230-232] Mechanical Illustrations 193 



so that 



du _ du 



(462), 



on substitution from equations (459) and (460). The general theory does 

 not lead us to expect to find a definite steady state except for small values 

 of K. 



Now as K decreases, the relative importance of the three terms of the 

 bracket (4e 2 + 2e/3i' \/K -f /3V) changes, until ultimately the term 4e 2 is of 

 preponderating importance. When these values of K are reached, the last 

 term in equation (462) becomes negligible, being of the same order of im- 

 portance as the term /3V K of the bracket, so that the equation assumes 

 the form 



du 



-TT = - 2ew. 



at 



It therefore appears that for these values of K the gas tends towards 

 a state given by u *= 0, or 



en = ^K* (463). 



H 

 In this steady state, = in the limit when K is very small, so that the 



K 



energy tends to become wholly translational. It is therefore clear that in the 

 appropriate generalised space of the kind defined in 227, the stream lines 

 will tend to form into a cluster, and that equation (463) is the equation of 

 this cluster. 



232. It will, however, be advantageous to examine the general arrange- 

 ment of these stream lines. 



If we eliminate the time between equations (459) and (460) we obtain 



2e H 



- (464). 



""^ pv VK ^H f K) 



Since we may properly write 



H=Rr (465), 



where T, r are principal and subsidiary temperatures, it is clear that equation 

 (464) may be regarded as the differential equation of the stream lines in 

 question. The case is one of extreme simplicity, partly because the necessary 

 space is only two-dimensional, and partly because the stream lines are true 

 hydrodynamical stream lines in the sense that no two lines intersect. We 

 shall therefore proceed to draw this system of stream lines. It is not possible 

 to solve the differential equation (464) in finite terms, but a graphical treat- 

 ment will give all the information we require in a convenient form. 



j. 13 



