2L2 M. F. Lindemann on the Forms of the 



'T 



y 



= A*g-«), y=-A(L-a)(T-0 



hold good. 



The string is deflected from the position of equilibrium in 

 the opposite direction, y becoming negative, while previously 

 (when A>0) it was positive. It consists of two straight lines 

 whose point of intersection passes along the parabola symme- 

 trical to (29) 



AT 



3/=-g=(L=*>, ..... (30) 



in the direction from x=h to x=0. 



Lastly, for t=T the last of the equations (27) subsists ; the 

 string has returned to its initial position. 



The parabolas (29) and (30) would be travelled in the oppo- 

 site direction by the angle of the string on the second of the two 

 suppositions respecting X possible according to no. 11 and, in 

 accordance therewith, the substitution of L— x for x in the 

 foregoing formulae. 



14. It is of importance for acoustics that the function y, 

 defined by (27), be expanded into a Fourrier series. Since ?/=0 



for £=0 and t= ~-, a series of sines will be chosen so that 

 yssSRi Bin nj-, 



> _2f 



>n ~TJo 



T . 2mrt „ ALT . mm 



L ' 



& n — ™l ysin-7p- dt= -jj-jsin 



If, in addition, by means of (28) the amplitude P of the cen- 

 tral point of the string be introduced, we shall get for y the 

 following expansion into a trigonometrical series : — 



8P ~ 1 . nirx . 2nirt /olN 



This is of the same form as the series which otherwise occur with 

 vibrating strings. 



According to formula (31), besides the fundamental tone all 

 the upper tones are contained in the sound of the string. But 

 if the stroking-place happens to be at a point marking an 

 aliquot part, those upper tones are wanting (according to the 

 observations of Helmholtz and Neumann) which have a node- 

 point at the stroking-place. This fact does not obtain expres- 

 sion in formula (31). Hence, although the above-described 

 motion of the violin-string agrees very well with Neumann's 

 stroboscopic observations, the hypothesis enunciated in no. 8 



