Thermal Conductivity with Temperature. 517 



responding values of 6 and x, the constants A, B, C can be 

 determined, or at least such of them as are required for the 

 determination of the constants k and m. In order to deter- 

 mine these constants we must apparently use some method of 

 successive approximations ; and the precise method adopted 

 will probably be a matter of taste. I may, however, suggest 

 the following as certainly applicable to the case of tc and m 

 positive, i. e. to equation (36), and as inferentially applicable 

 to the other case (37) if we can get some imaginary quanti- 

 ties to cancel each other. We will therefore proceed with 

 the general case, and not trouble about whether the quantities 

 are real or imaginarv. 



The quantity ~ — ^- is always pretty large, even 



when has its maximum value ® ; call it y. Then writing 

 sinh -1 y— log (y + ^ y 2 + 1), we see that, since y is large, y 2 + 1 

 is practically the same as y 2 , and therefore that sinh - ' y£± log 2y 

 to all intents and purposes. This approximation is always 

 very close; and it is perfectly accurate when ?\s = l, ?'. e. when 

 4AC = B 2 . That this is not likelv to be far from the case is 

 illustrated in the Table (§ 34). 



The quantity p — ^y however, is not large at all, but 



has a value not very different from unity. We had better 

 therefore take its sinh -1 in the logarithmic form, and write 

 (36) with the limits put in 



-g-+B 



m. 



Vi ,0 « 



s+» 



~V '° g 2C® + B + -J 4AC + 4BC0 + 40»e» = V (IW) * 



. . . (38) 



2A 2C 



or, writing -ry =/ >' and -d=- 9 > 



r r~\ 



1 + 6 \ V l + s% + ^(rs + 2s ® + s 2 ® 2 )l 7n <S ~c _ 1 </(**™\ x 



r * L 1 + sB + V (rs + 2*6 + s 2 2 ) J ~ *" ' ' 



L l + gJ ... (39) 



(l+ g)(l + s0 + y/rs + 2s0+s 2 2 )~« iy/r '=Ke»*; [19] 



