on the same axis far apart 379 



where b is the radius of the second disk, and p is a quantity corresponding to /i 

 in the first disk, then the most convenient expression for the potential due to 

 the first disk at a point (p) in the second, is 



. . 



-* {l - p) + -- - (I - p) - &c 



where U denotes the value of the potential at the axis, and where, after the 

 differentiations, va is to be made equal to c. 



To investigate the mutual action of the two disks, let us assume that the 

 surface-density on the second disk is the sum of a number of terms of which 

 the general form is 



-^W- ...... (18) 



The potential at the surface of the second disk arising from this distribution 

 will be the sum of a series of terms of the form 



The potential arising from the presence of the first disk is given in equa- 

 tion (17). 



Having thus expressed the most general symmetrical distribution of elec- 

 tricity on the two disks and the potentials thence arising, we are able to calculate 

 the potential energy of the system in terms of the squares and products of the 

 two sets of coefficients A and B. 



If W denotes the potential energy, 



W-*ffoV4s, ...... (20) 



when the integration is to be extended over every element of surface ds. 

 Confining our attention to the second disk, the part of W thence arising is 



Trb* I aVpdp, ...(21) 



./o 



and the part arising from the term in the density whose coefficient is B tn is 



f>toW4^ ...... (22) 



The part of the value of V which arises from the electricity on the second 

 disk itself is the sum of a series of terms of the form (19). The surface-integral 

 of the product of any two of these of different orders is zero, so that in finding 

 the potential energy of the disk on itself we have to deal only with terms of 

 the form 



77 1 (an!) 2 i . . 



2 " 4 b 2 4 " (n !) 4 4 + i ' 



The energy arising from the mutual action of the disks consists of terms 



