PEIRCE AND WILLSON. — THERMAL CONDUCTIVITIES. 29 



and Ai = 



4jun 



(I +/i)-sinliao— (1- ju)'"«>"'' ("o "'-^S) + (1 — ;U-)[sinh (ag -2a — 25) — sinli (ao — 2a)' 



This expression, though still sufficiently complicated, shows that for 

 properly choseu cases, as good for our present purpose as any others, the 

 computation is comparatively simple. 



If, for instance, the thickness of the lower slab is half that of the disk 

 formed of the two slabs and the intermediate sheet of metal, I' =z ^ I and 

 n(p = 2 u, so that 



A ^ ^/^" ns^ 



' (l + yit)2sinh«o-(l-/x)2sinh(«o-2S)-(l-/i.'^)sinh2 8 '^ ^ 



O 



. , ,, (1 — u)- . , ,_ (u— l)sinh'-S,, , ,, , . '.^ ^ 



sinhaf, 1 — — „ - sinh-5H — ; ^- 1 — cosh ao + i" (1 + cosh a,))] 



Z/u 4jusinli a() 



If we denote the denominator of this expression by sinh «o • (1 + ^)i ^^^ 



n 



note that, if we make u e(\ua\ to unity, we shall have Ai = -^-, — , cor- 



smh ixq 



responding to the case of a homogeneous cylinder already treated in 

 Problem 4, we shall see that T'l in the case of the heterogeneous cylin- 

 der can be found by multi{)lying each term of the series for 7\ by the 



quantity — ^— , and that in our problems the resulting series is usu- 

 ally more convergent than the original. 



In order to exaggerate the magnitude of the disturbing effect of the 

 tinfoil, we have chosen for computation a value of /x much smaller and 

 a value of 8 much greater than the proper values of these quantities for 

 most of our experiments, assuming that a = 5^= 10Z' = 500 (I" — /'), 

 and that /x = 0.002, so that A is nearly equal to 





^ Xj, tanh I «o - j^ ctnh ^ a, - ^^ , where «„ = -^ 



These values correspond in certain cases to large disturbances of temijcr- 

 ature on the axis of the slabs, as the results show. Consider, for instance, 

 the point z = /', r = 0, in a com|iound slab 2 cm. thick and 20 cm. in 

 diameter, built up of a slab of |)oorly conducting material 1 cm. thick, a 

 sheet of metal 0.2 mm. thi<'k, and a second slal) of the same material as 

 the first. Let the lower face be kept at the temperature 0° C, the other 

 face at the temperature 100'^ C, and let every point of the edge be kept 



