Dvtreet 
Netiood, 
416 FLUXIONS. 
56. atten ihe Oe ied Mac- du _ 
laurin’s theorem, quantities U, U’, &c. haveallfi- de ~~ (z—@ 
—— ore Cae ae 
an infinite value, we may conclude function general developement of " 
does not admit of being expressed by a series, the terms of the functiail) Se (e+h=ay 
of which are positive integer powers of x. The func- anes 
tion w =L(2) is of this nature; for, from it we get Waka — he him ke. - 
dw 1 dt te er (=—a)* rae) ay Ly 
=, dw >=~2 ae => In the particular case of «=a, every term of the ¢ 
7 din ¥: —. Mabe sgwesnts hep ad Sy bey 
position that x == 0 renders 7- = = an infinite quan- grerns eee or b - 
ae Sto quantity indicated by the developement is 77 = ~, 
tity ; in like, manner the quantities =, qar> &¢- an in which the exponent of A is ne 
become all infinite as well as the function u itself. As the developement contains only positive powers of 
h, it could not express the new value of the function 
The quaitities denoted by the symbols 4° $e, bee in this particular case. Waar ernest value, ° 
true. 
aly bscomejafinite, however, when perticalar valussure 0 6eneeslidevelopensent wOEue me EeiaE aS G7 
iven to x; therefre it ~will always be possible, by Ta : iva 
for's theorem, to develope any function f(z+A) into dual cases, it has been or. paen Saree 
that, although in general the function may be developed ee MA oa be 4 
: : : : ° si t, and ed that a be a 
into aseries, the terms of which contain only the inte- Giri so new state of the f s. : ie tree 
4 
i 
¢ 
Zz 
5 
. 
- 
Z 
H 
E 
5 
j 
a|3 
all the coefficients of the 
have 54 (x+h—a)® for the new value or the <a me ster ac in term, will te init." On 
= other when a particular a renders 
state of the function ; and because in this case [> ee Rotictente faGnite, we mas pare EK 
1 an 1 , &e, by Taylor's theo ent ought, in that particular case, to contain 
2a)? #  Aa—a)* or negative powers of h. ‘ 
rem, we have, F , f 
+ b4(xph—a) =b4-(2—a)"+ —h , Sc. 
+(«+ph—a) =b4+( (aa)? . Of Vanishing Fractions, § | 
je ; oa Fa . r 
Wt &e. ms 57. A vanishing fraction is.a fractional function of a 
+ (aa)! Mi : iable quantity ¢ of such a form, that its numerator 
In the particular case of x= a, or z—a=0, the denominator become both = 0 when a pariieular value 
-quantity to be developed, viz. b+(24h—a)?, becomes is given to x. Such, for example, is this, 
b A san ion containing a fractional er of P ps, es 
baw which haa the twofold value be/iemad bo h, Which when c=a becomes > ; however, by remark- 
because the si, a square root may be taken either jn; —a=(«—a) (x 1 appear th 
+ or —: ‘As the developement contains only positive pe Be mt BIR Ane lee - ne 
oan pees of dp SR ES quantity, itean >—Z = xa, so that the value of the fraction when 
ly have a sing’ ue, and therefore it is impossible ss imvfact hence it that the 
that it should ex the function in’ this parti 5==0, 30 IEEE OA : P 
ys: Dy, when x il a faction basin tht ose Fal angnabe value; and 
velopement, its first term u=b + (z—a)® Wecomes 5; we also see that it assumes the form > only be 
but the coefficient of its second term, viz. > ae flee eles Space have's-comanes 
s ia bat ity i xi—ata—rat+a3 
a(e—a)t > aot ee ee great, The fraction - re hae also the property 
and the same is true also of the coefficients of the third inp oe ee , 
and following terms. “Thus the analytical fact, chi ‘of Jooming  elaetedoath Kan ray sdigaerl 
the devslepereset penmet in thls, particule chew ropes. nominator have a common divisor z—a, and this being 
sent unction, is indicated its terms failing to Pe ta’ 
pga thing —s If, However, we give tox taken out of both, the fraction ya a: 
any value, except that of z=a, function is Upon the ition that c= quantity be- 
rectly expressed by its developement. * ned chais Toenticetly 0 Pher ps 
Next, let the function be = =" :im thee, ection SEA pecomasissO, 
8 . 
azc—a* 
Wy \ 
7A: : os 
ey 
- fe 
