208 Mr. 0. Heaviside on Electromagnetic Waves, and the 



at £=0 ; Gri, Hi, upon that at z — l ; the apparatus being of 

 any kind, specified by resistance-operators, making IV, IV 

 the effective resistance and inductance of apparatus at 2 = 0, 

 and R/, L/, at z = L G is given by 



G = 1 + (R /2 + L'V)- 1 [(F + Q 2 )(R ' 2 + L ' 2 n 2 ) 



+ 2P(R'R ' + L'Lo'n 2 ) + 2Qn(R 'I/-R / L '), 



from which H is derived by changing the signs of P and Q ; 



whilst G x and H^ are the same functions of R/> L/ as Gr and 



H are of R(/, L '. 



Now drop the accents, since we have only copper wires of 

 low resistance (but not very thick) in question, and the ter- 

 minal apparatus are to be of the simplest character. K/S>i 

 will be vanishingly small practically, so take K = 0. Next let 

 R/Ln be small, and let the apparatus at z = l be a mere coil, 

 Ri, of negligible inductance first. We shall now have 



P = R/2Lr, Q = nv, 



and these make 



«H 1+ r;> hS =( 1 -k)- 



Thus Ri^Lv makes H^ vanish, whatever the length of 

 line, and the terms due to reflexions disappear. 

 We now have 



n _ *o _r;/2l» v n. -| 



where Gc Q ~* expresses the effect of the apparatus at z = in 

 reducing the potential-difference there, V being the im- 

 pressed force, and the value of G being unity where there is 

 a short-circuit. 



Now to show that R, = Lu makes the magnetic force of the 

 receiver the greatest, go back to the general formula, let 

 €~ p/ be small, and let the size of wire vary, whilst the size of 

 the receiving- coil is fixed. It will be easily found, from the 

 expression for G l} that the magnetic force of the coil is a 

 maximum when 



V+*f*-(3ffi$f, 



where we keep in L x the inductance of the receiver. Or, 

 when R/Ln and K/S?i are both small, 



(Ri> + Wn*f = Lv, 



or, as described, R^Lm when the receiver has a sufficiently 

 small time-constant. The rule is, equality of impedances. 

 We may operate in a similar manner upon the terminal 



