356 PROCEEDINGS OF THE AMERICAN ACADEMY. 



to make the loss of heat through them as small as possible. Under 

 these circumstances there was a very rough approximation to a uniform 

 temperature gradient from the warm face to the cold one, at each edge, 

 but it was difficult to measure the edge temperatures accurately and the 

 areas of the faces were therefore made so large that the temperatures of 

 points on the axis of the slab (that is, the line which joins the centres 

 of the faces) would surely be the same within one one hundredth of a 

 degree of the centigrade scale, in the final state, whether the whole of 

 each edge was kept at the temperature of the warmer face or at the 

 temperature of the colder face. 



In anticipation of some further work of the same kind, I have been 

 led to compute the final axial temperatures in a square slab (a X a X c) 

 of thickness c, when one face is kept at temperature T while the other 

 face and all the edges are kept at the lower temperature 7\. The work 

 is straightforward enough, but the computation when the slab is rela- 

 tively broad is very laborious, and in view of the practical importance 

 of the wall method in determinations of the conductivities of poor con- 

 ductors of heat, it seems well to record some of the results. 



The problem just stated is solved (7\ — WTi + WT ) when one 

 has found x a solution ( W) of the equation 



dx 2 + df + dz* U W 



which is equal to unity when z = 0, and to zero when z = c for all 

 positive values of x and y not greater than a; and which vanishes 

 when x = 0, or y = 0, or x = a, or y = a, for all positive values of z 

 not greater than c. 

 A convenient normal solution of (1) is 



Jfcn-Z fcn-(2c— 2) 



A (e a — e a ) sin sin —*-, (2) 



where F = m 2 + w 2 , and it is evident that 



W (x, y, z) = 



m= ccn— oo 



2 2( 



m=\ n=l \ 



sinh — * • sin sin — ^ ) (3) 



. . irkc a a a I 



Tr'mn sinh — / 



a 

 where m and n are odd integers. 



x Byerly, Fourier's series, etc., p. 127. 



