712 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The final result (40) is the same as was found in formula (20). All 

 the terms in equations (38) to (40) are vector quantities, and are sup- 

 posed to be added vectorially. It is evident that the e. m. f. impressed 

 at A is unaffected by reflected waves ; while the e. m. f. at the distant 

 free end increases by jumps that are numerically smaller and smaller ; 

 but the phase relation may be such that the eftect on the resultant of 

 any one jump may exceed that of the preceding jump. 



In particular cases when the length L of the line is suitably chosen 

 with respect to the frequency and to the line constants, the final volt- 

 age at the distant end may greatly exceed the voltage at the sending 

 end, either on the basis of maximum cyclic values, or of effective, i. e. 

 voltmeter, values. 



Turning now to the current wave, or magnetic flux wave, when this 

 reaches B for the first time its value is I^ e-^'^, a vector quantity, hav- 

 ing a certain phase lag La^ with respect to the current entering simul- 

 taneously at A, and also undergoing simple harmonic variation. When 

 a current wave strikes a free end or discontinuity, it is reflected with 

 inversion ; i. e., with reversal of sign, or 180° (jr radians), change of 

 phase. It starts back, therefore, towards A as — /q e~^" amperes, and 

 the total current increment at B is nil. On arriving at A its con- 

 dition is — /q i.—-^'^ amperes. It then passes through the impressed 

 e. m. f. without change and is reflected from the groujid. A current 

 reflected from a ground or short circuit undergoes no change of phase. 

 It adds, therefore, a jump of —2 I^ f—'^La. amperes at A vectorially to the 

 current I^ impressed there. Whether this will involve an increase or 

 decrease in the magnitude of the current at A will depend upon the 

 relative phases of /o and of — 2/o e~2ia^ which in turn will depend on 

 the length of the line L for any given frequency. The current wave 

 then returns to B for the second time, reaching it in the condition 

 — /o e— ^-^* . It is again reflected with inversion or starts back to A as 

 /o €—2-^* amperes, adding nothing to the current at B. It keeps up 

 this battledore and shuttlecock action theoretically forever ; but prac- 

 tically until it becomes attenuated to insignificant dimensions. The 

 final resulting current at B is zero, which, of course, must be true at 

 an open end. This is due to the occurrence there of pairs of incident 

 and reflected current waves having equal magnitudes but respectively 

 opposite signs. 



The total current strength at A is 



/o from ^ = to ^ = 0.002 



/o - 2 h e--^^" " t = 0.002 to t = 0.004 



/o - 2 /o €-2^" + 2 /o €-^» " t = 0.004 to t = 0.006 

 and so on. 



