296 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 
Cor. The same is true of the formula for the nth difference of w:,,: for 
A Ug, y= (1+ Ag) (1+ Ay)—1} ag, y 
AU, yo {l+a, hee Ay Se Ye, y 
d d 
— + —_ 
or (07 4249-1)" 
and the same results are produced as when m is a whole number. 
Prop. 4. F (A) &*f (2) =e" F (e&*" 144-1) f(z) 
Let %=e'*; and %,=/(«) in Prop. 3. 
AM er® f(x) = fF (@) Ave +n Asan” ott se. 
= f (a) (e"—1)” &* +n Afa(e—1)"-1! ef? +7 + &e. 
=e"* (e"—-1+6 A)" fe) 
=e"* (el +A—1)"f (2) 
which being true for all values of n, shews that the following theorem is also 
true: 
F (4) . e°* f (2) =e" F (e” 144-1) . f(z) 
Prop. 5. AU,= ((1+5)?-1) w = {+a ie ate oe 1}, 
Let z=e’, and let u, be represented by ~ when e’ is written for 2, let also 
ay be the symbol of difference w.1—ws. Then by (C), when ~ is an integer, 

“=D (D- (D2 2)\< 7). Waare, 
a” 
d 
eee Oe ee ay oe ees ee 
Ree te Getr & +ra3 (3) +e.) a, 
“Grr oa 1) (D=2) + &e.) 4 
But A, u=(eP—1) wy o. ag =(1L+ Ay) 
or D=log (1+ Ay). 
1 
D log (1+ — 
Hence (1+ =) = (14.2) O04) (+5) 
u 
D log (14+ — 
and = Aa, = (+2) —1) w= { +4) ®( "1h w 
log (142) , 
Cor. Un n= (1 t+ Ag) ug 
