Temperatures of a Thin Rod. 221) 



We may arrive at these results by employing the well- 

 known solution in Bessel functions, of the problem of the 

 steady temperatures of an infinitely long cylinder whose base 

 is kept at a fixed temperature, and whose surface radiates to 

 a medium at a lower fixed temperature. This solution was 

 given in essence by Fourier *, although the Bessel functions 

 were not then so named. If x denote the distance along the 

 axis of the cylinder measured from the hot end, r the radial 

 coordinate, then v the temperature at the point m 9 r, must 

 satisfy 



i 3 / at-\ . a 2 



r ~dr 



('g) + i^° < 7 > 



at all points in the interior, and 



k^ +ev^0 ........ (8) 



dr 



at all points of the surface where r=a. 

 The solution satisfying (7) and (8) is 



i7=2A,exp(-X^)J (V)> • • • (9) 

 where the constants A 8 must satisfy the equation 



V=SA S J (V). (10) 



V being the constant temperature of the hot end, and X s a 

 root of the transcendental equation 



Xow 



2 2 2 2 4 : 



k°^±\ + eJ (\a) = 0. . . (11) 



O r , r=a 



J (Xr)=l — 52- + 92I2— ...; . . (12) 



and hence to approximate, suppose \r so small that second 

 and higher powers may be neglected. The justification of 

 this hypothesis will be shown later. We have therefore 



J«(xr)=i ; H5^|_ — T" • • < 13 > 



The surface equation therefore becomes 



* Theorie Analytique, chapter vi. 



