Direct 
Methed. 
—— 
428 FLUXIONS. P 
- Slteoes +6 ‘ad given pedi eos 29 cates g may bord 
ds (o—)" ae cxprtaned by 6 eeee eaeeeeenee ceo nee 
instead of dv wubstitute its value £—— dx, a8 given by parr strnpregdine! hen! Trae fata be assianteaby 
the, segs ‘equation, .and soxeover, pidion yor soesne of ie two-sgguetioiag mid Ug rseale wl! be sn 
* + (e—9)" as given equation, ation involving ind known quantities, 
Or dak dividing by 4% we get which will be an Sten or tos local of ths eoaire of 
deo r the circle of curvature. aoe 
de @—9 oa Festa (A) Of test artis nage bea 
i i i 1) and (2) in dius of curvature, when the nature o curve: 
hiegge senate ng eatin OT the Laden trae oan of the rectangular co-ordi- 
do_d lastly, ates: But let us now suppose, that its nature is de- 
circle, but as we suppose v=y and 5 = 5%, and y aed Wy it SOROS Si ee ee 
AE= perpendicu from 
T= = $4, the same three equations relatively to the axis B cient a ella ta? balk Ore BAE =u, 
ds dss which that perpendicular makes with the axis (Fig.18.) Fig. 18. 
curve ‘CPD ‘will stand thus: ! Draw the normal PN, and let 2 and y be the rectan- 
(p—#? + (y—9)'=" () gular co-ordinates of the point P, ‘The angle at N=w 
5% ae’? meee (2) 7 neni" SOR ee eee 
a es vt (3) ‘tangent makes, with the axis; hence 5% = cot. u (for 
im (gana ; é , the é ey os ; 
Food he tek util eStansd a Chins Sehltitiig we Sell Spee (1), OTs) Se OR ape ate eee 
rd (4) and y as functions of x, 2 nt “5. re it gen weet 
=: Mit 
(4x'-+-d9") (art, $4.) but cosee:* w= 1 + cotant v= 22 hI" 
-1= = (5) therefore dxd*y= —du (de +dy)=—dudz 
1 teat aye ja ( ing read Seri that d2*= 
and again, from . and equation (3), da4dy, art. 75.) and since by formula (A), last art 
dtd + pgrottieyben cn hay be 
ae | vt a. (A) = — Geter) 2 ES y therefore, = 
This value of r being substituted in equations (4) and 2M Bi ot Sue Vy ee eee 
(5), pS cae these results r= (A’). csdsthees 
+ Pa =" . . ae 
=s—dy——, B hav ; for the radius 
BEE Fe PD got Soret ae aadnc abcess te anarae 
+°9 fuuxion of the arc divided by the fluxion of the angle 
WY Fendgit pi eR acy: Redbone nerd 
The three constant quantities p, q, r, which enter into “And as we have found (art, 77.) that dz=pdu 
the general equation of a circle, being thus determined, d? p 
it may be concluded that no other circle can be- ee therefore : — 
tween Sy Sp eer ok gy is panto y few a &p n 
values » Gy Ts or it ible, let u = a *\. 53 
cudiinate in thét'cihde ittle caeieaondling 40 thie Bie = Pt Te (a) 
scissa 2, 7’ its radius, and p’, 
a 
the co-ordinates of its 
centre ; then its equation w 
be 
p—#)* + (u—q')* = 12, 
Now, that on! cincke Aspe ts between the curve 
and the other circle, it is necessary that u=y, fe = 
a 
ot = a - But these being the very same conditions 
they will give 
the same values for p’, q’, r’, as we have found for 
must coincide, ! 
osculating circle ; also the circle of curvature, because it 
serves to measure the curvature of the curve, The 
quantity r, the radius of the circle, is called the radius 
of curvature. 
87. The centre of the circle of curvature being dif- 
ferent for different points of a curve, there is a certain 
line LM, belonging to the curve, which is the locus of 
all the centres. Th uantities p,q are manifestly co- 
ordinates to any point H of this locus, and its nature is 
Pid : 
This is the expression for r which we proposed to in- 
vestigate, t : 
89. We shall now shew the application of the for- 
mulz we have found for the radius of curvature. 
Ex. 1. Let the curve be a parabola (Fig. 23.) and Fig. 28. 
let A, the vertex of the axis AB, be the of the 
co-ordinates AQ = 2, and QP =y. Let a be the pa- 
rameter of the axis, eat noe apse sate 
; Pw heer Peer 
=ax, hence =at«t, ndd =f and d?y= 
Rei dying A rh RATE Nah 
| ata? - ‘ada 
2" Pherefore, di? of: dot alate ae 
at ore, + ce ha haw 
) pi det pay (a 40) 
ro dad*y * Tiae ave “mn 
Ifx = 0, then r= 4a; this is the radius of curyature 
at the vertex. 7? wih in velar : 
As an example of the application of the third formula 
for the radii OF COLYER let 7 again take the para- 
