PROPAGATION OF PERIODIC CURRENTS 539 



— F'(m +) in the mode-a impressed voltage V'{y) = V^iy) — Vi{y) 

 at y = u = 5/2. (In what follows, the constituent Josix) will be 

 omitted because it is relatively unimportant.) 



Thus, for the formulation of the mode-a efifects the functions f(y) 

 and F(y), representing the electirc forces and potentials in equations 

 (41), • • • (45) and (50), • • • (54), have the following mode-a values: 



f(y) = Esiy) - E,(y) = £'(v), (128) 



^(v) = Vs{y) - V,iy) = V'iy), ' (129) 



whence, in particular, 



F{0) = r(0), (130) 



F(s) = r{s), (131) 



F(tc -) - F{u +) = V'iu -) - V'(h +). (132) 



Substituting these values into equations (50), (51), (52), (54), and 

 carrying out the indicated integrations ^^ when x = and when x = s, 

 and finally dividing each equation by the value of the primary current 

 /i(0) = E0/2K at X = 0, we obtain the following formulas (134), 

 • • • (137) for the four mode-a current ratios at x = 0, and the formulas 

 (141), • • • (144) for those at x = s. Also, there are included formulas 

 for J(x)//i(0) at X = and at x = s, J{x) denoting the sum of the 

 mode-a current constituents due to the impressed potential, that is, 



J{x) = /o(x) + J.(x) + Juix) (133) 



since Jos{x) is neglected. 



At X = the formulas for the four current-ratios are 



j-(0) _^K sz ll - e-^y+y'>^'-'J 



/i(0) ^ K' K (7 + y')si2 ' ^ "*^ 



l^ := - T— (135) 



/i(0) ^ K'' ' ^^^^ 



^= 2T^,e-^-^-'^^'-K (137) 



The sum of the last three is 



^= -T§,1\ - e-^y^y>^r^j, (138) 



