Put X equal to the numerator, and X' equal to the de-
nominator of the prpposed'fractiop, each being such a
function. of x that for some particular value of x they
both reduce to zero ; then give to x an increment, A, and
develop by Taylor's Theorem, gpd we have
_ (?X cZ^X A2 drsx A3

-^ +'^ ^+"^ 172+.;^^ n2T3+' ^^°-

■^ + <& ^+ (?a;2 1 . 2+ (?a;3 1.2. 3+' ^^^•
If we substitute for x its value a, both X and X' re-
duce to zero ; then, dividing both numerator and denom-
inator by A, we have

^ ^ _A_ ^ A^
dx'^ dx^ 1 . 2+ c?a:3 1 . 2 . 3+' ^'^°-
<^X' (^^X' A £3^ A^
c?a;+c?a!2 i.2+(?a;3 1.2.3+'^*°-

(^X

Now, making A=0, this fraction reduces to -^=7 ; and

W-A.
if, by substituting for x its value a, we find that each term
of this fraction becomes 0, we must expunge these terms,
and divide again by A both numerator and denominator,
until we obtain a fraction of which the numerator and
denominator do not both reduce to zero. Hence we ob-
tain the following



VANISHING FBACTIONS. 21^



Rule.
I^nd the differential coefficients of the numerator and
denominator separately, and if one or the other of these
do not redicce to zero lohen the value ofx is substituted,
the ratio of these differential coefficients will be the value
of the fraction ; but if both reduce to zero, we must dif-
ferentiate again, until we find a differential coefficient
which does not reduce to zero when the value ofx is sub-
stituted for x; then the ratio of these differential coeffi-
cients will be the value of the vanishing fraction,

Examples.
