156 BELL SYSTEM TECHNICAL JOURNAL 



When (2.22) are interpreted by Sylvester's theorem we obtain 

 v{n) = m(^r) [Kxr + \:)v{o) - \[ ~ ^" Bi(o) 



L Ar — Ar J 



- (2.23) 

 i(n) = m'(^r) - ^l" ~ ^" Cv(o) + i(X7" + K)i{o) 



L Ar — Ar 



where N'(Xr) is the transposed of N(^r) and N(^r) is given by (2.18) and 



the summations run from r — I to r = m. 



2.5 Results for Any Symmetrical Section Line — Active 



When the sections contain generators the equations to be solved are 

 those of (2.9). The solutions corresponding to the initial values v{o) and 

 i{o) are, for « ^ 1, 



v{n) = cosh nT v{o) — sinh nV Zoi{o) 



n 



+ J2 {cosh (n - p)T Bi°{p - 1) - sinh (n - p)Y ZoCv°(p - 1) } 



(2.24) 

 i(n) — —sinh nT' ZJ v(o) + cosh nV i{o) 



+ Y. {cosh {n - p)V' Cv°(p - 1) - sinh (n - p)r' Z7'Bi°(p - 1)} 

 p=i 



These may be simplified somewhat by replacing ZgC and Z7 B by sinh T 

 and sinh V, respectively. 



The series in the above expressions may be summed when the generators 

 are such that 



v°(n) = e-"H°, i°{n) = g-"V (2.25) 



where ^ is a scalar and i° and v° are column matrices whose elements are 

 independent of n. Thus 



v{n) = cosh «r v{o) — sinh nV ZoUp) 



+ W^ - e-H){/ - e-'I)-\Bi° - Z,Cv°) 

 + iie-"^ - e-n){e-'' - e-'I)-\Bi° + Z,Cv°) 



i{n) = — sinh nV Z~ v(o) + cosh uF' i{o) 



+ Ke""" - e-'"'/)(/' - e-ny\cv° - z-'Bi°) 



+ ^(e~"^' - e—n){e~^' - e-n)-\Cv° + Z'^Bi^) 

 provided that the inverse matrices exist. 



(2.26) 



