218 Prof. Thomson on the Dynamical Theory of Heat. 



if t, t', &c. denote the temperatures of the different localities 



where there is either emission or absorption of heat, and + Hi, 



+ H^/, + H^//, &c. the quantities of heat taken in or given out in 



those localities respectively. To prove this, conceive an engine 



emitting a quantity H< of heat at the temperature t, and taking 



t' 

 in the corresponding quantity -H< at the temperature t' ; then 



t' 

 an engine emitting the quantity jHt + Hf at t', and taking in 



the corresponding quantity t" ( 7^ + "TT j at the temperature t" ; 



/TT TT \ 



another emitting t" i~+~f) + ^< ^^ i", and taking in the 



corresponding quantity /'" ("7 +^' + "777-') ^^ ^" > ^^^ ^° ^^^ 



Considering n—2 such engines as forming one system, we have 

 a material system causing, by reversible operations, an emis- 

 sion of heat amounting to H< at the temperature t, H</ at the 

 temperature f, and Hj(n-2) at t^"~^^ ; and taking in 



^(n-D/H, _^Hi, _^ _ _^ ^^^) at the temperature t'»-'K Now 



this system, along with the given one, constitutes a complex 

 system which causes on the whole neither absorption nor emission 

 of heat at the temperatures t, t', &c.,or at any other temperatures 

 than ^^"~^\ t'-"^ ; but gives rise to an absorption or emission 



equal .0 ± [«.-.. (5.+ H'+.... + i^) +H/.-..] a. 



^"~'^, and an emission or absorption equal to +H^(n) at t^"K 



This complete system fulfils the criterion of reversibility, and, 

 having only two temperatures at localities where heat is taken 

 in or given out, is therefore subject to Law II. ; that is, we must 

 have 



h...=-,-£,P"-.(|'.^......'^).hH 



which is the same as 



t t' /("-'^ ^w • ' ■ ' \ )' 



This equation may be considered as the mathematical expres- 

 sion of the second fundamental law of the dynamical theory of 

 heat. The corresponding expression of the first law is 



W + J(H,+ H,, + .... + H,(„-,) + H,(„;)=0 . . (2), 



