386 Mr. W. B. Morton on the Effect of Capacity 



•\ A=y , 



— S- (A sin qc — B cos g'c) = S 2 w(A cos qc + B sin 9c) . 



Putting T n 2 « 



tan « = — — = LCgi*^ = ^ 9 



we get B = A tan (^c + a), 



and y = y o C°M?(o-*)+"t BinM<- 



COS (^C-j-a) 



77" 



The amplitude becomes infinite when qc-\-a = -~ > 



cot ^c = tan a ; 



2tTC 27T S 2 /1N 



«•'■ cot T = I'S (1) 



This is Cohn and Heerwagen's formula. 



Taking now circuit AB, suppose the impressed potential- 

 difference to act at A. 



Let Vi= (Aj cos qx-t B x sin qx) smnt between x=0 and x = a, 



Y 2 = (A 2 cos qx + B 2 sin ^a) sin w£ between x = a and #=&. 



The conditions to be satisfied are : — 



at x = 0, Vj = V sin nt, 



at A = a 7 Y 1 = Y 2? 



and C 1 -C 2 = S 1 ^- 



atx = a + b, V = 0. 

 Putting in the values we get 

 A,=V„ 

 A x cos qa + Bj sin qa = A 2 cos qa -f B 2 sin qa 



= --{( — A x + A 2 ) sin ^2 + ^! — B 2 ) cos qa}, 



A 2 cos q(a + b) + B 2 sin ^(a + 5) = 0, 



t has been put for —^ . 

 b 



On solving these equations for the constants we find 

 v __ sin q{a + b — x) —t sin q(a — x) sin qb _ . 



V 1 — ; 7 ; jr - : : f \ n Sill ^C, 



sm q(a + b)—t sin ^a sm ^6 



^ 7 sina(a + & — .t) T7 . 



Vo = ; rr = — • r V sm nt. 



sm q{a + 6) — t sm ^a sm qb 



