142 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1951 



where r is the co-ordinate of a surface of temperature T. The general solu- 

 tion to equation (1) is given by the equation: 



T = A^B\ogr-^, (2) 



where A and B are constants determined by the boundary conditions. 

 The temperature has a maximum value T ' at some radius r', at which the 

 temperature gradient dT/dr = 0. Substituting the expression for T given 

 by equation (2) in this condition, it is found that: 



r-=^. (3) 



If the expression for B given by equation (3) is substituted in equation (2), 

 and if A is taken as given by the resulting expression for T^ when r = r' , 

 equation (2) may be written in the form: 



. = r' + ^i^(i + 2iog-^-(,:J). (4) 



This equation gives the general expression for the temperature distribution 

 in terms of the radius r' at which the temperature has its maximum value 

 T' . In the special case in which the temperature Ti at the inner radius ri 

 is the same as that at the outer radius ^2 , substitution in equation (4) of 

 r = ri and r = r2 gives two expressions for Ti . From these there can be 

 obtained the same expression for the radius / of maximum temperature as 

 is given in Reference (1) for this special case. In the notation used here this 

 expression is: 



2 2 



r" = 'A:^^. (5) 



Substitution of this expression for / in equation (4) gives an expression for 

 T — Ti which is identical with that given by equation (18) of Reference (1). 

 Using the expression for T given by equation (4), integration of Zir 

 rTdr over the interval ri to ^2 , and division of this integral by it {r\ — rj), 

 the coil volume per unit length, gives the following expresson for the mean 

 coil temperature T : 



^,.A|iogp^-nogp ^.^A 



Experimental 



By means of equation (4), coil temperature measurements may be ana- 

 lyzed to determine the thermal conductivity and the rates of heat removal 



