374 FORM OF CURVES NEAR THE ORIGIN. 



This makes y vary as cc 2 , and therefore y 9 vary as x*. 

 Hence the neglected term by 3 varies as x 6 , while the terms 

 retained, x* and ayx*, vary as x*. But by taking x small 

 enough x 6 can be made as small as we please compared with 

 x*, and therefore near ihe origin one branch of the curve may 

 be found approximately by neglecting 6y 8 . The branch we 

 thus obtain, being determined by the equation a? = ay, is 

 a portion of a parabola having its axis coincident with that 

 oiy. 



Next, assume that near the origin the term ayx* may be 

 neglected in comparison with the others. We thus find 



= ; 

 therefore y varies as or. 



Hence the neglected term ayx* would vary as x t+ * ; that is 



as x 3 , while the terms retained would vary as x*. But since 







x 3 can be made as great as we please compared with x* 

 by taking x small enough, we do not obtain an approximate 

 branch near the origin by neglecting ayx*. 



Again, assume that x* may be neglected near the origin ; 

 then 



therefore by 2 ax 9 = 0. 



Hence y varies as x ; the terms retained vary as x* and the 

 rejected term varies as x 4 ; and thus an approximation to the 

 curve near the origin is given by 



The figure shews the nature of the 

 curve near the origin ; AB is the para- 

 bolic branch, and CD, C'D', are the two 

 branches found by neglecting x*. 



