82 PROCEEDINGS OF THE AMERICAN ACADEMY. 



and (3)-(3), and assuming that we have to do with free electrons, 

 electrons (B), only, we should have to conclude that p is 1. To 

 determine the algebraic sign of a we must look to the relative magni- 

 tude of p and p. According to equation (15), p being 1, we have 



<r>0, if ^<3; <r = 0, if v = 3; a<0, if *>>3. 



From (15') we should get 



<r>0, if v<4; a = 0, if v = 4; a<0, if i>>4. 



Turning back now to equation (2), and assuming that we have to 

 do with (B) electrons only, we get as the specific conductivity 



K b =k b Fo(8)T^-^ + ^, 



where Fi (8) is a factor which increases with rise of temperature. 

 Accordingly, K b must increase with rise of temperature, unless 

 "<2(p+1)> — that is, unless v< 1, if p= 1. But we have just seen that, 

 in order for a metal to have a negative and proportional to T, v must be 

 greater than 3, and so v — \{p + 1)>2, which will make K b propor- 

 tional to some power of T higher than the second. Metals commonly 

 represented by straight lines on the thermoelectric diagram, with <r 

 positive for the conventional conception of current and therefore 

 negative for the electron conception of current, are copper, silver, zinc 

 and antimony. There may be inaccuracy in taking the lines for these 

 metals as straight, but it is unlikely that this inaccuracy is great 

 enough to account for the absurdity here indicated regarding con- 

 ductivity. It seems difficult to avoid the conclusion that the concep- 

 tion of free electrons acting alone is insufficient to account for the 

 phenomena of both electric conduction and thermo-electric action. 



Accordingly we must presently return to the consideration of those 

 electrons which take part in the action (A). 



The antagonism between v and p which is to be observed in equation 

 (15) and (15') is rational. Great concentration of free electrons at 

 the hot end of a wire, corresponding to a large value of v, must tend 

 by gas-pressure of the electrons to give a large value of F at the cold 

 end, so that the term (/ -r- /3) in the value of a (see equation (14)) 

 will be negative. Rapid increase of R, on the other hand, with rise 

 of temperature, corresponding to a large value of p, involves large 

 thermal capacity of the electrons and tends to keep <r positive. 



The increment of F for any rise of temperature (IT along the wire 

 is readily found from equation (8) to be 



clF = - (1 + q)RdT = -(14- p + v )RdT, (16) 



