12 



c^D = -4a 2 [3a 3 - 37a 2 + 120a - 96 + 24a 2 + 24a (3a 2 - 1 5a + 1 6 - 4a.,)] [33] 



c D = a[l5a 3 - l68a 2 + 512a - 384 + 96a 2 + 48a(5a 2 - 24a + 24 - 6a.,)] [34] 



c 2 D = -4 [(a - 4) 2 (a - 1) + 4aJ [35] 



where 



D = 2 (9a 3 - 94a 2 + 272a - 1 92) + 8 [(a - 4) 2 (a - 1 ) + 4a J In a 



+ 96a £ - 2a (15a 3 - 264a 2 + 944a - 768) - 384aa 2 |> [36] 

 - 96a 2 (5a 2 - 24a + 24) + 576a 2 3 £ 

 and a is a root of the seventh -degree polynomial 



where 



A + a B + a C + a a D + a„E + a a„F + a a G + a a H = 



1 2 12 2 1 2 13 13 



A(a) = a(a - 4) 2 (5a 4 - 83a 3 + 288a 2 - 368a + 128) 



B(a) = 72(a - 4) 2 (5a 3 - 25a 2 + 40a - l6) 



C(a) = 4a(a - 4)(53« 2 - 148a + 128) 



D(a) = -288(a - 4)(5« 2 - l6a + 16) 



E(a) = -96a(3a - 4) 



P(a) = 1152(2a - 3) 



G(a) = 48a(3a - 8) 



H(a) = -1152 (a - 3) 



[37: 



> [38] 



/ 



The solution gives, for the initial doublet strength at x = a, 



m(a) = -^-[(a - 4)(a 2 - 12a + 16) + 48a(a - 4)(a - 2) + l6a £ - 96aa 2 )] [39] 



When a , a , a , ... are all small in comparison with unity, an ap- 

 proximate solution for a is 



