76 DYNAMICAL THEOEY OF SOUND 



interval the string is carried forward again, and so on*, the 

 complete cycle taking place in the period of vibration. 



In order to obtain data for mathematical analysis Helmholtz 

 began by an experimental study of the character of the vibration 

 at various points. The device was an optical one, of the kind 

 employed by Lissajous ( 18), by which the rectilinear vibration 

 of the point examined is compounded with an independent 

 vibration at right angles, whose period is commensurable, or 

 nearly so, with that of the string. A microscope whose axis is 

 horizontal is directed to the point to be studied, the string 

 itself being vertical. The eye-piece of the microscope is fixed, 

 but the objective is carried by one of the prongs of a tuning 

 fork and vibrates in a vertical direction. When the fork alone 

 vibrates the image of a bright point on the string is drawn 

 out into a vertical line; when the string alone vibrates the 

 appearance is that of a horizontal line. When both vibrations 

 coexist the result would be a closed curve if the periods were 

 exactly commensurable. For example, if the period of the fork 

 were exactly commensurable with that of the string, and if the 

 vibration of the point examined were simple-harmonic, the 

 result would be one of the corresponding series of Lissajous 

 figures ( 18); whilst if the relation between the periods were 

 inexact, the curve would pass in succession through the various 

 phases of the series. In the actual circumstances the forms of 

 the curves are modified, and it is possible from the result to 

 make inferences as to the true nature of the vibration studied. 



Fig. 30. 



The interpretation is facilitated by the ideal representation 

 of the successive phases as orthogonal projections of a curve 

 traced on a revolving cylinder. It was found that the space- 



* In order that work may be done it is necessary to suppose that the 

 frictional force is greater in the first stage than in the second. This is 

 consistent with the known law that friction of (relative) rest is greater than 

 friction of motion. The remark is due to Lord Rayleigh. 



