32 



y 



+ 3 



222 Dr. A. C. Crehore on the Theory of the 



The values of fij and fi s " have been given in (33) and (34), 

 by which /jl/ — fi s n = 2n s ' ; hence 



^°{l^'L\2 */ ; ""("2"" 1 / V J sm T 



..... (61 

 This equation gives the complete motion of the string in 

 terms of the four physical constants T h p, Z, and k. If the 

 external circuit is open fe=k, and 



As the tension of the string is increased the values of 

 nj become imaginary one after another, and the oscillatory 

 case applies when s is greater than a fixed value, while the 

 non-oscillatory solution applies to the fundamental value of 

 <s( = l) and all others less than the fixed value. 



The equation (61) reduces to (9), when £ = 0, and evidently 

 represents the arc of a circle. The motion of the string- 

 shadow at the central point is not a simple case of logarithmic 

 decrement where the successive ratios of the deflexions in 

 equal time intervals are constant. Prof. Einthoven * remarks 

 in referring to this point that a distinct difference was often 

 found between the ratios of the deflexions at uniform time 

 intervals, so that he was often obliged to calculate an average 

 value. A knowledge of the constants above given should 

 make it possible to calculate these discrepancies. 



It is not necessary to cite other particular cases which are 

 included under the general equation, but it is worth re- 

 marking that in every instance there are harmonics present 

 in the motion of the string shadow which are directly 

 traceable to the form of the curve assumed by the string 

 when deflected by a steady direct current. It is undesirable 

 to have these introduced into the record as they are, simply 

 because of the construction of the instrument. If the form 

 of curve is such that all the coefficients in the equation 

 of the string (9), when developed by Fourier's series, vanish 

 except the first one, this is evidently the best form for the 

 curve of the string since all the higher harmonics are 

 absent. Equation (54), for example, reduces to the first 



* W. Einthoven, "Weitere Mitteilungen iiber dasSaitengalvanometer," 

 innalen der Physih, 4 Folge, vol. xxi. p. 490 (1906). 



