LINEAR THEORY OF FLUCTUATIONS 135 



the parameters necessary to complete the description of a contact between 

 given substances at a given temperature show up impHcitly only through 

 /?. According to our theory the factor a- W -42 does not depend in any unique 

 way upon R; it matters by what means R is varied. If the resistance R is 

 changed by altering the contact area A^ , keeping other parameters fixed, 

 we would find that RA2 is constant so that the factor in question would be 

 proportional to R^, that is, 6 = 1. However, if R is changed by varying the 

 contact pressure, the effect would show up through the factor a^, (/3 also, 

 to some extent, perhaps) and, since one would expect a to increase somewhat 

 with pressure whereas R decreases with pressure, the factor of interest would 

 probably depend upon some power of R between 3 and 4, that is, 1 < 6 < 2. 

 The theory formulated here suffers from the difficulty that the integral 

 of the power spectrum with respect to frequency is logarithmically divergent 

 at and 00 , that is 



/ S{(S) d<j} = I do) oj= log (CO2/CO1) — > 00 as coi — >0 and 



J 01 1 ''oil 



C02- 



The divergence at 00 does not bother us as much as the divergence at 

 since, with only a divergence at 00, the self-covariance Ri{t)Ri(t + u) exists 

 for all non- vanishing values of u; whereas, with a singularity at 0, the self- 

 covariance does not exist for any value of m. For this reason we cannot con- 

 sider the self-covariance here. In Part c of this Section we consider a possible 

 way of removing the divergence at 0, and consequently, then, we are able 

 to calculate the self-covariance for non-vanishing values of u. 



c. Refinement of the Theory of Part b. Here we propose a simple modifi- 

 cation of the model of Part b, removing the divergence of the integral of 

 S{u)) at CO = 0. The modification considered here, although it is one of several 

 possibilities any one of which is sufficient for removing the divergence 

 (See Section 6.), is perhaps the only one that is sufficiently simple to treat 

 in a memorandum of this scope. The results of this section are thus intended 

 to be only provisional and suggestive. 



Let us reconsider the statistics of the function h{r) giving the separation 

 between the surfaces near a point r on the mid-plane. The distribution of 

 /?'s considered in the last section is open to several criticisms: (1) it possesses 

 no characteristic length parallel to the mid-plane; and (2) the self-covariance 

 h\{r)h\{r'Y^^ does not exist for any value of r — r' . 



To correct partially for these difficulties we replace Eq. (5.22) by 



\h^fl= ^^ , (5.25) 



(e) 



where / is a new characteristic length. The self-covariance h\{r)lh{r') 

 based upon (5.25) now exists for all values of r — r' except 0. Thus we still 



