PLATE 
cCCcLUL 
Pig. 14 
2=(¢-+a) Cos, ~ +b Cos. =, 
: . 5: 
y=(c+e) Sin. = +4 Sin. ae 
Put Cos. 5, = pand Sin, = = ¢. Then, by the 
Anrrumetic of Sines, (Art. 7 , 
Cos. <= 2p'—l, Sin. = =P 
Cos. ganesh ener 
Sin, ae =5q—20q) + 169! ; 
(e+e) (2ptel) +b (1—12 +16 *) 
z=(c+a)(2 pi— q*), 
y=2e-+a Pat ioe— 20g + 10g"): 
these equations, and the equation p*+q°=1, p and 
- may be e " » and the’ result: will be an 
Bai equation equation, involving z and y only, which will 
petael te care 
teal tn ineitary tan spicy ill Bé'en 
bers, the epicycloid will be an 
eri Alda petedalnts aaa ( Scholium to 
Def.) If, however, a and c be incommensurable, the 
of z produces an equation of an infinite from 
number of terms, and therefore in this case the curve 
is transcendental ; and in this case also it never returns 
into itself. 
8. As the order of the curve depends upon the ratio 
of the radii of the fixed and generating circles, it may 
be worth while to: seeing one horse 
"First let us take the case of a circle EPC, (Fig. 14.) 
— rolls on the inside of another AEH, fe edape 
pass through its centre C. erat b=a=—te, 
berwase «lies det in a-cont ; therefore, 
the co-ordinates to the line CA, drawn through 
A, the point of contact of Yo. twocrees, we have 
by the formula (B), ' 
« 8 ye a +8-\ 
Sin. ¥ Sm. (-=) 
Now, if'in_ the formule for the cosine and sine of 
a— 6, a and 6 being any arcs, (Anitumetic of Sines, 
Art. 10.) we suppose a=0, and. observe that then 
Art 10.) Sin. a=0, we shall have Cos, (—b) = Cos. b, 
Sin, (—b)=—Sin. b, and therefore Cos, (— =) = = 
Cos. a iar therefore 
z=c Cos. — =,y=0. 
This value of y shews,. that the int is 
in the axis CA, pee Yr es on ‘zis evi-. 
the cosine of the are z, or AE, the distance of 
the generating from the centre at any time, is the 
cosine of the arc that has then been e@ over. 
in this case is therefore AB, that diameter of 
EPICYCLOLD.  . 
BOR icneinle double the 
; and 2ACE= 2 
nny but FOE 
At we AR. Sm 
= MEAS ; there. ; 
fore arc PE=are AE, and so ‘3s a point 
that would be described b ane ‘the circle E 
pecs 
on the inside of the circle AEH 
4. Next, let us suppose that the circle EP rolls on 
the outside of another AEH, of the same “magnitude, 
Fig. 15.) and that the generating point departs from 
Wk foe pols Ut ciniines 6 dis Sea AMM TASTE. 
In this case, b=a=c, and we have by formula (B), Fig. 15. 
CR=2=2eCos. = —¢ Cos. =, 
PR=y=2c Sin. —< Siny . 
tions i) 
and Sin. 4 v=2 Sin. v Cos. », (Anirameric of Sines, 
=2c Sin. v (1— Cos. v) 
Ay x _ 1—Cos.* » 
Let us put 2—c=z’, so that instead of making C the 
mong it ne 
From these two equations, let by 
the curve ; which Hea lay 
ral remarkable : For , ifany straight. 
and VP’ are each 
A be dra the ew points 
ts wn to curve at i 
Pe, they wl fom orm a right angle at X fete tabs 
pay 4 
case (Fig. 16. pid ty ras ein ve 
from a cylinder or circle AEH, round which it was 
Let v=, then observing that Cos. 20=2 Cos v1; 
Art. 14.) we-have, after substitution, &c. _- 
2—c=2c Cos. v (1—Cos.v) . 6.2. e+ ae ee (1) 
Sin.* v 4 ; 
ri G@—cy Costv™  Cos#o ate tae 
po mare: of the abscisse, we are now to reckon them 
Aj and let Cos.v= >, and we have frm equas 
2cp—2cp* a hie 
Poe +y) i Font 
on method, (Avceana, Sect. Pa} ew 
ition ht aoe (2° 2c2’ +y*)*, 
of tet fousth ocer ’ 
This curve has been called the cardioide; it has seve- 
line be drawn through A, to meet the fixed circle « 
in V, ere ee ake eee 
Secin thaGecrt’ ace, sitais gi 
in the curve, . are , an 
number of points ‘in i By hippy any 
y As a third icular case, let us 
tO bp inthe cireuinieranioouh the 
circle, and its radius shes peg +4 great. 
ag ey roth ond 
E Se of ona is to caesar 
pic ad ap aw PERL, bey ettadiges: 
wound. ; 
In this case, we have b=a= an infinitely great 
The, Suantty; and because, in general, Cos (F-4=) = 
Zz . oe - {z . 
Cos, = omg Bing Seige + 5) 
= Sin. = Cos, = s+ Cos. = Sin. = (ArireMetic. 
pata of Sine) when sisal great then Cou (444), 
