and A w»=(e7*—1) ue; it fol- 
lows that Ue tn= (1+ A)” Ugs . 
We proceed now to apply it to the demonstration of the theorems which con- 
nect together A” u,4;, and we+p, &C. 
d 
d d 
ae Ce (n—1) — ae (n— 2) 
Cl): Anu,=(e4*—1) u,= (e d2_ ne aay paar ee —&e.) Ux 
n (n—1) 

—Ugen—eUzen-1 Yo Uy +n XC. 
Corn. I. if »=—t1; AW! ty = Uy 1 + Uy 9+ Uy + he. 
Ge BM, =U, _1 + Uy_9+U,_3+&c. together with an arbitrary constant ; 
or Uy =A Un 1 +A Ug gt AU, 3+ Ke. 
Cor. 2. Ifm=—2; Yu,=uy_9+2t,_3+3Uz_4 +&e. together with A+B«. 
d d 
d 
whe as Ne 
Q). an u,=(—1)" (1—oF Yu, =(—" dnt" Ne a2_bou, 
=(-1)"(u, —nu,..+" 
-1), 
n 
Tg “+2—8) 

Cor. 1. Ifm=—1, a7? u,=3u,= — (vp +Un41+%r42+&e.); to which we may 
add an arbitrary constant. 
Cor. 2. If n=—2, Pu,=u,+2u,41+3uU,42+&e. together with A+Bz. 


nd = a n ad = os 
(3). A u,=e** (‘ — ) tigeed® - ) Un 
ole ed e_y 
d d 
—n 
=e 42 (1+¢%—-1-1) Un 
