FLUXIONS. 455 
ataialad qriantty, end the nie may be found by the common _Invene 
ie - —~ 
its form is circular, en a ee 
d, he we have shewn how to\pass from any proposed primi- equations of 
add- tive equation to its estinalacnation 6 any order, We the first or- 
or eae ates ta Re reverse problem, ; 
. which to find e — itive : tion I } to « a 
of this a fluxional equation ‘ot the first order, cilitaltting two ite 
(Act. Lip, 168. It Eaten iain dAdh 49), that, in deducing 
ave a solution, from a primitive equation its fluxional equation of the 
and David first ic. ine we can always eberahaabe ay one of the 
‘until the year constant quantities contamed in the primtitive equation ; 
: therefore, that this last may have the most general form 
On a ver- possible, it ought always to contain a constant but arbi- 
centre of the trary quantity, that does not appear in the fluxional 
AHC, CKB, equation. ; . 
esphere, and _169. Every fluxional equation, which involves only the 
; then, semicylinders simple powers of dx and dy, has this form, Mdx4+Ndy 
ill, when produced =0, M and N being supposed functions of any two vari- 
‘form four openings, able quantities x and y; and it expresses a certain relation 
sane oe between the ent teas x, its function y, and the 
ee ees. figxinsl poeticions <2. The method which analysts 
a? ; their s ax —4a°; first thought of employing, todiscoverthe primitive equa- 
spherical surface of the dome is 2a**; tion, was to separate the variable santitiill Wd tanive 
perfectly qua- it, if possible, the form Xdz 4+ Ydy=0, X being a 
2 ‘ ter of the base. _function’of x alone, and Y a function of jalone. 
solution, but without a demon- primitive equation was then f’Xdx4 f'Y dy =c, and 
to geom es 
gsomctia a pte sp Repstennl seston le 
Traités de Cal. Diff. vol. ii. mydxtnzdy=0; 
vol. i, Part 2d, divide the terms by « y, and it becomes 
mdz, ndy 
y oh ey = 0. 
Rectification Curves « Double Curvature. Hence, taking the. fl of the terms, we find 
' & of . ml, (x)-+nl. (y)=L (¢), or 1. (a™) +1 (y")=L (c) ; 
Rectifica- 166. The nature ofa line of been and passing from s to numbers, 
tion of in Conve Lives, (Art. 51). Let CPD be st of ay =e; 
ance cur--gueve of eis Kind, (Fig. 42 to ‘thireé co-or- and this is the primitive equation. 
‘Y. From every point 170. The variable ies may always be separa- 
in =a ee _PP’, &e. be drawn to ted, when. the Acciaeal enmetins is homogeneous, In 
Saree <a 3 aa ek too oe this case, elie equation BLS -4-NudyoaO ben the form 
Palin Wilcke the projection wf the cd curve, (Ay* 2h 4 Byk—n okt" 4 &e.)dx 4(Dy!tra'—P 
Again, from P’ draw P’QY, P’R’ perpendicular to AX. + &c.)dy=0, : 
AY. Pat AQ, or PRY = 2; AE oe PQ oe the sum of the exponents in each term being / +/. 
Toit eres 5 oes je ion C’P’=v’. Then Put m=h+hk, then, dividing all + pe terms by x”, any 
we if we suppose 
cin suice Ae exalted upce & plata, term as A yt. becomes A (2) ; thus Mand N be. 
curve, and its into a straight line, each come functions of 2; 80 that if We divide the equa- 
of the curve v, and tow Nd. BE oe 
have d v=4/ (ded #5 therefoce,by substituting for on Tet N Ga by M, and the fraction ay By #* 
dv’ its value d x*4dy*, we find Sn pret weareane¥ will be a function of = 
doen (dna rede), : d Sade. rating saga danas, 
i f for the rectification a Diy aha es . 
fine of double carvature.” B By uae gy the tee equ, therefore, #d 244 (2-+2Z)de=0, and hence 
tions which the nature line, » " dz, dz | rye FEF 
of the line, w nee Ft Gz Ho andes. (7 =C. 
uxion of the curve will contain only a single variable © Examrre 1. Let (aw-by)dy+ (fx+gy)de=0. 
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