794 



the accidental deviations in a cube, the side of whicli is equal to 

 the wave-length of the light used in the experiments on opalescence. 

 Now there is a difficulty with this formula, to whicli, indeed, 

 lead also the considerations of Einstein as well as statistical mecha- 

 nics when worked out in an analogous way for the critical point. 

 In all these cases the mutual independence of the elements of 

 volume is presupposed. Now, let there be given for the element of 



volume V the mean square of deviation viz. {n — ?iy. Consider p 

 equal contiguous elements of volume Vi,h\,etc., in which n^,n^, etc. 

 particles are situated, n^, n^ etc. indicating the mean values of these 

 numbers. 



Hence in the volume }'':^i\-\-v^-\-... there are X^=n^-\-n^-\-... 

 particles. 



For the mean value of iV we have 



ÏV = ïï; + ÏÏ", + . . . 

 subsequently 



{N— Ny = {{n, - nj + (n, - n,) + . . f =/, (n - n y 

 since, the elements of volume being supposed independent of each 

 other, the means of the double products vanish. So we tind for the 

 deviation of density that the product of volume and mean square 

 of deviation must be a constant. 



Indeed the above-mentioned formula of probability for the devia- 

 tions of density is so far inexact, as the terms of higher order 

 appearing in it are at variance with the mutual independence of 

 the elements of volume, which underlies the deduction of the fre- 

 quency-law. In fact this deduction is only valid for such large elements 

 of volume that these terms are no more of any influence. It is 

 easily seen that this limit, above which the formula is valid, in- 

 creases indefinitely in approaching the critical point. This explains 

 also mathematically the wrong dependence on c found for the mean 

 deviation in the critical point itself. 



Now one could try to deduce the formula to a farther approxi- 

 mation. However, also the supposition of independence of the ele- 

 ments of volume is inexact in case these are small, and it would 

 thus be impossible to ascertain how far the formula would yet differ 

 from reality. ^) 



1) A deduction of the inequalities in which the inexact terms of higher order 

 do not at all appear, is given by Zeenike in his [thesis, which will shortly 

 appear. As this deduction too uses the independence alluded to, the objection men- 

 tioned holds here also. 



The remark of Einstein (I.e. p. 1285) that there would be no principal dilFiculty 



