EPICYCLOID. 183 
like the first. Indeed, they may be considered These two equations express generally the nature of Epicyclote. 
Epicycloid. 
a continuous curve, which will go on con- all epicycloids, whether exterior or interior ; because, al- 
=—— ws 
1 
PLaTE 
ec_Lill. 
Fig. 12. 
een ads will come again 
ff 
the circumferences of the and 
or their radii, are incomm a eeeahas be 
that case, the two circles will never come 
into contact at the same point. If, however, the 
commensurable, it is evident that, after a cer- 
genes of revolutions of the generating circle, the 
to. the ts A and 
&, from, which they eet out ; and thus curve will 
return into itself. 
If the point that describes the A pabiaaye be pee 
on the Sislstoke to te circle, as at curve a w 
to the curtate oad: but if ft be with- 
4 Ege ee circle at p’, then the curve ‘a’ p' dB will cor- 
respond to the prolate cycloid ; and lastly, if the ge- 
Fst gl) pat be at P, the curve APDB will be more 
and will iecoapold to the common cycloid: 
Baer. E iawn E 
ce 
e he te nod checks, ea ish Geto also 
let F’H be the circle when it has made ex- 
half a peorrt Then its centre O’ will be 
axis C e t will be at 
Lapa wires generating poin 
now that the 
e arc HE, wafcant + sake 
© GO, mdi the reveléi radius from the position O’D 
to the position OP, while the 
Sar pene See motien. the 
will be the t of the erati 
eno gh erdnders! er gat seg rid 
 Tadtow the ee fe of the. fixed circle origin 
centre as the 
of the co-ordinates, 
though in investigating 
generating circle to be without the fixed circle. By a 
well known principle i in mathematical analysis, we have 
only to change the sine‘of a and } from + to —, there- 
by silicate that the lines which these letters repre- 
sent, are to be considered as having a contrary direc- 
tion to that which they had in the former case, and the 
aig will be adapted to the case of interior epicy- 
In the preceding ms ations, the co-ordinates are 
expressed in terms of the arc, which the gen 
circle has rolled over, reckoned from H, the middle of 
the base, (Fig. 12.) but it will be convenient to have 
them also expressed by the arc described from the be- 
“motion.” Draw a straight line from C 
5 A, 
circles, and et us suppose, 
rolled along 
which was at first atA’, has described the epicycloidal 
A’P. Let Pc meet the circle in N, then the arcs EN, 
EA will be equal. Produce PO, CO to Land F; draw 
PQ, OR icular to CA ; and OT dicular 
to QP. the abscissa CQ=2’, the ordinate QP=y’, 
the are AE=~’ ; and, as before, put CE=c, 
PO=}. . Then the angle OCR, or FOT, is = radius 
being unity, and the angle NOE, or FOL=~, theres 
: av es oe ap 1\,, 4 
foreTOL== += = (44 ae 
Hence CR=(c+p.a) Cos. =, 
OR=(c+a) Sin. z . 
1 BOze—sd Gon: (14+ =) 
Put CK, the abscisse, . . . adherens vam, i 
ORR Ae panting, 2195220 5% ay Ps 81h. cag, PT=b Sin. (+4 >)» 
CE, the rad of fixed cixel CEDDEDED DLL Sg and since s’=CR+TO, and y=OR—PT, we have 
OE, the rad. of ee Pay 
gen. circle, . 
OP, the dist. of gen. point from the centre, : 36: 
Then = is the are of a circle, whose radius is unity, 
which measures the angle ECH= FOM ; and, in like 
manner, 7 is the measure of the angle LOE, or POF ;, 
= (cpa) Cos. —5 Cos.(—+ =) 
y=(c+a)Sin.=— é Sin. (+2) 
From this solution we may deduce the following con« 
uences : 
: 1. These two sets of formule (A and B), enable us 
hence = 4.2 = (14 ~) zis the measure of the an- by the help of de te fa uy ehicrelie Cr eatle 
gle POM: Hence by the help of the trigonometrical tables, To do ths this, 
vt €6 =€O x Cos. BCH =(c42) Cos.£, 
OG=CO x Sin. ECH=(c-+2) Sin. = 
_OM=PO x Cos, POM= Sin(— + + th. 
we must give particular values to the angle <, then 
we must find from the tables the values of the sines and 
cosinesof —, andof (e+ =) =t* = jandfromthese, 
the coordinates of poinjs in dhe 
and, in these calculations, regard 
hs tale dy cand 
“PM=PO x Sin. POM=5 Cos. (++ 1), must Bel of the sines and cosines, as is 
2 explained in a4 mee e Arnirumetic of Sines, 
2. Ife and a be Soames indeterminate 
Now r=CG-+OM, and vy ree therefore 
A) 
. 
c 
1 
a “ 
Seo | Te = +b Sin. i +=): 
_ example, if c 
arc z may be eleminated from either of the formulz 
Oy @)s oon thence an equation may be found, which 
expréss the relation of z toy in finite terms. For: 
sa:t 9% so that 4 <= 
from formula (A), ais 
githen, 
OE=a, 
them, we have supposed the y 
Fig. 13.) the first point of contact of the two prars 
, that the generating circle has CLUE. 
the are PAE, while the: generating oe Fig, 13. 
