376 Sir W. R. Hamilton on the Calculation of the Numerical 
It is easy, if it be desired, to translate these to other known 
notations of factorials, but they may suffice on the present 
occasion. 
[2.] With the notations above described, it is evident that 
Legh Ski a i, ha RA Sy 23) 
and more generally, that 
Li'= 
m+n 
=n] oe aon 
Hence results the series, 
(14+1,4+-174.J1=(1-I,)‘l=e'; = eee) 
and accordingly, we have the finite relation, 
Geta ye: a Oe 
The imaginary equation, 
év-1=(1— V—1],)"1,_. . . (8) 
breaks up into the two real expressions, 
cance (2 pA) a, vac t OAR ey 
sin FU, 12). 2 Oe 
The series of Taylor may be concisely denoted by the formula, 
fle+)= SLD fag ee ee) 
and accordingly, 
LD, f(@+h)=Lf(e+)=fler+)—fe. . (9)! 
And other elementary applications of the symbol I; may easily 
be assigned, whereof some have been elsewhere indicated. 
[3.] The following investigations relate chiefly to the function, 
Per ea) neon Le ene) 
or 
F,, f=I(1+412)"ft, . . » » (10) 
where 
PSE pi EAT P) Fe) aa. ee 
Developing by (5) and (6), and observing that 
* ef —1)"=(—1)"[2m]”, . . (12) 
and that therefore 
2s)" [o}- poy“ =(—1)"((0]-™)*, = Ay 
we find the well-known series, 
. t 2 72 i 73 2 
f=1—(5) + (75) -(p-e3) +8» 3) 
