developement of exponential functions , &c. 
41 
n 
A u 
r a. — 4- & c * 
'x, y, z, 8cc. L ~ ^ 1 x, y, z, &c, * * * * * ^0 
Hence the function to be developed is 
p. V'. 
ill 
Sec. 
} 
n 
n being a positive or negative integer 
In the former case, the coefficient of t*. t' y . Sec. is 
|(i + a). (i + a'). &c. — x o*. o'y. &C. 
1.2. ...XX 1.2 . ... y X &C. 
that is, developing the numerator 
| ( 1 + A )". (i + A') n . &c. — ~( I + A ) n T 1 &C.+ &c. | 0 *. o'K 8 c C. 
1.2 JT X I • 2 . . . X &C* 
Now, ( i-f-A) w o* = ra*, (i+a')* o ly —n y , Sec. and thus the 
numerator of this expression becomes, 
n x+y + 8c ^_n^n— 1 ) X+y+8£C ‘-J r &C. 
— , A w &c ’ 
and the coefficient of t*. V y . See. therefore becomes 
,*+>+ &c. 
A (28). 
: — n) 
A" o'" 
i . 2 . . . .x x i . 2 . . . . ;y x &c. x, y. See. 
In the latter case, where the exponent is negative (: 
the function to be developed is 
{*■+'+ &c 
the coefficient of t x . t ,y . Sec. in the latter part of this expres- 
sion, is 
flog. { (I + A )• ( I + A' ). &C. "I ] n 
1 i L i o* o fy Sec 
|(I + A)(X + A / ).&C.-I j 
3. 2 ..... AT X I. 2. ...,^X See. 
Now, let us for an instant suppose the expression 
flog. |(I + A)(I + A'). &c. } n 
y • • • • . ^ ^ ^ . 
< 
p(i + A)(i + A')- &c. — i 
G 
MDCCCXVI. 
