go Mr . Herschel on the 
to the well-known property of the functions A x + l o x > 
A x+z o x , &c., each of which is equal to zero. Our series 
then becomes 
{/(i) +2X1 i2a+&c. } o * = f ( i+a)o* 
and we have therefore 
/(/)=/( 1 )+ 4 */ ( 1 + A ) 0 + rr / ( 1 + A ) 0 " 4 - &c * • — ( 2 ) 
In applying this series to any particular case we have only to 
develope/^i-j-A) in powers of A: then striking out the first 
term, as well as all those where the exponent of A is higher 
than that of t, to apply each of the remaining ones imme- 
diately before the annexed power of o, and the developement 
is then in a form adapted to numerical computation. This 
formula may be also farther compressed into 
/(/) ” /( X + A) /' ; .(3) 
by simply writing it as follows : 
/(/)=/(!+ a){i+t+tt+ & <4 
I shall notice one more form in which the same Result may 
be exhibited. If we continue the series ( 1 ), as before, to 
infinity, and add the term 1 at its commencement, it becomes 
{ a+ AS + A£!+ &c,} «*/(>)■ 
f A V./( 1) 
whence, we obtain 
y(/)==/(i) +7 ./ ,D o./(i) + — • ^ AD ° 2 */( 1 ) + & c - 
or, attending carefully to the application of the symbols 
/(«)=f iD { 1 +r + £ rr+ &c - 1 
=/ • D+ “-y(i) ; (4) 
We will now proceed to apply these results to the actual 
1 
