(437 ) 



V 



*l 



Let us seek the points of this curve for which =0. Such a 



il.V 



point lies for the branch of the small volumes on the right of the 



dv 

 point lor which — = 0, and on the left for the branch of the larger 



dx 



volumes. i;_v differentiating the equation of the curve in the last 



r 



mentioned form with respect to x, and keeping constant, we find 



b 



as a condition : 



v'\*dA dB 



i ) , ! / r=I "- 

 b / dx dx 



or 



r 



+ 



V dx 



/dA 



dx 



dA dB 



trom which it appears that — and must have the samesicnfor 



dx <l r 



dA cja^l-*) 1 -a t x*\ , dB 



such points. Now = — - J and for we find the 



dx a 2 d,r 



value — — — — . So \ / — rails * SO together 



(l — x -\- >i,ry x s> v 



-■'■< •'■ <r .-' <- i i /!+*, 



with O— ; or > and ^> — / must hold 



x s> n L — x^> n l—x /> n V l+ f 3 



dB dA 



at the same time. What it depends on whether -and -is positive 



dx dx 



v 

 or negative, is seen when the value obtained for is substituted in 



b 



the equation of the curve. It' we write this as follow-: 



we find : 



dB dB 



— A B 



dx \ i / dx 



dA ~ ) ~ ± V ./.I 



dx dx 



or 



