TROCHOIDAL CUEVE3. 389 



x = 3b cos + b cos 3d = 46 cos 8 0, 

 y = Sbsin0-b sin 30 = 45 sin 3 ; 

 therefore x% + y^ = a$. 



If in the hypocycloid we suppose a 2b, we obtain 

 x = 2b cos and y = ; 



thus y is always zero and x may have any value between a 

 and + a ; therefore the curve reduces to a diameter of the fixed 

 circle. 



3G3. If in Art. 360 the describing point, instead of being 

 on the perimeter of the rolling circle, is on a fixed radius 

 of that circle, but either within or without the perimeter, the 

 curve generated is called the epitrochoid when the rolling 

 circle moves on the outside of the fixed circle, and the hypo- 

 trochoid when the rolling circle moves on the inside of the 

 fixed circle. In the former case we have 



x (a+b) cos mb cos j 0, 

 sin mb sin 0, 



and in the latter case 



x = (a b) cos + mb cos j 0, 



y = (a b) sin mb sin j 0, 



mb being the distance of the describing point from the centre 

 of the rolling circle. 



364. If a circle roll along a straight line the curve traced 

 out by a point in the perimeter of the rolling circle is, as we 

 have already stated, called the cycloid. If the describing 

 point be inside the perimeter the curve is called the prolate 

 cycloid, if outside the curtate cycloid ; the term trochoid is also 

 used to denote both the prolate cycloid and the curtate cycloid. 



