CUSPS. CONJUGATE POINTS. 323 



we see that y is impossible so long as x is less than a, and 

 that when x is greater than a there are two values of y for 

 every value of x. When x = a the radical in y vanishes, 

 and the two values of y become equal ; at the same time 



-r- has only one value, namely - . Hence there is a cusp. 

 In the figure, A represents the cusp ; the straight line OA 



uOT 



has for its equation y = ; and 



since of the two values of y given 

 by equation (2), one is greater and 



O3C 



the other less than , it is obvious 

 c 



that the two branches lie on op- 

 posite sides of OA, and the cusp 

 at A is of the first species. Generally the cusp is of the first 



d^y 

 species if the two values of - indefinitely near to the point 



are of contrary signs, and of the second species if they are of 

 the same sign. 



Cusps of the first species have been called " keratoid cusps," 

 and of the second "rhamphoid cusps." 



Conjugate Points. 



302. DEFINITION. A conjugate point is an isolated point 

 the co-ordinates of which satisfy the equation to the curve. 

 For example, let 



Here the values x = 0, y = 0, satisfy the equation to the curve, 

 but no branch of the curve passes through the point thus 

 determined, y being impossible for all other values of x com- 

 prised between a and a. Hence the origin of co-ordinates 

 is a conjugate point in this curve. 

 In the above example, since 



we find that the value of ~- is impossible when x = ; but -g- 



Y2 



