444 Herschel on various points of Analysis. 
/D(f) [x) = a. {D<p(x)|"' + 6. + &c. 
This rule does not extend to the signs D, A, accord- 
ing to the remark in IV. i. 
These rules will suffice to explain whatever may appear 
obscure or capricious in the following sheets. We shall now 
proceed to their practical application. 
If (?> (^) be a function of t, developable in a series 
A_ 00 ^ + Aq -1- hf -f . . . . AJl" + • • • A ^ ^ 
(p (^) is said to be the generating function of A^, and it may 
be said to be taken with respect to t. To this we shall appro- 
priate a peculiar symbol G/, as follows : 
(p (/) = {Ax} 
When only one symbol t is used, the index below the G may 
be understood, and our equation will be 
? (0 =G {A^. 
If (p (^, t') be a function of t, developable in a double 
series 
+ 
— 4 iy 
+ + . 
+ . • • ^ . 
A,,, + 
,X 00 
K.t -t' 
A„,v^“-^'^+ ••••A , 
CX3» J 0O> 00 
(p (t, V) is said to be the generating function of A with 
y 
respect to t, t\ and may be thus expressed ; 
and so on, if there be any number of symbols t, t'\ &c To 
