( 212 ) 



has the same value. But until the relation between a 1} a 2 and « 12 is 

 known, such a simple rule cannot be considered as valid. 



The calculation of the temperatures at which the two curves 



d"\p .dp . , 



— - = and — = are in contact cannot be performed on account 



dx* dx 



of the very intricate forms to which this problem leads. For this 



purpose we might solve the value of v — b from the quadratic form 



of page 201, which is then a function of x, and substitute then the 



dp d*ty 



value of v, either in — = 0, or in = 0. We have then a formula 



dx dx* 



in which T is expressed in the values of x of the points of inter- 

 section of the two curves. If we seek the maximum value of T by 

 differentiating with respect to x at T = constant, we find a relation 

 in x, from which the x of the points of contact would have to be 

 calculated, and the value of T max or 7 T m ,„ by substitution of this 



B 1 



value of x. But even in the special case of — = , or k = n, in 



C n — 1 



db x (1 — x) , i . , , « . , . , 



which v — b = , this calculation leads to a form in which 



dx 1 



— r+* 



n — I 

 x rises to the third degree. In all other cases the equation is much 

 more intricate. The value of x, v and T for the maximum and 

 minimum temperature might of course also be calculated by the aid 

 of the three eqations: 



d*ty dp , d*\b d*p 



* — 0, - = aüd - 



_(drp\* 

 -{dx*)' 



dx* dx dx* dxdv 



d*p 



which last equation expresses that the two curves touch. As 



dv dx 



dp d*\p 



is certainly negative in the points of the line — = 0, — - must 



certainly be positive for the contact, which had already been repre- 

 sented in the figures 24 of these contributions. 



