514 - 
of v\ and take MFe 
vy 
FLUXIO N S, 
: and when AM is convex perty of the curve PL or u 
whence MF — 
cy 
M 
2y—v 
towards L, write —nj inftead of -J-o. 
2. Or find u from the nature’of the curve, by help of 
wnich expunge u out of the equation MF=— 7 —r* 
2 yu—uy 
Ccr. The curve FyTI paffing through all the point's F, 
or touching all the refledted rays MF, mf is called the 
catacauftic or cauftic by re¬ 
flection. In which any por¬ 
tion H/F of tlte curve is 
— LM +MF-LA-AH. 
For drawing niL infinitely 
near ML, and Mo, mr , per¬ 
pendicular to mV, ML ; then 
fince Mo—Mr, therefore L m 
+ ffl/=LM+M/, or LM+ 
M/’—L?;z— tnf—o ; and ad¬ 
ding /F, LM + MF—L m — mf—fV ; but thefe moments 
are as the fluxions, and therefore the fluents thereof 
will be equal, that is the curve HF=zLM-j-MF—LA 
—AH. 
Ex. r. Let AM be a right line, then 
vy vy 
v is infinite, whence MF—— ; — =—— — 
’ 2 y—v —v 
Or thus : u is a Handing quantity and 
u—q, therefore MF= — 
, 2 yu—uy u y 
— y: whence perpendicular PF=PL, and F is the focus 
of the reflected rays. 
Ex. 2. Let MD be a circle, C the center; 
vy 
then MF=—-—. And when y is infinite, 
2 y—v 
MF=|» = iME. And if MD be very 
LDxDC 
cryy 
V 2 ry—yj 
and u sz 
—7 
fmall, MF=- 
Andif LC—CD, 
2LC+CD 
N l then MF=|o=fME=JLM. 
Ex. 3. Let MD be a parabola, and let the rays be 
rp parallel to the axis DB, then y is infi- 
vy 
nite, whence MF 
2.MCE? 
■ = hu. . But v 
2 y—v 
(putting rz= latus re£tum) 
$ alfo QTrz;———, therefore QT=rt/, and 
r 
QF=|a=MF, and therefore F is the 
focus of the parabola. 
Ex. 4. Let DM be a parabola, and let all the rays be 
perpendicular to the axis DQ. Let 
DP=x, PM=z, MQ= 25 , and rx—zz : 
_ 4 "* ___ 
rr 
by the nature of the figure : whence 
vy 2 2 3 2 zx 
MF= — = *»=-H* = -- 
2 y—v rr r 
C +ji. 
Ex. 5. Let DM be the log. 
fpiral, the center L the luminous 
vy 
2j y—v 
D T 
\ 
point: <z >—y, whence MF: 
=y> 
Ex. 6. Suppofe DM to be an elliufis, L the focus, 
the tranfveile —zr } conjugate =2c; then by the pro- 
2 U 
uyy 
yyf‘ 
cy 2 ) X 2 ry—yy 
o ,3 y 
2yu—uy 
2r—y, confequently (fince the 
angle LMP-FMT) the point F is the other focus. 
And in like manner it will be found that rays 1 (Thing 
from one tocus 0/ an hyperbola, and reflected by the 
curve, will diverge from the other focus. 
Prop. LXVII .—The natwmof the rcfratting curve AMn, 
and the luminous point L, being given ; to find the focus F, or 
the point inhere the nearef ref ratted rays MF, «F, concur-, 
and where they meet the axis of the fgure. 
142. Suppole the arch Mn to be infinitely fmall, and 
let C be the center and CM the radius of curvature in M ; 
and let fall the perpendiculars CE, Cc, on the rays of 
incidence LM, L«, and the perpendiculars CG, Cg, on 
the refraCted rays MF, »F ; and let the fine of incidence 
CE to the fine of refraction CG be as tn to n. On the. 
centers L, F, defcribe the fmall arches Mr, Mo. Now 
fince Ce exceeds CE, therefore Cg exceeds CG, they 
being in a given ratio ; whence MF, n F, interfeCtbeyond G. 
T. he figures GCME and onMr are fimilar ; therefore 
MGx Mr 
and- LM : LQ :: 
ME : MG :: Mr : Mo— 
LQxMr 
ME 
Mr : Of— 
^ LM 
m : n :: (C e : Cg : 
n X LQ x Mr 
eQ : Sg—- 
my, LM 
F 0 M, Frj; Mo—S g 
my MG 2 xLM 
and by the property of refraCtion ? 
CE : CG:: Ce—CE : Cg—Cg-.-.) 
and by the fimilar triangles 
M 0 :; MG : MFt= 
m X MG x LM— n x LQX ME 
Produce Ml- till it imerfeCt the axis of the curve in 
O, and let LA —d, AO—7, LM —7, MO—s, AH=x, 
MH — z t then y — , and s—^Jzz-\-f—x . 
:: rn : on :: m : 1 
nzz-\-ndx-\-nxx 
and ny——ms or 
mfx—m xx — mzx. 
And it is y : — j 
ny— — ms, that is 
-f zz-\-d-\-x zz-\-f—x 
Therefore, 
1. To find the focus F, let CM be the radius of cur¬ 
vature, CE perpendicular to LM, and CG to MG, 
ME=y, MG —u. 
LM—y, m and n 
the fines of inci¬ 
dence and refrac¬ 
tion. Then find v 
and u, and take MF 
mu tty 
muy — nvv — nvy’ 
If AM is concave 
towards L, write —v for v, and —u for u : and if the 
rays converge when they fall on the curve AM, write 
—y for y, 
2. To 
