442 
mide(l 4a); 
—— We now make 1 + a2 =2', from which = 
fs") by raising both sides to the — 6th power, 
and taking the fluxions, we get 
a(—1), 
a* * 
Hence, by substituting, the fluxion is transformed to 
| ; . 
—_pl—=)ds 
of which the fluent is Ke 
1 v4 24 
eA (8$ 1) =¢— sap 
In like manner, the fluxion 2° dx (a? 4 2) be- 
comes | dz (=*—a*z'), by making a* + 2*=23; hence 
the fluent is 
3/(a® 4+ 2*)*(420°—Sa?) 4c. 
128. When OS entices lo not satisfy one of the two 
conditions specified in art. 126, the fluxion cannot be 
rendered rational by any known method; we may, 
however, reduce it to the most simple form of which 
it will admit, by means of the formula fu d¢ = wt 
— ft du (art. 114. rule IV.) 
us put u=2P and dt=a"dz, then du= 
rvde=— 
+1 
pz? dz; and (= 5 hence, 
r™+12P Pp P11, m1 
m1 Jaf z dz. 
Bat z=a 4 bx", anddz=nb.2"'dz, therefore, 
1 
fren 2=Z LS wendin. zPol 
wdz.2?= 
Again, because 2? = 2?—! z = 2?—! (a4.5.2"), there- 
fore 
fordz.2=aferds. P46 [Pl etd e (2) 
These values (1) and (2) being put equal, we find 
b(m +1 np) fP- erde= 
xe 2P—a(m+41) fz? xd x (3) 
Change now p—1 into p, and m 4. into m, and we 
have 
wf) 
preva —a(m—n+1) far? d x 
6(m+1+4+2p) 
and putting for the last term of equation (2) its value 
as given by (3), we get 4 : 
(B) ‘ 
frrdzs?= xPgeri + anp fz" dx. ‘ 
129. We shall now 
m-1l+np 
in which it must be 
shew the use of these formule, 
recollected that 
z=a-+ bz". 
1. Formula (A) makes the fluent of a 2? dz, 
pend on that of 2»—" 2?d'z, and again this last on the 
fluent of x2 ="d x, and so on; it therefore serves 
to diminish. the mt of x out of the binomia 
@; at last to bring she hasnt to eco 
a Pd 2, i being any whole positive number. 
2. Formula (B) serves to diminish the exponent p 
FLUXIONS. ? 
by 1, 2, 3, &c. units, and thus to make the fluent of 
d on that of 2" 2?“ ds. " 
the fost in tee gedpad member os phe vie or eo 
[rasa 
_frades? aE Hating i frrind S 
de- Es cay V7 =e2*)’ 
SFeH 
depend on that of 
a 2? dx 
equation, then substituting’m—a instead of m, in the 
hot Geel ae instead of p in the second, we 
. , 
ons, Pt 1 pnt p+n+1 Yfomerehd. Py 
(D)" (m41) ; 
an(p1) 
These formule serve, on the contrary, to increase the 
exponents of «, out of the binomial, and in it, and are 
useful, when the one or the other is ve. 
- 4. These formule shew the law according to which 
the terms of a fluent are formed : thus, it is easy to see 
that the fluent of ; 
ada 
het Ny Bb: rseed, Vv (1— 2%). Pe 
B i e fluxion of thi ion, we 
the co-efhicienta, A, B,C, by the meth method of indleter- 
minate co-efficients, with less trouble than by applying 
the general formula. j 
130. We shall now indicate a mode of finding fluents, 
remarkable on account of its simplicity, and nume- 
fons faatasde in Wiech 1 sadly Wage. Taking the 
fluxion of a*—'s/(1—22) we have d fe-yu—«)} 
ee 
ov (1—z*)’ 
the first term of this 
= (n—1) 4 /(1 — 2°) dz — 
By multiplying and divi 
flaxion by 22) my after taking the fluents 
and transposing, ®) 4 veer 
ede em tY(isa) nlp mide 
70—*) n a J 7a—y 
By treating the expression 2*—!4/(«?=+1) in the same 
manner, we get 
ae psiloc Il PO dar 
n aS J (el), 
find the fluent of every 
wdx 
V@=D 
z 
form Fatma) © (tata) 
ced at last to the m7 (eae) ” Va 
By dividing the numerator and denominator by a, the 
radicals in these expressions may be changed into 
hen oil and 4/ («* == 1), ahd ‘Chati Sottitglas (E) 
and (F) make the fluent at last depend on 
rdzx xdx 
or 
if n is odd; 
dz 0 f Ezy if n is even. 
The two first fluents come immediately under Rule 
II, art.114, by putting 2*=£1, or I=t-z*=z, from which 
adx=or—xdx=d z; the third has been 
in art. 124; the fourth is the arc, of which the sine is a. 
For example, we have. 
4 
