FLUXIONS. 461 
L cao As fPdz=Pr—fedPa2fXde—feXdz; dy Inverse 
“Method. i st; dp=Ydy, therefi => : 
F— therefor y= 2fXde—feXdeporten .° as Nidyp an pf mR git; Ueenehone pg.) Se 
; Hemiaand aes den twa scenstmnhcommartinnaytnas '=54/(0-42 (Tidy) and a 
dy—ard£f= 0, here X= az, in this case, =f +7; 
eld ada ages She enter 7 (c+2f Ydy) 
n.the very same way, the primitive of the. uxional, where ¢ and c’ denote any two constant tities. 
sigs tall 1 apace aeipnete: eet TEs, "Hles the equation’ te ys powenndil or 
under the form aaa Xdz, then we have ie = ed =a+4y. Here Y=a+y, and 2f¥ dy=2ay-+y* 
fXdz=P +c, where P denotes the fluent /X dr. hence 
perp eprint ng aghemanyert soe td beeper pas dy "; 
rest the operation is the same as has been explained. “7 Jetiagty) + 
The primitive will contain three constant corrections : 
alike fluxional-equation of the fourth order would cons = = 1. fa +yt+ ve+2ayty)} +c’. (123) 
tain four, and so on. , 
ji ‘dy d?y eg — 187. When the equation contains os and x, 
aeiad dz dx?” # ut 9 it may be transformed to a fluxional equation of the 
d i 
put =p, then, dz being regarded constant, =~ 
= 22; the equation will now involve p, dp, dz, and 
constant ities, and it will be of the first order in 
respect of p and x: we may thence find dz = Pd p, P 
being put for some function of p; and since dy = 
pdx=P pdp, we have 
2=fPdp, y=fPpdp 
These fluents being taken, and a constant quantity add~ 
ed to each, by elimi p, we get an equation 
expressing the relation between 2 and y. 
dy - d , 
Ex. Let aft sy (145% * — 0; when pis put 
d. dp, d’ : ad 
for 5% and Fp for 5 this equation becomes —~? + 
+p)* =0, hence 
OE ES ead LO 
(+p) (4p 
end takitg the Huents 
i i metelnnen expression 
problem : ae Ned Balen nde onan Me ead 
quantity. e primitive equa- 
tion just now found shews it to be a circle, which in- 
deed is sufficien! tly evident. 
frees 
first order, hentinae hated yond. Ses * for 
dy: then, if we can primitive of that fluxion- 
al equation, and thence the value of p in terms of z, 
we may have the value of y from formula y = 
Spas, or else, if we have the value of « in terms of 
p, then because fp dx = px—fxdp, we shall have 
y=pr—fedp. 
Ex. Suppose the equation to be 
_@e+ayh = y 5 OHryide _y 
dedy  ~— dp " 
where X denotes some function of # then 
<< Op and [= sh » 
OES aA 
2 Vv 
Let V represent f/'S, then p= Ja=v) and, y= 
n Vda 
vaU—VYy 
As the first member: of the proposed fluxional equa- 
tion the radius of curvature of an cee: 
its primitive equation expresses the nature of a curve 
whose radius of curvature is a given function of the 
abscissa. 
188, If the fuxional equation contain 5, 
and y, we may, as before, put p= 9, from which we 
got OY dP _ REP the -equation will now involve 
dp, dy, pandy only. When the primitive equation 
can be found, cdl Cette “lar Vitias be in terms of y, 
we may find z by the formula «= “Y; but when y 
is expressed. by p, we may then em loy the formula 
Shaul 
dy 
dx” 
to the same 
imitive equation. In fluxional equations 
of the 
order, however, it must always be under« 
