96 THE STUDY OF SPEECH CURVES. 



It is necessary to have some idea of the accuracy with which an 

 actual analysis gives the values required by the formulas. The result 

 depends on the number of members used and on the number of significant 

 figures. With four significant figures used in the analysis oi y = 10. sin 3^^, 

 the results rounded off to two decimal places give: for 6 members (12 

 ordinates) 1.43, 2.12, 6.22, 6.68, 2.65, 2.17; for 12 members (24 ordi- 

 nates) 1.85, 2.57, 6.72, 6.08, 1.89, 1.09, 0.76, 0.60, 0.50, 0.45, 0.42, 0.41; 

 for 18 members (36 ordinates) 1.92, 2.64, 6.80, 6.00, 1.81, 1.00, 0.67, 0.49, 

 0.39, 0.32, 0.27, 0.24, 0.22, 0.20, 0.19, 0.18, 0.18, 0.18; for 36 members 

 (72 ordinates) 1.97, 2.69, 6.81, 5.96, 1.76, 0.95, 0.65, 0.47, 0.34, 0.27, 0.22, 

 0.18, 0.16, 0.14, 0.12, 0.11, 0.10, 0.09, 0.09, 0.07, 0.07, 0.06, 0.06, 0.06, 

 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.04, 0.04, 0.04, 0.04, 0.04, 0.04. A com- 

 parison of these figures with those obtained above by calculation shows 

 some inaccuracy for the analysis into 6 members, and a steady increase 

 of accuracy as the number of members is increased. 



Helmholtz and Hermann suggest an analogy between the results of 

 a harmonic analysis and the effect of a maintained sinusoid vibration (such 

 as produced by a steadily vibrating fork) on a harmonic series of resona- 

 tors. As explained in works on mathematical physics and acoustics, the 

 resonators will respond to the vibration a.sin pt with the ampUtudes 



a a 



a 



Ci- 



my(^fk'—py+4ph,'' 



where m is the vibrating mass of the resonator and e is the factor of fric- 

 tion. If we consider the factor of friction to be so small as to be 

 negligible and the resonators to all have the same mass, unity, we obtain 

 as the general formula 



p 

 or, if we put 7=7., then 



c^=fiJF=7i' ^^^^ 



^^-Fl^tff- (3^) 



The amplitudes of the responses thus depend on the frequency of the 

 fundamental. When the experiment is repeated with a higher set of 

 resonators and with a maintained sinusoid proportionately higher, the 



