. 
—— 
FLUXIONS. 
177. Let M be any function of two quantities x, y,, 
and let u=fM dz, the fluent being supposed taken 
that x is variable and y constant; 
ry to find the fluxional co-effi- 
cient of u relatively to y; or in other words, to find the | 
fluxion of fMd'r relatively to y, without previously 
finding the fluent of the expression in respect of x. . Be- 
ae du. @u dM 
cause w= {Md 2, therefore — = Vedas ay’ 
Pu d?u 
@u dM en 
dydx dzdy’ dady~ “dy wanda ey 445 
OM SeeGrrt: ; 
dy 4% and taking the flaent relatively to x, considers 
—dz 
| dy 
will apply to the ealeulation of in art. 174, with- 
out previously taking the fluent P= f'Md=. It was 
invented by Leibnitz, and was considered as an im- 
portant discovery in the calculus. (See Bossut Traité 
du Cal. Diff: &¢. vol. ii..p. 58:), ‘The whole fluxion of 
Siete G eiice eas bbe 
Mdz+4 1 fz a} dy. 
Observing that the fluent in the parenthesis is to be 
f walone tobe variable. 
L178. a fluxional equation involves the second 
or higher powers of dz and dy, as in this example 
dy — ads? =0, we may find the value of “Y , by re- 
ei ee equation. In the present case, 
52 ==ta, s0 that dy + adz=0, and also dy—adz=0; 
hence y+- az 4+-c =0, and y—ax + c’=0, are tw 
shitive’ equations, from either of which the dodo? 
SS may be derived, and also ftom their pro- 
(y+42-+4¢) (y—azr+c’)=0. 
_ 179. When the ion contains only one of the va- 
triable quantities, « for example, we may deduce from. 
d 
it St = X, a function of z; and hence y=/X dz. But 
oie) <5 Kig 1525 EET equation ‘in respect. of 
», then, putting 5% =p, we may find «=P, some- 
function of p, and hence dz = d P ; and since dy=pdz, 
therefore, dy = pdP, andy = fpdP= pP —{Pdp. 
‘The relation between x and y is ‘now to be found by 
eliminating p, by means of the two equations 
we oP, FPP ae es ee 
Let the equation be rdxady=b4/ (dz*+ dy’); 
d 
Making p = 7. we have 
_ powers of ¢ 
nate i 
459 
ra=bJ/(l+p?)—ap=P, - 
y=bp/(1+ p)—dap?—b fdp (1 + p’)- 
The fluent of d p 4/(1 + p*) may be found by art. 123, 
180.” When the primitive equation cannot be dedu- 
ced from a fluxional equation by any of the known ar- 
tifices of analysis, then, as alast resource, recourse must 
be had to approximation by infinite series. 
Ex. Let the fluxional equation be dy, + ydz= 
m 2 dx, and let us suppose it to be known that when 
z=a,theny=6. Assume «=a +t, and y=b-+-u; 
then when t= 0 we have u=0; we have also dx=di 
and dy=du; by these, the proposed equation is trans- 
formed to 
du+ (b+-u) dt = m(a+t)" dt. 
We next assume _ 
us Ate Betl cette &e. 
«, A, B, C, &c. being indeterminate quantities which 
are to be investi, From this assumption, d «= 
f. Ao epi 4(e$2)C% t+ &e} dt. 
The terms of the equation being now brought all to 
one side and put = 0, and these expressions for « and 
du being substituted init, we have 
aAfm 4 (241) Be +(e 42CKt! + &e. 
Hy lig A+ Batty softs 
—ma—m 2 gett mars 2—&e: 
If we suppose « = 1, the terms placed vertically be- 
come similar, and then as ya meyeer like 
=O agreeably to e theory of indetermi- 
coefficients, there results Gn 
A+6—ma*=0, 2B+4+A—mna!=0, 
n(n —1 
b emaiee 
TC. a’—2= 0, &e.. 
3C+B—m 
Hence. A = m a” — 3, 
B-™2 a1 ma"+b 
2 + : 
c _-mn(n—1)a"—?—mna"—! +-mar—b 
a 2, 3 > 
&c. “ 
These values being substituted in the series, we —_ 
u expressed. by ¢ known quantities ; we may then 
put «—a for ¢, andy— 6 for u, and the: result will, 
uns relation between x etard 
e might have proceeded wi original equation 
dy+ydx=ma"dx exactly as we have done with 
the transformed equation, assuming y= Az*4Ba*t! 
+&c. But as the result would not have contained a con- 
stant correction, it would only have’ given the relation of 
yand x upon the i :y=0, when z=0, The’ 
transformation, serves to introduce the constant correc- 
tion. 
181, In the assumed series w= A" Bi**) &e. 
the exponents of ¢ form an arithmetical. progression, of 
whic the common difference is 1. In many. cases,, 
ever, the common difference will be a fraction, as. 
in this example (dz + dy)y = dw; here we may as-. 
sume 
Invesse: 
Method. 
