390 EXAMPLES. 



The equations 



x = a (1 m cos &), 



y = a(Q + m sin &), 



will represent a prolate cycloid, a common cycloid, or a 

 curtate cycloid, according as m is less than unity, equal to 

 unity, or greater than unity. See Art. 358. 



EXAMPLES. 



Trace the following curves : 



1. f = ax* - x\ 2. / = a 3 -a- s . 



3. y*(x-a} = (x + a)x*. 4. afy' = a 2 (x* - f). 



5. ^(aj-4fl)=aa?(a?-3a). 6. (x* + yj = 4aV/. y^ 



7. 2/ 2 (2a - a?) = x 3 . (The cissoid.) 



8. a?tf = (a* - f) (b + yY. (The conchoid.) Transfer the 



origin to the point (0, 6) and then change to polar 

 co-ordinates and we have for the equation 



r = b cosec a. 



9. (a;* + 2ry = 2 ( a;2 -/)- (The lemniscata.) 



10. r = a0sin0. 11. r = a (0 + sin0). 



12. r sin 6 a cos* 0. 13. r=logsin#. 



14. r 2 cos 6 = a 2 sin 9 3d. 15. r 2 cos = a 2 sin 3 0. 



16. r (9 - sin 0) = a (0 + sin 6}. 



17. r = a (1 cos 0). (The cardioide.) 



18. rd = a. (The hyperbolic spiral.) 



19. Find the equations to the tangent and normal at the 



point P in the epicycloid. See the Figure to Art. 360. 

 Shew that the normal at P passes through B. 



20. Trace the curve determined by the equations 



x = a (1 cos 0), y = a<f>; 

 this curve is called the companion to the cycloid. 



