162 DR. J. BOTTOMLEY ON THE CHANGING 



Differentiating (18) twice, 



df'x d % x/X ! X\ X d z x. 



m <w 



d % x(V , aA X d % x t , s 



substituting from (20), we have 



where 



and 



5 + A$+B*=0, (22 ) 



. 2 rs! X Xm\ 

 A=-(- + T + 7-), 



L lmm l 



To solve the above equation, assume 



Differentiating this twice and four times, substituting in 

 (22), and dividing by a common factor, we have 



/a 4 + A/^ + B = o. 



Hence 



A 



n 2 



±X /|'-B. 



The quantity under the radical sign will be positive, 

 zero, or negative according as 



'X' . X . Xm\ z _ X r Xm 



according as 



( 



L / ImJ L lm x 



VL + 7 ImJ + ^m x l z 



2 

 •4 >=<o. 



The expression on the left is essentially positive ; there- 

 fore the quantity under the radical sign is positive, and 



v 





