Molotropic Potential. 433 



replace (93) and (91). Thus <£ is an area, and L ... are 



ratios: and <£ = I dXlv'J, L = t ad\/\/J. 



J ° . . . J ° 



§ 25. The connexion of the divisors 10 and 12 for volume 



and surface distributions deserves a word of comment. The 

 difference depends in the main on the fact that the field- 

 integral (i.e. j( SXX^t) is for a conductor limited to the 

 space outside : for the volume distribution there is no such 

 limitation. Tims for the sphere in the isotropic case, charge 

 gives the same potential and the same field-integral outside 

 the sphere, whether it is uniformly distributed over the sur- 

 face or through the volume, and the extra fifth part is the 



internal integral for the latter case. With <£ ?: = /o( — — - ■ j 9 



i a P a * i 

 and <j} e -=if-— i we nave 



or 



J W dT < = 4ir> and 2j (-£-)*•= 9 ■ 



r=0 r—a 



The ellipsoid is not itself a potential surface with an even 

 volume distribution, and the ratio of the inner to the outer 

 field-integral turns on the ratio of a quadratic function of 

 L,,. . . to (j) () (vide infra). 



For the maintenance of the ratio 6 : 5 of the totals, simple 

 reasons can be given. For a cone of angle day the potential 

 at the vertex is pdw\r 2 drj r = pr <2 dwj2. Applying this to the 

 potential of an ellipsoid at its centre, and noting 7' d d(o=pd$, 



have \ 9 = I ™ This last gives the potential at 



the centre, of a surface-charge with element pjxIS/2, and 

 therefore total charge Spr /2 or 3e/2. Such a distribution 

 makes the surface equipotential, and its potential is that of its 

 centre. Thus the potential of a volume distribution at the 

 centre i^ 3/2 times that of an equal charge distributed on the 

 surface so as to make it equipotential. ( lombine this with the 

 fact that the mean of the potential for volume distribution 

 taken through the ellipsoid is 4/5 of its value at the centre. 

 as follows from (95), and we have the explanation of the ratio 

 <i/5. In seolotropic work we have (ui>)» orR in the denominator 

 vide infra), and it- ratio to /■ is the same for different distances 

 in one direction, so that \ pdeo \ r 2 '/rj\\=p \/>dH/2\i. 



In seolotropic potentials the surface w P =eA f has trie special 

 characteristics which belong ton sphere in isotropic potentials. 

 In the solution (83) write P/eA p for a,... ; then 



A= I > i e 2 \, Aa=l.'e r; A P , J=(l+\/e) 3 , and <z = P/A p (e + X). 



we 



