PEIRCE AND WILLSON. THERMAL CONDUCTIVITIES. 23 



when z ~ I, whatever r is ; and (3) has the uniform value F (0) when 

 r =^ a, whatever z is. The value of this function F (6) is evidently 



F(e) . (1 - 2 T,_^- 2 T,) + 2F(eo) . r,_,+ 2 F(e,) . T;, (33) 



or, for points on the axis, 



F{e) {1 - 2 S,_^- 2 5J + 2 F(eo) . St_,+ 2F(e,) . S^; (34) 



that is, 



F{e)-F{e,) = 2 s,[F(e,) - f(0o)-] + [F(e)-F(Oo):\ (i - 2 5,_,- 2 *s;). 



(35) 



z I 



In the case of an infinite lamina, where - = , - = , 



a a 



F(6) = F (6,) + ~ (F (6,) - F {6,)). (36) 



The difference between the values, at any point, of F {0) in the case of 

 the infinite lamina and in the case a = 5 I, is 



[F{ei) - i^(6o)] [2 '^. - 7] + [^ W - ^ W] [1 - 2 S,_,- 2 S,-] . 



It is easy to prove that for given values of? and a, 1 — 2 Si_^ — 2 S^ 

 has its greatest value when z = ^ /, and if a -^ / is as great as 5, it is clear 



from Table V. that neither 1 — 2 Si_^— 2 S^ nor [2 aS^ — j) can for any 



point of the axis be nearly so great as 0.00001, so that whatever is, the 

 value of F (0) is surely equal, within less than one ten-thousandth part 

 of the greater of the quantities F (0,) — i^((9o), F(d) — F {6^), to the 

 value which it would have at the same point on the axis if the disk were 

 infinite. By exactly what amount the temperatures themselves would 

 differ in the two cases cannot be stated unless we know something of the 

 nature of the function F. 



For certain substances, experiment seems to show that within wide 

 limits F {d) can be expressed as a linear function of 6, as Fourier as- 

 sumed. In the case of any one of these substances we may say, for 

 example, that the final temperature at a point on the axis of a disk the 

 radius of which is at least five times its thickness, if one face is kept at 

 100° C. and the other at 0° C, cannot be changed by nearly so much as 

 0°.01 C. by altering the temperature of the edge of the disk from 0° C. 



