134 PROFESSOR W. THOMSON ON THE 



in each metal respectively. It is easily shown (as will be seen by the treatment 

 of the subject to follow immediately) that if the values of <r v <r 2 , &c, depend 

 either on the section of the conductor, or on the rate of variation of temperature 

 along it, or on any other variable differing in different parts of the conductor, except 

 the temperature, a current might be maintained by the application of heat to a ho- 

 mogeneous metallic conductor. We may, therefore, at once assume them to be, if 

 not invariable, absolute functions of the temperature. From this it follows, that 



if (p t denote any function of t, the value of the sum, I <p ttrdt, for any conducting 



arc of homogeneous metal, depends only on the temperatures of its extremities ; 



2 a, 

 and therefore the parts of the sums 2 a t and — , corresponding to the successive 



metals in the principal conductor, are respectively 



— I (T l dt, —I <T 2 dt, — / <7 n dt, —I (T 1 dt, 



J T, J T 2 J T« J T 



and -f'Eidt, -p!xdt -r^^dt, -F'^dt. 



«- / T [ t J T 2 t J 'l n t J T t 



Hence the general equations (7) and (9) become 



F=J { n, +n.,+ + n„— / <r l dt— (r 2 dt / cr n dt—l o\dM..(10) 



^ + 7^+ + ™ — / -f dt - -rdt- -/ ~r dt - -±dt = . (11) 



which are the fundamental equations of thermo-electricity in non-crystalline con- 

 ductors. In these, along with the equation 



P + F 



JB 



(12) 



which shows the strength of the current actually sustained in the conductor when 

 an independent electro-motive force, P, is applied between the principal electrodes 

 E, E', we have a full expression of the most general circumstances of thermo- 

 electric currents in linear conductors of non-crystalline metals. 



114. The special qualities of the metals of a thermo-electric circuit must be 

 investigated experimentally before we can fix the values of u v ri„ &c, and 

 cr v <r 2 , &c, for any particular case. The relation between these quantities 

 expressed in the general equation (11), having, as we have seen, a very high 

 degree of probability, not merely as an approximate law, but as an essential 

 truth, may be used as a guide, but must be held provisionally until we have suf- 

 ficient experimental evidence in its favour. The first fundamental equation (10) 

 admits of no doubt whatever in its universal application, and we shall see {§ 123 

 below) that it leads to most remarkable conclusions from known experimental 

 facts. 



