( 523 ) 



For a concave beginning (i. e. turiietl towards the A'^-axis) 



dx* 



{ dT \ d'T dT 



is ahvavs neoative tor ^— becomes laröer nesative . Hence : — 



"" \ dx Ö fe y ^^^., ^^^ 



jwsltire. Oil the other hand this quotient will ha negative ïov a convex 



hegiwitiig. In the same way for 7'=/'(,i''). 



d^T ' fdT 



With a concave end will again he negative -- becomes 



\ d'T dT . , 



smaller positive , so : — neqative. For a convex end this 



/ dif dij ^' 



quantity ^vill be positive. We have therefore the following transition 



conditions. 



I For 7 = /(.r) ^''"'^''''^ I beginning 2(^,-ry,)+(7,-4r0 (/^ - 1) >0 



convex ) \. 



II For r=/(.f) '''"''''''' I end -2(7,-7,)-(9,-47;) (1 -.-^<0| 



convex ] ^ I 



III For r=:/(.i-') '*''"'''''''' ! begmning2(7,-9,)-(7, + 47;)(l-.r'^) >0| 



convex ) \ I 



IV For T^f{x') ''''''''^^^^ I end _2(7,-7,)+(7,+47;) (/' - 1) <0 



convex \ ^ 



vtv in another form : 



\ 2(7,-47',) 



The different regions with their limits, which occur in these 

 conditions, are represented in fig. 8 (Plate). The tigui-e holds for 

 7'„ = V'., 7\, the values of <i^ and 7^ are expressed in multiples of J\. 



Let us subject the limiting-curves to a closer examination (see fig. 3). 



a. Curve I, viz. 



According to (8) all the curves T = /{x) with a concave heg in?ving 

 will lie above this curve, with a convex beginning beloiv it. For 7, 

 must then be respectively larger or smaller than the values given 

 by the second member. 



