460 FLUXIONS. 
Method. yr Aae + Bay tC ae + KO 
yr" By proceeding as before, we find 
Avast? "yABos® F844. ACaa’ 4-&e. 
+ABee ty | pst 4 be. re 
facet yee | 
—! +A" +32 +e, 
Hence 2a— 1=0, a4+-b— l=a, a+c—1=4, &e. 
Therefore a= 4, 6=1, c=}, &e. 
Again, Ate=1, AB (a+-5) + A=0, &e. 
Hence A =,/2, B= — }, C=y'54/2, &e. 
and y= 2446/2 — dat y..t 28/2 —Ko 
This is the primitive equation upon the hypothesis 
that y=@ when r=—0. 
On the resolution of fluxional ee - 
mation, consult Euler Jnstit. Cal. nleg vol. i. sect. 2; 
Lacroix Traité du Cal. Dif: Partii. Chap. 6. 
Of Fluxional Equations y the Second and Higher 
‘Se 
Of fusion. 182. Let f(z, y, c,c’) denote any function, or expres- 
al equations sion com) of the variable quantities x, y, and two 
ofthe s- constant quantities ¢, c’, besides, any other constant 
cond and = quantities. Then 
higher or- = 
at —— 3g uatio u aking the 
may y ve equation. Dy ; 
fluxion, (as explained Art. 45—50,) we obtain its, flux. 
ional equation of the first order, which will contain 
in addition to the other quantities, and may be expres- 
sed thus, 
S le 9 -, ¢,¢)=0.', (2) 
By taking the fluxions a second time, an equation 
will result, involving the fluxional co-efficient of the se« 
2 
cond order, which may be expressed thus, 
dy d* 
jes (x, » 7 ae Cc, e’)=0. (3) 
As these three equations will all hold ‘true at once, 
‘we may exterminate the two constant quantities c,-c’, 
and the result will be a single equation 
P(=y, % $2) =0, (4) 
in which the quantities c,’ are not found. This will 
the constant quantities c,c’. 
ad 
We may arrive at the very same equation (4) in two, dz 
other ways. 
1. We may give the primitive (1) these two forms, 
oa e)=e', (2, » c’)==e. 
pe pt ae per result of the first, nor 
¢ in the result of the second. These results may thgre- 
fore be put under the form 
After eliminating a and 4 by these three equations, we 
get 
2y— 229 4 ot SY x0, 
a fluxional equation of the second order, of which the 
complete primi ve equation is 2%—' az = 
We may otherwise put the primitive ples rhems or 
Taking the flaxions, and arranging the results, so that 
the constant quantities may stand alone, we have 
d 
at Qey— ne wet 
=2b, = 2a. 
ay = ay" 
I~ ae Yas 
taking the fluxion of either of these equations, the 
pe bien quantity in the second member disappears, 
and we find 2y— 29% 4 2 LY, the same as 
da? 
before. 
183. As two of the constant quantities contained in 
the fluent of X dz, then taking the fluent ofboth mem- 
bers of the equation, and considering that, in the first 
member, dz in the denominator is constant, we have 
afxae= Pc, anddy=Pde4cda, — 
Taking now the fluents a second time, — 
y=fPdepoate 
+ 
a 
» 
¢ 
-. 
1 gg menses = 4 
ae 
2-99 
Pe er 
hte he 
