FLUXIONS. 
becomes 
x—a out of the numerator and denominator, the frac- 
tion is changed to —“-; and this expression when 
ra becomes an infinite quantity. 
Hence we see that a vanishing fraction may in some 
ea eee a others may be 
nothing, or infinite ; but in every case jue ma 
be determined by freeing the mumerator and denomi- 
then, by Taylor's theorem, P becomes p+ oh + 
2p fp? dQ aQ h* 
Tey tSe and Q becomes Q4 FS i 4 TYE 
+ &c. and the fraction becomes ) , 
dP dP he ; 
2 Cat lk alee 
dQ, @Qh 
Qt ae hase th 
E dP @P 
Let us denote the fluxional coefficients 7—, ie 
qQd 
briefly by P’, P*, &e. and 52, TR, ge by @, @, 
&c. then observing that when x = a, P= 0, Q=0, 
have, after dividing by h, pen ~ oh 
P’443P"h+&e. 
TOTP 
for the new value of the Vy sae If we make h=0, 
this expression becomes simply &, which must be the 
value of the fraction slit wai liislite it is evi. 
dently the same thing to sw first, that z= 2+h, 
and afterwards Ridess h=0, as to cappoan' ti 
once that z=a. If it happen that one of the two quan- 
tities P’, Q’, is equal to 0, then the fraction —-is either 
nothing, or infinite ; but if both are =0, then, after re- 
jecting P’ and Q’ from the neral expression, and di- 
viding again by h, we have @ for the value of the frac« 
tion, in the case of xa; and so on. Hence this rule. 
To find the value of; 4 fraction which becomes © 
when r=a. Divide the fluxion of the numerator by 
that of the denominator, let the result be ai then if 
this expression does not beeome +, when a is substitu. 
ted for x, it is the value sought ; but if it does treat this 
in all respects as the ether was treated, dedu- 
VOL, IX, PART 11, 
417 
Direct 
cing from it a new fraction Z, and proceed in this man- Method. 
ner, until an expression be found which does not become 
© by the substitution of a for «; and the first expres- 
sion that occurs havin this property is the value sought. 
Ex. 1. The sum of terms of the series 14-24 .2°+4- 
—tl 
23+&c, is 
z—1 : Jf we suppose x= 1, this expres 
o- 
sion becomes —-. It is required to find in that case its va~ 
lue. Here P=2"—1, Q=x—1, therefore d P=n2"—) dz, 
dP nat pr 
dQ=dz, and aa" a oO 
Q” 
when 1 is put for x, becomes + =n, which is the value 
This expression, 
sought, as is otherwise sufficiently evident. 
ax*+actk—2acr ¥ 
Ex. 2. Let the fraction be b—2bcap bor? Wed 
0 
becomes —- when x=c. In this case, 
Spe amisddas dQ= (26 2—2 bc) dx, 
dP _az—ac 
dQ ~ ba—be~ BY 
This fraction =, becomes also, when «= c, there. 
fore, ing as before, dP’=adz; dQ’=bdz; 
and Jey => Which is the value sought. 
a —}* 
Ex. 3. The fraction 
Here 
dP= fa", (a)—8" 1. @} dz; dQ=dz; 
al. (a)—b' 1, b)_ Pr 
dQ~ Q” 
Pr a ‘ 
when # = 0, 7; becomes L. (a)—1. (8) = 1+) which 
se ae if 
59. tule of Jast article will not in iy 
it Taylor's theorem does not give the dev: gl 
the ions P, Q in the case of =a. When this hap~ 
ro ey substitute a+-/, instead of 2, in the frace 
tion Q° and develope the numerator and denominator 
into ascending series of the form A k"4.B k*4.Ch? 4 
&e. A’A™ 4 Bh” + Ch?’ 4 &. We have then, 
instead of Z, this other fraction, 
AI4BR+C iP &e, ' 
Al} 4B" 4ChP’ +. &e,” 
or, dividing the numerator and denominator by jm’, 
Ale” + Biles CHP" 4. Be. 
A+B "5 CH=" > Be, 
Under this form, it is to see, that if m be greater 
than m’, the supposition A=0, makes the. frac. 
tion =0 ; weparaeteten > 6 the same supposition re. 
duces the fraction to 473 because whatever be the va- 
lue of h, h™—"’ = h°=1; lastly, if m be less than m’, so 
that m—m’ is ive, then, when h=0, the fraction 
becomes infinite, Hence this rule, 
A 36 
becomes ©- when «= 0; 
