:i25] 



54 



a ( a _ 4) 2 (5oe 4 - 83a 3 + 288a 2 - 368a + 128) - 96a^a(3a - 4) 



+ 4a a (a - 4)(53a 2 - 148a + 128) + 1152a x a*(2a - 3) 



+ 72a x (a - 4) 2 (5a 3 - 25a 2 + 40a - 16) + 48a x a 3 a(3a - 8) 



- 288a,a (a - 4) (5a 2 - 1 6 + 16) - 1152a 2 a a (a - 3) = 



Corresponding to a solution a of [125], c , c , and c 2 can be obtained from 

 Equations [121], [122], and [123]- The solution of the latter equations gives 



c Q D = -4a 2 [3a 3 - 37a 2 + 120a - 96 + 24a £ + 24a(3a 2 - 15a + 16 - 4a 2 )] [126] 



c x D = a[l5a 3 - 168a 2 + 512a - 384 + 96a 2 + 48a(5a 2 - 24a + 24 - 6a g )] [127] 



c 2 D = -4 [(a - 4) 2 (« - 1) + 4a 2 ] [128] 



where 



D = 2 (9a 3 - 94a 2 + 272a - 1 92 ) + 8 [(a - 4) 2 (a - 1 ) + 4aJ log a + 96a,, 

 -2a (15a 3 - 264a 2 + 944a - 768) - 384aa £ - 96a 2 (5a 2 - 24a + 24) 

 + 576a 2 a 2 [129] 



The initial doublet strength at x = a is 



m(a) = c + c,a + c a 2 + . . . 



1 2 



or, from Equations [126] through [129], 



m(a) = -jf [(« - 4)(a 2 - 12a + 16) + 48a(a - 4)(a - 2) + l6a £ - 96aa £ ] [130] 



Equations [125] through [130] determine the end points of the distri- 

 bution and its Initial trends. In general, Equation [125] will have more than 

 one real root. In this case the initial trends corresponding to each of the 

 roots should be examined, and that root chosen which appears to give the 

 "simplest" trend. 



