EXAMPLES OF CURVES. 391 



21. Obtain in an algebraical form the equation to the epi- 



cycloid for which a = 2b. 



Remit. 4 (cc 2 + f - a 2 ) 3 = 27y. 



22. Shew that when a = 6 the epicycloid becomes the car- 



dioide. 



a 



23. Trace the curve whose equation is r=acos-; and 



3 



shew that if A be the point where the curve meets the 

 prime radius produced backwards and PSQR any 

 chord drawn through the pole S meeting the curve 

 at P, Q, and R, the angles PAQ and QAjR are each 

 60, and the angle ASQ equal to thrice the angle 

 APS. 



24. Shew that the equations 



r = a a tan and 2a = r r tan 



represent the same curve in different positions, and 

 that the radii vectores to the points of intersection 

 bisect the angles between the tangents at those points. 



77 C / C\ 



25. Trace the curve - = sin - log I m sin - ) , (1) when m is 



c a \ aj 



greater than unity, (2) when m is equal to unity, 



(3) when m is less than unity and greater than the 

 reciprocal of the base of the Napierian logarithms, 



(4) when m is less than the reciprocal of the base of 

 the Napierian logarithms. 



