PRINCIPLES OF THE MECHANICAL THEORY OP HEAT. 261 



(1.293 kilograms of air) in order to raise it to a temperature of 273°, while, ^\•itll 

 pressure uuchanged, it is expanded to double its volume, is 

 273 • 1.293 • 0.2377=83 units of heat, 

 since the specific heat of the air imder constant pressure is ecpial to 0.2377 ; 

 hence we have S3 = U'+24.37, or U'=83-24.37=58.63. 



Thus, of the §3 units of heat which we supply to the 1.293 kilogram of air, in 

 order to raise its temperature from 0° to 273°, while with unaltered pressure it 

 undergoes expansion to twice its original volume, 58. G3 units of heat liave gone 

 over into this air, while the remaining 24.37 units of heat were expended in 

 doing the work involved in its expansion. In order, then, to raise the tem- 

 perature of 1.293 kilograms of air from 0° to 273°, while the volume of the air 

 remains unaltered, and so no external work is done, only 58.63 units of heat 

 are necessary. The specific heat of the air under constant pressure stands, there- 

 fore, to the specific heat of the air under constant volume as 83 : 58.G.3, or as 

 1.415 : 1 ; while this ratio has been found, in another manner, to be as 1.421 to 1. 



We have here supposed the mechanical equivalent of heat to be known, and from 

 this derived the ratio of the specific heat of the air under constant pressure and 

 constant volume, while in § 4 the inverse process was followed, inasnmch as we 

 assumed tliis last ratio to be known, and from thence derived the mechanical equiv- 

 alent of heat. 



The quantity of heat q which must be supplied to a body in order to raise its 

 temperature from t to t-\-t', to increase the heat contained in it from U to U-fU', 

 and to enlarge its volume from v to v-\-v'j is b}' no means the same under all 

 circumstances ; for, with a like condition at the beginning and the ending, the 

 work done during the transition from the first to the last may be very 

 different. The equation I is properly constructed only for a special case ; for the 

 case, namely, in which the pressure p remains unaltered while the volume of the 

 body enlarges from v to v-{-v'. When the pressure jj is variable the equation I 

 can only so long be recognized as valid, as the augmentation v' of volume is 

 small enough to be regarded as the differential of space ; as is the case with 

 the differential equation I a corresponding to the equation I. 



But when, with a variable value of p, the enlargement of volume v' is some- 

 what considerable, the work done during the expansion from v to v-\-v' can 

 no longer be expressed simply by the product ^^f'. Here the case presents itself 

 W'hen a higher method of calculation must indispensably be ])ut in practice, if 

 the object be an exact expression for the work done. With elementary 

 expedients we can, in such cases, only attain, by special calculations, to aj)proxi- 

 mative values. 



Let us procec<l, in order to make this more intelligible, to the consideration of 

 a special case. We have above calculated the quantity of lieat q which is reciuisite 

 to raise a cubic metre of air of 0° and sustaining atmospheric pressure, to 273°, 

 while the air expands luider an unvarying pressure to doul)leits original volume. 

 Uere is the final condition : two cubic metres of air of 273° temperature and an 

 elasticity of one atmosphere. The same final condition can, however, be also 

 reached, beginning with the same incipient condition in another manner. Let 

 the piston (Fig. 9) be again in its original position K, and, under it in the cylin- 

 der, one cul»ic metre of air at 0° sustaining atmospheric pressure, the burden of 

 the piston being thus 10333 kilograms. If this weight be now slowly and regu- 

 larly diminished to one-half, the air will gradually expand and push the piston 

 upwards; thus there should, in the first place, be so much heat supplied to the 

 air that, witli an unchanged temperature^ of 0°, tlie i)iston is lieaved u[i\vards (»ue 

 metre, and the vcdnmc of air therefore doubled. 



Tlie quantity of heat necessary for this is only to be detennincd by Iiighcr 

 processes of calcuIaLion, but an approximate value may be olttained in an 

 elementary way. Let us conceive the pressure ^>, which weighs ujion the piston 



