Theory of the Beats of Mistimed Consonances. 281 



not only possible, but certain that it must be so. When we 

 speak of pendulum-vibrations we are apt to forget our history, 

 and to think of the circular pendulum as if it really executed 

 perfect harmonic motions. But this is only true of the ideal 

 cycloidal pendulum, which never has been realized, and is 

 never likely to be. And the difference between the two is not 

 negligible : we need only think of the correction for arc 

 which the periodic time requires in pendulum-observations of 

 accuracy, to see that higher harmonic terms must enter into 

 the motion ; and in all cases in which the fundamental vibra- 

 tion is not very small indeed, these higher terms will certainly 

 not be negligible. 



With any other experimenter than Konig I should be dis- 

 posed to point to a number of influences which lead to trans- 

 formation. One may possibly have escaped him : it is neces- 

 sary for his purposes to be sure that the bending of the spring 

 calls into action a force strictly proportional to the displace- 

 ment. For large displacements it is improbable that this con- 

 dition is satisfied. Further, any unsteadiness in the stand of 

 the instrument leads to transformations of very considerable 

 extent ; but this is not likely to have misled Konig. 



The remaining investigations will now be more easily dealt 

 with. 



On pages 860 & 861 the argument rests on the idea that the 

 harmonic in the source of excitation must be developed to the 

 same actual magnitude as the excited harmonic vibration. 

 But this is not at all the case. According to both theory and 

 practice, the excited harmonic vibration can be developed 

 by an extremely small corresponding vibration in the exciter; 

 and it seems to me quite probable that a large fundamental 

 may contain sufficient harmonic for the purpose of excitation 

 without showing any trace of it in its curves, as is apparently 

 the case on page 863. 



I do not say that this is the sole explanation of the results; 

 but it is sufficient to prevent me from accepting Konig' s argu- 

 ment as a proof. 



There is much difficulty in the question of transmission 

 through the air. When we think of the complicated series 

 of currents and vortices that must surround the prongs of a 

 tuning-fork, it seems very difficult to be sure that there is no 

 transformation there. But certainly Konig's experiment with 

 the phonautograph (p. 864) seems to refer us back to the ex- 

 planation above stated. 



With respect to the transmission of the vibrations through 

 threads (p. 865), I examined this point some years ago by 

 means of a little apparatus shown to the British Association 



Phil Mag. S. 5. Vol. 12. No. 75. Oct, 1881. Y 



