Theory o/Crookes's Force. 23 



hypothetical distribution Clausius assumed is not at all ad- 

 equate to represent the actual one. The pressures obtained by 

 this formula are so insignificant that it is not worth while 

 giving the details of the method by which it is deduced. That 

 Clausius' hypothesis is by no means adequate, can also be seen 

 by the consideration that it is only after the Clausian laws for 

 the conduction of heat have ceased to apply, owing to the 

 rarefaction of the gas, that Crookes's force becomes remark- 

 able, as well as by considering what the distribution tends 

 towards, as has been done by Mr. Stoney, in his paper pub- 

 lished in the December Number of this Magazine. He shows 

 that the distribution lies between one which could be repre- 

 sented by two unopposing streams of molecules, moving one 

 towards the heater and the other towards the cooler and un- 

 polarized gas. With such a distribution the laws of conduc- 

 tion of heat would, of course, differ somewhat from those de- 

 duced from Clausius' distribution. 



I shall now calculate the result upon an arbitrarily assumed 

 distribution, which, however, probably represents the actual 

 one more nearly than Clausius's. I shall assume that the dis- 

 tribution of velocities can be represented by the formula 



v = v Q (l + a. cos + /_? sin sin (f> + y sin cos cf> 



+ a cos + b sin 2 sin 2 <j> + c sin 2 . cos 2 <j> + 2/. sin 2 sin cos (f> 

 + 2g sin . cos . cos </> + 2A sin . cos . sin (f>, 



where 



cos 0=/JL. 



This is equivalent to saying that it is represented very nearly 

 by the radii drawn to the surface of a slightly elliptical ellip- 

 soid from a point near its centre. I shall assume that «, /_>, y y 

 a, b, c y f, <7, h are all quantities whose squares and products may 

 be neglected. For the number of molecules moving in the 

 given direction 0, </>, 1 shall assume that it varies inversely as 

 the velocity of the molecules moving in that direction, so that 

 nv = ~Nv . This evidently satisfies the condition A^O. By 

 these assumptions we obtain* approximately nv 2 = ~Nv . v and 

 nv* = ~Nv . v 2 , and hence 



( [l+afi+ft^T— //, 2 sin (/> + y \/ 1 — /uu 2 . cos <£ 

 _J_wJ W + KW^mV + KW 5 )-™^ 



71V — — - JJN t'_ < ______ 



+ 2/V 1 — -fju 2 sin (/> cos + 2gfi >/ 1 — fi 2 . cos <f> 



+ 2Vs/l-^ 2 .sinc£], 



