37G TRACING OF CURVES. 



Again, reject y* ; thus 



ay*x x t =0, 



x 3 



therefore if = . 



J a 



Hence y* varies as x s ; the rejected term varies as x e 

 and the terms retained vary as 

 x*, and consequently we obtain 

 an approximate branch. v^ 



The branch to the left of the 

 axis of y is that given by w 2 = ax, 

 and the cusp to the right of the 



x 3 ^ 



axis of y is that given by y 2 = . 



(Z 



In this example, y z may be found 



in terms of x and the whole curve traced. 



345. We may observe that in the examples of the pre- 

 ceding Articles, the supposition which was found inadmissible 

 near the origin, will be admissible for points at a very great 

 distance from the origin. Thus if 



y* + ay*x - x 4 = 0, 



when x and y are indefinitely great, ay"x may be neglected 

 in comparison with y* and x* ; and y* x 4 , or y = + x, will be 

 an approximation at points remote from the origin. If we 

 find the asymptotes by Art. 277, we shall have 



to which y = + & 



may be considered an approximation when x and y are inde- 

 finitely great. 



346. Kequired the nature of the curve 



y" + xy 3 + ax*y - bx 3 = 

 near the origin. 



Assume ax*y bx 3 = 



