384 Mr. W. Sutherland on Thermal 



from one end of the frustrum to the other I -j- dx, we have 

 from (15) J ax 



*» p dv dJl 



Rl va\vc(AB? + 2BR+l)' 



where 



I6170 *> W 



f 



J : 



which, if we regard dv/dx, p, and v as having constant average 

 values throughout the short length L, may be written 



p dv 1 1_ 1 R 2 A + B-(B 2 -A)* R^ + B+ffl'-A)* 



v' dx' c' 2(B*-A) 0g R 2 A + B + (B 2 --A)* , R 1 A + B--(B 2 -A)* 



or 



p dv 1 1 w f 1 , 2(B 2 -A)I(R 2 -Ri) 



v'd.v'c'2(B 2 -A)i ° g \ i " f "AR a R 1 + B(R a +R 1 ) + (B 3 -A)*(R 1 -"R. 



The form of this expression suggests that we should expand 

 the log by the approximate relation log (I + z)—z, which 

 yields 



& 



(Bj-RO/c 



v dx ARsRi + BfRs + RO + (B 2 -A)*(R 1 -R 8 ) + 1 ' 



For a frustrum pointing in the opposite direction we should 

 have to interchange R 2 and R x and change the sign of c, 

 which would give us our last expression with only the sign 

 of R l — R 2 changed in the denominator; thus for a pair of 

 frustra oppositely directed, we get 



2 pdp R 2 -Ri x 



v dv c 



AR 2 R 1 + B(R 2 + R 1 ) + 1-(B 2 -A)(R 1 -R 2 ) 2 /{AR 2 R 1 + B(R 2 + R 1 )+1} 



or confining our attention to cases where (R x — R 2 ) 2 may be 

 neglected, and remembering that (R 2 — R 1 ) = Lc and that 

 2hdv/dx is equal to the difference of velocities at the two 

 ends of the double frustrum, we see that an approximate 

 integral solution of the differential equation of thermal trans- 

 piration through a series of oppositely directed frustra in 

 which R 2 and R x are not very different (R 2 not to exceed 



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