Conceive now the point P’ to approach P, then O will 
to H, and ‘ine OG SiN in mag- 
nitude to HP=r, which will be its limit ; ee 
tion to which OG is equal, will approach to POP haw 
art. 73.) Hence, remarking that the arc PP’, and the 
<' P', are the corresponding increments of and 
x, and passing to the limits, we have (art. 23% = r. 
Bat this value of r, is the very expression 
have found for the radius of curvature, (formula (A’), 
art. 88.) Therefore, ris the radius of curvature, at P ; 
tre of the circle of curvature ; and the evolute LOM is 
the locus of the centre of that circle Hence, if AQ=a, 
pact be the co-ordinates of P, any point in the 
curve PD, and AK=p, and KH=q, the tes 
of H, the corresponding point in the evolute, by art. 
86. formule (B), (C), its equation will be, 
dy. dattdyt) 5 datpdyt 
P= — ds a =9 t dy 
For example, if the curve CPD be a parabola, van 
putting a for the parameter, in this case y*=a 2; and 
hence, dy=H*, doa > — 
ans Me: 
p=3r+ 4a, I=——’ 
As 9 comes out negative, the parabolic arc kd its 
delaioticien sides of the axis. Moreover, since 
ay om = amma (4 ne | “1\iayse x, we easily find, 
27 a q'=16(p—}a)s. 
to the semicubical parabola. _, 
evolutes, one of the most, elegant 
ath in hie Hovolog: , who hand- 
in 
F 
Fe 
a F 
Ee 
nt 
j F 
| 
i 
‘t 
eg 
i 
8 E 
I 
FLUXIONS. 
in eunerianny ahethdeaen eager: to: fatiis , 
Fe ree Tan oak cera 
onc griele Svea 
7 
oF changing the Independent Fri oni 
95. We have all along supposed z to 
manner whatever, and estimated the ceo 2 ey 
place in y, any function of 2, by it to the 
ay tan pee Son SD 
t variable quantity. It is sometimes 
at tin ecmapehiot ate he hypothesis, and pass 
oy being a imation of , to that 
pepe, netion 
ire em aa 
upon the quantity that is regarded as the i 
variable, Thus ‘'ar€°@7, we have fouhd, if s de- 
note the eub-tangent SESCEYG 0d 8/408 MANERA 
dinates, then «== y ; and thisis true, whether y be con- 
Giclee ay & Palio A 9 ee: From 
this expression, regarding Fanchiong ? 
(oo tak Ue Wemmetanet nant? ite we 
Rea say) = dz— sie nit 
If, however, we sie z as functions of y, og 
asdy must be now constant, we have ded. 27%, 
an expression quite different from the former. We are 
now to inves ah sacri by be pra eg Aca 
=o ), and let tae whats 
), us su} it w az bes 
en y becomes spins Taylor's theo- 
pa oY Bg SY 
det at & SY Rtk. (1) 
But %, us now sup} that from the equation 
y=f(@), we deduce z=F( (9); 00-that w ton finctioe of 
eas she rpee 
rkt dz : 
hah ge dp Dt ayiagt® . @) 
Let the value of 4, as -the first ion, 
be substituted in the secon ; and, witha view to abridge, 
let us put > ' 
dy dy. da 
y for, ee also for 7, 
dry. 
, &e. and we 
dy’ 
hav hty Sty” nt &e.) 
acs tye ty" + &e.)* 
+ yh sy ety" E+ 8c) 48. 
oferta nate 
Ox(e ¥—Ihb (ely! hatySe 
aly" Baty! ype yf) $8. 
comes Pak 
rem, 
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