obtained by Means of imaginary Quantities. $7 
X 
Hence y — (1 + An -f- B n % + Cn 3 + &c.) tt 
= 1 +f (A» + B «*+ &c.) + i^jO( A « + B n*+ &c.)* 
= 1 +x(A + Bra + &c.) +^— (A +B»+&c.)*. 
Now, since n is arbitrary, and ought, by the nature of the 
function y, to disappear from the expression of the function, it 
follows, that all terms multiplied by each power of n must de- 
stroy each other; neglecting, therefore, the terms which ought 
of themselves to disappear, whatever n is, we have simply, 
y — e = l + Ajt + — + —y + & c. 
ifA=i) = i+ x + - 4 L + + &c.* 
This demonstration for the developement of e° is general, 
whatever x is, provided it is always the sign of a real quantity ; 
but e x 1 can never be proved equal to 1 x \/ — 1 — ■— 
— X "1. + & G -'f What then is to be understood by e x ^ ~ 1 ? 
merely this, that e x V ~ 1 is an abridged symbol for the series 
of characters 1 -f- x V — 1 — -fj — &c. not proved, but as- 
sumed, by extending the form really belonging to e x to e x V ~ \ 
In like manner, e~ xV “ 1 is an abridged symbol for 
* This demonstration is due to M. Lagrange. 
f In all treatises, after the demonstration for the developement of e* , e* ^ i s 
put 1 = x V — 1 — — See. as if this case was really included in the general one 
of/. 
MDCCCI. 
o 
