22 Mr. G. F. Fitzgerald on the Mechanical 



so that if Iv 2 be expanded in spherical harmonics, K depends 

 only upon the spherical harmonic of the second order. Simi- 

 larly, if Iv z be similarly expanded, it is easy to see that 



Q^Mm^Wfid/j, 



can only depend upon the spherical harmonic of the first order 

 in Iv 3 . 



If now we turn to particular hypotheses as to the character 

 of the distribution of velocities and numbers, the first that 

 claims our attention is Clausius's. He starts from the assump- 

 tion that the distribution of velocities among the molecules 

 that have just encountered one another in any given layer may 

 be perfectly represented by supposing a small constant velo- 

 city in the direction of the transference of heat to be super- 

 posed upon a uniform distribution. This is the same as sup- 

 posing that these velocities in various directions may be repre- 

 sented by the radii drawn to the surface of a sphere from a 

 point slightly displaced from its centre. It is worthy of 

 remark, in connexion with what I mentioned before with refer- 

 ence to the way the quantities in the various spherical har- 

 monics are related to one another, that, supposing the sphere 

 to be an ellipsoid of even greater ellipticity would not have 

 affected his results ; for it is easy to show that the ellipticity 

 of an ellipsoid of revolution only enters into the spherical har- 

 monics of the second and higher orders ; so that it would not 

 enter into the equation giving the quantity of heat, except when 

 multiplied by terms of at least the order of the quantity of heat. 

 Thus, even though the square of the ellipticity were of the order 

 of the displacement from the centre of the point from which 

 the radii representing the velocities are drawn, nevertheless 

 that would at most only have introduced terms depending 

 upon the product of these two, which would not have mate- 

 rially affected his results. Hence we see that Clausius' suc- 

 cess in calculating the quantity of heat conducted is no proof 

 that his hypothesis is by any means a sufficient representation 

 of the actual distribution for the purpose of calculating the 

 resultant stresses ; and that it is not is proved by calculating 

 what the Crookes's force would be upon his hypothesis. If this 

 be done with the help of the quantities he gives in his note (see 

 Phil. Mag. [IV.] vol. xxiii. p. 526), we get 



W ' P ''PT' 



and the pressures deduced from this formula are very much 

 smaller than those observed ; so that it seems certain that the 



i 



