459 
Sir John Herschel has given the following theorem, which 
enables us to develope F (e") in a series of ascending powers 
of h when such a development is possible :— 
h ? 
P(e) =r (l)t7R (It Alo +e r(I + A)o? + &e. 
Comparing this with the one given above, we obtain the 
following theorem : 
(28's (w) = F[a(1+ A)]o", 
by the help of which we arrive at a still more general one, 
f(2Z) e@) = Fle +a] FO). 
Sir William R. Hamilton wished to be allowed to remind 
the Academy that he had communicated to them, in 1831, 
another extension of Herschel’s Theorem, which was pub- 
lished in the seventeenth volume of the Transactions (page 
236), namely, the following : 
VIE OO) =SU +4) V' (HO)? 
where the accents in the first member might have been omitted, 
and where y’ denoted any combination of differencings and dif- 
_ ferentiatings, performed with respect to 0’, and generally any 
operation with respect to that accented zero, of which the sym- 
bol might indifferently follow or precede (1 + A), asa sym- 
bolic factor. By making y (0’) =<”, and y’ = pv”, where 
PDs a the theorem of Herschel is obtained. A much less 
. general formula was cited as ‘‘ Hamilton’s theorem,” in the 
last Number of the Cambridge and Dublin Mathematical 
Journal, namely, the following : 
S(#) =f(1 + A)2°; 
which had, however, been also given in the same short paper 
; of 1831. 
