268 Mr. J. McCowan on the Heating of 



which is the initial value of U. Hence, finding the general 

 value of U, we have finally for $, 



^ = l; C (Y 2 -V 2 )-U 



i-Wi #\ i- Trite* (~) n - i9m+ 2F* (2n + l)irs 



This series is very convenient, owing to its rapid conver- 

 gence, so rapid that for many purposes it might be sufficient 

 to take 



which is correct initially and finally, and always fairly approxi- 

 mate. 



It is interesting to examine the effect of surface-emissivity 

 on this solution. From Section 2 it is clear that unless the 

 bar is very thick, the temperature may be taken as approxi- 

 mately constant throughout any cross section ; making, then, 

 this assumption, the thermal equation will be 



dt dz 2 ' I 



where h is the ratio of the perimeter to the area of the cross 

 section, and e the emissivity. Hence for the steady state we 

 have 



5_ J c ^o 2 f i cosn z n/ (he Ik) \ 

 Phe \ cosh l^/(he/k) J ' 



Proceeding almost exactly as in the previous case, we find, 

 first, 



coshz\/(he/k) _±he (—) n cos(2n + l)7rz/2l 

 coshlS{helk)~vk*2n+l *(2n + l)V 2 /4Z 2 + /^/F 

 then generalize for U, and obtain finally 



^ = jcV^( 1 coshz^/{he/k)\ 

 Phe \ coshls/hejk 5 



JcY 2 4 ( — ) n COS (2n + l)7Tz/'2l -{(2n + ))*iiS*/4*2+A«}</* 



Ph 7r 2 2n + l , (2n + l) 2 7r 2 /4/ 2 + /^ € 



We see that the effect of the emissivity is negligible when 

 £Phe]7r 2 k is negligible compared with unity. We may take, 

 adopting the usual C.G.S. units, ^=1/4000 for polished sur- 

 faces, &===1 for copper, and 1/6 for iron : hence, for a bar of 

 circular section, radius a and length L cms., 



4Phe/7T 2 k = '000,05 L 2 /u for copper, and «0003L 2 /a for iron. 



