98 
PHYSICS: E. H. HALL 
Proc. N. A. S. 
No single point M 0 need be preferred. If u{M) is summable with re- 
spect to h(M 0 ,M) on the frontier of T when M Q is some definite point in 
T, it is summable on the frontier of T with respect to h(M h M), Mi being 
any point of TV 4 
1 Proc. Amer. Acad. Set., Boston, 41, 1905-06. 
2 Borel, E-, Legons sur les fonctions monogenes, Paris, 1917, p. 10. 
3 Evans, Rend. R. Accad. Lincei, (Ser. 5), 28, 1919 (262-265). 
4 For example it is sufficient to consider merely rectangles. See also the class Y below. 
5 de la Vallee Poussin, C, Integrates de Lebesgue, Paris, 1916, p. 57. 
6 Volterra, V., Legons sur les fonctions de lignes, Paris, 1913, chapt. II. It is to be 
noted that the definitions of continuity in the case of functions of curves and in the 
case of functions of point sets do not correspond at all to the same situation. 
7 The discontinuities need not be of the first kind. 
8 By } $(M) \ 2 is meant the quantity { ^(M) j 2 + j $ y (M ) | 2 . 
9 Bull. Amer. Meth. Soc, 25, November, 1918 (65-68). 
10 de la Vallee Poussin, C, loc. cit., p. 98; Trans. Amer. Math. Soc, 16, 1915 (493). 
11 This identity is given for the special case of a rectangle, where the integrals (19) 
do not have to be used, by W. H. Young, Proc. London Math. Soc, (Ser. 2), 16, 1917. 
12 Osgood, Lehrbuch der Funktionentheorie, Leipzig, 1912, p. 151. 
13 Osgood and Taylor, Transactions of the American Mathematical Society, 14, 1913 
(277-298). 
14 A complete exposition of the theorems of this paper will appear in an early issue 
of the Rice Institute Pamphlet. 
THERMO-ELECTRIC ACTION AND THERMAL CONDUCTION 
IN METALS: A SUMMARY 
By Edwin H. Hall 
Jefferson Physical Laboratory, Harvard University 
Communicated January 5, 1921 
I shall now undertake to show in brief what I have accomplished in 
the series of papers which for some years I have been publishing in these 
Proceedings in relation to thermo-electric action and thermal conduction 
in metals. 
Starting with the hypothesis of dual electric conduction, that is, con- 
duction maintained in part by the passage of electrons from atomic union 
to atomic union during the contacts of atoms with positive ions, and in 
part by "free" electrons in the comparatively weak fields of force called 
the inter-atomic spaces, I derived 1 an equation giving the conditions of 
steady state in a detached bar hot at one end and cold at the other, and 
two equations for the conditions of equilibrium at a junction of two metals. 
For the detached bar the steady state is a current of associated elec- 
trons up the temperature gradient and an equal current of free electrons 
down the temperature gradient. This involves a freeing of electrons from 
atomic unions, ionization, at the hot end of the bar, with absorption of 
heat, and re-association at the cold end, with evolution of heat. Evi- 
dently this is a process of conveying heat, and the question at once arises 
