726 PROCEEDINGS OF THE AMERICAN ACADEMY. 



the result of superposing two relatively independent components, (a) the 

 5th, 10th, etc., partials, always in the Helmholtzian proportion, which 

 dominate the whole solution near the point f, but are completely absent 

 at that point, and (b) something else which changes more or less con- 

 tinuously near the point § and is the whole solution at that point itself. 

 In the surfaces for -fy an< ^ T% tn ^ s ^ component has not yet been entirely 

 masked by the increasingly important a component — hence their similar- 

 ity. The same thing can be noticed in each of the preceding conver- 

 gences, and, in general, in any pair of convergences corresponding to a 

 formula of the form p m/(q m + 1) ; so that, although the complete 

 solution, regarded as a function of the point of excitation, is discontinuous 

 in every interval, nevertheless it is possible to get a general idea of its 

 changes ; for as the point of excitation moves along the string, one set of 

 partials after another becomes infinite, jumps through zero to the oppo- 

 site infinity, and decreases again, dominating the whole solution through a 

 brief interval — which is very brief indeed for all but the lowest partials 

 — and then disappearing into the ever changing but relatively continuous 

 b component of subsequent cases. 28 The significance of the points which 

 have been arbitrarily chosen as the successive steps in the convergences 

 of Table IV seems to be that each of them is in its turn the simplest 

 rational point between its predecessor and the goal, and therefore shows 

 the gradual rise of the desired set of partials, while avoiding as much as 

 possible the dominant intervals of extraneous sets. 



One other interesting fact is brought out by these graphical solutions. 

 From the method of construction, it is evident that every surface is 

 necessarily symmetrical with respect to the diagonals of its fundamental 

 squares, so that any curve y =■ <f> (t) for a given point x is identical with 

 a curve y = if/ (x) for a corresponding time t = xT/l; that is, any one 

 of the y t curves which are reproduced in this and in Krigar-Menzel's 

 paper is a configuration which an " infinite string " would actually take 

 on once in each complete vibration. The variety and irregularity of 

 these configurations show vividly how complicated the motion of a bowed 

 string sometimes is. 



Summary. 



The results of this investigation may be summarized as follows : 



1. The envelope of a string which is rubbed either transversely or 



26 Of course this has nothing to do with the interesting question as to what 

 happens when the point of excitation actually does move along the string, as, 

 for instance, when a wire is stroked with a bit of rosined chamois. 



