234 Mr. J. W. Peck on the Steady 



approximation it is necessary to have a comparison of quan- 

 tities o£ similar dimensions. This is done here by expressing 

 the k, e relationship in the form of a length, and for a series 

 of rods we have thus a method of proportioning the dimen- 

 sions so as to get results of the same order of exactitude in 

 all the cases. 



Since the Fourier solution begins to be inexact when we 

 get down the list say as far as bismuth (on account of the 

 impossibility of thinning down the bars to the proper size) a 

 secoud approximation becomes desirable. It may be arrived 

 at as follows : taking one additional term in the values of 



J (X^) and — ~r , we have 



or 



J„(\ r ) = l-^, ....... (18) 



*¥*- - X M« <»' 



The surface equation (8) therefore becomes 



ka*\ i —ka(U + ea)\* + 16e = 0, 

 or 



_ 9 2(2k + ea)±2V4k 2 +f 2 a 2 



k 

 [: [ Replacing - by L we have, after some reductions, the two 



values for X 2 . 



16L 2 + 4aL-f a 2 4aL-a 2 



(20) 



2a 2 ]J ' 2a 2 L 2 



Denoting these by X/, X 2 2 ; we have as solution 



i?=Aexp (-X^) (l-^j+B exp (-\ 2 x) (l-^p)(21) 



To determine the constants A and B we have 



: v=a (i-^) + b(i-^) 



and, therefore (since this must hold for all values of r when 

 x is zero) 



A + B = V, AV + BX 2 2 = 0. 



