204 M. F. Lindemann on the Forms of the 



Donkin* and Lord Kayleigh enunciated the proposition that 

 the inclination of the string to the position of rest takes two 

 constant values alternately at the two ends. But this is not 

 wholly correct; e.g. for #=0 we have, according to the 

 above : — 



f = h - for0^<^. 



dx a 2L ' 



dy _ 5(L-2a) aT 



dx~2a(L-a) " 2L ' 



dy_ _b_ aT T 



dx~~ L-a " 2L< =«2 5 



&c. Between each two leaps of the function -~- there is there- 

 fore a moment when the inclination takes a third value, as at 

 that moment the line (9) itself forms a part of the string. 

 This is in accordance with the circumstance that at a place of 

 discontinuity a trigonometric series represents the mean value. 

 7. From the foregoing it follows that the series (1) for 



T 



= t ="9 * s constantly a complete linear function of x and t, 



and, indeed, according to the quantity of t, like the functions 

 occurring in the equations (8), (11), (13), (15), (16), (17). 

 Therefore series (1) in fact satisfies the differential equation (2), 

 since simultaneously 



dp > dx* u * 



That the conditions for the possibility of an angle in trans- 

 versely vibrating strings are fulfilled in the solution by means 

 of a Fourier series was already remarked by ChristofTel I. c. 

 It is easy to verify this in the present case. Here the expres- 

 sions 



Ot ot OX 



where a 2 signifies the quantity oocurring in (2), and 



(37 .(ty + dy 

 ot "dx "dt 



have each the same value on both sides of the angle 7, S. We 

 have to form these expressions for the several time-intervals 

 above distinguished ; an easy calculation then shows that they 

 in fact give identical values on both sides of the angle 7, B. 

 The like holds good for the angle y f , 8'. 



* Acoustics, Part I. p. 83 : Oxford, 1870. 



