542 Mr. 0. J. Lodge on a Mechanical Illustration 



•\-e 



Thomson's relation , Ll\ 



§ 36. It is easy now to find the total electromotive force of 

 a complete thermo-electric circuit — that is, the electromotive 

 force acting in a ring made of two metals A and B, one of the 

 junctions being at a temperature lf the other at a temperature 

 2 ; for we have simply to add the difference of the two Peltier 

 effects to the difference of the two Thomson effects, (23) and 

 (27), and we get 



E = I^-e.-II, + @ S =A(A'„-A" )\" la ^ dd 



-B(// 4 -//' i )£^-V- • • (31) 

 which is evidently equal to 



4d8, [17] 



7 2 



as Thomson has shown it must be. 



It is evident also from the form of II, viz. 



tnat n AB +n EC +n CD + .... + n YZ + n ZA =o, . (32) 



provided 6 is the same for all — or that the electromotive force 

 in a ring formed of any number of metals all at the same tem- 

 perature is zero. The values found for the effects therefore 

 satisfy all the conditions laid down for them. 



§ 37. So far I have written down the results of our hypo- 

 thesis without any approximation ; but simpler and nearly 

 accurate values will be obtained by expanding them all in 

 series and taking the first term or two, or, what comes to the 

 same thing, by using the first two terms only of the expres- 

 sion (5) for r. Thus, writing 



we get at once from the hypothesis all the results in the form 

 given to them by Tait, and experimentally verified by him for 

 moderate ranges of temperature. The contact-force at a junc- 

 tion (22) becomes 



n^{A(« - a ' )-B(/3 -/3' ) 



+ 0[A(«-«')-B(/3-/3')]}, . (33) 



* The numbers in square brackets refer to Thomson's equations so 

 numbered in his papers on " The Dynamical Theory of Heat. Part VI. — 

 Thermo-electric Currents," reprinted from the Trans. R. S. Edin., in the 

 Phil, Mag. for March, April, May, and June 1856. 



