on the same axis far apart 381 



We have next to calculate the energy arising from this distribution on the 

 first disk, together with a corresponding distribution on the second disk, the 

 coefficients of the harmonics for the second disk being B, B 2 , B 4 , &c. 



It will consist of three parts, the potential energy of the first disk on itself, 

 of the first and second on each other, and of the second on itself. 



The first part will involve only terms having for coefficients the squares of 

 the coefficients A , for those involving products of harmonics of different orders 

 will vanish on integration. 



The third part will, for the same reason, involve only squares of the coeffi- 

 cients B. 



The second part will involve all products of the form AB. 

 Performing the integrations, putting a = ex and b = cy, 



<4 a 



a 4 * a 4 2 2 . 5 * a 4 2" 



+ R2 1 1L + B 2 - - * i- B 2 - - 

 ' b 4 4 ^ b 4 2* . 5 4 ^ 4 b 4 2 



+ (* 8 + V 8 



^ * * 2 ' 3 ' 5 





*) I / / / 



- 4^ - 3 . 4y - 1^-5 ^y _ l^ y + &c J 



4-5 

 3 -ii 



^5 %2>;2 _ &c J 



3-5 -7 II 



*V _L_ r x _ 4_7 ^ 2 _. 2 &c 1 

 c 3-5 -7\- " 



c 5-7 



In this expression for the energy of the system the coefficients A 2 , A t , B a , B 4 

 are treated as independent of A and B. To determine the nearest approach 

 to equilibrium which can be obtained from a distribution defined by this limited 

 number of harmonics, we must make W a minimum with respect to A t , B t , 

 A 4 and B 4 '. 



