CONDUCTIVITIES OF METALS AND ALLOYS AT LOW TEMPERATURES. 389 
Thus tanh a b will vary between the limits 0 - 997 and 0'936, having the former 
value in the case of the worst conducting rods at low temperatures, and the latter in 
the case of good conducting rods at high temperatures. The error in the estimate of 
b', if we write tanh ah = 1, will even in the latter case only amount to 3 per cent., 
and the error in k due to it will not exceed 0’04 per cent. 
We have, then, the simplified equation 
v= i k —b+- i T l 
qk+q'W qk+q'k'a 
or 
V = _ qk - b+ ' l ' i ' r 
7 J ~u 
qk + q'hf qk + q'k'L t" \qk q'k'J J 
'pK (1+A. 
- 1/2 
qk 
( \"ff qk \ m 
qk + q'k ,b+ \qk + q'k') \pk" 
■ • ( 10 ) 
The first term on the right is obviously the correction due to the increase of cross 
section at the sleeve, and the second that due to the ends of the sleeve, at which the 
flow does not immediately take advantage of the increase of section. 
We have seen that in the above equation k varies from "04 to l’O according to the 
rod under test, while k' varies from 0T7 at 100° to 0'27 at 300° absolute. Hence qk 
varies from 0‘011 to 0'27 according to the rod, and q'k' from 0‘015 at 100° to 0’024 at 
300° absolute. 
The variation of k' with temperature has, however, only a very small effect on the 
value of b', as will be seen from the following table of values for k' = 0T7 and 0'27. 
Values of b' in centimetres. 
k 
T—1 
II 
o 
pH 
For k = 27. 
For mean li 
•03 
•176 
•161 
•168 
•05 
•213 
•195 
•204 
•07 
•235 
•219 
•227 
•09 
•250 
•235 
•242 
•11 
•261 
•249 
•255 
•14 
•274 
•262 
•268 
k 
For k =17. 
For k = 27. 
For mean k. 
•17 
•282 
•272 
•277 
• 22 
•291 
•283 
•287 
•27 
•297 
•290 
•294 
•50 
•312 
•308 
•310 
1-00 
•321 
•318 
•320 
If for simplicity of calculation we take the mean value of V as holding throughout 
the whole range of temperature for any single bar, we make an error in the calculation 
of k which will not exceed 0'4 per cent, even in the case of the worst conductor tested, 
Lipowitz’s alloy, and will be insignificant for most of the other materials. 
Thus we have the following values for x A , sc B and x Q of the formula (4) for k :— 
