6 It turns out that the necessary and sufficient condition for the existence of the complex coefficient X simultaneously satisfying Eqs. (16) and (17) is: ( , ), ( , ), 1 2 2 1 1 1 g R A B f R A B (18) where the two functions ( , ) 1 f R and ( , ) 1 g R are defined by , ) ) 2 ( (4 4 ) 2 ( 1) ( 1) ) ( (1 ) 2( 1) )( ) ( 1) 2 (1 ( ) ) 2 ( (4 4 ( 1) ( 1) ) 2 2 ( 1) ( ) ) 2( (4 4 ( 1) ( 1) ) 2 ( ) ( ) 2( 1) )( ) ( 1 (1 4 (4 ) 2 ( , ) 1 1 4 1 2 1 1 2 1 1 1 2 1 1 2 2 3 1 2 1 2 1 2 1 1 1 3 2 1 3 2 1 1 6 1 2 1 1 4 1 1 1 2 2 1 1 1 2 3 3 1 1 2 1 1 2 1 1 1 2 4 1 1 1 2 6 3 1 3 1 2 1 2 1 1 1 2 3 3 2 1 2 3 3 4 1 2 1 R R R R R R R R R R R R R R R R f R (19) . ) ) 2 ( (4 4 ) 2 1 ( 1) ) ( (1 ) 2( 1) )( ( 1) ( 1 2 ) ) 2 ( (4 4 1 ( 1) ) 2 2 ( 1) ( ) ) 2( (4 ) (4 ( 1) ( 1) ) 2 ( ) ( ) 2( 1) )( ) ( 4 (4 ) 1 (1 2 ( , ) 1 1 4 1 2 1 1 2 1 1 2 1 1 1 2 2 3 1 2 1 2 1 2 1 1 1 3 2 1 3 2 1 1 6 1 2 1 1 4 1 1 2 1 2 1 1 1 2 3 3 1 1 1 2 1 2 1 1 1 2 4 1 1 1 2 6 3 1 3 1 2 1 2 1 1 1 2 3 3 4 1 2 2 3 3 2 1 1 R R R R R R R R R R R R R R R R g R (20) If the two conditions in Eq. (18) are met, the internal stress field of linear form within the elliptical inclusion is given by , ) ( ) 2( ) 2( , ) ( ) 2(3 3 2) 2( , ) ( 2(3 3 2) ) 2( , ) ( 2 ( ) , ( ) 2 1 2 1 2 1 1 2 1 1 2 1 2 1 1 2 1 2 1 2 2 1 2 1 1 1 2 1 2 1 2 1 2 1 2 2 2 1 2 1 1 1 2 1 1 2 2 1 2 1 2 1 2 2 1 R R R R R X x R R X y R R R R R X x R R X y R R R R R X x R R X y z R R R X z z R X z xy yy xx 1 z S (21) where X1 and X2 are the real and imaginary parts of the complex number X, which is determined by either Eq. (16) or Eq. (17). For example, it follows from Eq. (16) that
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