388 
PROFESSOR C. H. LEES: THERMAL AND ELECTRICAL 
Let q, q' be the areas of cross section of rod and sleeve respectively, p, p' their 
outer perimeters, and k, k' the conductivities of their materials. 
The thickness t" and conductivity k" of the thin layer of olive oil between rod 
and sleeve are small enough compared to those of the rod and sleeve to allow the 
heat conducted through the layer parallel to the axis of the rod to be neglected. 
The heat conducted across a length dx of the layer from rod to sleeve in 1 second is 
equal to pk" (v—v') dx/t 
We have then the following equations for the steady distribution of temperature 
in rod and sleeve :— 
In rod 
In sleeve 
7 d 2 v pk" , A 
(0 
+ ^ ( v ~ v ')-p' hv ' = 0- 
( 6 ) 
The last term of the second of these equations is small and may be neglected. # 
Taking the temperature at the central transverse section to be zero, we have as the 
solutions of equations (5) and (6) 
'(qk + q'k')v = H„(iC+ Hi l su pq 
' qk a cosii ab 
(0 
(qk+q'k') v' = H, 
X- 
l'k! 1 sinh ax 
qk a cosh ab 
■ • ( 8 ) 
where H 0 is the flow of heat through the central transverse section of rod and sleeve, 
and 
a" 
pk' 1 
t" 
J_ _L 
qk + q'k'; 
If V — the length of free rod equivalent in thermal resistance to the length b of 
rod enclosed by sleeve, we have 
V = b 
qk 
qk + q' k' 
q'k' tanh ab 
qk ab 
(9) 
To determine whether any further simplification is possible, we note that in the 
apparatus used the constants have the following values :—• 
b = (P33 centim., q = 0‘2G8 sq. centim., qk = (P011 to 0'27 according to the rod 
used, q' — 0 - 088 sq. centim., k' — 0'17 to 0'27 according to the temperature, 
q'k' = 0'0I5 to 0’024, k" = 0'0004, t" — 0'00I2 centim., and p — 1’84 centims. 
Hence ab lies between 3‘2 and 1*7. 
* The error introduced by neglecting li in estimating b', the length of free rod equivalent in thermal 
resistance to the length b of rod enclosed by sleeve, is about 1 per cent, in the worst case, i.e., that of the 
shortest rod and lowest conductor. The error produced in the determination of k is about * 1 per cent. 
