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PROCEEDINGS OF THE AMERICAN ACADEMY. 



§ 2. Aliquot Points — Experimental Work. 



The observations to be described in this section were made with the 

 help of glass balls from 0.1 mm. to 0.2 mm. in diameter, which were 

 formed by melting the end of a fine glass fibre in a flame, broken off, 

 and fastened on the wire with a touch of shellac. When illuminated 

 from the side by a Nernst filament behind a small hole in a screen, and 

 observed through a stationary microscope, such a ball, acting as a spheri- 

 cal mirror, gives a bright point image of the source. When the wire is 

 vibrating this point of light becomes a line, and, in the case of the rubbed 

 string, this line is always bisected by the position of the point when the 

 wire is at rest; half its length is, therefore, the amplitude of the vibra- 

 tion at the point in question. If the motion were transverse, the curve 

 obtained by plotting the amplitudes which correspond to a single speed 

 of the rubbing wheel, against distances along the wire, would be one 



o D c u A 



Figure 1. The envelopes fur the aliquot cases \, \, \, and A. 



boundary of a region actually swept over by the wire during each vibra- 

 tion, that is, a region which one would see if a white string were vibrat- 

 ing over a black background. In other words, this curve would be the 

 envelope of the various configurations of the wire. In the case of longi- 

 tudinal vibrations these geometric notions are inapplicable, but it will 

 still be convenient to plot amplitudes as though they were transverse, 

 and to call the resulting curve the envelope of the corresponding vibra- 

 tion form. Such envelopes were obtained for each of the aliqnot cases 

 L , ^, i, and \, and were found to consist of 2, 3, 4, and 5 straight lines 

 respectively, the corners corresponding to points which divide the wire 

 into 2, 3, 4, and 5 equal parts respectively. The shapes and relative mag- 

 nitudes of these four envelopes for the same bowing speed are shown in 

 Figure 1, the amplitude of the point bowed increasing directly as the 

 distance of that point from the more distant end of the string, 8 and the 

 other corners being determined by the intersections of two sets of rays 



8 In other words, the points A lf B lt C lf and D x are collinear. 



