59? 
OPTICS. 
Let C (fig- 9-) be the centre of the propofed arc; CB, 
that radius of the circle which is parallel to the incident 
rays ; and ACM, the diameter which is perpendicular to 
CB. Suppofe GH, one of the incident rays, to be re¬ 
flected in' the direction HD; join CH ; bifeCt CH in I, 
and HI in O ; with the centres C, O, and radii Cl, OH, 
defcribe the circles EIF, HKI ; and let K be tlie inter- 
feCtion of HKI and HD; join OK, IK.; and from C diaw 
CD perpendicular to HD. 
Then, fince the angle HKI, in a femicircle, is a right 
angle, the triangles HKI, HDC, are fimilar;_ whence, 
HD : HK :: HC : HI :: 2 : i ; therefore K is a point 
in thecauftic; (Prop I. Cor. 2 ) Alio, the 
IHK—2^CHG = i / /ICE, (Euc. 29. i.) and, lince circu¬ 
lar arcs are as the angles which they fubtend at their re- 
fpeCtive centres, and their radii jointly, The arc El : the 
arc IK :: 1X2 : 2X1- Hence it follows, that the locus 
of the point K is an epicycloid, generated by the rota¬ 
tion of the circle HKI upon the circle EIF, in the plane 
of incidence AHM. 
Prop. III. To find the nature of the cauftic, when the 
reflecting curve is a circular arc, and the focus of inci¬ 
dent rays is in the circumference of the circle. 
Let AHM (fig. 10.) be the reflecting circle, C its centre, 
A the focus of incident rays; draw the diameter AM ; 
and let the ray AH be reflected in the direction HIC ; 
join CH, and divide it into three equal parts, Cl, 10 , OH ; 
with the centres C and O, and radii Cl, OI, defcribe the 
circles EIF, IKH ; let K be the interfection of the re¬ 
flected ray HK, and the circle HKI; join OK. Then, 
fince the /OKH= the /OHK= the ,/CHA= the ^ 
CAH, the triangles HCA, KOK, are fimilar, and HA : 
HC :: HK : HO; alternately, HA : HK -.: HC : HO 
:: 3 : 1 ; therefore K is a point in the cauftic ; (Prop. I. 
Cor. 4.) Alfo, fince the /HOK22 the ^/HCA, the 
ICE— the ^KOI; and the radii Cl, OI, are equal; 
therefore the arcs El, IK, are equal; and the locus of the 
point K is an epicycloid, generated by the rotation of the 
circle HKI upon the circle EIF, in the plane of the reflect¬ 
ing arc AHM. 
Prop. IV. To find the nature of the cauftic, when the 
reflecting curve is a common cycloid, and the rays are 
incident parallel to its axis. 
Let AHM (fig. 11.) be the reflecting femi-cycloid, 
whofe bale is AD, and axis DM ; GH a ray of light in¬ 
cident upon it at H ; BHI the fituation of the generating 
circle, when the point which traces out the cycloid is at 
H ; and let I be the point in contaCt with the bafe. Take 
O the centre of this circle, and draw the diameter HON ; 
join OI ; bifeCt the line OI in C, and, with the centre C 
and radius Cl, defcribe the circle OKI, cutting HN in K; 
join IK, CK. Then, fince OI is perpendicular to AD, 
or parallel to HG, the ^OIH= the ^/GHI; and the 
XOIH— the ^OHI; therefore,the /OHI= the ^GIII, 
and the ray GH is reflected in the direction HOK. Alfo, 
fince IK is perpendicular to HK, and .TH is half the ra¬ 
dius of curvature of the cycloid at H, HK is one-fourth 
part of the chord of curvature in the direction of the re¬ 
flected ray ; and therefore K is a point in the cauftic ; 
(Prop. I. Cor. 2.) Again, fince the ^KCI=22^NOI, 
and OI=2lC, the arc IK= the arc IN —ID; therefore 
the locus of the point K is a common cyclcid, whofe bafe 
is AD, and generating circle OKI. 
