104 BELL SYSTEM TECHNICAL JOURNAL 



Therefore, since gs = s\As when .v = Xq, 



R, = (3/2)^0-^(5/4)^1- - A,A,-], 



Rs = - 2Ao-'i'lA^o' - SAoA.Ao + 2A,'2- 



Substituting for Ao, Ai, A2 and A^ the expressions derived by giving 5 

 the values 0, 1, 2 and 3 respectively in equation (11), we obtain for 

 i?i, R'l and R?. the functions of n and c given on page 2. 



For values of 5 greater than 3 the direct evaluation of d^v^'+^/dx^ 

 by successive differentiation becomes very tedious. It will be found 

 much more practical to use the following procedure,^ where P is a 

 symbol of operation, A = Ao and b = Ai. 



Ao-"'Rs= (I/5!) 



d^g- 



-(s+l)/2 



dx^ 



= r^ 



^^-(.+l)/2 



D^-'b + 



+ 



lid A 



^2^-(m)/2 



2\dA'^ 



D'-W- + 



r d'-'A 

 L(5-l 



Db'-' + 



or 



(12) 



Rs = ^0^/^ E 



)\dA'-' 



^s^-(s+l)/2 



sldA' 



m Id A " 



(Z)'-'"6"0. 



The following equations give the details requisite for the formation 

 of Rs to Rs inclusive; As can be computed from equation (11). 



Db = A2, D'~b = A3, D'B = A,, D*b = A,, 



D'b = Ae, D'b = A-!,D^b = As, 



Db- = lA^Ao, 



D^b^ = lAiAs + Ai', 



DW = 2AiAi + 2^12^3, 



D'b- = 2AiAr, + 2^2^ 4 + A,-, 



D^b^ = 2AiAe + 2A2A,-{- 2A3Ai, 



D'F~ = 2AiA^ + 2^2^ 6 + 2^3^ 5 + ^44-, 



D¥ = 3AM2, 



D^¥ = SAx^Az + 3^i.4o2, 



D^¥ = 3 AM 4 + 6A1A2A, + ^2^ 



*See DeMorgan's "Differential and Integral Calculus," 1842, page 328, art. 214, 



