( 828 ) 



(/"if' . . . , , . . . . 



which is negative or positive, hikes place in the points ot the 



dx* 



d*\b . . . . . . . 



curve = with nnixinium or minimum volume or tor which 



,/•> d 3 ip d'p 



- = 0: and the transition of the points for which = — — 



dx 3 dvdx* dx* V) T 



is negative or positive, takes place in the points with maximum or 



or minimum value of X. From all this follows that the curve 



- = contracts with rise of T, and has contracted to a single 

 dx* 



point for certain value of 7' = T,,. It is now necessary for our 

 purpose to determine the value of T,,, and also the value ,/;,, and r,. 

 of the point at which this locus vanishes. This means analytically 

 that we have to determine the values of T, as and v, which satisfy: 



' — 0, ^=0 and = — = 



dx* dx' dvdx* dx* 



or the equations: 



db\* d ' a 



.dx) ) dx* 



MET — + ^-4^ = - v 1 ) 



x(l-x) ' (v— bf\ v 



'dbV 

 \d~x) 



2x*(l—xf {<-by 



fdby d*a 



(2) 



\d.rj 1 dx* 



and ■ ] " /T l-^ = W • • • • (3) 



If (1) is divided by (3), we get a relation between x and v, 

 which in connection with (2) may lead to the knowledge ot x, 

 and Vg. 



Then we get: 



{v—bf 



'dbX 



){ Tx) 



= v -f b = (v - h) + 2b 



fdb\2>x\\-xf 

 and as {v—bf = — —i — jr- - we find 



V''*/ t — Z.r 



b _ x(l-x) __ 1_ j 2x*(l-xf j Vt 

 dx 



(4) 



