( J 55 ) 



2a b f b 



we get 



MRT= , 



b v \ v 



1 da ( 1 da 



a a d.v \ 2a dx I 



MRT = 1 — (7) 



b 1 db J 1 db l 



b dx ' b dx 



da 2 db 1 da 



For values of x for which — = and — — -= — — or — - — = 0, 



dx b dx a dx dx 



the value of T=0. Thus the same value of T always belongs to 

 a couple of values of x which approach each other. x\nd at the 

 maximum value of T the two values of .r have coincided. By differen- 

 tiating (7) we get an equation which may be written in the form : 



d*a (day 



3a 2 — , 



2a db dx- \dxj da 



TJx~~ ~cPa 7<ïay~ ** 

 tt dx* ~~ \cLv) 

 from which, taking (6) into consideration, we obtain again (4). 



It appears from the foregoing that putting a x ^ = a, a 3 comes to 

 the same thing as putting m = 2. For mixture:; with minimum plait- 

 point the value of m differs much from this value, as it is then 

 smaller than I, and so ay will have to differ pretty much from 

 a x a % . If we put ay = l'a x a„ in which /'<!, we find from: 



d'a (da\* 



(2-m)a-4 = 4(l-Z')a 1 a s 



dx 2 



or 



1 — 2" 1 («i + «» — 2a ls ) a = (1 — P) a, a s 



It may be derived from this equation that m may lie near 2, 



even when / differs comparatively much from 1. 



m 

 The value of a varying with x, also the ratio of 1 and 1 — /' 



a 



will vary with x. If we make a increase with x, which probably 

 will be in general the case, then a x is the smallest value of a and 



