24 PROCEEDINGS OF THE AMERICAN ACADEMY. 



to 100° C. The effect of radiation or conduction from the edge is 

 therefore of no consequence. 



Most experimenters have been able to reproduce mathematically the 

 results of their work on thermal conductivities by assuming that in every 

 case the conductivity, /c, is a linear function of 6, say k (1 + 2 hO), where 

 b is small (usually less than .OOo), so that F {B) — C + k 6 {} + h6). 

 On this assumption the temperatures within an infinite disk would be 

 given by the equation, 



k' ^ (1 + ^» e) = ^ {0, (1 -\-he,)-eo{\ + b 6,)) + K e,{\ + b o,), (37) 



or b (0- - 0,') + {B- Bo) = ^- {Oi -Bo+b {Bi" - 6„-)}. 



Except in instances where near certain temperatures some great chem- 

 ical or physical changes take place in the materials concerned, experiment 

 appears to show that k always changes slowly with the temperature, and, 

 whether or not we know the exact nature of the connection between the 

 two, it is easy to get a superior limit for the effect on the final tempera- 

 tures at points on the axis of such a disk as has just been described, of 

 changes in the edge temperatures. Neither in our own experience nor 

 in any published reports that have come to our notice have we found any 

 substance in which the change of k with 6 is so rapid that in a disk, 

 where a > 5 I, made of it. with its faces kept at 0° C. and 100° C. 

 respectively, the final temperatures of points on the axis could be affected 

 by nearly so much as 0°.01 C. by changing the edge temperature fiom 

 0° C. to 100° C. We are here concerned merely with the magnitude of 

 a possible error, and in every case to which we need to apply our theory 

 we shall be well within bounds if we assume that the error is not greater 

 than twice the error which would be found if 6 and f(B) were identical, 

 as Fourier assumed them to be. We have, therefore, tabulated for a 

 numerical example the final temperatures computed on Fourier's hypoth- 

 esis at several points on the axis of a disk of radius a and length /, when 

 one face (z =^ 0) is kept at the uniform temperature 0° C. and the other 

 face (z = I) at the uniform temperature 100° C. on two or three different 

 assumptions with respect to the edge temperatures. If the face temper- 

 atures are B^ and Bf, and if the temperature has the same value, B, at all 

 points of the edge, the final axial temperatures are given by the equation 



e = B(l -2S^-2S,_;) -\-2BoS,_,+ 2B,S„ 



