362 



II. The motions of a linear elastic series form but a case of the 

 preceding problem. It was shown that the vibration of a series of 

 n equal bodies are compounded of n — 1 distinct vibrations, per- 

 formed in times which are proportional to the secants of the mul- 

 tiples of the nth part of a quadrant. These times, then, are all 

 incommensurable, so that a perfectly elastic series of n bodies could 

 never again return to its original state ; nay, not even two of the 

 bodies could ever again be simultaneously at the corresponding parts 

 of their orbit. 



This incommensurability of the periodic times presents a great 

 obstacle to a theoretic estimate of the velocity with which an im- 

 pulse is transmitted, since it is difficult to decide what phenomenon 

 should be defined as constituting the transmission ; and since the 

 equations to be evolved contain the sines of angles of which the 

 ratios are incommensurable. Thus, although the equations enable 

 us to compute the state of the system at any prescribed time, we are 

 unable to resolve generally the converse question, — At what time is 

 any one body in a given state ] 



One very important deduction is, that a blow on one end of an 

 elastic series evokes every oscillation of which the series is suscep- 

 tible, and that, therefore, no pure or musical sound can ever be pro- 

 duced by a perfectly elastic body. A simple oscillation can only be 

 produced by the concurrence of twice as many initiatory conditions 

 as there are particles. Now there is no doubt that the vibrations 

 of elastic bodies do resolve themselves into simple or very slightly 

 complicated vibrations, so that the viscidity, imperfect elasticity of 

 the parts, or some analogous quality of the material, must ope- 

 rate. 



The time needed for the transition from an infinitely confused to 

 a simple vibration, and the manner in which that transition is ac- 

 complished, may lead to the explanation of consonant sounds ; and 

 the existence of some of the higher classes of vibrations with that 

 vibration which gives the musical pitch, may occasion the peculiar 

 phenomena of vowel sound. 



