PHYSICS: E. H. HALL 
11 
THE VALIDITY OF THE EQUATION P = T dv/dT IN THERMO- 
ELECTRICITY 
By Edwin H. Hall 
Jefferson Physical Laboratory, Harvard University 
Read before the Academy, November 21, 1917 
The equation P = T dv/dT, in which P stands for the Peltier effect and v 
for the Volta effect between any two metals, was first obtained, I believe, by 
Lord Kelvin, comparatively late in his life. He derived it from a course of 
argument involving the imagined performance of a certain isothermal cycle. 
It was derived later by O. W. Richardson through a series of steps including 
his theory of thermionic emission and the imagined performance of a rever- 
sible cycle taking electrons from a metal at one temperature and putting them 
back into the same metal at a different temperature. 
When Kelvin came to test his formula by the experimental data available 
to him he found no support for it. In fact, it is said that "the experimental 
verification failed by a thousandfold." 
During the last few years experiments made by various skillful investigators 
measuring dv/dT in high vacua have perhaps reduced the ratio of the two sides 
of the professed equation to fifty; but it still remains so large that it would be 
considered a complete refutation of the equation's pretensions, if it were not 
possible to find an escape from this conclusion by dwelling upon the great 
difficulty of the experimental investigation. 
I have not, until recently, given much attention to this equation; but two or 
three months ago my colleague, Professor Bridgman, who has lately been 
studying thermo-electricity intensively, put into my hands for criticism a 
manuscript in which, gbing over with modifications of his own the Kelvin 
method and the Richardson method, he had in both cases reached their con- 
clusion. Studying Bridgman's manuscript I am, on the contrary, strongly 
inclined to the opinion that in each method the proof fails. As Processor 
Bridgman is now too much occupied with government work to give adequate 
attention to the question at issue, he has given me permission to present this 
question here. 
I cannot reproduce at length either of the arguments. I can only point 
out what I believe to be the weak link in each. In the first, which was sug- 
gested by that of Kelvin, Bridgman assumes that when, by an applied e.m.f., 
electricity is carried along a metallic path from metal A, constituting one plate 
of a condenser, to metal B, constituting the other plate, the only heat phe- 
nomenon, except the negligible resistance effect, is the production or absorp- 
tion of heat at the junction of the two metals, the ordinary Peltier effect. 
This assumption, which is vital to the argument, I question. 
Let n e be the number of free electrons per unit volume of a metal and 
