| t 
: 
; 
Investigation of the Epicyeloid. 281 
2. It is apparent from the above, that the generating circle, while 
it moves around upon the circumference of the fixed circle, revolves 
at the same time about its own center. Any point, therefore, in its 
circumference, as B, may be considered as having two motions—one 
about the center C of the generating circle, and the other about the 
center D of the fixed circle. 
3. If the generating circle be moved around upon the circumfer- 
ence of the fixed circle, without revolving upon its own center C; 
or, in other words, if it be carried around upon the fixed circle in 
such a manner, that the same point A in its cireumference shall al- 
ways touch the circumference of the fixed circle; the velocities of 
the several points A, C and B, will evidently be as AD, CD and BD, 
their distances from the center of motion D respectively. BD may 
consequently be taken as the measure of the motion of the point B 
about the center D; which motion being in the direction of a tangent 
BG drawn to the extremity B of the radius DB, will be correctly 
tepresented by the line BG, that line being made equal in length 
to BD. 
4. Again, if the generating circle be allowed to revolve upon the 
circumference of the fixed circle, the point A in the circumference 
of the generating circle, (which before moved around the center D 
of the fixed circle, with a velocity proportional to AD,) will, by the 
impinging of the two cireles, cease to move about the center D, and 
will take up a new direction around the center C of the generating 
Grele, with a velocity proportional to AD, the same with which it 
before. moved about the center D of the fixed circle; and further- 
Nore, as Bisa point in the same circumference with A, it will have 
0 equal motion about the center C; consequently, AD may be taken 
asthe measure of the motion of the point B about the center C, which 
Motion being in the direction of the tangent BI drawn to the extremi- 
'B of the radius CB, will be correctly represented by the line BI, 
that line being made equal in length to AD. 
- As BG and BI are respectively proportional to and in the di- 
"ction of the two motions by which the point B is governed in de- 
“nbing the epicycloid, if the parallelogram BHIG be constructed, 
'S diagonal BH will represent the resultant of the two motions; that 
* the point B will, by the influence of the two motions BG and BI, 
describe the direction BH, with a motion proportional to BH, and 
“ousequently, BH is a tangent to the epicycloid at the point B. 
Vou. XXI.—No. 2 36 
