" 



Theory of Metallic Conductors. 415 



will be equal to the difference of ne and the expression (3), i. e. 



where <f) = fa + fa has been written for the resultant potential. 

 If the conductors are neutral and if there is no external 

 field, we have p = Q, <£ = 0, and therefore C=ne. Thus, 

 ultimately, the equation for the unknown density of distri- 

 bution p = p(a, y, z) becomes 



log (l-p/ne) = ^= £(*. + *,■)• . . . (4) 



If p' be the density in any element dr f and r' the distance 

 of dr ! from the point a?, y, z, then, taking the charges in 

 rational units, 



1 Cp'dr' 



integrated over the volume of all conductors of the system. 

 Thus the equation (4) becomes 



log(l-,/« e )=^ + ^(^.. . . (5) 



This is an integral equation of the second kind, with 

 fa = fa(x,y, z) as the given, and p as the unknown, function. 

 Since, on the left hand, p appears through the log, our 

 equation is a non-linear one, and thus differs from those 

 hitherto studied by Fredholm, Hilbert, E. Schmidt, and other 

 mathematicians. 



Owing to its non-linearity, the solutions of this rigorous 

 integral equation would obviously be deprived of the classical 

 property of superponibility. On the other hand, we know 

 from experience that this property does hold, at least — it 

 would seem, — with a good approximation. If so, and if the 

 assumptions of the electron theory of conductors are essentially 

 sound, we can draw from the experimental facts the con- 

 clusion that, at least for such inducing fields, charges, etc., as 

 are at our disposal, the left-hand member of (5) has to 

 become sensibly linear — that is to say, that p is a small 

 fraction of ne, i. e. that the defect or the excess of free 

 electrons in a given volume is but a small fraction of the 

 normal number of free electrons contained in that volume *. 



* If the degree of accuracy with which superponibility holds were 

 ascertained by experiments especially undertaken then one could form 

 an idea of the upper limit of p/ne (with tue greatest attainable p, say) 

 and therefore of the lower limit of n y the number of free electrons per 

 cm. 3 . I do not know whether such an (electrostatic) estimate of the 

 lower limit of n has ever been contemplated. 



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