AND OF A DISK OF SENSIBLE THICKNESS. G79 



The distance of the given point from the axis is 



r = aap (l), 



and its distance from the plane of the lower disk is 



z = a(a 2 -l)i(l-^)> (2). 



If we write ajp 2 = a 2 r 2 (3) ; 



then, if A 2 is the charge of the upper disk, distributed as when undisturbed 

 by the lower disk, the density at any point is 



A, 



cr = 



.(4). 



If A t is the charge of the lower disk, also undisturbed, the potential at 

 the given point due to it is 



\l> = A 1 a~ 1 cosec" 1 a .............................. (5), 



or, if we write a 2 = y 2 +l .................................... (6), 



........................... (7). 



We have next to find the relation between p and y when the given 

 point is in the upper disk, and therefore z = b. 



Equation (2) becomes b" = ay (1 -/3 2 ) ................................. (8), 



and p=l-a*p ....................................... (9), 





Since the given point is on the upper disk, and since & is small, p must 

 be between 1 and 0, and y between- and (-) ; and between those limits we 



Cb \ / 



may write, as a sufficient approximation for our purpose, 



We have now to find the value of the surface integral of the product of 

 the density into the potential taken over the upper disk, or 



{ 



J 



(12). 



