386 
PROFESSOR C. H. LEES: THERMAL AND ELECTRICAL 
l&Tx 
\qk) 
1 \qkl 
1/2 
x c , 
( 1 ) 
Theory of the Apparatus. 
If a thin rod of uniform cross section is entirely surrounded by a vessel kept at 
constant temperature, and has heat supplied to it near one end, while the other is in 
good thermal contact with the wall of the vessel, the distribution of temperature 
within the rod, from its point of contact with the vessel to that at which the heat is 
supplied, will he represented by the equation 
Q 
V ~ (phqk) 112 
where v is the excess of temperature of the rod at the cross section situated x centims. 
from the end of the rod in contact with the vessel, over that of the vessel, p is the 
perimeter, and q the area of cross section of the rod, h the heat lost per second from 
1 sq. centim. of the surface of the rod when its temperature exceeds that of the 
enclosing vessel by 1° C., k is the thermal conductivity of the rod, and Q is the 
amount of heat which crosses the section of the rod at any point x c per second. 
In calculating Q from the energy supplied and the heat lost from the surface of the 
bar beyond x c , we may, if the point x c is near the free end of the rod, and the heat is 
supplied to the rod at a uniform rate between x G and the free end, take the mean 
temperature of the surface of the bar between x c and the end to be identical with the 
temperature v G , which would be observed at x G ,, a point which divides the distance 
between x G and the end, in the ratio 1 : 2 if the above equation for v held throughout 
the heated portion of the rod. # If s is the area of the surface of the bar beyond x G , 
and H is the total heat supplied to the bar, 
Q = H— shv G i . 
Hence 
fphf 2 
v = H sinh m 12 
qk) 
x 
(phqk) 1 ' 2 cosh ' 
x G + hs sinh 
qk) 
Xr 
( 2 ) 
(3) 
If the temperature excesses v A and v B at two sections x A and x B are observed, then 
H 
Vr 
■v> 
= (phqk ) 1/2 cosh 
'ph\ 
qk) 
1/2 
x G + hs sinh 
'ph\ 
qk) 
1/2 
Xr 
sinh 
'P^Yx— ftinli (Y\ 
qk 
J / 
sinh 
, 1/2 
qk) 
X. 
Hence 
H 
Vy 
■V 
A L 
X 
" s:nh( $r 
Xx 
1 ph 
qk. 
1/2 
Xx 
—x A sinh (^-jj 
1/2 
X, 
\ 1/2 
X, 
and 
f'nhY 2 
= qk cosh i'~j) x c + hsx c 
'ph\ 
qk) 
sinh (& 
qk 
1/2 
X c 
\qk) 
1/2 
X c 
k = 
H 
y B -v A 
— ]isx G & G i 
q cosh (j—)) x G , 
\qk) 
(4) 
* This ratio is exact if the loss of heat from one square centimetre of the surface of the bar may be 
neglected in comparison with the heat generated per square centimetre. 
