Note 20: capacity of a thick disk 409 



also of one or more linear parameters, the values of which if determined by 

 experiment* would be absolute physical constants, such as might be employed 

 to give us an invariable standard of length. 



Now although absolute physical constants occur in relation to all the 

 properties of matter, it does not seem likely that we should be able to deduce 

 a linear constant from the properties of anything so little like ordinary matter 

 as electricity appears to be. 



NOTE 20, ART. 272. 

 On the Electric Capacity of a Disk of sensible Thickness. 



Consider two equal disks having the same axis, let the radius of either 

 disk be a, and the distance between them b, and let b be small compared with a. 



Let us begin by supposing that the distribution on each disk is the same as 

 if the other were away, and let us calculate the potential energy of the system. 



We shall use elliptical co-ordinates, such that the focal circle is the edge 

 of the lower disk. In other words we define the position of a given point by its 

 greatest and least distances from the edge of the lower disk, these distances 



( + j8) and a (a - )8). 

 The distance of the given point from the axis is 



r=aap, ...... (i) 



and its distance from the plane of the lower disk is 



z = a (a 2 - i)* (I - p*)l. ...... (2) 



If AI is the charge of the lower disk, the potential at the given point is 



ifi = Aa~ l cosec" 1 a, ...... (3) 



or, if we write a* = y 2 + i, ...... (4) 



...... (5) 



If A 2 is the charge of the upper disk, the density at any point is 



A 



a = 



(6) 



where p* = - 2 (a 2 - r 2 ) = I - 2 j3 2 . (7) 



Putting z = b in equation (2), 



b 2 = a 2 y 2 (i - 2 ) or j8* = I - -i ( 8 ) 



6 2 6 2 



Hence p 2 = -^- - y 2 + . (9) 



2 y 2 a? 



* [This implies that such parameters are of sensible magnitude, and not deter- 

 mined by the dimensions of the electron.] 



