56* CONDUCTION OF HEAT 463 



56*. Conditions for the Stationary State 



Since the two magnitudes N and k are simultaneously variable 

 together, either can be considered as a function of the other. For 

 the complete solution of this problem the determination of their 

 mutual relation is necessary. This can, as Clausius 1 has 

 taught, be obtained from the condition that the transfer of heat is 

 not bound up with a simultaneous transfer of mass. 



If we make the further assumption that the state of flow of 

 heat has become stationary, the following three propositions hold, 

 according to Clausius : 



(1) The mass of gas which passes in unit time through the 

 7/2-surface in the direction of increasing x must be equal to that 

 which passes through it in the reverse direction, i.e. of decreasing 

 x ; for otherwise the density would alter with the time. 



(2) The momentum which passes in the positive direction in 

 excess of that which passes in the negative direction must have 

 the same value for every section, and thus be independent of x. 

 For if through two parallel planes there did not pass equal 

 quantities, on the one hand, into the space lying between the 

 planes, and, on the other, out of this same region, the mass in it 

 would increase in momentum and so in speed. 



(3) The energy which passes through any section must, just as 

 the momentum, exhibit at every position of the section, i.e. for 

 every value of x, the same excess of quantity crossing in the 

 direction of increasing x over that crossing in the opposite 

 direction. 



These three propositions may be expressed in the form of 

 equations, each of which contains an integral of the form of that 

 just given. The three integrals differ in that for the first the 

 factor \m^ is absent ; for the second it is replaced by mu cos s ; 

 and for the third it remains as it is. 



We satisfy Clausius ' second and third propositions, at least 

 with sufficient approximation, by the assumption that N and k 

 are linear functions of x. For then the differential coefficients 

 are constant, and as, according to the hypothesis of slow variation 

 assumed before, these are small in value, and are, moreover, 

 multiplied by the small quantity r, the variations of N and k may 

 be neglected in their coefficients. 



1 ' Ueber die Warmeleitung gasformiger Korper,' 16, Pogg. Ann. 1862, 

 cxv. p. 32 ; Abhandlungen iiber Warmetheorie, 2. Abth. p. 303, 1867. 



