126 PROFESSOR W. THOMSON ON THE 



a joint communication to be made by Mr Joule and myself to the Royal Society 

 of London before the close of the present session. 



101. A corollary from the second General Law of the Dynamical Theory stated 

 above in § 98, equivalent to the law itself in generality, is, that if a material 

 system experience a continuous action, or a complete cycle of operations, of a 

 perfectly reversible kind, the quantities of heat which it takes in at different 

 temperatures are subject to a linear equation, of which the coefficients are the 

 corresponding values of an absolute function of the temperature. The thermo- 

 metric assumption which has been adopted is equivalent to assuming that this 

 absolute function is the reciprocal of the temperature ; and the equation conse- 

 quently takes the form 



H* H.' JhL" „ „ 



T + l- + -f +&c -= ' 

 if t, t\ &c, denote the temperatures of the different localities where there is 

 either emission or absorption of heat, and =*= H<, =*= H,-, =*= H t », &c, the quantities 

 of heat taken in or given out in those localities respectively. To prove this, con- 

 ceive an engine emitting a quantity H f of heat at the temperature t, and taking 



in the corresponding quantity - H t at the temperature t' \ then an engine emitting 



H /TT TT \ 



the quantity - H t + EL, at if, and taking in the corresponding quantity f ' I -j ■ + •— j 



(TT TT \ 

 -~ + -j ) + H, at t", and taking in the 



(TT TT TT \ 



~r + ~F + ~F~) a t*'"; an( i so on - Considering n-2 



such engines as forming one system, we have a material system causing, by reversible 

 operations, an emission of heat amounting to H t at the temperature t, H,, at the 



temperature *',... and H (( „_ 2) at £ ( "- 2) ; and taking in t {n - l) (y + 7^+ • • • • +^£?) 



at the temperature ^ n_1) . Now this system, along with the given one, constitutes a 

 complex system, causing on the whole neither absorption nor emission of heat at 

 the temperatures t, f, &c, or at any other temperatures than t {n ~ l) , f n) \ but giv- 

 ing rise to an absorption or emission equal to ±1 * (n-1) (-~ + -J : + . . . . + -^Sr ) 



+ H,(n-i) at £ (n_1) , and an emission or absorption equal to ± H <( «) at # n \ This com- 

 plete system fulfils the criterion of reversibility, and, having only two tempera- 

 tures at localities where heat is taken in or given out, is therefore subject to 

 Law II. ; that is, we must have 



TT **> T^n-n / H * Hf H t (n-2)\ „ ~l 



which is the same as 



5i + S ? + H^n-l) Hk») _ ft m 



-7 + a + ■••• +i(S=ir + iw-- .... (i). 



