Substitution of equations B-33, B-36, B-37 and B-40 into 111-16 yields: 



0)2 b 



A r co 2 b 



_ (* 2 * ) u 

 V3p 0w c w 2,/ 



W 



r ifg _ lc * 1 u- fi-^lw 



[ 3p 0w c w 2 p 0w o)b2j [ (ob J 



(B-47) 



or 



where 



and 



"A - 



(B-48) 



/__a)2b__\ u _ w (B-49) 



V 3p 0w c w 2 ) 



z -(*■)[( Kir) u -«] ,B - 50) 



The remainder of the simplification process for equation 111-16 will be conducted in several steps. In each 

 step, the order of magnitude of the various terms will be compared and terms which cause an error of less than 

 ten percent will be neglected. This procedure will be done in such a way that no term is neglected which could 

 become important due to a later subtraction. It should be mentioned that most of the terms which are 

 neglected result in errors of much less than ten percent and in those cases where the errors approach ten 

 percent, that error is approached only at the limiting values of the terms involved. In order that the 

 simplification process can be checked by the interested reader, the results of each step will be given, rather 

 than just the final result. 



The next step is to simplify the terms in the parentheses in equations 111-17 and 111-18. The results are 



s 22 s 33 - s 23 s 32 = (^_) - (ep^)( 11 +4^ ) ■ ( B - 51 > 



^23^42 ~~ ^22^43 = 



(B-52) 



S 13 S 32 - S 12 S 33 = ( 3Pof 2 Cf a2 ) ( 1 ~W) ~ ' (2p 0) 3 C| 3a2 ) ( 1 ~~W ) ' 



(B-53) 



and 



cob \ / co^ 2 b 



S 12 S 43 S 13 S 42 - (3^-) + ( p 0( 3 C( 3a3 ) 



- ( Cl ^ + (—^—\ f11 +^- \ (B " 54) 



V Po( 0)b2 ) + ^6 Pof 2 Cf b2 )(" 33 )' 



The next apparent step would be to simplify the expressions for U and W. This, however, can lead to errors 

 when the subtractions indicated for Y and Z are made. Hence, the next step is to simplify Y and (Y-Z). Before 

 writing the simplified expressions for Y and (Y-Z), several intermediate steps will be given in order to assist the 

 interested reader in checking the final expressions. 



44 



