FORM OF CURVES NEAR THE ORIGIN. 375 



The conclusions in this case may be verified by solving the 

 given equation with respect to x*. We thus find 



Expand \f(a* 4&y) in powers of y by the Binomial Theorem, 

 and take the upper sign, then 



x* = ay approximately ; 

 with the lower sign 



a; 2 = -y* approximately. 



In this manner, or by transforming the equation into a 

 polar form, we may complete the tracing of the curve. It will 

 be found that the branches extending from the origin to 

 and B respectively, unite, thus forming a loop. The branch 

 from the origin to D' extends to infinity, and has no recti- 

 linear asymptote. The curve is obviously symmetrical with 

 respect to the axis of y. 



344. Determine the nature of the curve 



2/ 4 + ay 2 x x 4 = Q 

 near the origin. 



First, if we neglect x* we have 



therefore y 1 = ax. 



Hence x varies as y 2 ; the rejected term varies as y 9 , while 

 the terms retained vary as y*, and therefore we have in the 

 parabola y z = ax an approximation to the given curve near 

 the. origin. 



Next, reject the term ay*x ; thus 

 2/ 4 -* 4 = 0, 

 therefore y = x. 



Hence y varies as x ; the rejected term varies as a?, and 

 the terms retained vary as x* ; hence this does not give us 

 an approximate branch. 



