CONDUCTIVITIES OF METALS AND ALLOYS AT LOW TEMPERATURES. 387 
where S A , &c., are written for the functions sinh (ph/qJc) 112 x A /(ph/qk) 1/2 x x , &c., whose 
values for small values of the denominators do not differ much from 1. 
From this equation k may be calculated if an approximate value of it is known and 
is substituted in the hyperbolic functions which involve it. In the experiments to be 
described these functions have values not differing greatly from unity. A sufficiently 
close approximation to the value of k for this substitution may, in fact, be found by 
puttin 
sinh ( ph/qk ) 1/2 x 
'ph\ 
qkl 
1/2 
X 
=cosh (&y x =i. 
The effect of temperature on the dimensions of the rods is neglected, as it does 
not appear likely to influence the value of the conductivity found by so much as 
1 part in 300. 
In the actual apparatus the heat was supplied to the rod by the passage of the 
electric current through a platinoid wire wound on an independent sleeve placed on 
the bar, and the temperatures were measured by the resistances of platinum wires 
wound on similar sleeves. # It is therefore necessary to determine the effect of the 
sleeves on the result previously given. 
Effect of the Sleeves. 
Let v be the excess of temperature of the rod and v' that of the sleeve at points in 
a plane perpendicular to the axis of the rod, distant x from the central transverse 
section of the sleeve, the isothermal surfaces being assumed, as in the previous 
calculation, to be planes perpendicular to the axis.f 
* See note, p. 382. 
f The distribution of temperature v throughout a rod of length 21, radius R, and conductivity k, which 
receives heat at a uniform rate h per square centimetre per second through a strip of breadth b of its 
curved surface at one end, and loses it at a corresponding strip at the other end and at no other point, is 
given by the equation 
v = 
8hl 
771 = 0 
(-If 
1 
(2m + iy 
• / , Tib . . irx -r /-7rr\ 
sin (2m+ l)^j sin (2m+ 1)^ I 0 ( 2m+ 1 
I 
Ii 
-- 7tR\ 
2m+1 2r)’ 
where x is the distance of a point from the central transverse section of the rod, r its distance from the 
axis, and I 0 and L Bessel functions for unreal arguments. 
The mean temperature v over a strip of the surface of the rod of breadth b, whose centre is a from the 
central transverse section, is given by 
v = 
16H/ 2 . 
h r 4 6 2 R' 
: (~ (2m+ip sin ( 2m+ ir sin ( 2m +v >ti siu ( 2m+x ) 
ira 
Yi 
Io 2m +1 
f)/ 
Ii 
-7tR 
2m+l YT 
where H = h2-n-bli is the total flow of heat. 
If the heat were generated at a uniform rate throughout the material of the rod within the distance b 
of the end, the flow of heat would be linear and the temperature at a would be given by 
16ffi 2 
hr W 
: ( ~ l)m (2m+iy sin (2w + l) Yi sin (2m + l) YV 
A comparison of the numerical values of the first few terms of the two series, in the case of the shortest 
rod used, shows that even in that case the error committed in assuming the isothermal surfaces in the rod 
to be plane is not so large as 1 in 500, 
