Thermal Conductivity with Temperature. 513 



This may be written 



whence, by help of [4] &c. ; 



dx** m + e\dx) fc'm + ' ' L J 



32. To integrate this once, make the substitution I -j-j = £, 



when [11] becomes 



cfo 2z __ a 'n + 6 a 



which is of the form 



whose integral is 



Hence the first integral of [11] is 



the limits being taken to suit a rod whose length is unlimited 



(that is, one in which 6 and — vanish together) — a condition 



which is necessary for simplicity, as is explained in § 17. 



We may now insert the value of 6 from (5) and write the 

 above thus, 



= S{M + (mn-M)0+0*}-M0, . [12] 



where for shortness the letter M is written instead of the con- 

 stant 



m + n 2 



inn-— 



log a (log a)' 



33. So far we have proceeded with perfect accuracy ; but 

 in order to integrate the equation any further it is necessary 

 to expand a e — 1 and neglect higher powers of 6 log a, as was 

 done and justified for all probable values of 6 in § 12. The 



