Temperatures of a Thin Bod. 227 



An inconsistency seems to arise when this result is regarded 

 from the point o£ view of the isothermal surfaces and the 

 tubes of flow. For the above solution gives 



*=A (2) 



as the family of isothermal surfaces (where A is a parameter), 

 i. e. a series of planes transverse to the axis of the rod. 

 Consequently the tubes of flow are rectilinear filaments 

 parallel to that axis. This, however, contradicts the initial 

 hypothesis of emission of heat at the lateral surface ; and it 

 therefore becomes of interest to trace the cause of this 

 discrepancy. The difficulty is not avoided by supposing that 

 the lateral radiation is negligible ; for it is this radiation 

 alone that is responsible for the drop of temperature along 

 the rod. If the lateral radiation were entirely negligible the 

 equation 



tl=fv, (3) 



Car ks v 7 



on which the above solution is founded would fail altogether, 

 and would in fact reduce to 



SH> w 



giving a solution of the type 



*; = A# + B, (5) 



thus giving isothermals the same as in (2). 



If, therefore, we are to retain the drop in geometrical pro- 

 gression as opposed to that in arithmetical progression, a further 

 consideration is necessary. 



The physical explanation is that the tubes are not rigidly 

 parallel and rectilinear ; they bend slightly away from the 

 axis and increase in cross-section as we proceed along the rod ; 

 so that all the heat which is delivered at the hot end where 

 they start parallel to the axis is ultimately sent out at the 

 lateral surface. The amount of this bending is very small for 

 good conductors, but is considerable for the badly conducting 

 metals and for the non-metals. This will be shown below by 

 deriving the Fourier result as a first approximation from the 

 rigorous solution in Bessel functions ; and in the same way a 

 second approximation will be arrived at which will give as 

 the isothermals co-axial paraboloids of revolution, and as the 

 lines of flow logarithmic curves (the rod having a circular 

 cross-section). With this result the difficulty mentioned above 

 is avoided, and the position of the Fourier solution in the 



Q2 



