SINGULAR POINTS. EXAMPLES. 327 



2 P4 .1 



4. If y=<f> (x} + (xa) ^ F(x], when x = a, there is a cusp 



of the first kind if ~^ - be greater than I and less 

 2q 



than 2, and a cusp of the second kind if -^ - be 



2q 



greater than 2. 



5. The curve, y* = (x of (x c) has a cusp of the first 



kind at the point x = a. 



6. The curve (xy + I) 2 + (x I) 8 (x - 2) = has a cusp of 



the first kind at the point x = 1. 



7. The curve y b = (x a)' + (x a)* has a cusp of the 



second kind at the point x = a. 



8. The curve x 4 2ax"*y axy* + a?y* = has a cusp of the 



second kind at the origin. 



9. The curve x* ax z y axy* + cfy 2 = has a conjugate 



point at the origin. 



10. The curve x 4 - 2ay* - Sa*y* - 2a V + a 4 = has a double 



point when x = a, and -^" then = + A/| ; also a double 



point when y = a, and -j- then = + Vf . 



11. If a-y 2 = (x a) 2 (x F), when x = a there is a conjugate 



point if a be less than b, a double point if a be greater 

 than b, and a cusp if a = b. 



12. Shew that the curve ay* x 3 + bx* = Q has a conjugate 

 point at the origin, and a point of inflexion when 



13. Find the points of inflexion in the following curves 



14. Find the singular points in the following curves: 



x 4 = 0; 



