346 BELL SYSTEM TECHNICAL JOURNAL 



and in case of equality the sum may be found by a limiting process. Since 

 we are interested in this order of approximation only for the diagonal terms 

 we set i = j and obtain for the sum 



jn— 1 jti jn 



ndi di — a. 



di — ds {di — d^y ' 



I 9^ s 



n(n — 1) n-2 



Thus the contribution to the ith diagonal element of f(D + E) from terms 

 of type (A2-8) is 



fn*,H.r.,.^[,^,-'M^'] («-« 



where the prime on S indicates that the term s = iis to be omitted. 



Thus, to summarize, we may say that the first approximation to the non- 

 diagonal term in the ith row and^th column (i 9^ j) oif{D -\- E) is 



E./'^j-y (A2-10) 



di — dj 



and the second approximation to the diagonal term in the ith row and ith 

 column oif{D + E) is 



, f, p ^ r m) _ m - m i ^^^""^ 



where the primes on / denote derivatives and the prime on 2 indicates that 

 the term 5 = i is to be omitted. 



Two results obtained from (A2-10) and (A2-11) are of interest. For 

 the first result we set/(2) = z~^ and get the following approximations to the 

 elements of (D + £)-!: 



- Ei^idi di)-\ i ^ j 



r 1^ -1 (A2-12) 



dT' - dT' \Eii - E Ei.Esrd::'\, i = j. 



For the second result we set f{z) = z^'^ and obtain the following approxi- 

 mations to the elements of {D + eY'^: 



E,i(dr + d)'Y\ i^j 



dT + Jd7"' [£« - E E,.E.,{dT + dTr'\ , 

 In (A2-12) and (A2-13) the summations include the term s = i. 



(A2-13) 

 i = ;. 



