JEolotropic Potential. 455 



(1— 2/>-)/(l — U/V) for a given translation, and for that 

 translation reversed. The factor can be traced, and proves to be 

 the connecting link between plane-wave and source forms of 

 the argument. The plane-wave form is %lc — (V — U)f, the 



source form y/(l - S/f )2* 8 + (2/wp) 2 -f 2p#- V* (1 - 2^)* H 

 ^nifi of the first form is the normal to the wave-surface in the 

 second form, we readily rind with 1{ for the radical 



(l-tJ/V>(R+2^)=R(l~V) <>.(133) 



leading to 



R+2^-V«(l-2p 2 )={2^- (V-U) t }(l-S/)/(l-U/V) 



It seems clear then that this form of the integral (for 

 potential or for energy) is related to (I think we may say based 

 on) plane wave forms, and that when translation exists a 

 modifying factor is needed to pass to the source form. The 

 essential difference between the two is that the first shows a 

 period varying with direction if %lx — (V — \])t multiplied 

 by a constant is the argument ; the second shows a fixed 

 period if R + 2p# — t(l — 2p 2 ) multiplied by constant is the 

 argument : the factor of transition has the function of allow- 

 ing for change of period due to the motion. 



The external potential is got by distributing an equal 

 charge over the volume of the reolian on which the point xyz 

 lies, i. <?., we must write for the above 



he C day r k a {lx + my + nz)* -\ _^ 



JA a = A«, and JA a = A + p\A f( , . . . ; therefore 



+=Jw ( *! r \i-*4£±m±l!*F] m (134) 



Interpretation with reference to the ellipsoid u a in the 

 first shape of the formula shows that the range of values of 

 (I inn) is complete, for xyz being on u a every plane cuts the 

 ellipsoid. 



The values given to <£ L ... by (129) are 



_ 1 k\lco , f _A„ £Pda> T ,_A„ L*mnd<o 



*- t*Ju p uj K ~ ^)u p (u A )r^ - 27rJ u p (u K) r VM) 



They satisfy the condition 2(AL + 2A / L ')=A o £ in an 

 obvious way, and 2(y>L + 2//L/) = 2 requires 



o—4? |* da} - y/AlT dto 

 2irJ(^)i ' 2tt hu A )V 



