52 DYNAMICAL THEORY OF SOUND 



the cylinder. This is illustrated by Fig. 22, where the dotted 

 branch may be regarded as the projection of that part of the 

 sine-curve which lies on the rear half of the curved surface. A 

 change in the relative phase in (1) is equivalent to a change in 

 the angle a, and may be represented by a rotation of the cylinder 

 about its axis, of corresponding amount. This, again, may be 

 illustrated from Fig. 22 by starting the curve one step further to 

 the right or left. When the ratio of the periods is nearly, but 

 not exactly, that of two integers, the orbit gradually passes 

 through the various phases of the commensurable case, in a 

 recurring cycle*. Thus in the case of approximate unison, or 

 approximate octave, the cycle includes the phases shewn in 

 Fig. 21 or 22, followed by the same in reverse order. The same 

 result is obtained by a continuous rotation of Lissajous' 

 cylinder. 



19. Transition to Continuous Systems. 



The space which we have devoted to the study of dynamical 

 systems of finite freedom is justified by the consideration that 

 we here meet with principles, in their primitive and most easily 

 apprehended forms, which run through the whole of theoretical 

 acoustics. In the subsequent chapters we shall be concerned 

 with systems such as strings, bars, membranes, columns of air, 

 where the number of degrees of freedom is infinite. Mathematic- 

 ally, it is sometimes possible to pass from one of these classes to 

 the other by a sort of limiting processes when D. Bernoulli (1732) 

 discussed the vibrations of a hanging chain as a limiting form 

 of the problem where a large number of equal and equidistant 

 particles are attached to a tense string whose own mass is 

 neglected. In any case, there can be no question that the 

 general principles referred to retain their validity. The main 

 qualification to be noticed is that the normal modes are now 

 infinite in number. It is usual to consider them as arranged 



* In Lissajous' method the vibrations which are optically compounded are 

 those of two tuning forks. The figures obtained when the tones sounded by the 

 forks form any one of the simpler musical intervals give a beautiful verification 

 of the numerical relations referred to in 3. In the case of unison, when the 

 tuning is not quite exact, the cycle of changes synchronises with the beats 

 which are heard ; see 10. 



