4?4j8 Mr, Herschel on vanous points of Analysis, 
a'A =A . Ax+i_,+ A — &c;...(8) 
X jr -f Z I 1.2 X-\-l—2 ’ V / 
But to proceed. We have, 
^ = s(A^x'.i^}=lil(f 57-4(0 
or which is the same thing, 
^ ^ ? H~' ■■ i"g^} 
Now, 
jj^log ^ logf} D (9 log ^): log^ X ^ log / _ , \ • Incr / 
<fiog^ — iTIoil - J.iogr 
and so on. Thus our equation becomes 
G{x'A^.}==D'(? log-'): logt 
and, iff (x) = ax“ + bx° + &c, we see that 
G {A^./(x)} =(<zD*+6D®+ &c) (?log-‘): logi 
= if- D) i<P log—) : log; ; (9) 
If/(x) be a rational integral function of Xy the second member 
of this equation will require only the ordinary rules of the 
differential calculus for its formation, and of course the first 
may be rigorously obtained. 
Conceive f{t) and F (F) to be developed into the two series 
consider the double 
series 
First, <r may be expressed as follows, 
<r = s4a,t".S^|A^.{/i'tV}} =S, 
But by a similar mode of reasoning, we should also find 
