438 MR. S. CHAPMAN ON THE KINETIC THEORY OF A GAS 



distribution of velocity held exactly. This is because we shall only be considering 

 slight disturbances from the uniform steady state. 



Thus from MAXWELL'S law of distribution* we find that q = (2hm)~ l , 



+ W 3 ) = 5g a . Putting Q = u (u* + v 2 + iir>), and taking ? = t = w = 0, we 

 find that = 5q, r-^ = =-* = 0, so that the equation of transfer takes the form 



or (since, as we shah 1 see later, on p. 447, the left-hand side of this equation is of a 

 lower order than the terms on the right) 



(10) 5tf |Z = ^.u(u'+v i +w i \. 



We next put Q = u 2 and Q = uv in order to consider the phenomenon of viscosity ; 

 in this case, of course, we must take into account the mass velocity of the gas. The 

 substitution of these values of Q in the general equation is quite straightforward,! so 

 that we set down the results at once : 



(11) _| 



\ dx dy 



fit->\ /3v n . 8mA 



(12) vq ( 3-* + --^) = 



1 \9aj oyj 



We now turn to the calculation of Aw 2 and &u (u* + v 2 + w 2 \ ; the value of A?> will 

 not be calculated directly, but obtained from Aw 2 by transformation of co-ordinates, 

 and the value of A 18 (tt 2 +v a +ti^) will ultimately be eliminated from our equations, 

 and so need not be calculated. 



First of all we shall calculate the values of ^ 12 Q from equations (2)-(5) for 

 substitution in (8). Since, however, several terms of 12 Q, disappear on integration 



with respect to e, it will be most convenient if we immediately calculate <5 12 Q de. 



Jo 



3. Values of ^Q de. 



Jo 



In calculating ^ 12 Q we shall find it convenient, partly for immediate brevity, but 

 much more for the sake of a subsequent transformation of the variables of integration 

 in (8), to write 



(13) (m+m')(U'-U) = X, (w+m')(V'-V) = Y, (m+m')(W-W) = Z, 



(14) mU+m'U' = X,, mV+m'V^Yi, mW+m'W = Z,. 



* In which /(,,) = * 

 \* 

 t See JEANS' treatise, 336, 338. 



