DIRECTIONS OF THE TANGENTS. 377 



as an approximation near the origin. Hence 



ay bx, 

 therefore y varies as x, 



the terms retained vary as x 3 , and those rejected vary as x*, 

 and we have therefore an approximation to the curve at the 

 origin. If we examine all the six cases which present them- 

 selves by retaining two of the terms of the given equation and 

 rejecting the other two, we shall find that the only other 

 allowable supposition is, that xy 3 and bx 3 can be rejected, and 

 we obtain for an approximation 



?/ 4 + ax*y = 0, 

 or tj ~~ ctt.L . 



It will be easy to draw the branches we have found; the 

 equation y 3 = ax* gives us a cusp, the two branches being on 

 the two sides of the negative part of the axis of y. 



347. If in any examples we wish only to find the direc- 

 tions of the tangents at the origin, we may arrive at them 

 . immediately, as shewn in Art. 195. 



Suppose y* + xy 3 + aa?y bx 3 = 0, 



fv\ 3 y 



therefore ( y + x} I - 1 + a ' & = 0. 



' \xj x 



Hence, when x and y vanish, we have 



the limit of - = - . 

 x a 



11 



Besides this, the limit of - may have an infinite value, and 

 u? 



this can be determined by examining if - has zero for a limit. 

 The given equation may be put in the form 



fx\* ( x\ 

 . . y + x+ (-} \a-b-[ = Q. 



9 w r yJ 



