78-81] Rotating Gas 71 



Rotating Gas. 



80. An interesting example of the treatment of a permanent field of 

 force is supplied by the case of a gas and its containing vessel rotating 

 together with an angular velocity o> about the axis of z. We can reduce the 

 problem to a statical one by supposing the gas acted on by a field of force of 

 potential ft) 2 (# 2 + y 2 ), and therefore have as the law of distribution (equation 

 146) 



y = Ae~ 2h:E+hm<a2 (x * +y2) ........................ (167), 



where E stands for \m (u? + v* + w 2 ) + . . . and u, v, w are the velocities 

 referred to axes fixed in the containing vessel. To transform these velocities 

 to axes fixed in space, we must write u my for u and v + tox for v. The 

 value of E, referred to fixed axes, must therefore be taken to be 



E = E + mco (xv - yu) + ma> 2 ( 



where E is formally the same as E, but the velocities are now referred to 

 fixed axes. Equation (167) now becomes 



/= Ae-* h (-Bo-w to>-yu)) ........................... (168), 



giving the law of distribution of velocities referred to fixed axes. This equa- 

 tion is, as it ought to be, a special case of equation (150) in which the same 

 law of distribution was found by another method. 



Calculation of Effective Density. 



81. If we wish to calculate the probability of finding the centre of a 

 molecule within a specified small element of volume of dimensions com- 

 parable with the size of a single molecule, we must proceed as was done in 

 the case of hard spherical molecules in Chapter IV. This probability was 

 there calculated from an " effective-density " which became identical with 

 the true density when the molecules were infinitely small, but which in 

 general depended on the size of the molecules. 



In the former instance the size of the molecules entered into our calcu- 

 lations by excluding a certain region of the generalised space. In the 

 present case the size of the molecules is not a very definite conception, 

 any more than is a collision between molecules (cf. 71), but there is, as we 

 shall now see, no want of definiteness as to the regions of the generalised 

 space which are to be excluded. 



In equation (133) we expressed the total energy of the gas in the form 

 =E a + E b +...+3> ........................ (169), 



where E a , E b} ... are the energies of the separate molecules, and <& is the 

 total potential energy of the intermolecular forces, and is therefore a function 

 of the positional coordinates only. In the case of the elastic spheres, 4> 



