212 Prof. A. Gray, on Electric 



Thus 



2tto-' ( Mfdf= ±TT*a<r'ba f Ijfaa) J fab) Q . . (37) 

 Jo Jo x 



is the potential energy exhausted in superimposing con- 

 centrically an already constructed circular disk, of radius a 

 and surface density a\ on a previously existing disk of 

 surface density a and radius b (>«). 



Comparing this result with (34) we obtain 



' °° dX k > 



Jfaa) Jfab)^=^^b 2 + a 2 )E-(b 2 -a?)F\, (38) 



j: 



where E and F are, as before, the elliptic integrals to 

 modulus a\b. 



The whole potential energy exhausted in constructing 

 the two disks separately, and then putting them together in 

 the manner described, is thus 



2ttw( V(^)^ + 2ttV 2 « 2 f Vaa)^ 

 Jo K " Jo x 



+ 4*Wa& I J x (\a) J\ (X&) ^ 



Jo A - 



= %[a 2 b 2 + a' 2 a z + 7raa'b{(b 2 + a 2 )E-(b 2 -a 2 )F}]. (39) 

 If a = a' this gives 



,^»J Ji 2 (Xa) g + ft« f Ji'(X6) d ~-h2ab Cj fact) J fab) d £ 



= .^[// + a 3 + 7r^{(// + 6r)E-(6 2 -« 2 )F}]. (40) 



7. It is possible to express the Legendrian elliptic integrals 

 in various ways as Bessel function integrals. Let us calculate 

 for a uniform thin ring (of unit line density) the potential 

 produced at a point P on a coaxial circle of radius 6, and at 

 an axial distance c (fig. 1). We find by (12) that it is 



V = 2w \ e~ Xc J Q (\a)Jfab) d\. 

 Jo 



Calculating directly we get easily 



