DAVIS. — LONGITUDINAL VIBRATIONS OF A RUBBED STRING. 717 



considered in this paper as well as for the transverse vibrations of a 

 bowed string. 



§ 5. Integral Surfaces. 



It is usual to solve problems connected with the vibration of bowed 

 strings by Helmholz's method and to express the results as trigonometric 

 series in x and t. A solution for the case 2/5 can easily be obtained in 

 this way, and is 



. n 77 x . n 77 1 

 sin — - — sin — - , if ^ 5, 10, etc., 



where J 7 is the velocity of the bow — a form which is entirely satisfac- 

 tory for the theory of music, where the important thing is to know the 

 intensities of the various harmonics produced, but one which gives very 

 little information of any other kind. In the case of an aliquot point it 

 is possible, with considerable labor, to work out from the trigonometric 

 series a complete description of the configurations and motion of the 

 string itself, because the various series in x or t which present them- 

 selves are easily recognizable standard forms ; 1Q but even in this, the 

 simplest of all the non-aliquot cases, such a fortuitous proceeding would be 

 very difficult, if not impossible,17 and the only other obvious method of 

 handling this form of the solution, namely by direct computation, is im- 

 practicable. The desired information can, however, be obtained by a 

 simple graphical process, which it is the purpose of this section to 

 describe. 



An integral surface is a three-dimensional graph which represents the 

 displacement u of any portion of the string at auy time, as a function of 

 x and t. Such surfaces can be handled in two dimensions by plotting 

 their contour lines, u = a constant, on the basal x t plane, just as would 

 be done on an ordinary map. For a string of length I, one is concerned 

 only with that part of the x t plane which lies between the lines x = 

 and x=I, and the displacement, u, is zero along each of these lines. 

 Also, since the motion of such a string is necessarily periodic with a 

 period 2T, it is necessary to cover only so much of the above mentioned 

 strip as lies between the lines t = and t = 2T. If / and 2* are taken 



16 See section 3 of this paper. 



17 At least without the assistance of a complete set of experimentally determined 

 y t curves. 



