530 Prof. Thomson on the Dynamical Theory of Heat. 



and near the narrow passage. Let Q be the quantity of heat 

 (which, according to circumstances, may be positive, zero, or 

 negative) emitted by a pound of air during its whole passage 

 from the former locality through the narrow passage to the 

 latter; and let S denote the mechanical value of the sound 

 emitted from the " rapids." The only other external mechanical 

 effect besides these two produced by the air, is the excess (which, 

 according to circumstances, may be negative, zero, or positive) 

 of the work done by the air in pressing out through the second 

 part of the pipe above that spent in pressing it in through the 

 first j the amount of which, for each pound of air that passes, is 

 of course p V —pu. Hence the whole mechanical value of the 

 effects produced externally by each pound of the air from its own 

 mechanical energy is 



JQ + S+p'u'-pu (15). 



Hence if <f>{v, f) denote the value of e expressed as a function of 

 the independent variables v and t, so that $(u, t) may express 

 the mechanical energy of a pound of air before, and <£(«', v) the 

 mechanical energy of a pound of air after passing the rapids, we 

 have 



<j>{u',t') = <}>{v,t)-{JQ + S+p'u'-pu} . . (16). 



95. If the circumstances be arranged (as is always possible) 

 so as to prevent the air from experiencing either gain or los3 of 

 heat by conduction through the pipe and stopcock, we shall have 

 Q=0; and if (as is perhaps also possible) only a mechanically 

 inappreciable amount of sound be allowed to escape, we may 

 take S = 0. Then the preceding equation becomes 



<j>{u',t') = <f>{u,t)-(p'u'-pu) . . . (17). 

 If by experimenting in such circumstances it be found that /' 

 does not differ sensibly from /, Mayer's hypothesis is verified for 

 air at the temperature t ; and as p'u' would then be equal to pu, 

 according to Boyle and Mariotte's law, we should have 



(f>{u',t) = (}>(ti,t), 

 which is, in fact, the expression of Mayer's hypothesis, in terms 

 of the notation for mechanical energy introduced in this paper. 

 If, on the other hand, t' be found to differ from t*, let values of 



* If the values of fi I have used formerly be correct, t' would be less 

 than t for all cases in which t is lower than about 30° Cent. ; but, on the 

 contrary, if t he considerably above 30° Cent., t' would be found to exceed /. 

 (Sec ' Account of Carnot's Theory,' Appendix II.) It may be shown, that 

 if they are correct, air at the temperature 0° forced up with a pressure of 

 ten atmospheres towards a small orifice, and expanding through it to the 

 atmospheric pressure, would go down iu temperature by about 4°'4 ; but 

 that if it had the temperature of 100° iu approaching the orifice, it would 

 leave at a temperature about 5 0- 2 higher, provided that in each case there 

 is no appreciable expenditure of mechanical energy on sound. 



