Absolute Thermal Conductivity of Crystals fyc. 113 



IK 

 S X 



where ^W ...;.■ (4) 



hence 



d© 



— — = q (©' cosech qz — © coth g^) 



and 



— — — q (©' coth g2— © cosech bz). 



Substituting these values in equations (1) and (2) and elimi- 

 nating, we get from the first and second of each set 



dT f dT dT' dT d 



e^L + © / ^i = T^+T / ^-=^(TT / ); 

 dx dx ax ax ax v 



also from the first and third of each set, 



~ k (dT , dT ,, \ 



© = — \-t— cosech qz— j- com qz ) , 

 wo \ ax ax / 



„, k (dT ,, dT , \ 



©' = — I -7— com qz — cosech qz ) . 



Xq\dx * ax / 



Therefore, combining all these, 



V £2« / \dx ) 



£2 s inh^ = -^- ^11- (5) 



| (TT/) 



Hence ^ is determined in terms of q, which itself contains it 

 together with the radiation-coefficient h! (which must be sup- 

 posed known). We may write the last equation thus, by (4), 



(dWY_ (dT\ 2 

 sinhqz ^ sk \dx / \dx / .„. 



~^~~W T ^ +T ,dT ' ■ ■ ■ • W 

 dx dx 



which shows that this method is not satisfactory for determi- 

 ning q when the product qz is very small. 



4. Now there are three special cases, depending on the value 

 of A':— 



(1) When the crystal has its natural surface, and the value 

 of li r is determined by special experiment on its rate of cooling. 

 In this case the above equation remains as written, and may 

 be treated, as qz will generally be less than unity, by expand- 

 ing the left-hand member, 



/, q 2 z 2 gV \ 



and then solving for q by successive approximations. 

 Phil. May. S. 5. Vol. 5. No. 29. Feb. 1878. I 



