FLUXIONS. 
499 
abfeifla correfpotiding to the point of contrary flexure. 
And to determine for any value of x, whether the curve 
be concave or convex, fubftitute that value for x into the 
expreflion for;!, the x being fuppofed conftant, and if it 
come out pofitive, the curve is convex to the axis j if nega¬ 
tive , it is concave. 
Ex. i. Let the equation of the curve be;'=3x+i8x 2 
— 2x 3 . Here y=3x+3^xx —6x 2 x, and y—36S 2 —i2xx 2 — 
(it x—1) 36 — izx. 
Cy' \ Now make 36 — 12X 
~o, and x=3 ; take 
therefore A B=3,and 
draw theordinateBC, 
and C is the point of 
contrary flexure. If 
x be between o and 3, 36—12X is pofitive, therefore the 
part AC of the curve is convex to AB ; but when x is 
greater than 3, 36—1 2* is negative, and therefore the 
curvejs concave towards the axis. 
Ex. 2. Let GCV be a curve of fuch a nature, that if 
GA (which is perpendicular to AB) be produced to any 
Gf point P, and PC be 
drawn to any point of 
the curve, vC (hall 
always be equal to 
AG. Put AB—a", 
BC—y, PA=a, AG 
—b ; then by fim. 
trian. PAtt, BCz>, a 
(PA): x—3 /b 2 —y 2 
(AB — B»):: 7(BC): 
( Bt> ) ; 
a+yX 
3 / b 2 — y 2 
hence, xy z 
V b 2 —y 2 
oyy + y 2 y 
3 / b 2 —y 2 
y 3 +b 2 a 
y 2 Vb 2 -J 
2 —b v 3 —3 b 2 ay 2 
take the fluxion, and yx-\-xy — j-f b 2 —y 2 — 
fubftitute for x its value, and we get x —— 
X y ; now make y conftant, and we have xz=z 
-xy 
y2 which put =0, in which cafe the 
b 2 y 3 — y^xy/b 2 - 
numeraior=o ; hence,;- 3 + 3ay 2 — zb 2 a ; from whence y may 
be found, and then x, which will give the point of con- 
trary flexure. This curve is the Conchoid of Nicomcdcs. 
Ex. 3. Let the equation of the curve be ;-=:iSox 2 — 
1 iox 3 -f- 3OX 4 —3X 5 . Here j> =3 6oxx— 3 3 ox 2 x+ i 20x3.x— 
j 3 x 4 Xj and y— 3 6ox 2 — 66oxx 2 -E 3 6ox“X 2 6ox 3 x-z=o, or 
_x 3 +6x 2 —1 ix + 6z=o, whole Ample factors are 1—x, 
2—x, 3—x, and the roots are i, 2, 3, the abfciiftc cor- 
refponding to the points of contrary flexure, of which 
therefore there are three. As — x 3 + 6 x 2 — iix + 6=i —x 
X 2 _v x 3 — x, when x is lefs than 1, this quantity is pofi¬ 
tive, and therefore the curve is convex to the axis ; when 
x is between i and 2, it is negative, and the curve is con¬ 
cave ; when x is between 2 and 3, it is pofitive, and the 
curve is convex ; when x is greater than 3, it is negative, 
and the curve will then continue concave. 
72. If by making y—o, the equation has 2 equal roots, 
then; * 1 pafles through o without changing itsfign; in this 
cafe therefore, the point found is not a point of contrary 
flexure. And this will always be the cafe, when the 
equation has an even number of equal roots. 
73. To find the point C of con¬ 
trary flexure of a fpiral, it is mani- 
feft, that as long as the point A ap¬ 
proaches to C, the perpendicular Sjy 
upon the tangent mu ft increafe ; 
and after A has palled through C 
towards S, the perpendicular will 
then decreafe ; theiefore at the 
point C it is a maximum ; hence, 
if we make the fluxion of the per¬ 
pendicular —o, it will give the 
point of contrary flexure. 
OF DETERMINING THE RADII OF CURVATURE, 
AND THE EVOLUTES OF CURVES. 
74. A curve ^OH is faid to be the evolute of another 
curve ARB, when it is of fuch a nature, that a thread 
ROH, coinciding therewith (or wrapped upon the fame), 
being unwound or difengaged from it, by a power a fling 
at the end R, fhall, by that end, (the thread continuing 
tight,) deferibe the given curve ARB. 
From the point O, where the right line RO (called the 
radius of curvature) touches the evolute pOH, let the 
7 femicircle SRD be 
1 deferibed ; which fe¬ 
micircle, having the 
fame radius with the 
given curve, at R,\vill 
confequently have the 
fame degree of curva¬ 
ture. But the curva- 
x ture in two curves is 
the fame, when, the 
fluxions of their ab- 
feiflas being the fame, 
both the firft, and fe- 
cond fluxions of their correfponding ordinates Rn and 
R m are refpeCtively equal to each other; for, the firft 
fluxions being equal, the two curves will have, at the 
common point R, one and the fame tangent tRh ; and, if 
the fecond fluxions be likewife equal, the curvature, or 
deflection from that tangent, will alfobe the fame in both ; 
becaufe thefe laft exprefs the increafe or decreafe of mo¬ 
tion in the direction of the ordinate, upon which the cur¬ 
vature entirely depends. 
This being premifed, let the abfeifla Sot of the femi¬ 
circle (conlidered as variable) be put —iv, its ordinate 
Rw— v, R rzzzzzv, 
a tangent to 
rhzz 
the 
zv, and 
circle at R, 
: then, R/z being 
the triangles R/tr 
and ROot will be equiangular, and therefore zv (Rr) : 
z (RA) :: v (R»«) : RO=^; which, becaufe the 
zv 
radius of every circle is a conftant quantity, muft be 
invariable, and confequently its fluxion 
vz vz 
=0 
Whence v is found =3 -Tl—— 
z — 
conftant, and w 2 -f v 2 z=z z 2 , 
2 vv — 2 zz, and fo-Ll_— 
(becaufe, w bein': 
we have, in fluxions, 
Therefore fince v is 
= Ro( 4 ) 
, . _ , z 3 v 2 A-zv 2 
we alfo get SO : ~ ~ 
_ w v ZV - ZV V - ZO V 
Which laft is a general expreflion for the radius of any 
circle, whatever, in terms of the fluxions of its ab- 
fcilfa (a;) and ordinate (»). But, by what is premifed 
above, thefd fluxions are refpeCtively equal to thofe of 
the abfeifla An (x) and ordinate R?t (y ) of the pro- 
pofed curve ARB. Therefore, by writing x, y, and y, 
inftead of zv. 
and v, we have 
JEEE^ fz= 
—xy \ —xy ) 
for the general value of the radius of curvature, RO. 
The fame otherwife.—If the radius of the circle be 
put R, and every thing elfe be fuppofed as above; 
then (by the property of the circle) we fhall have v 2 
(R»i 2 ) — 2Rw — w 2 (SmX Dot) : whence, in fluxions 
(making at conftant) we get ivv—zRzv — zww, and zv 2 vj- 
2vv ~ z w 2 : from tlie laft of which equations v is 
found 
