82 



EKDAHL AND KEELING 



where U = ico + *A. The term *X comes into the expression for *Z a (co) as a 

 result of the terms *X*n; in Eq. A. 4. In the six-reservoir model, the lower 

 atmospheric (troposphere) transfer function is written 



: Zi(co) = *^(U)/*D 6 (U) 



(A.8) 



The coefficients of the *nj and *n;' in Eq. A. 4 form a square array (matrix) of 

 elements. These elements are identically the elements of a determinant which, 

 for the six-reservoir case, is 



"D S (U) = 



— */ 



I — *l 



*l l U+ *k 3 + */ 2 + */ 4 + */ 



— *k 

 K 3 







u + *k< 



(A.9) 



In place of the transfer coefficients, *kji', of Eq.'s A. 3 and A. 4, we have here 

 written *l\ and *k; to coincide with the notation of Keeling" and that of the 

 next paper where explicit expressions are given. For the five-reservoir case, 

 *D S (U) is obtained from Eq. A.9 by setting */ 6 = in the 5 by 5 principal minor 

 of U + */ s . Explicit algebraic expressions are 



I 4 j. * A TT 3 j. * 



D 5 (U) = (IT + *AU J + *BIT + *CU + *D)U 



(A. 10) 



: D 6 (U) = (U + */ 5 )*D 5 (U) + */ 6 (U + */ 3 )(U + */, )(U 2 + *aU + *b)U (A.ll) 



where 



4 6 



*A = I */j + I *kj 



j=l j=3 



*B = (*/, + */ 3 )(*k 3 + *k 4 + *k 5 + *k 6 ) 



+ (*/ 2 + */4)(*k 4 + *k s + *k 6 ) + *k 3 (*k 5 + *k 6 ) + *k 4 *k 6 

 + */,*/ 3 + */,*/ 4 + */ 2 */ 3 



*C =(*/, + */ 3 )(*k 3 *k 5 + *k 3 *k 6 + *k 4 *k 6 ) + (*/ 2 + */ 4 )(*k 4 *k 6 ) 

 + */!*/ 3 (*k 3 + *k 4 + *k 5 + *k 6 ) 

 + (*/,*/ 4 + */ 2 */ 3 )(*k 4 + *k 5 + *k 6 ) 



*D = */,*/ 3 (*k 3 *k 5 + *k 3 *k 6 + *k 4 *k 6 ) + (*/,*/ 4 + */ 2 */ 3 )*k 4 *k 6 



> (A. 12) 



