PEINCIPLES OF THE MECHANICAL THEORY OF HEAT. 2G7 



prcbSure to 2), and tlie vapor will again be fully restored to its original state, 

 (water of t°.) In this last part of the operation the work d, represented l)y the 

 rectangle Dr/a A, is expended, and thereby the quantity of heat q is again sup- 

 plied to the body under experiment, a (piantity which it had lost during the 

 expansion from () 13 to O C 



While the water has Iteeu thus fully restored to its original state, the work 

 iyu-\-d will, during the operation cited, have been performed thnntgh the expend- 

 iture of the quantities of heat r and (Z; ^^^'^ thereupon the quantity of heat r'-\-q 

 have been gained by the expenditure of the work Xi' u-\-d. The sum of the work 

 gained is thus, 



Xm-\-d—p'u—d= <> t(, 

 a work which is represented by the shaded parallelogram abed, (Fig. 11.) 

 The heat expended in producing this work is, 



r-\-q—r'—q=:)'—r'. 

 But the quantity of heat requisite for the performance of the work >!> u is A<1> u; 

 we have then the equation 



A <^ii=r—r'. 

 Now, in this equation the values A and r are already known ; only r', there- 

 fore, is wanting to enable us to determine that of u. And as the rigorous solu- 

 tion of this i)r(jblem is not possible without the aid of the higher analysis, we 

 must here content ourselves with an elementary process of approximation. 



The quantities of heat Apu and Ax)'u stand evidently in proportion to the 

 tensions p and^'; and to these we may assign as proportional, since the ques- 

 tion regards only slight differences of temperature, the density of the vapor of 

 water at t° and at (^—1)°. But to these densities are also proportional the (juan- 

 tities of Avafer which, at the temperature t°, are evaporated during the expansion 

 through the volume u, and at the temperature (^—1)° are condensed during the 

 compression through the volume u. Whence, therefore, we have p : [j'=-x^ : p', and 

 p-\-Ap) u : fj'-{-Ap'u=p : p^j or r : r'=j> : x^'- 



But since, within such naiTow limits of temperature, the saturated vapor may 

 be assumed as following the law of Mariotto Gay-Lussao ; therefore 



p ■.p'=lJ^at: l + a(^-l) 

 consequently, also, 



r : r'=l + at: l-fa(^-l) 



1 + at 

 , { , l-\-a(t—l) ) 

 / 1+at V 



, ra 

 r—r 



l + at 



or, if we divide numerators and denominators by a, and take -=:«, 



if T denote the absolute temperature* which corresponds to t° C. Thus we have 



*Let p be the elastic force of a confineJ mass of air at 0"^; then, according to Mariotte 

 Gay-Lussac's law, this elasticity at P C. is equal to ;* (l-f-O.OUtJfJaO ; the clastic force 

 of the enclosed mass of air is thus null, if l-f-O.OtKilin^^O; that is to say, if <=— 'i7'.P C. At 

 this temperature, which is 27:5'^ below the froeziiif^ point of water, the pases lose their power 

 of expansion ; and it is this point which we indicate as the tihsolule zcro-puitit. It is the 

 temperature counted from tins point onward, according to the Celsius degrees, T=■27:>-(-^ (if t 

 be the temperature counted onward from tlie freezing point of water,) which is denoted as 

 absolute temperature. In the second vertical series of our foregoing table are given the 

 absolute temperatures which correspond to the temperatures of the lirst column measured by 

 the thermometer of Celsius. 



