4 83 
FLUXIONS. 
done, as r has been continually approaching to s and only 
sow coincides with it ; therefore when r comes to C, the 
BT z=?~: 
therefore fet off BT join T and C, and 
y ' y 
TC will be a tangent to the curve at C. If y decreafe 
whilft x increafes, then y becomes negative by Art. 16. 
a fid confequently —, or BT, becomes negative, which 
y 
fhews that T lies on the other fide of B. See Algebra, 
vol. i. p. 318. The line BT is called the fubtangent. 
Ex. 1. Let the curve AC be a parabola, that is, a 
curve vvhofe abfcilfa varies as any diredt power of the 
ordinqte ; to draw a tangent at the point C. The equa¬ 
tion exprefling the relation between x and / is ax—y\ for 
then x : y" :: 1 -.a, a conflant ratio. Take the fluxion of 
both (ides of the equation, and we have ax=ny’ : — 1 y, 
y ’ 1 
becaufe — —x. 
x nyf — 1 yx ny n 
hence,-, .•. BT =-— —-^——vx. 
y a y a a 
If n—z it becomes the common conical parabola, and 
BT— 2x. 
Ex. 2. To draw a tangent to the ellipfe ACPDE, at 
any point C. Let AD and PE be the two axes; put AO 
=<z, PO— b, ABrj, BC—y, then BD=2 fl —x; and by the 
property of the 
c 
- \ 
c 
p \ 
r a\ B 
0 J 
ellipfe a 2 : b 2 
2 ( 2 - 
b 2 
-XX x : y 2 — 
X 2 ax — x 2 ; 
a- 
take the fluxi- 
|TX b 2 
iD ons, and —X 
<22 
2ax — 2 XX— 2 yy, 
multiply both 
fides by di¬ 
vide by 2 which is common, and alfo by a —x, and x— 
E 
As this value of TB is independent of or PO, if we 
take pO— AO, fo that ApD may be a circle, and produce 
BC to c, cT will be a tangent to the circle. If B be be¬ 
tween O ajid D, fo that whilft x increafes / decreafes, 
then / becomes negative by Art. 16. and confequently 
yx . 
— is negative, which (hews that the fubtangent BT lies 
y 
the other way from B. 
Ex. 3. To draw a tangent to the hyperbola AC, whofe 
major axis is A.D. Bifed ^\D in O; put AO=ai, the femi- 
line W t, ceafing to cut the curve, muft become a tangent, 
and confequently WCr will then coincide with VCT. 
Now whilft the abfcifla AB by increafing becomes AD, 
the ordinate BC becomes Dr; hence, the increment of 
the ordinate BC is Er; and, by fimilar triangles, the in- 
crement^CE of the abfcifla : the cotemporary increment 
Er of the ordinate :: CG : Gin. But when r arrives at 
C, WC coincides with VC, and confequently m muft 
coincide with n ; hence, the limiting ratio of the incre¬ 
ment CE of the abfcifla to the increment Er of the ordi¬ 
nate, is that of the finite lines CG : Gn, which (by fim. 
trian.) is the ratio of CE : Es, taking DEs in any fitua- 
tion before its coincidence, with BC ; hence, by Prop. 2. 
Cor. 1. if CE reprefent the fluxion of the abfcifla, Es 
will reprefent the cotemporary fluxion of the ordinate. 
Put AB=x, BC=/, then BD=CE=x, Es=y ; and as 
BC is parallel to Es, and TB to CE, the angle TCB=a 
CsE, and CTB=sCE, confequently the triangles TBC, 
CEs, are fimilar; hence, y (Es) : x (CE) •.:/ (CB) : 
b 2 
the hyperbola, a 2 : b 2 : 2< 2 + xXx : v 2 =;—X 2ax y x 2 > 
a 2 
which is the fame equation as for the ellipfe, except that 
2 AX -J- X ^ 
the fign of x 2 is here pofitive ; . •. BT=-• 
0 v ’ a+x 
Ex. 4. To draw a tangent to the cifloid of Diodes, whofe 
x 3 
equation isj 2 =r-— -(Algebra, p.319.) Take the fluxio^ 
2x 2 xX< 2—x-(-x 3 x 3 ax 2 x — 2x 3 x . x 
and 2 yy=- - -- ! -= -—--i hence, -r = 
,2 y 
a—x 
_. t CT yx 2 V 2 x a ~- x __ 2 X 3 
3«x 2 —2x 3 ’ y 3«x 2 -— 2X 3 
2 yXa—x 
a—x 
X 
2x X a—x ; 
3<ZX - 2X J 3 a- 2X 
Ex. 5. To draw a tangent to the catenary curve. The 
equation of this curve is z 2 =i2ax-\-x 2 ; hence, zz— 
a-\~x 
ax-j-xx, and z— -Xx; but y 2 =.z 2 —x 2 (Prop. 22.) = 
j4-x 2 <z4-x 2 — z 2 „ 
~2 X A ' Z —x 2 = ——- XX 2 : 
L> and y — — : 
J z 
hence, BT 
y a a 
24. Draw CN perpendicular to the tangent, and it 
is called the normal, and NB the Jub-normal. Now the 
triangles TBC, 
NBC are fimilar; 
hence, ^(TB) : ^ 
y (BC) "f: y : BN 
— the fub-nor- 
mal. Alfo CN 2 
=y 2 +4f=y 2 f 
normal. 
Ex. Let the curve be a parabola. Her e ax=y; . • 
. x ny l — i yy a 
ax\ny n — A y y and BN— ——-—In the 
y 
ny n 
common parabola, where n— 2, BN—-, a being the la- 
Q? VV X efi V yX #2 ^2 I r 
77X—=7rX---5 hence, BT=—=; — x-- tusre&um. Alfo, CN=J/ 2 -f-« 
b 2 a—x y b 2 a—x y b 2 a —x ’ 4 
2 ax —x 2 
a —x 
b 2 
by fubftituting —X 2 ax — x 2 for y 2 . 
25. If two quantities begin together and increafe un- 
formly, one by x and the other by ?nx } m being conftant, 
1 then. 
