762 PEOFESSOE BOOLE ON THE COMPAEISON OF TPvANSCENDEXTS, WITH 
fix) 
Now/’(^) being as above, let F(^) = - .^_ where m is an integer. Then 
/(«) 
{x — aY 
+ 
[x — aY‘ 
y(m-l)(g) ] 
m- 
Tl x — a 
I X (^— «) 
TW 
TTi 
■ &c. 
( 2 -) 
Here we obtain, as before, a development of F(^r) in ascending powers oi x — a. but the 
development begins with negative powers of that quantity. To this species of develop- 
ment we shall have frequent occasion to refer, and no doubt after this explanation will 
arise as to what is meant when we speak of the development of any function F(j’) in 
ascending powers of x — a. 
These things being premised, let the symbol 0 be thus defined, viz. ^®(x)f(x) he any 
f unction ofx composed of two factors <p{x) and f(x) whereof (p{x) is rational, let 
denote the result obtained by successively developing the function in ascending powers of 
each distinct simple factor x — a in the denominator of talcing in each development 
the coefficient of adding together the coefficients thus obtained from the several deve- 
lopments, and subtracting from the result the coefficient of^in the development of the same 
function (p(x)f(x) in descending powers ofx. 
It is seen from the above that the interpretation of 0 is relative. It directs us to 
obtain certain developments, but the nature of these developments, if we except the last 
of them, depends upon the nature of the function Avit h i n the brackets []. Thus while 
in the expression Q\jp{x:)']f{x) the operation of the symbol 0 extends over the entii’e func- 
tion <p(x)f(x), the interpretation of 0 , by which the nature of that operation is defined, is 
derived solely from the factor <p{x). 
Thus to take an example of some generality, let it be required to deduce an expres- 
sion for 
^\_{x-a){x-b)^f^^^' 
where y’(^) denotes some function of x which does not become infinite when x-=.a or b. 
The distinct simple factors in the denominator of the function Avithin the brackets [ ] 
are x — a and x — b. If we make x — a~z, or x=a-\-z, aa'c liaA’e to dcA elope 
[a-\-zY ^ 
in ascending poAvers of z. The coetficient of 7 in that expansion is 
aj{a) 
Again, making x—b=z and x—b-\-z, Ave haA^e to dcA^elope the function 
{b + ^)^ 
z^{b — a + z) 
0 -) 
