Mr. Herschel on the 
SB 
i / jj (1 + and thus we obtain 
H x * 1.2 ..... x 
/ {( 1 + A)“}o* = «*/(! + A )o'; (18) 
From this equation it is easy to derive the two following 
0 = {/C 1 + A ) +/(i^)j° 2 *“ I; (i9j 
o= {/(i + a) — (so) 
Let/(/) be a rational, integral, finite function of and 
suppose it to contain the powers of t , t p , t r , &c. ; it is evi- 
dent then that we shall have, by (14) 
/( 1 + A)o* = o; (21) 
in every case except where x is equal to either of the num- 
bers p } q , r, &c. The following forms off satisfy this condition 
/(O = ( lo g- 0" 
/(0=*L(i) + *L|f} 
/(0=’ ! L(x + i) + (-i)': ”L| 1 + -L]. 
/(0 = ”C(i) -(-!)» »c{-i} 
or, lastly, the sums, powers, or products of any of these forms, 
any how combined.* The excepted values of x, are— for 
the first of these forms, x = n — for the second, x = 2, 
and for the third and fourth, x = n,orn — 2, n — 4, &c. Also 
from the general theorems delivered by Mr. Spence, we find 
for the value of/(i A )o»— ** (which comprehends all the 
excepted cases) in the third and fourth of the above forms re- 
spectively zx Jj ( 2 ) and z *+ I C(i). 
It may not be uninteresting to descend to a few more par- 
ticular applications of these general theorems. If we suppose 
• Logarithmic transcendents, pages 45, 69. 
