Investigation of the Epicycloid. 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 circumference 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 BDj 

 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 

 represented by the line BG, that line being made equal in length 

 toBD. 



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 circles, cease to move about the center D, and 

 will take up a new direction around the center C of the generating 

 circle, with a velocity proportional to AD, the same with which it 

 before moved about the center D of the fixed circle; and further- 

 more, as B is a point in the same circumference with A, it will have 

 an equal motion about the center C ; consequently, AD may be taken 

 as the 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- 

 ty B of the radius CB, will be correctly represented by the line BI, 

 that line being made equal in length to AD. 



5. As BG and BI are respectively proportional to and in the di- 

 rection of the two motions by which the point B is governed in de- 

 scribing the epicycloid, if the parallelogram BHIG be constructed, 

 its diagonal BH will represent the resultant of the two motions; that 

 is, 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 

 consequently, BH is a tangent to the epicycloid at the point B. 



Vol. XXI.— No. 2. 36 



