T= T q . Let us also imagine — = as surface construed on the 



( 204 ) 



the value of v is much smaller for the point discussed here than for 

 the point in which the curve — - = disappears. 



We should have to expect a priori that these two points differ. 



Let us imagine = for different values of T as a surface with 



dx* 



the axes x, v and T, and let the T-axis be the vertical axis. Then 

 this is a surface with the whole v,;c-p\ane as basis, with top at 



dp 



dx 



same axes, then the sections of this surface pass to larger x and 



larger v with ascending values of T. Now this latter surface cannot 



d*q 

 pass through the top of — - = without yielding other points of 



intersection, a curve of intersection. If it can touch in the top, then 

 it would be possible that this curve of intersection did not exist. 



But a tangent plane to — = runs parallel to the a?,v-plane, and 



dx' 



dp 

 and such a tangent plane — = has not. 



dx 



d?ih 

 That a tangent plane in the top of — = runs parallel to the 



dx* 



.r,y-plane, follows from : 



d s xb d*\b d*\p 



Z_ dT A - dx A — dv = 0. 



dx*dT T dm* dx*dv 



In the top — - and are both equal to zero, whereas — -— 



1 dx 3 dx*dv dx\lT 



dT dT 

 is not equal to 0; hence — and-- is zero, or the tangent plane 



dr dv 



parallel to the v^-plane. 



As soon as — — — is not only smaller than n, but also smaller 



«12 "«I 

 tt' — 1 



than 1 -f 8 , the two curves do not intersect at any tempe- 



rature. If on the other hand — < n, and >l-f 8 - — -—- , the 



a l% —a t (w-h3) 2 



d*ü) dp 



two surfaces = and — = intersect in such a way that the 



dx* dx 



projection of the section is a closed curve; the section itself lies then 



