A.— MATHEMATICAL AND PHYSICAL SClKxXCES. 25 



Bv oliminating a.,' — cr," we obtain 



all of which symbols are related to the temperature T.^ (T, being kept 

 constant). 



Similarly, by making the change at the lower temperature, 



Also, by eliminating tt., (or tt,) 



All this leads one to realise that ( ^_ | is identical with ^^'^ where 

 v., is the loc:l e.m.f. contributed by the junction itself ; and that 



Controversialists may be divided between those who believe that the 

 difference of potential (V) at any junction is measured by the heat taken 

 in thereat and those who do not consider that this assertion is justified, but 

 that it must be replaced by the above equation. It of course does not 

 folJoiv that there is any difference between the two assertions in particular 

 cases ; it remains to be found out whether there is any difference between 

 them or not. 



Though this is so, yet of course it may not be assumed ab initio that 

 heat entry and external work done are equal to one another. This is the 

 assumption that is actually made by those who write 7r.j--V.,, both being 

 expressed in terms of the same units. 



It is instructive to enquire what the relation connecting these quantities 

 ie for other phenomena. 



- Example : — It is found that in many cases E is given by a parabolic equation 

 which can be written 



E = a(T,-T,) 



rp T, + T, 



whence by keeping Tj constant and differentiating with respect to '£., 



7T3 = aT.,(To-T.,). 



Similarly, by differentiating with respect to Ti, the heat removed at Ti is found 

 to be 



_ Ki = aTi(T„-Tj). 



■ Also a/- 02"= aT., 



and O]' — Oi" = aTi. 

 Tliese are terminal values onlj' and are consistent with taking 



o' —a" = aT at any intermediate temperature, 

 or o' = a'T and a" = a'T 

 where a = a'— a". 



