( 193 ) 



to 0. For still higher value of x the second factor becomes equal 



. _ , dT 



to 0. Between these two special values of x, — is negative — and 



das 



for values of x which are larger than that for which also the second 



dT 

 tactor is zero, — is again positive. So the value of T presents a 



dw - 



maximum and a minimum. 



In general we must now put two cases as possible according as 



1 da 1 db 

 the value of x for which ——=——, is smaller or larger than that 



a dx b dv 



fda\ 2 d*a 

 for which { — = — a — - The intermediate case in which these two 

 \dxj 6 dar 



values would coincide, might be considered as a third possibility. 



Let us call the maximum value of the temperature Tm, and the 



minimum value T m . For a value of T below T m there is only one 



dp d^p 



point of intersection of — = and — — = 0, namely at small value 



do dvdx 



of x. For values of T above Tm there is also only one point of 



intersection for large value of x. But for values of T between T m 



and Tm there are three points of intersection. Of these three points 



of intersection there is always one, the middle one, which lies at a 



value of x lying between that which makes the first factor equal 



to zero, and that which makes the second factor equal to zero. 



To give a survey of the course of the points of intersection of 



dp d^p 



— = and - — = at different temperatures, and so of the circum- 



dv dvdx 



dp 

 stances for which — = has a maximum or minimum volume, we 

 dv 



shall have to separately treat the cases for the different situation 



* * . . , 1 da 1 db , fda\ 2 d"a 



ot the two values of x, for which = , and ( — = — a — . 



a dx b dx \4<cJ 3 dx* 



Let us first take the case for which the value of x for minimum 



de- 

 value of — is the smallest. This case is the simplest, and was dis- 



bx 



d*p 



cussed by me already before. Then a curve = indicated in 



dvdx 



dp 

 fig. 34 by a, passes through the double point of — = 0. For lower 



dv 



dp 

 T, a has assumed the position ft, and — = the position y, so that 



dv 



there are then two points of intersection (1 and 2) to be found. 



