Snide 
Method. 
ay 
i al 
x 
FLUXIONS. 
441 
Then, squaring and suppressing the common fatter, m41 - 
ye Mave etsicss).°3 hence ae (2a) _ ‘ 
; Baz _ 2(a—8)zdz 4 mel" 
BF pant | Got) vedi Sehcae tt 
Wile 468 0) (onin) 8 eran) PY ebeietions the prapoved Harien bevatres 
These functions, as expressed by z, are all rational. B Ste—6) 8 ad 
# nba ; 
UA dah se oS ER then, 
making the above 
f 1 tore? 
ip >> the fluent of which is — 2 are (tan. = 
dz s—zx 
Pr oy TCT 2are fen. = /=} 
If we suppose a= 1, b= 0, then, because 1 —x* = 
+ s)G—s), we have «=—1, 6=1; in this 
case, the ula becomes 
ic (aro hae See: 
The fluent of this fluxion is otherwise expressed by 
arc (sin. = 2) 4c’, (Art. 114, Rule V.) 
If dy = dz /(a* — x*), by applying the transforma- 
ij ving that «= —a,&—=a, we 
 —Satz2dz 
dy= J 
j= +2) 
The fluent may now be found by the rules for ra- 
tional fractions. 
The same mode of transformation will apply to. the 
es pa ee rman e Bs 4620 
are real. 
125. The radicals 4/(a+-b2-+4-2*), and ,/(a+-b2—<2*) 
may also be transformed ing «= z— 46 inthe 
first case, and «= z 4+ 46 in the second; then 
Soiree eee Leone) In the 
case, the irrationality may be removed by making 
7 (a = 2) = a—uz, for then 
2au ?@=e1 
a= Pag tet a8 Seis 
‘ x —dz 
It is ~ that Tata) ene Alay 
by ere lepers the fluent may now be found 
by - art, 114, Again, making 4/(5?— 2?) = 
6— uz, the fluxion is changed 
an a fluxion of which the fluent has been repeat- 
edly assigned, 
from its second form to 
Of Binomial Flusions. 
Of binomial 126, We propose now to find the fluent of 
fluxions, pe 2 
K2"de(ab2"). 
m,n, p, being any numbers whatever, whole or frac- 
, positive or ve, 
In the first place, we put 
zmatbn; 
i 
Hence, = (=“)’: Raising now both members 
of this i the i, and taki 
h ages power m + taking the 
VOL. IX. PART It. 
tution, it is immediately trans- 
Now if =i, bad “2, -0hiih: anion “his “HG” Bich 
K’ 2? dz, and its fluent may be found by Rule II. art. 
114, 
If as —1=some positive whole number r, the flux- 
ion has the form K’ (z—-a)"z” dz. This expression may 
be developed, by the binomial theorem, into a series of 
a finite number of terms, each having the form A 2! dz; 
their fluentsmay therefore be found as in the former case, 
and thence the fluent of the proposed fluxion will be 
known. 
If a bea negative whole number =— r, the 
Kids ~ ; this expression 
(z—a)” : 
may be transformed into a rational fraction, by assuming 
z= ul, g being some whole number, such, that pq is 
also a Whole number, as has been shewn (art. 122). 
Therefore, if the exponent of x out of the binomial, in« 
creased by unity, be divisible by the of x in the 
fluxion will have the form 
binomial, the fluent may always be found by the rules al- 
read: ined. 
This, wever, is not the only case in which we can 
find the fluent. If the part of the fluxion in the bino« 
mial be divided by 2", and the part without the bino« 
mial be multiplied by z”?, which will not change the 
value, the fluxion will be expressed thus 
Kat" (6 aan Yds a 
and putting z= 4 +ax2-*, roceeding, as in 
Seimei vale. Gis Bixien be ttuiheoerto 
From this expression we Iearn that the fluent may be 
found when leaky ha Fda tae or rather when i + 
—n n P 
is a whole number. Hence it appears that when = : 
n 
is not a whole number, if p be added, and the result be 
a whole number, the fluent may still be found. 
127. When p is a fraction, (which is the most im~ 
portant case,) and g its denominator, it is most conve- 
nient to assume a + b2"='2’. For example, let it be 
required to find the fluent of 
5. 
a~*dz(a+23) 5. 
e m+1 Par 
In this case, —~— = — }, to this, if we add p= 
— $, we find —2; therefore we must multiply and 
divide by (23) or a~*, and then the fluxion be- 
comes 
3K 
