on certain Equations in the Analytical Theory of Heat. 439 



reduced to about 1*16. The varnish was, however, slightly 

 softened at the highest temperature, so that the character of 

 the surface would change somewhat during the cooling. 



The much larger square nickel-plated bars used by Mitchell 

 in his repetition of Forbes's experiments on conductivity give 

 for the cooling experiments = 1*26 (seep. 441). This increase 

 of n may be due to change of form of section, or to change 

 in dimensions, as both these circumstances affect the stream- 

 lines* produced in the air by the presence of the heated bars. 

 It seems probable, however, that the only part of the loss of 

 heat which is altered by alteration of the nature of the surface, 

 is that part due to radiation. 



From these facts we conclude that the loss of heat from an 

 element of surface of heated bar, due to the effects of radia- 

 tion, conduction, and convection into the surrounding air, is 

 proportional to the nth power of the excess of temperature 

 of that element above that of the surrounding air. 



The fundamental equation for the state of heat along such 

 a bar becomes then : — 



cp -dt aA w g c } 



It is evident from this equation what a great effect the outer 

 conductivity has on the nature of the solution of the problem 

 of motion of heat in a bar. The solution in terms of expo- 

 nentials for the steady state used by Despretzf , Wiedemann 

 and Franz |, and others is replaced by a power of the tempe- 

 rature, and the solution for the " steady periodic " state first 

 given by Angstrom § no longer holds. The above solutions 

 neglect also variations of k, and we proceed to consider the 

 effect of this. 



Taking the case in which the temperature state is steady, 

 we have the equation 



|_(^|£)_£ /tB » =0 ; .... (4) 

 Ox\ ox J q 



or taking k v as a linear function of the temperature, thus, 



k v = k + k'v, 



we have, on expanding the equation (4), 



(fc+ttog+^'-f^-a • • (5) 



* See for stream-lines, Lodge, Phil. Mag. xvii. p. 214 (1884); 

 Rayleigh, Proc. 11. S. Dec. 1882. 



t Ann. de Chim. et de Phys. xix. et xxxvi. 



\ Pogg. Ann. lxxxix. 



§ Ibid, cxiv., cxviii., cxxiii. 



