170 EVALUATION OF DIFFERENTIAL COEFFICIENTS 

 Hence, y'^)' + 3'(f)' -4a-?-a' = 0. 



If now ^ have any finite limit, the term y*( -) will vanish 

 x \xJ 



when v = 0, and we have for finding the ultimate value of ^ 



x 



the equation 



or 



^ r 



therefore ^ 



a; 



'' : *{*H-* 



y 2 + V7 



If - have an infinite value, then -- has a value zero 

 x y 



putting the given equation in the form 



y 



57 



we see that - = ultimately would not satisfy it. Hence 



. y . 



has not an infinite value. 



Again, suppose ay 8 bx*y + x* = ; 



y - 



therefore a - 6 - } + x = : 



\xJ \x/ 



when x vanishes, we have -]af-) J [ = ; 



a; [ \a;/ ) 



therefore ^ = ultimately, or - = + A /- 

 a; x ~ l \/ a 



Again, suppose a; 4 + ax*y + Ja;^ 2 y* = ; 



therefore x + a + b ] y ] = 0. 



x \xj * 



