3> 



String Galvanometer of Eintlioven. 217 



waves represented by the separate terms have the same 

 period and may be combined into a single wave of this 

 period, and differing* in phase. 



If the impressed force were not a simple sine-wave but a 

 complex wave, a similar expression to (43) might be written 

 for each component wave in the complex impressed force. 

 There would then be a sine-wave in the resulting motion of 

 the string corresponding to each component in the impressed 

 force, but each differing in phase from the impressed wave 

 by a definite amount depending upon the physical constants. 

 These phase-angles differ from each other among the waves 

 of different frequencies, and, as a result, the record cannot 

 be a strictly faithful reproduction of the impressed force ; 

 but the amount of departure from it may be small, and 

 differs according to circumstances. 



If we do not introduce the supposition of a uniform 

 magnetic field, but suppose that H varies from point to 

 point along the length of the string, the value of <E> 5 in (16) is 



s = i( l B.sm~d.v (44) 



Jo l 



And if K denotes the integral, we have as the second 

 member of (12) 



|*-=f «=« ( 45 > 



where h now stands for the more general expression 



^= z -| Hsin^^. . . . (46) 



If the impressed force is harmonic, as in (24), the general 

 solution applying to any field distribution for the periodic 

 part is 



t f A| . , A v . 7tx h 2 . , . v . 2irx 



y = L< - — sin 6, cos (coi — eijsin-T- + : — sin eo cos (g)£— e 2 ) sin — — 

 Lh'iO) I fc 2 (o ' I 



+ ~ sin e 3 cos (tot — €3)5111—,- + f . (47) 



Some consideration will be given to the best form of 

 field distribution along the length of the string to give the 

 simplest results, that in which the motion best represents the 

 impressed force, and the force generates the least motion 

 which does not occur in the original electromotive force, but 

 this is deferred to a subsequent section. 



Referring to the solution (43) for the caso of a uniform 

 field and harmonic impressed force, it is to be expected that 



