Mdotropic Potential. 441 



in which J,'= [a 2 + X) (A- + \) (c 2 4- X) , and so <£ L . . . belong 

 to the isotropic solution. 



Thus E is as much above the statical value E^ or S s , as S 

 is below it : or the total energy associated with the nucleus 

 is raised, while the part interpreted as radiant energy is 

 diminished by an equal amount. 



The p-component of momentum is 



c/S i rfS 



R=-^- = 



lw 



v y r =^(i+^N /(/) )/v 2 . 



Since y EL v (r = <f) . the possible range of such a fraction as 

 Nofi 8 0„ is to 1 : or R/w ranges from E,/V 2 to 2E S /V 2 , the 



mY 2 



latter form giving E,,= — with R 



The special values for a much elongated form of ellipsoid 

 and for a much flattened form, say the needle and disk 

 limits, maybe noted as extreme cases. If two axes are small 

 compared with the third c, 



N [t r- = O , S = E. s ( 1 — r- — hp 2 — i<f) ; 



and thus P/u=Q/»=E,/V 2 , and 1 R/w=2E s /V 2 for the needle 

 limit. If one axis c is small and the others equal 



L o a 2 =M o 6 2 =0 o /2 3 S = E,(1- [ } - pi 2 — pA ; 



and P/*/ = Q/r = 3E,/2Y 2 , R/w=E,/V a for the disk limit. 

 When r only exists but is finite S = E S (1 — r 2 ) for the needle 

 limit, and = E„ A / 1 — r 2 for the disk limit. 



§ 30. The solution when density is a linear function of osyz 

 may be dealt with constructively. First we verify that 



* I= 4 ['^-+7'.y+£'~-Xl-<0 • (112) 



is the external potential for a density p(ax + dy + t/z), and 

 that the internal potential is got by taking for the lower 

 limit of the integral (u a as before being %cue? + 2a?yz). 

 We have 



Phil. Mag. 8. 6. Vol. ".♦. No. 52. April 1905. 2 (i 



