440 DR. CHARLES H. LEES ON THE EFFECT OF TEMPERATURE ON THE 



The temperature of the apparatus then rose, and during the rise observations were 

 taken at frequent intervals with the heating current alternately through one and the 

 other heating coil. The current used in the bridge circuit was occasionally reversed 

 in order to see that no serious thermal electromotive forces were present in the circuit. 



Observations were thus obtained up to about the temperature the material had at 

 the commencement, and the agreement of the observations with those previously 

 taken was a test of the constancy of the conditions under which the observations 

 were made. 



Theory of the Apparatus. 



The complete theory of a finite straight-line source of heat within and parallel to 

 the axis of a finite cylinder of conducting material, surrounded by a concentric shell 

 of a second conducting material whose outer surfaces are exposed to a gas, requires 

 the evaluation of a number of definite integrals involving Bessel functions, and has 

 not been worked out. 



In the apparatus used, however, the concentric shell surrounding the cylinder is of 

 metal whose thermal conductivity is 200 or 300 times that of the cylinder. The 

 shell may, therefore, with a close degree of approximation, be taken as an isothermal 

 surface. 



Let a cylinder of conducting material, of thermal conductivity k, of length 2/, and 

 radius n, have within it a steady straight-line source of heat parallel to the axis of 

 the cylinder, and distant c from it, and let its external surfaces be maintained at a 

 constant temperature. To find the distribution of temperature throughout the 

 cylinder, given that of the source. 



If r is the temperature at any point of the cylinder distant ;c from the mid-cross- 

 section and r from the axis in an axial plane inclined at 6 to that of reference, V the 

 temperature of its surface, the following equations must be satisfied 



3V 3V 1 9t> 1 3V ,, ,. , , v 



-2+^-5+- + -3^ = throughout the cylinder. (1). 

 So- 2 cr 2 rcr r 2 38* 



r = V at r -a. . . . . ..... . (2), 



v = V at ./ = 1 . . . . ' ..... (3). 



If the source is symmetrical with respect to the mid-cross-section of the cylinder, 

 and its intensity is zero at x = +/, it may be expressed by a Fourier series whose 



general term is H B cos HIT . , where n is an odd integer. 



1 



A straight-line source of strength H B cos-^- in an infinite solid of thermal 



ZlV 



conductivity k produces at the point xp a temperature 



