79-5 



2. Now, ill order to avoid the difficulties mentioned, it is necessary 

 to take into account the influence of deviations in the one element 

 on the state in another. Let ns divide the system into infinitely 

 small elements of volume. A molecule is considered to lie in the 

 element when its centre is situated in it. We consider an element 

 dv,, in the origin of coordinates. Around this element we imagine 

 the sphere of attraction i.e. the region in which a molecule must 

 lie when it is to have any influence on the state in dv^. We determine 

 the numbers of molecules for the elements of the sphere of attract- 

 ion in giving the deviations v^,v^ etc. from the mean number of 

 molecules per unit of volume. 



We suppose the mean value of the density r„, when \\ etc. are 

 given, to be a linear function of the deviations i\ etc., i.e. we put') 



Taking the mean value of r„ over all possible values of r,, it 

 appears immediately that 6'==0, hence 



ï'o =/i»'if^y, + /;M''2 + • •.-... (2) 



The coefficients ƒ denote the coupling of the elements, they only 

 depend on the relative coordinates, i.e. here, on x y z. That the in- 

 fluence of an element, when the density is given, must be propor- 

 tional to its size is immediately seen by considering the influence 

 of uniting two elements in (2). 



We shall now write the sum (2) as an integral. For the density 

 in the element dx dy dz we put v_,y~; further, we can dispose of ƒ 

 in such a way that / [0,0,0) =zO. Then for (2) we get 



ih>,:f{'V,y,z)(hvdydz (3) 



-ƒƒƒ'■ 



The integration may be extended here from — oo to -|- oo, ƒ 

 being zero outside the sphere of action *). 



in extending his deduction to a further approximation, is therefore mistaken. On 

 the contrary, the consideration of higher terms so long as the independence is 

 made use of, will not lead to anything. 



1) Putting things more generally, we could write a series in v^ etc. instead of 

 (1). However, for the purpose we have in view, (1) is sufficient. 



') The quantity v can only take the values 1 — adv and — adv, hence r is a 

 discontinuous function of the coordinates. One might be inclined therefore, to continue 

 writing a sum instead of the integral (3) and to solve the problem dealt with in 

 the text with the aid of this sum. In doing so one gets sum-formulae which are 

 wholly analogous to the integrals we used. However, we prefer introducing the 

 integral, as the discontinuous function v has entirely disappeared from formula (6) 

 only the function g appearing in it, which is continuous when the function f is 



