during its Expansion under different circumstances. 401 



of heat must be consumed ; by the friction, hoM-ever, heat will 

 be again developed; and although a portion of this is conducted 

 EMay by the surrounding mass, still the remaining portion com- 

 municates itself to the vapour. It is here, however, evident that 

 the effect of friction does not, as Mr. Thomson supposes, exhibit 

 itself in a gain of heat, but, on the contrary, in a loss of heat ; 

 the latter, however, not corresponding to the entire quantity of 

 work expended in overcoming the friction, but only to a portion 

 thereof. AYe will neglect this, and also the loss arising from the 

 second circumstance alluded to, which is undoubtedly incon- 

 siderable, — in this way the calculation is rendered very simple. 

 It is here necessary to subtract from the amount of heat 



y'hdt found in the former case, the heat which corresponds to 



the difference of the quantities of exterior work produced in both 

 cases. Let p be the tension of the vapour for the temperature t, 

 and s the volume of the unit of weight belonging to this tempe- 

 rature. Farther, let/jj and p^ be the values oi p, and 5j and s^ 

 the values of s at the commencement and at the end of the ope- 

 ration, p,2 being, according to our assumption, the pressure of 

 one atmosphere ; the exterior work then is — 



'pds, 



- *i 



in the second case = / 'pads. 



The corresponding amounts of heat are obtained by multi- 

 plying these quantities by the equivalent of heat for the unit of 

 work, which equivalent I have formerly denoted by A. If Qg 

 express the quantity of heat sought, which is required by the 

 unit of vapour as it issues, we must set 



Q^= T'/idt—A /pds + A / "'pods. . . . (2.) 



It is, however, evident that 



and 





' p^ds=p^{s.2- s,), 



I pds=p^{s-a)-p^{Si-a)-/ '{s-a)dp 



=p^{s - a) -;;, (s, - a) -J^ ' {s-a) -^j dt, 



where o- is an arbitrary constant ; for this we will set the volume 

 of a unit of weight of water, the alteration of which with the 

 Phil. May. 8. 4. Vol. 1. No. 5. Maij 1851. 3 E 



