FLUXIONS. 
then the equation . 
amis dy + by dxr=ar"da. — 
“When n=1, the equation belongs to the class we have 
last arti 
considered in article. When -n= 2, the equation 
porehihee ae oe Lelyader 
James Bernoulli was the first that considered this 
however, did not ap until the year 1744, almost 
ve, 
i ioe sree emt 
; particular cases, he pro- 
it to the consideration of geometers. ( . to 
Lei; Cc pteeh ty uk rf r- 
H the son “ots be varrh-aenaeny how-to 
lutions rete sgl pee ye 
which equation is ; x=adzxz, 
d 7. 
dz= ali . 
ing from this case, geometers have succeeded in 
Soediiaghle variable quantities when m is any num- Sun 
—4i |. 2 
ber of the form Bice? * being supposed any inte- 
ger number. The problem, however, remains 
unresolved, and indeed its solution is as much a deside- 
ratum analysts as the qua¢ e of the circle 
wis sinaing the gecneters of ancient tines, _On Ric- 
cati’s equation, see Euler Inst. Cal. Integ. vol. i. sect. 2. 
. 1. Lacroix, T'ratté du Cal. Dif. vo ii. p. 256, &e. 
“if the separation of the variable tities general] 
be a problem of i 2 in so 
its fluxional may be the result left 
after the fluxion of the pri has been divided by 
‘its terms. Thus, if the pri- 
ing the common factor = xdy—ydx=0: This ex- 
pression «d y—yd<« is not an exact fluxion, but it may 
VOL. IX. PART II. 
_ as independent of 
457 
1 - 
be rendered so, by restoring the factor ry for then it 
becomes *2¥ = ¥4 ‘the fluxion of 2, 
Food x 
174. In let u be any function of two vari- 
able quantities « and y, then, whether these be regarded 
shenithabandiienaimditenatioset 
du du : 
the other, we have fem afte ay Oh (art. 103.) in 
this expression * means the fluxional coefficient of the 
function u, taken as if x were the only variable quan- 
tity contained in the function, y being of course con- 
sidered as constant ; and 5 is to be understood in a 
similar sense in regard to « (art. 100.) Put $“ = M, * 
= N, then, te sa Sait ta it has been 
du u 
proved (art. 102.) Pets de dsdy’ therefore 
aM _aN ps 
dy — az 
Hence we may conclude, that if M and N are such 
ctions of two variable ilies x and y that M dx + 
Ndy isa fluxion, the condition expressed by 
the equation (1) will always be satisfied. 
On-the contrary, if M and-N are such functions of x 
and y, that i = SN, then Max-4Na y shall be an 
exact fluxion, which in every case be found. 
To prove the part of the proposition, let us 
su See eee eee the hypo- 
thesis, that in the function M, « is vari e, and y con- 
stant; and let the fluent be P+-Y, where Y is any 
function whatever of y, which serves as the constant 
correction of the fluent, and P is a known fiction of” 
«and y, which results from /"M d « relatively to x only, 
so that M= SP. The complete fluxion of PY is 
dP aP 
dP 
ay ttt WF dy+d Y (103.) or Md x dg net 
dY ; by comparing this with Md2+Nd y, we see that 
the two expressions will. be identical, if we can. give: 
such a value to Y, that Nd y=-5 dy dY, or 
dY= (x— Say (2y 
and then the fluent of Md.x4-Ndy will be PY. 
Now, by taking the fluxion of M= 5° in respect 
a 
aM_ @P dM 
y 
of y, we have it 2 but by hypothesis on 
a&P 
dN or. dN_ dP 
ge and dydz™ dzayo™ ntherelone ery 
dN &P : dP\ 
Avie Tidy = that 18s a(N—4-)=0, 
the fluxion being taken, supposing 2 alone variable ; 
therefore N — is constant in respect ef x, so that 
it is a function of y only ; hence the possibility of find- 
cl lama 
Inverse 
Method. 
