Sars hkae: tion 
measure the fluxion of the other quantity, We have 
supposed y to be a function of x, sa. that d is consi- 
diese wo conctent, bet, to poss Sune ie of y 
a function of x, to that of za con- 
? 
art. 96, and put 999° foe #9." A similar remark 
Oy a ee ee 
regarding = and y as functions the one 
of the other, they may both be referred to some other 
variable quantity ¢, by means of the formula above 
4 a 
ey 
Exampte. Let the equation be (a4z) * ~— ar 
+2 49 0, in which de is constant te ho 
the above substitution, it becomes (a+) ian 
t a tay? in which dy is regarded as constant. 
190. gg amainath ia ete pe which 
have this form, 
like ion of the first order, Fndecd, except in 
partodiar cases, there are no kniown methods of redu- 
i to the finding of the fluxion of a Ga 
variable quantity, that is, to ‘the quadrature of 
If R=0, in which case the equation is 
a ad 
Fo +P 5% +Qy=0, 
to 
a 
i be reduced to a fluxional equation of the first 
order by a very simple transformation. Putting ¢ 
for the num umber of which the Nap. log.=1, assuine 
y=e/*4*, then, taking the fluxions, by art. 26. rule 
©. considering d x as constant, 
dy=udzel*@+, diy= e/uds (dudx+w'd x*), 
The values of dy and d'y being substituted in. the 
equation, and the common factors rejected, it becomes 
du (u'+ Pu+Q)dz=0. 
If P and Q were constant nem Me u might have 
been 4 constant quantity, we then have d u=0, 
and to u we have 
u+ Pu+Q=0. 
Let a and } be the roots of this quadratic uati 
then w=a and u=6, and hence these two values Spe 
ed Athy . 
Faby nine 
or putting C for e°, and C’ for e° ; 
FLUXIONS.: 
’ as one of the 
~ 
> 
y=Ce™*, y= 
lara es ony pr  of  ee, 
const quantity, bat by a owege 
if. rd y=Ce® Peo Rag 
or the complete primitive ara F To 
i th cml : re 
¥ =a Ce 
age Be 
= +iCe * Bt cwcd* uC . 
T=7v7'\ 
From thi and the prinitve equation, fer elimina 
ting C and C’, we have : 
Papeete -- ions 
TR mom Hd il 
This will agree with the proposed proposed equation, i we ove 
a and 6 such values, that a +4 =p | 
If a and 6 come out impossible q 
ponents of ¢, Se Se will have the form 
“++ 8s/—1, but then the exponential e® "1, “may be! 
expressed by circular functions, Art. 123, and pe 
METIC io Sang pe” 
191. wealeiile oueiibindiin. 
tos os, then fy and ae two. Pip of y, which. 
each mitra Sechide + Qy=0, 
we may take > WY 32 
yaCu poy, 7 wr 
for the com equation. _ " 
_ sisviaattaad Sue Far the “3 
al equation becomes 
(Fr + Ps ose 
WA Pes: ‘hy whe ster 
therefore the identical, d > the value of 
inte he Maes and 80 ue 
The property which we have china: fo, be 
linear equation of the second order applies to nae 
equations of all orders whatever. See Lagrange Theorie 
des Fonct. Anal. 65—70.. 
192. As an exam —— of the manner of resolving a 
fluxional equation gig ie order by approxima 
tion, let the 
yt anyds =O, 
batters = see 
y= (ALB ott ae 2**4 ps JTS 00), 
thence we deduce 7% = Bui cy 
room yas Epa mb eyodng NBS Fade 
ax*y= aAx*t "+ &e. 
Fepeniste: mibsissing. in de usipond egeetie, 
we get _ ; 
~a(a—1)A=0, 
Wk “ch 2+ 2 PNB SPARED, be, , 
“The first uation ppt ils sper 
corrections that enter into 
then 
the value of y; that 
have a = 0, a 8 Vetter 8 oe 
may be the case, we must - 
