REFLECTIONS FROM CIRCULAR BENDS 345 



where a„ is a scalar and 



{D + £)2 = (D + E)(D + E) = D^-\- DE+ ED + E' 



and so on. The sum of the terms independent of E is f(D). The terms 

 of order £ are 



EmD-\- E 



1 {D + Ef 



(A2-5) 



DE + ED in {D + Ef 



Dm + Z)£Z) + £Z>2 in (Z) + £)' 



2 Z)^£Z)'" in (Z> + ET 



where the summation extends over the non-negative integer values of I 

 and m for which / + w = n— 1 . The element in the ith row and^th column 

 of D ED^ is cTiEijd^ and hence the corresponding element in the summa- 

 tion in (A2-5) is 



£,,-S^^= ^_^ (A2-6) 



\n di , I = j. 



Thus the terms of order E in the ith. row and yth column of f{D + E) are, 

 from (A2-6) and (A2-4), 



^' - ^i (A2-7) 



£»/' {d,), i = j 



where the prime on/ denotes its first derivative. 

 The terms of order £' in {D + Ef are 



S D'ED^ED"" = E [^i£i;]MMf£t7^]Af (A2-8) 



where the summations extend over all the non-negative integer values of 

 k, -C, m for which k-{-^+m = 7i— 2. On the right [diEij\M denotes a 

 square matrix whose element in the ith row and yth column is diE^ . 

 Likewise the second factor in brackets is a matrix having diEij(fj in the ith 

 row andyth column. The element in the ith row and jth column of (A2-8) 

 is, from the rule for the product of two matrices, 



If t, s, andy are unequal the sum in k, I, m is 



1 r^*^ — d^ _ dj — (/" I 

 di — dj \_di — d, dj — (i» J 



