322 CUSPS. 



and from this we see that for values of x comprised between 

 and 1, both positive and negative, y is possible, and that 

 there are two values of y for every value of x. When x = 

 the two values of y become equal ; but since 



we see that when x = there are two values of -% , Hence, 



instead of clearing an equation of radicals so as to bring 

 it into a rational form, and then applying the method of 

 Art. 298, we may often detect a multiple point more easily 

 by observing what values of x make one of the radicals in the 

 value of y vanish. 



Cusps. 



301. DEFINITION. A cusp is a point of a curve at which 

 two branches meet a common tangent and stop at that point. 

 If the two branches lie on opposite sides of the common 

 tangent, the cusp is said to be of the first species ; if on the 

 same side, the cusp is said to be of the second species. 



Since a cusp is really a multiple point, if a cusp exist in 

 the curve <j> (x, y) = at any point, we must have 



, 

 ay 



at that point. To distinguish a cusp from an ordinary mul- 

 tiple point, we must trace the curve in the vicinity of the 

 point in question. 



Example. Let (cy - bx)* - (a? ~ a) " = ............ (1). 



Of 



Here when x a and y = we have the equation to the 

 curve satisfied and also 



ay 



Putting the given equation in the form 

 bx 1 /((x-a} 



