1877 J 



the Motion of Vibrating Bodies. 



159 



the mirror were not vibrating. When this figure is produced the number 

 of points passing in a given time is twice the number of vibrations of 

 the mirror; this is the most useful form for observation, as it is very 

 easily recognized. By still further increasing the rapidity of passage of 

 the points a triple figure is again produced when the rate of rotation is 

 exactly double that which forms the similar figure mentioned previously ; 

 or, calling the velocity producing the single figure 1, the triple is observed 

 with a velocity of 1| and also 3. When the rate is 4, a figure composed 

 of four overlapping waves is formed. 



It is thus seen that, like the single wave, the triple figure may be pro- 

 duced by more than one velocity — in fact it has been observed with four 

 distinct velocities, namely, 3, -§, f> and J-, taking the velocity producing 

 the single figure as 1. Similarly the double figure has been obtained 

 with a velocity of 2, J , and § ; and the quadruple with the velocities 4 

 and -§-. It is obvious that these are not the only rates capable of form- 

 ing these figures. Theoretically each may be obtained by an infinite 

 number of velocities ; and the relation between the number of vibra- 

 tions and the passage of the points may be expressed in the following 

 terms : — 



A single wave is formed when a whole number of vibrations takes 

 place in the time of the passing of a point over one interval. 



A double figure is formed when a whole number of vibrations takes 

 place in the time of the passing of a point over two intervals, provided 

 that this whole number is not divisible by two. 



A triple figure is formed when a whole number of vibrations takes 

 place in the time of passing of a point over three intervals, provided 

 that the whole number is not divisible by three. 



Lastly, a quadruple figure is formed when a whole number of vibra- 

 tions takes place in the time of passing of a point over four intervals, 

 provided that the whole number is not divisible by two or by four. 



Besides the mirror other means may be employed to view these figures : 

 for instance, they may be observed through a lens attached to one of 

 the prongs of the fork, or a real image of the figures may be produced 

 by the lens and observed with or without a fixed lens. Instead of a 

 tuning-fork a reed may be used to carry the mirror or lens. 



From the foregoing it will be seen that the formation of these figures 

 may be employed for determining the speed of revolution of the disk or 

 cylinder, if the period of the fork or reed, and also the number of points 

 on the rotating body, are known. 



For some velocities, indicated by whole numbers per minute, the circle 

 on the disk or cylinder must be divided into equal intervals ; thus in the 

 case of a fork vibrating 60 times a second, or 3600 times a minute, it 

 is necessary that 7200 points should pass in a minute in order to form 

 the double figure. If the disk is rotating 100 times a minute, it is clear 

 that the circle must contain 72 equal intervals; for a velocity of 96 



VOL. XXVI. M 



