: ae , 
FLUXIONS. 
stant alwa disappear in taking its fluxion ; so that « 
a variable, and c constant, the fluxions of 
diced anton expression du. 
on the contrary, the 
an any, Ah ony ders the coh cre 
This constant quantity is common! = by writers 
on Fluxions, the Correction of the 
114. By reversing the principal rules-of the direct 
wethod We find as many for the inverse method. 
I. The fluent at agg consisting of several terms, 
isthe sum of the facts of its terms, each r its 
sign, and coefficient (art. 27.) nog deer ee du, 
=av—bu+te, for the constant 
quantity, or eae be ay ag 
IL. As the fluxion of x* 4+ cisnz"-* dx; (Rule (A 
Art. 26.) ‘on the contrary, th Ment of 9 = de wi 
be «* + c; that is, we must increase the exponent by uni~ 
pS Nnetgd ste “ind divide bythe ‘exponent 
Az'tl 
[eet +c 
As particular examples, 
SS afar de=— har fen— ate 
Allee, Siz aavess 
applies to all fluxions which can be redu- 
cid bata? Pee Fer eset let the fluxion be 
ax" dz(b + e2* )™: as the fluxion of 4 + e2* is 
qreords, We er the first factor a 2"—' dx to 
this form, dice in ay in the constant factor ne. 
» putting z = 6 4 e 2", we have 
< xnexidz(b+ea" =< Sande; 
, 
Therefore the fluent is 
” eh a . 
it << ne(m+1) Crear Pinte. 
The transformation which has introduced + is not abso- 
me, rer er nae in such a Sieemes may with advan- 
av 
focercore baa hie 
TH. The foregoing rule fails when n=—1, because 
then fz" diss = 4, an expression of which the first 
term is infinite ; but this because the fluxion 
belongs to another kind of It appears from 
Art. 26, Rule (B), and inore ge- 
5 fea Miro 40 
meralor is th futon ofthe Spee Me 
so ong pad — lar denominator. In this 
10 the lsent under the orm l(c), 
fu ci 4 523d x . 
435 
rule by a eee of its constant factor : thus, 
Se aur 12 so {h(Se47) 41 (ot 
=5 1 fo(ss4+7)} 
IV. We have found, (Art. 29.) that uw and ¢ being 
any functions of avarable quantity, d(u #)=ud¢-+1d 
therefore, 
ut=fudt+fidu 
and fu dt= ut—ftdu. 
This manner of expressing a fluent is of great im: 
ance : waiters oA the, digerenttsl ke pate 
thud of integrating by parts. As an example, let the 
fluxion be 1, (aie then, putting dt=d 2x, and w=1. (2), 
so that d um, we have 
fi. @draal(2)—2fe. 
The rule indicated by the formula has the advantage of 
making the fluent depend upon’ another, which, 
analytical address, may frequently be more easily sd 
V. From rule (D), art. 26, if the radius of a circle 
be unity, we have 
rp idz 
va) 
Sms = are. (cos. =z) 4c. 
By these expressions are meant the are, of which the 
Poy pe open Again, from art, 35, 
SGs= are (tan.=z)-+e. 
We may also suppose the radius =r, and we shall have 
= are. (sin, =) +c, 
rdz 
Ae apni fe 
To find the fluent of epbacrstes eam 
m al 
pat ols age 
Resta Spe Therefore, ——— Fa) are (tan. = ¢) 
is the fluent 
sought, supposing the radius unity, and 
dz ‘ 
JSagie= Tat’ ahs (mises) pe: 
ee eee er 
fi. mdz 
J 7@=by= Ve 
are Gin. ==) +e, 
Inverse 
calculus call it the me- 
The direct method furnishes other rules, which willbe — 
noticed in the sequel, 
Decamposition'of Ratiinat Fractiont: 
115, Let it be proposed to’ resolve the fraction 
ast! y into two others, of which t shall be the 
Ce rma. teat tl , that the 
fractions sought. may have the form —*— and > 
A and B being indeterminate coefficients, which are in« 
