40 On the Multiple Sounds produced by Vibrating Bodies. 



case in which the string is pinched. The initial figure and the 

 primitive velocities are not arbitrarily given ; they depend on 

 the shock, and can be known only when the shock is exactly 

 known. 



The difficulty presented by the analytical theory of the bow, 

 and even of the isolated shocks produced by pinching a string, 

 is in fact that general and hitherto insurmountable difficulty, 

 of subjecting the phaenomenon of the shock to an exact ana- 

 lysis. It is known that only one very particular case of this 

 phaenomenon has been able to be approached, that only which 

 Poisson has developed. 



M. Duhamel does not consider the action of the bow under 

 the point of view which has just been indicated. He regards 

 tie friction of the bow as equivalent, not to a series of shocks, 

 but to a system of constant forces. A centre of attraction 

 placed beside the string, far enough off for the displacements 

 to be comparatively imperceptible, might therefore, according 

 to this view, be a substitute for the bow. This seems at first 

 sight scarcely probable. Be it as it may, if the new theory be 

 exact, it should explain all the effi-cts of the bow, and more- 

 over its consequences should agree with the results of experi- 

 ment. 



M. Duhamel cites an experiment of verification, from which 

 he has found that a circular bow, or rather a friction-wheel 

 acting upon a stretched string, causes it to deviate from 

 its position of equilibrium, and brings it in a very short time 

 to a new position of equilibrium in which the friction main- 

 tains it, without the string continuing to produce a sound. 

 This fact, the result of M. Duhamel's theory and experiments, 

 may be regarded as negative with relation to this very posi- 

 tive fact, that hurdy-gurdy players sustain the sound of a 

 string for whole hours, if desired. If M. Duhamel has obtained 

 a negative result, this is caused probably by his not having 

 placed the' wheel in the conditions in which it acts in the 

 manner of the bow. 



In fine, it is not easy to see how M. Duhamel's theory 

 accounts for the four principal effects of the bow, and how it 

 has reference to the vibrations which the hairs of the bow con- 

 stantly execute when they are passed over a sonorous string. 



