Vibrations of twitched and stroked Strings. 207 



Therefore either a = b, or — r— is continuous. 

 7 ax 



Further, in order to preserve the continuity of the string, 



the condition 



a(Z + X)+h=b(Xo + X) + k 



must be fulfilled. From this, in an analogous manner, it fol- 

 lows that -^— | ' 3 and consequently also t-, is continuous as 



soon as a is different from b. But if we assume that a = b, the 

 last condition gives h=k, and the one before made use of 

 gives h = aT + k ; therefore must a = b = 0, as we have already 



remarked (at the commencement of this no.). Hence -j- can 



become discontinuous only at a node-point * ; but we have ex- 

 cluded the occurrence of nodes. By this the above assertion 

 is proved. It was necessary to use this demonstration, since 

 in ChristoffePs investigations the continuous progression of 

 the place of discontinuity is presupposed. 



In consequence of these discussions the above conditions of 

 discontinuity can be applied, and, taking (18*) into conside- 

 ration, can be written in the following form: — 



^©.^=»-'©-«^ • « 



Hence it follows that at no place in the string can - ° 



vanish ; for otherwise a would necessarily be = b, which is 

 impossible (conf. supra). Therefore 



The function Xo + X has in the internal from % = to »^=L 

 neither a maximum nor a minimum; it increases continuously , 

 or it decreases continuously. 



10. Hence in the interval mentioned there is only one value 

 of -a? which satisfies the equation £ = £o + £. That is, at any 

 time t the string has only one angle. Or, since y, according 

 to no. 8, is a complete linear function of a, 



At any time whatever the string consists of two straight lines. 



From this immediately follows : — 



All the points of the string pass simultaneously through the 

 position of equilibrium. 



On account of the latter circumstance it is advisable to 

 select as the point of time £ = that in which all the points x 



* In a node such discontinuities actually occur : conf. infrd, no. 15. 



Q2 



