Mr. Herschel 07 i various points of Analysis. 445 
denote the sum of all the terms of a series, we shall use the 
sign S, thus, 
(p (^) = S I , and in like manner <p {t, /') == ^ 
.f 
Yox t\e\.ht be written, and we obtain 
(p {ht) = S {a^ ~ ^ ■ r] = G [aX| 
Thus, if the generating function of A^be (p (t), that of 
will be (p {Jit). 
Let this equation be multiplied by and we get 
r’’ p {ht) = s [aji^ . r~''] 
= S { A^^., .r]=^G { A^^., . /;"+’■}. 
If then the generating function of A^ be (p (^), that of A^^^ 
will be t~~^ . (p {ht). 
Again, it is easy to see that 
a.C {a^} +6.G {b4 +&c. = G { a A^ + + &c. } 
and thus we have 
{at~“ + bt~^ + cr"' + &c.) . Ip {ht) = G [aA^^^ /»■'+“ + 
and, if /i = 1 
-h 6/“"^ + &c.) . (p (t) =:G^aA^_^^-\-bAx^0 + &c j;(i). 
Let us express the function aA^.^^ + bKx^& + &c by the 
symbol vA^ and let 
at + ht^ + cf + &c. =/ (0 
vA then will be the same as^ (A^), provided that in the 
MDCCGXIV. 3 M 
