116 On the Absolute Thermal Conductivity of Crystals. 

 of temperature down it, by § 1, 



t = A cosh jpa — B sinh^ ; 

 that is, 



t=T coshpx— (ToCoth^Z-TiCosech^sinh^ ; . (11) 



similarly down the second rod, reckoning x from b, the curve is 

 t' = A! cosh px — B 7 sinh^, -^ 



where 

 and 



,,_ T 2 smhpZ — T 3 sinhp?! 

 sinh pi 



p,_ T 2 coshffZ — T 3 cosh |?^ 

 sinh.pl 



(12) 



Thus 



We can now at once express the values of the " known quanti- 

 ties" which occur in equations (1) and (2), and in the right- 

 hand members of equations (5) and (6), viz. 



T, T, §i, and ~ 

 ax ax 



T _ T t sinhjpZ — T sinhpZx 

 sinh pi 

 - ( dt\ _ Ticoshp? — T cosh^>Zi 



\dx/ x-i ^ sinh pi 



sinh pi 

 <W_(dl\ ___ B >_ T 3 cosh ph-% cosh pl 

 dx ~~ \dx/x=o~ " ~~^ sinhpZ 



The symmetry of these expressions is visible in the following, 

 where for shortness sh is written instead of sinhpZ , ch x for 

 coshpl^ and so on : — 



dT 



dx 



(13) 



Tr^:l= 



pax 



sh sh x 1 



ch chj 





sh sh x 



T„ % \ • 



T T, 



* 



ch chi 



(14) 



8. We can now write down the value of the right-hand 

 member of equation (5) thus, 



\dx) \dx ) = ^^(T,ch 1 -T a ch ) 8 -.(T 1 ch -T oh 1 )'f . (1S) 



T^+oy— " p(T/r 8 -T T 2 )sinh P Z A 



dx dx 



