CONDUCTIVITIES OF METALS AND ALLOYS AT LOW TEMPERATURES. 431 
per minute, a correction equal to one-tenth the rate of rise has been subtracted from 
the observed temperature. 
The determination of the resistance of the rods depends virtually on the measure¬ 
ment of the current through them and the difference of potential at two points 
4 centims. apart. As the rods are about 7'5 centims. long, and the current was 
introduced through a strip about 1 millim. broad, situated at the ends of the curved 
surfaces of the rods, the current may, to the degree of accuracy aimed at in the 
present paper, be assumed to be uniformly distributed throughout the cross section 
of the rod in the interval between the two knife edges at which the potential differ¬ 
ence is measured.* If R K is the resistance found, p the resistivity of the material, 
A the area of cross section of the rod, and 2 a the distance between the knife edges, 
we may therefore write p — R R A/2«.f 
* The distribution of potential v throughout a rod of length 21 and radius R and resistivity p, through 
which an electric current flows which enters and leaves by strips of breadth b of the curved surface at each 
end, and is of uniform density c at each strip, is given by the equation 
v = 
8 dp 
I (-1)" 
n =0 
sin (2« + 1) ^ sin (2« + 1) ^ lo 
(2 ™ + uff)/n (< 2 »+1) 
/' 
-R 
"21 
where x is the distance of a point from the central transverse section, r is its distance from the axis of the 
rod, and I (r) is the Bessel function with unreal argument. 
Hence the potentials at the points ± a, R on the surface at which the knife edges touch the rod differ 
from that at the central section by 
= 1 (_i)* 
7r " n = 0 
1 
(2 ft 4 - 1 ) 
2 sin (2n+ 1) ~ sin (2n + 1) ~ I 0 
7tR 
(2 “ +1 >t')/ Il ( (2,1+1) 
7T R \ 
or, if C is the total current (= 27 tR be), 
Va = 
4C Ip v 
7T 3 5R n={ 
. <" I> ” (2A1? d " (2n + i) t si " (2n + 1} 1 1 
< 2 » +1 )f)/ I '(( 2 “ + 1 >f 
In the apparatus used l = 3 • 75, a = 2, b = - 1, R = ’29 centim. 
Hence 
11 b C\® 4 ^ ft 4 0° 1 1 R 
Ti = 46 ’ vr- 
and the equation reduces to 
The above statement is therefore correct to 1 part in 500 
v a = to within 1 part in 500. 
n R“ 
t If R is the radius of the rod, and 2 a the distance apart of the knife edges at 0° C., at any temperature 
9 0 C. we have 
p = R E A/2 a [1 + (2a - /3) 6\ 
where a is the mean coefficient of expansion of the material of the rod, and /3 that of the copper frame 
holding it, between 0° C. and 9° C. 
The values of the mean coefficients of expansion of metals at low temperatures are only known in a few 
cases, but from these cases it does not seem likely that the value of the correcting factor 1 + (2a - /3) 9 
exceeds 1 by more than 1 part in 200, in any of the experiments. 
