188 PROCEEDINGS OP THE AMERICAN ACADEMY, 



(69) A, = hr K <i 



(to the degree of approximation to which we restrict our calculationsj. 

 Substituting the expression (66) for (s) in (65) we obtain 



K K K 



-^ A. ■ . •'^ A, .^r^ TA 



■^ A, , n'^ J A, . 



^.(••-2)! ^^('-1) 



that is 



'^U.,-iA.--2-TA^i-, = 



^ 



for every possible value of s, where the first term of the multiplier of -r, 



is to be omitted for i = k and i =: k — 1 and the second term goes out 

 of itself for ^ = 0. Therefore, the necessary and sufficient conditions 

 for the satisfaction of (65) by (66) are 



A,^, = 2 f T + ^\ ■ A, lor ^ i ^ K - 2 



and 



T+ ^) ■ A, = ior i = K OT K — 1; 



that is 



(70) A,^2fr+ ^-^) • A-2 for 2 ^ z 5 K 



and 



for A^ with the greatest even suffix and A, with the greatest odd suffix 

 that occur in (66). On replacing i in (70) by 2i and 2 i -\- 1, respec- 

 tively, we have, for the ^'s with even and odd suffixes separately, 



A,, = 2{T+i-l)- A,,_, for 1<^<^. 



^2,+^ = 2(T+i-}^)' A,,_y for 1 ^ t ^ "-^ , 

 and, by repeated application of these formulae, 



(72) A,, = 2'(r+t- 1)'" -A, for 1 ^ * < ^ 

 and 



(73) A,,^, = 2'{T+i-hy'^ ■ A, for 1 = j' ^ ^ll ; 



