2. To find the point O where the refracted ray meets 
the axis, of the curve, let LA=r d, AO —f, AH—a-, 
Mll=z, then by the nature of the curve expunge x or ss 
nzz-\-ndx A-nxx mfx — mxx—mzi 
out of the equation — 
FLUXIONS. 
mvuy 
muy — nvv- 
muy 
-nvy 
And in or near the vertex 
mry 
^ J .a 
MF— 
j 
ZZ-\~ d-\-X 
and by reduction find f: and when the C is concave to¬ 
wards L, write — x, for -\-x, & c. 
Cor. The curve N f F palling through all the points 
F, or which touches all the refraCted rays KN, MF, is 
called the dia- 
caufiiic or cauf- 
tic by refrac¬ 
tion. And any 
portion of it 
NF = FM-f - 
ML—NK-KL. For fuppofing Mn infinitely final!, 
m 
and drawing Mo, Mr, perpendiculars on ft, n L ; then 
n 
by the nature of refraction m : * :: rn : on —~rn s there- 
m ’ 
m—n y)'—nu m—n yy 
And to find where 
the refradted ray 
meets the axis of 
the fphere, zz—zrx 
— xx, and zz—rx 
xx, by which ex- L 
punge z and i out of the equation 
.mfx — mxx — mzi. 
A H C 
nzi-\-ndx-\-n xx 
4 
zz+f—x 
rnf-—vi r 
-, and there arifes 
zz-\-d-\-x 
nrfnd 
•j/ 2 rx zdx + dd 
by redudtion of which f is found. 
V 2 rx — 2 f x +JJ 
And in the vertex A where x is o, 
mar 
mf—mr nrfnd 
—j- — 
d 
or 
/: 
for eon—- rn—o ; that is MF— vf x E/z —LM — o. 
m in ’ 
m — nyd—nr 
Ex. 5. If rays fall on the concave furface of the 
and adding F f, we have Fy=MF— vf -x L«—LM; 
1TI 
but thefe moments are as the fluxions; whence the 
u 
fluents will be equal, or FN = FM— KN— -X 
m 
fphere AM, then MF= 
-., mvy 
vertex, M f~ 
muuy 
■muy—-nvv-f nvy 
And near the 
LK—LM 
Ex. 1. 
finite, whence 
Ex. r. Let AM be a plane furface, then v, 11, are in- 
muvy muuy 
—-——— -. Now . let . 
muy — nvv—nvy —nv v 
the infinite radius of cur- 
vature, the perpendicular 
I.Arr/i, then by the fimilar 
f triangles LAM, MCE, v— 
rp 
mry 
n—my y— nr 
To find where MF L 
cuts the axis, here zz— 
■zTx—xx ; put the equation into fluxions, and write 
for x t —x for x, —r for r in the equation 
d L 
y 
; alfo mmyrr — uu—nny 
mfx — mxx — mzi. 
and we have 
nzi-fndxf-nxx 
^ zzfd-\-x 
nr—nd 
rr — w, by the right angled 
mm—nn n n 
triangles MEC, MGC : whence uu—— -— rr-i- w ; 
& mm m 2 ’ 
1 c n,TT- — muu y mm — nn wy VI 2 — n 2 
therefore MF——--=- yrry -- 
nvv mnvv m mnp 2 
y 3 — -7. And near the point A, A f— — - p. 
Ex. 2. Let parallel rays fall on the convex fide of 
X t ^ le *P^ ere AM ; then y is infinite, 
^ zz+f—x 
— mr—mf 
VdJ-\- 2 J x + 2rx 
mrd 
or f: 
and in the vertex 
f dd —2 dx-fzrx 
nr—nd — mr—mf 
f 
£f~~-«F and MF-- 
muuy 
muy — nvv—nvy mu—nv 
1 ^ and near the vertex A, u—v—AC ; 
, my AC 
whence A/—-. 
m—n 
Ex. 3. Let parallel rays fall on the concave fide of 
the fphere AM; then y is infinite, 
and v, u, are negative; whence 
n—my d—nr 
Cor. Hence we may find if the fphere in this cafe has 
a geometrical focus, or fuch a one where all rays are re¬ 
fracted accurately to one point. When this happens, 
AO muft be a determined quantity for ail parts of the 
curve, the fame as at the vertex ; and therefore the 
powers of x muft vanifh out of the general equation ex- 
preffing AO; confequently the coefficients of each 
power, being equated, muft deftroy one another. Thus 
the equation for AO being n 2 yr—d X JJ + 2 J X + 2 rx — m2 
y r+f ydd — idx + zrx, then putting n 2 y r—d X ~r+ 2r 
.F MF — 
muuy 
y.\~m 2 yr-fftyir —2 dyx, after reducing the equa¬ 
tion, there will be found n 2 d—n 2 r — m 2 r — m 2 f~n 2 r —• 
m 2 r- _ vi 2 y 1 ' Vl \ _ ; from whence will be-found d— 
nd — vid—nr ' 
n—m n—m 
— muy — nvv-\-nvy nv—mu 
and at the vertex A, v—u—AC, 
AC. 
mfn 
therefore f- 
m-\-n 
which being negative, 
Ex. 4. Suppofe the-rays proceeding from L, to fall 
on the convex fide of the fphere AM; then Mf—. 
.11 m 
fliews that the focus O lies on fame fide as the radiating 
point L. Therefore w hen the diftance of the radiating 
point is fuch that -t-r, then F or O will be a geo- 
n 
metrical 
