5 EPICYCLOID. 
jicycloid. wn ee pee ee phat 
Se = cos. £—+ Sin. =, because = being an indefi 
PLATE 
ccLill. 
Fig, 17. 
Fig. 18. 
Fig. 21. 
i ine i to radius, and its 
SR aes cart can ei Mediates 
rac Cos.—+ Sin. =, 
y= Sin. — 2Sin.— 
ions of the curve. © 
By the squares of « any, we ge T= < 
z, an ion expressing a curve, 
+ in equation property 
is called the involute of a circle. 
Prop. II. 
A tangent to an epicycloid at any point is perpendi- 
dilaton vissight lars, deatn from it; to the point of 
contact of the generating and fixed circles. 
Suppose two A 2, 3, 4, E, &e. N 2, 3, 4, E, 
umber of sides to be described about 
of the proposition 
Cor. 1. Let O’ be the point in which the centre of 
the circle crosses the axis (Fig. 18.), and 
DVI a circle on O’, with a radius equal to OP, 
ee LAT at tT tic sets ok. the 
ie generating circle), and H the point in which the 
generating circle ies the fixed circle, when its cen- 
tre is at O’. Then, if a circle be described on C as a 
centre to meet the epi id in any point P, and the 
circle DVI in V; and PE be drawn perpendicular to 
prctn he meet the fixed circle in E ; the nor- 
mal (Mh nier ebeguadsidettiagy acl 
toH. Let O be the centre of the generating circle 
seatamaritiiecns Fan 2 22 
centre, ) 
join OP, O/V; and because =CO, O’V=OP, and 
V=CP, the ce og COP are equal ; there- 
fore the angles HO’V, EOP are equal ; now HO’=EO, 
ler ee hid dala 
Cor. 2. wfc be FM g Magica 
ference of the generating circle, a tangent to the curve 
at any point P (Fig. 21.) will pass through F, the ex- 
trestity of the age ers Ey ik 
Sr. Pt See the two cir. 
Scuouwm. From this proposition it appears, that if 
185 
parallel rays of 
be 
cle FPE, which ro a circle AEH, havi 
Se ae 
volving circle ; e generati int being sw 
to set out from A, the middle’ of the perpendi- 
cular radius = let me generating ag 3 
any position , P being the erating point, an 
CHF that radius of thecircle DFR’ which pastes through 
pt yeah ow he niet ged Aa tae sg 3 
di bp cen nyh hs set cut tae ge 
touch epicycloid at P, + e angle EF 
at the circumference is half an ang! sh Cie peace, bth 
the same arc, and therefore is measured by 27777 = 
are PE 
are Fe, that is by MCE but this last arciis also the 
measure of the angle ECA, or CFG; therefore the an« 
gles PFC, CFG are equal; and hence, if GF be the in« 
cident ray, FP is the reflected ray. See Optics. 
Hence it that in this case the epicycloid is 
the catacaustic curve, or the curve which passes through 
the intersection of any two and contiguous 
lg after they have slop et 8 
bowl, by the reflection of the sun’s rays from the po- 
lished concave surface which rises above the surface of 
* the milk. 
When rays diverge from one end of the diameter of Fig, 20. 
acircle, and are reflected from the inside of the cir« 
cumference, in this case also the eatacaustic curve is 
an epicycloid, viz. the cardioide es Prop. 1). For 
SS aikins ths teks  dopating ame By. vel gues 
AED, int i + genera~« 
ing the epi id APH, through E, the point of con-~ 
cast of thes chads, ature CHE, “er susat he i 
circle 
dius, 
ting 
circle HFd, which meets the epicycloid. Because AE 
at the centre is double the 
rence. But because CF=CH, and conseq 
CHF=CFH, the angle ACE is double the. angle 
CFH ; therefore the angle CFH is equal to the angle 
CFP; pt cones ing any incident ray, 
FP is the ray ; moreover, FP is a tangent te 
the epicycloid. (Cor. 2). 
_ Nore. in perenne werctioy Se 
ways su generating point to be in the circum~ 
ference 
Prop. III. 
Let H be the middle of HPA, an. exterior epicy~ Fig. 21. 
cloid, C the centre of the fixed circle, and e 
erating circle when its centre is in the axis CH, 
C Gabe Se ne of'9 Crcie to pies thitomes © 
point-in the epicycloid, and meet generating circ! e 
in V, and jin HV, Sedsherd HY Sy eet 
cloidal arc HP, as the radius of the immoveable ci 
to the sum of the diameters of the immoveable and ge« 
nerating circles. 
Take p, a point in the epieycloid, mdefinitely near 
to Ps deeribe the are pt dnd join Hv; let FPE, Je 
+ ro 
ys fall on the concave circumference Epicycloid, 
of a circle DRd (Fig. 19.), they will, after reflection, “=v” 
ts to an gare DPA, generated by a cir- Pratz 
i the CCLII. 
same centre as the circle DRd, and its radius the Fig: 19. 
