28 PROCKEDINGS OF THE AMERICAN ACADEMY. 



the temperature at no point within the cylinder of radius I co-axial with 

 the disk differs by more than 0°.02 from the temperature at the corre- 

 sponding point in an infinite disk of the same tliickness and same face 

 temperatures. In practice there is always a gradual fall in edge temper- 

 atures as z increases from to /, and in such a case we may consider a 

 guard ring of width 4/ amply large- enough to make the final temperatures 

 within a right cylinder of radius / and thickness I sensibly ecjual to those 

 within an infinite slab of the same thickness and same face temperatures. 



In our experimental work we have sometimes found it desirable to 

 introduce between two slabs of low conductivity a thin sheet of tinfoil of 

 comparatively very high conductivity. It is evident that under con- 

 ceivable conditions such a layer of metal might seriously affect the final 

 temperatures in the slabs near their common axis. To investigate the 

 disturbances that miglit arise from this cause, we may apply the solution 

 of Problem 5 given above to the extreme case where the uniform edge 

 temperature is equal to one of the face temperatures, and where ki — k^. 



If we attempt to compute numerical values of the series 



S-.-^y)- frO 



p = 



by using the expression for A^ given in equation (32), we shall find the 

 amount of labor involved enormous ; we will therefore change the form 

 of the expression so as to make the nature of its dependence upon the 

 dimensions of the cylinders and upon their conductivities more evident, 

 keeping in mind the fact that the ratio of ^o to ki is very large. If we 

 denote the denominator of the second member of (i32) by D, and write 



«=sinh~^s', « -f 8 = sinh~ ^ s", and Mo^sinh^^s, 



we have 



Z> = S {(/. 5" 2 _ c" ') (fX C' "-s"-)-{l- /x)- C' S' C" S"} 



+ c{\~fj.) {s" c" (/. c' 2 - ,' ■') - s' c' Ox c" ■' - /' ')} 



= 1 s{(l _/x)2cosh 28 + (/x2 - 1 ) [cosh 2 (« + 8) - cosh 2 «] - (1 + /x)^} 

 — ^e(l— ;ut)-sinh28 + |c(l — /x2)[sinh 2(«-f 5)-sinli 2m] 



= i {(1 - fxY smh («„ — 2 8) - (1 + fiY s\nh «„ 



— (1 — /x-)[sinh («o — 2« — 2 8) — sinh («o — 2(t)']}, 