Prop. V. When a fmall pencil of homogeneal rays falls 
obliquely upon a plane refraCting furface, and in a plane 
which is perpendicular to that furface, having given the 
focus of incident rays, and the angles of incidence and 
refraCtion, to find the geometrical focus of refraCted 
. raysv 
Let BAC (fig. 12.) be the refraCting furface; QA, QB, 
the extreme rays of the oblique pencii, incident in the 
V01.. XVII. No. 1200. 
plane of the paper; <j-A, qB, produced, the directions in 
which they are refraCted ; q the interfeCtibn of the refraCted 
rays. From Q and q draw QC, qc, at right angles to BC ; 
and from A draw A a, A b, at right angles to QB, q B. 
Take S and s to reprefent the fines of incidence and re- 
fraCtion of the ray QA ; C and c their cofines ; T and S 
their tangents. Then, fince the angles AQC, BQC, are 
equal to the angles of incidence, and A qc, Bqc, to the 
angles of refraCtion, of the rays QA, QB,BQA and Br/A 
are contemporaneous increments of the angles of incidence 
and refraCtion of the ray QA; and therefore the £ BQA: 
the BqA : T : t -.-. — : t. Alfo, The ZBQA : the Z 
C c 
„ , A a A b 
B?A :: - : -; 
QA q A 
the angles of incidence and refraCtion to the radius BA, 
C . _c_ 
QA ' q A 
— ; whence, q A 
c 
Cor. 1. Since QA ; QC 
rXQC 
and fince Art, A b, are the cofines of 
C 
the /_ BQA: the /_ BqA :: T : t :: ~ v 
S _ j_ 
Ta ' c 2 ' 
: C, w-e have QA 
: ’ x <fl - ; therefore 
QA 
C <: 
r (radius) 
In the lame manner, qA~ 
t'X qc rxOC 
_ : _ ; whence, qc 
T t 
QC _ : L 
C 2 c 2 
c C U c 
Cor. 2. In the fame manner it may be proved, that Q A 
. »'X AC j a rxAc , »XA c rx AC 
; and qA=: _; whence, 
S 
S 
T 
~C 
— ; confequently, Ac : 
c 
AC 
T 
SxC 
t 
iXc 
1 
c 2 
Prop. VI. When a fmall pencil of homogeneal rays falls 
obliquely upon a fpherical refraCtor, in a plane which 
pafies through its centre, having given the focus of 
incident rays, and the angles of incidence and refrac¬ 
tion, to determine the geometrical focus of refraCted 
rays. 
Let QA, QB, (fig. 13.) be the extreme rays of a fmall 
pencil incident obliquely upon the fpherical refraCtor 
ABC, in a plane which pafies through its centre E; and 
let Aq, Bq, be the refraCted rays. Draw Er/D, EgG, and 
Brt, Bb, at right angles to QA, q A, or to thofe lines pro¬ 
duced ; and, when the arc AB is diminiftied without limit, 
E d and Eg- are at right angles to Q cl, Bq. Take I and R 
to reprefent the angles of incidence and refraCtion of the 
ray QA. Then, fince ED ; EG : fin. I : fin. R :: EtZ : 
Eg-, we have Drf : Gg- :: fin. I : fin. R (Euc. 19. v). Alfo, 
B«, Bb, are the cofines of I and R to the radius AB. 
From thefe two confiderations, and the fimilarity of the 
triangles QDif, Q :t B ; and qbB, <fGg\; we obtain the fol¬ 
lowing proportions ; 
Dd : Gg- :: fin. I : fin. R ; 
Gg- : :: Gq : bq ( Aq ) ; 
Bb : Brt :: cof. R : coll I ; 
Brt : Del :: Qrt (QA) : QD ; 
by compounding which proportions, we have fin. I X Gq 
X cof. R X QA=fin. R X A q X cof. I x QD ; and there¬ 
fore, Aq : G<7 :: *‘ n ' 1 X QA : X QD :: tang. I X 
cof. I cof. R 0 
QA : tang. R X QD- • 
Cor. 1.' The diftances qA, qG, muft be meafured in the 
fame or oppojite directions from q, according as QA, QD, 
are meafured in the fame or oppojite directions from Q. See 
Prop. I. and Cor. 1. 
Cor. 2. When the incident rays are parallel, QA=QD, 
and therefore Aq : Gq :: tang. I : tang. R. 
Cor. %. On the foregoing fuppofition, when the rays 
pafs out of a rarer medium into a denfer, and the angle of 
incidence becomes nearly a right angle, tang. I is indefi- 
7 N nicely 
