/ 
developement of exponential functions , &c. 
3 1 
developement of equation ( a ). And, first, in the case where 
n is a positive integer, we have 
/(/)=(/-!)”; /(*.) = 
consequently, 
f ( l+A)=( 1 -j-A — l) = A 
wherefore the equation (2) becomes 
f t t .71 . t* t 3 A n X I o 
(f - 1 ) =y- a °+— - a 0 + 7 x^ a ° 3 + &c -; (5)- 
of which the first n — 1 vanish of themselves. 
Let us next consider the formula (/ — 1 — n being a 
negative integer. As this function, when developed, must 
evidently contain the negative powers of t, as far as t— n , we 
first throw it into the form 
, ° r its equal t~ n ** j” 
supposing then/(/) = { I we sha11 bave b y a PPty in g 
the equation (2) 
{i^]”=»+fd i H ±A T 0+ ri- 1M— }”»‘ + 
&c.; ( 6 ). 
All that now remains to be done is, to develope the func- 
tion in powers of A. When n = 1, the deve- 
lopement is well known to be 
J. £ 4. £.* _ & c 1 
I 2*3 3 ' 
Hence, if we suppose 
t 
£ ‘ — I 
we shall have 
* 
B 
log. (I+A) 
0* 
X A 
JA'i. 
I * 3 + ± Z+iS’ 
( 7 )- 
