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BELL SYSTEM TECHNICAL JOURNAL 



contact with other hills C and compressing them according to equation 

 (3). The position of C is conveniently defined by its distance X from 

 the plane A. 



Fig. 25 — Schematic representation of a rough surface used in mathematical analysis. 



Any continuous distribution of hill positions, typified by C, which 

 would be encountered through a small compressional movement, may 

 be approximately represented by the expression. 



Nx = const. .Y", 



(4) 



where iVx is defined as the number which multiplied by dx gives the 

 number of hills coming into contact with the plane when it moves 

 from X to .r + dx. The exponent « is a constant which for convenience 

 we may call the distribution constant. 



For a total compression D the N^dx hills will be compressed an 

 amount D — x, and hence the total force of reaction F is given by 



F = const 



. rx"{D - 



Jo 



xyi^ix, 



which integrates to the form, 



F = const. Z)"+"^/2 = const. Z)^". (5) 



The constant here includes a summation of the individual constants 

 of equation (3). It is clear that if the hills have different radii the 

 constant only will be affected, so that equation (5) may be regarded 

 as general in this respect. 



