252 Prof. Clausius on the Application of the Mechanical 



remain at its maximum density for the increased temperature 

 T + rfT. For an increment of temperature dT, we will represent 

 by h . dT the quantity of heat which must be imparted to the 

 unit of mass of vapour during its contraction, in order that at 

 every density it may have precisely that temperature for which 

 this density is a maximum. The value and even the sign of the 

 magnitude // is at present unknown. The quantity of heat ne- 

 cessary in our case will therefore be 



mhdT. 



(3) During the elevation of temperature, a small quantity of 

 liquid, represented generally by 



— dT 



becomes vaporized, for which the quantity of heat 



is necessary. Herein, according to equation (7), 

 dm _^ v — Ma- du M da 



dT" 5?~'^~'^*Zr 



_ m du M. da- 

 u'dT'^U'df' 

 so that by substitution the last expression becomes 



,n.iiT .^*«. J^duMda\^^ 



^.!>if».M. ><:<.. r.w, ^u dT u dT/ 



Equating the sum of these three quantities of heat and the 



former expression -7™ dT, we obtain the equation 



13. As indicated by equation (III), the above expression for 

 -^ must be differentiated according to T, and,t^e.qxpreasipii f(^ 



-~ according to v. > The magnitude M is constant, the magni- 

 a X 



tudes M, <r, r, c, and h are all functions of T alone, and only the 

 magnitude m is a function of T and v, so that 



du 



J^'^ !.,M.L . (10) 



L_^/^\ 1 dr__r 

 dT\dv J ~ Ui' dT u 



til r ' R t i I 



.ot9j 04 iBffp» yyilvKdT/ ~ \ ^^^u^dT/dv* <ti^i hsio^tud 



