DAVIS. — LONGITUDINAL VIBRATIONS OP A RUBBED STRING. 727 



longitudinally at an aliquot point (1/k) is not Helmholtz's parabola itself, 

 but k chords inscribed in that parabola ; and for a given bowing speed, 

 the relative magnitude of the envelopes of the aliquot cases is that indi- 

 cated in Figure 1. These facts are proved both theoretically and experi- 

 mentally. The envelopes of the simpler non-aliquot cases have also been 

 worked out theoretically, but it has not seemed worth while to verify 

 them experimentally. 



2. Helmholtz's velocity law (see page 695) has been accurately verified 

 for the simpler aliquot cases. 



3. Krigar-Menzel's law (see page 696) has been verified for the sim- 

 pler rational cases, both aliquot and non-aliquot. 



4. A graphical method has been developed by means of which the 

 differential equation of the vibrating string can be solved under any set of 

 boundary conditions which the theory of the rubbed string can present. 

 The advantage of this method is that it gives directly the motion of any 

 point of the string, or the configuration of the string as a whole at any 

 time, — information which it is difficult, even in the simplest cases, to 

 obtain by means of trigonometric series. 



5. Each of the solutions for the non-aliquot cases § , f , and f obtained 

 by this method, has been verified at a number of typical points along 

 the string. 



6. It has been shown that all rational cases can be grouped into a 

 great number of series or convergences, of which the series of aliquot 

 cases is the simplest example. The solutions belonging to any one of 

 these series are so related to one another that it is possible to predict 

 with considerable accuracy the general characteristics of an unknown 

 solution, and, in particular, to get its envelope, merely by determining to 

 what convergence it belongs. Also these convergences are interesting 

 in connection with the observations of Young and of Krigar-Menzel (see 

 page 694) on the prominence of those partials which have nodes near 

 the point of excitation. 



Jefferson Physical Laboratory, 

 Harvard University. 



