18 Prof. B. Osgood Peirce on the 



a: 



We inferred from these results that if the faces of a homo- 

 geneous disk, the radius of which is at least five times as 

 great as its thickness, he kept at temperatures of 100° C. and 

 0° C. respectively, the temperature in the final state, at ever) 7 

 point within a distance from the axis equal to the thickness 

 of the disk, would be the same within about one hundredth of 

 a degree, whether the curved surface were kept at 0° or at 

 100°. Since in practice there is a gradual fall of temperature 

 at the curved surface of such a disk from 100° to 0° in 

 passing from the hot to the cold side, the approximation is 

 really much closer than o, 01 ; and it is evident that, for the 

 purposes of the present problem, a disk of such relative dimen- 

 sions, if we consider only the points near the axis, has prac- 

 tically an infinite radius and the isothermal surfaces in this 

 portion of the disk are sensibly plane. 



(2) The radius of the base of a right cylinder of revolution 

 of height I is a. The centre of the lower base is used as the 

 origin of a system of columnar coordinates (r, 6, z), the axis 

 of the cylinder being the axis of z. The temperature, V, con- 

 tinuous everywhere within the cylinder, has the value zero on 

 the curved surface and on the lower base, and the constant 

 value V/ on the upper base. The planes z~ l\ z = l" divide the 

 cylinder into three portions (1), (2), and (3), in which V is 

 represented analytically by three harmonic functions, Y h V 2 , 

 V 3 , respectively. The thermal conductivities of the three 

 portions of the cylinder are k lt k 2 , k 3 respectively. If, when 



and when z = I", k 2 ^— = k 





then 



3* 3 *z' 



Y 1 =|A 1 .J (f).sinh(^), 

 Y^T J «(f)[A 2 sinb(f) + B s co S h(^)], 



V 3 ^ J „(f)[A 3S i n h( ? ) + B 3 cosh( ? )] > 



where A 1? A 2 , A 3 , B 2 , and B 3 are subject to the conditions 

 A, sinhff ) = A s sinh (£) + B 2 cosh ffi, 



