TRACING OF CURVES. 373 



From x = a to x = 2a, y is impossible. 

 When x = 2a, y = 0, ;r=. 



8*6 



When x is greater than 2a, y is possible. 



When x is negative and between and 3a, y is impossible. 



When x = 3a, y = oo , -~ = oo . 



When # lies between 3a and oo , y is possible. 



From (3) we see that the equation to the curve when y. 

 is very great is approximately 



lla 2 



_ 

 and whether x be positive or negative x 3a + -r is 



fSK 



numerically greater than x 3a. Hence the curve lies above 

 the asymptote. 



342. In the above examples the value of y is given 

 explicitly in terms of x. In a similar manner we may pro- 

 ceed if x is given explicitly in terms of y. But if the equa- 

 tion connecting x and y does not admit of easy solution we 

 must abandon this method. In such cases we may find the 

 asymptotes by Art. 277 : we may determine the nature of 

 the curve near the origin by a method exemplified in the 

 next two Articles ; from these results we may obtain some 

 idea of the form of the curve. By transforming the equation 

 to polar co-ordinates we shall sometimes be enabled to trace 

 it more accurately. 



343. To determine the form of the curve 



x* - ayx z + by 3 = ..................... (1) 



near the origin. 



First, suppose that near the origin the term fo/ s can be 

 neglected in comparison with the other two terms in (1) ; in 

 that case we should have 



x* ayx* = 0, 

 therefore x* = ay. 



