500 FLUXIONS. 
found = ^- 4 —-== —r» and confequently RO f~) = 
—V —V \w J 
ceed to examples, it will be proper to obferve, that the 
right line A p, denoting the radius of curvature at the 
vertex A (to be found by making x, or y, —o) mud al¬ 
ways be fubt rafted from RO and Ae to have the true 
length of the arch p O, and its oorrefponrlingabfcifla be 
75. Ex.i. Let the given curve ARB be the common para- 
bola, whofe equation is y—afx* : then willj— 
-—rr~ — --, the fame as before. 
—w v x y 
Otherwiie without the circle.—Let RO and rO be two a > 
rays perpendicular to the curve, indefinitely near to each 
other; and from their , , 
interfeftion O, let OF — y> ant3 (making x confiant) y —— i.^.r. a 2 JC -2 JC 2' 
be drawn parallel to zx 2 
An, cutting Rm and AF 
(parallel to R«) in E = —1'.: w 
and F. Therefore, fup- 
poiing R£=r, An—x, 
xx 2 
__a 
3 . 
4 * 2 
whence z (\/x+y 2 \ — - / 4A '+‘ g 
2 \ x ’ 
lRn —y\^ x c ’ ( asbef ' ore ) and the radius of curvature RO (— 
we fit all have, by firm- \—xy) ~ 
lar triangles, as RP ( v) .... 
: P q(y):: RE (v) : EO 
— and confequently 
which at the vertex A, where o, will be '=±a— 
Ap. Moreover Ae (x-3- ? \ — x a-\-z,x, and there¬ 
by./ 
FO (Ak + EO) =x + 
vy 
zhich value (as well as 
fore fe (Ae— Ap) — 3*, the abfeifla of the evolute : 
", r , , Hkevvife Oe f— -— y) — 1 — the ordinate of the 
that of AF) continuing the fame whether we regard the J V" 
radius RO, or the radius rO, its flux ion mu lt there- evolute. 1 herefore, q 7 ) 2 X« being in a conflant ratio 
to p? , namely, as 16 to 27, the curve is, in this 
cafe, the lemi-cubical parabola: whofe arch p O 
_3. 
(RO— Ap) is alfo given — c_b 4 *l 
fore be equal to nothing; that is, x■ 
vy-\~vy X x — vyx 
(?) 
; whence v — - , and confequently RO 
yx—xy 
x 2 z+vyz _ x 2 z-)-y 2 z _ z 3 
23/ a 
yx—xy 
yx—xy yx—xy 
1 -\-n 2 a 2 x , RO 
fuppofed conflant, or x— o, will become -—-, as above. 
£ 3 
But if y be fuppofed conflant, it will be —. And, 
yx 
if z be conflant, it will then be ~r : for, fince x 2 -\-y 2 •- 2 > 
Ae (x+ZJ-) 
76. Ex. 2. Let the curve ARB denote a parabola of 
which, if .v is any other kind: then, becaufe yz=.ax" is an equation to 
all kinds of parabolas, we have y—nax"~ «x and y— 
71 X n — 1 X ax’" 2 x 2 : therefore z { 3/ x 2 +j 2 ) — 
xj 
4 
x-\-n 2 a 2 x 
1 zzzn 2 a 2 x 
—z 2 , by taking the fluxion thereof, we have 2xx-\- 
—n x n —1X ax 
—1 
—, Oe 
2yy — o ; whence y — — — ; and therefore RO (= 
z 3 v z 3 vz 3 yz , . 
-e ) =-r-rrr = - — - — as before. 
yx — xy) ... . x 2 x ,‘, 2 j_ 42 v/U x 
2 ) 1 - 
( 5 - 
i + is-iXssh , „ n 2 a 2 o 
- v"";" '. and Ap — —-- 
yx- 
y 2 -p.v- X* 
—n —i x nax 
which, if 71— 4, "ill become 
■; but, if n be 
y 
Now from the feveral values of the radius of curvature 
RO, found above, the correfponding values of Ae and 
eO will likewife be given. Thus, if x be made conflant; 
<2)3 
then, RO being — we fliall have Ae (An-\-Om=.An 
—xy 
4 -xRO) =.v + ^-r, and eO (R»z—R»=. xRO—R«) 
z —xy z 
< 2)2 
= —;.— y. But, if j be made conflant, then, RO being 
2)3 <2)2 X e Zp‘ 
•==~r.y we fliall have AEzzxH——, and eO zzz—rr; — y. 
yx x yx 
Laftly, if z be fuppofed conflant; then RO being = 
yz 
— y we fliall have Ae 
x 
=, + £. a„d t O = 2 
X X 
— y. Which feveral 
expreflions will ferve 
as fo many general 
x theorems for deter¬ 
mining the quantity of 
curvature, and theevo- 
lutes of given curves .- 
but, before we pro¬ 
greater than a, it will be =0 ; and, if n be lefs than 
it will be infinite : whence it appears, that the radius 
of curvature at the vertex will be a finite quantity in 
curves whofe firft (or lead) ordinates are in the fubdu- 
plicate ratio of their abfeiflas, and in all other cafes, 
either nothing, or infinite. 
77. Ex. 3. Suppofe the given curve to be an ellipfis; 
whofe equation (putting a and c for the two principal 
diameters) is a 2 y 2 —c 2 y.ax — x 2 . 
Here, by taking the firfl and fecond fluxions of the 
given equation, we have 2 a 2 yy — c 2 x X a — 2 *» and 
2 a 2 y 2 2 a 2 yy— c 2 x X — 2x — — 2 c 2 x 2 ; whence y — 
c 2 x X a—zx .. a 2 y 2 -\-c 2 x 2 
-7-, and —y— -: which, by fub- 
2 a 2 y ’ y a-y 1 
ftituting the values of y and y , will become y — 
12 
cx'ya — 2 x 
2 a 3 / —* 2 
cx 2 
and — y 
a 2 c 2 x 2 X a—2x\ 
4 a 2 X ax—xx X ac 3 /ax —x 3 
x 2 a —2.1 j -|-4X< 2 *— x 2 _ 
a\/ax—x 2 a 4 Xax—x 2 -\/ax—x 2 4X«.v—.v 2 !? 
therefore a (vj 2 +* 2 ) — ^ ~2 a z X ^=== ^ + 4 ' 2 
X ax — x 4 
