198 M. F. Lindemann on the Forms of the 



is much rather to be specially established * ; for the twice re- 

 peated differentiation of each term of (1) leads to a series 

 which is not convergent. The desired proof is carried out in 

 nos. 2-6. In no. 7 it is shown that the conditions laid down 

 by Christoffel f are fulfilled, under which the occurrence of an 

 acute angle is compatible with the subsistence of the diffe- 

 rential equation (2). 



The remaining part of this essay is occupied with the stroked 

 string (violin-string). In nos. 8-14, therefore, the form of 

 vibration of the violin-string is discussed without making use 

 of those series. Christoffel's above-mentioned conditions of 

 discontinuity present the necessary means for that purpose. 

 The formulae found do not fully agree with those of Helmholtz, 

 as is set forth at the end in no. 19; nevertheless the motion of 

 the string proceeds, on the whole, as described by Helmholtz. 



As is known, the results thus obtained do not accord with 

 the observations ; on this account w r e have endeavoured, in 

 nos. 15-18, to settle the theory of the stroked string in a more 

 general manner. The results therein gained correspond very 

 well with the observations for the case that the stroking-place 

 is a point marking an aliquot division of the string and lies 

 pretty near one end (distant, at the most, but little more than 

 one fourth of the length of the string from the end, as other- 

 wise node-points are readily formed). 



The twitched String. 



2. We will suppose the string to execute transversal vibra- 

 tions in one plane. In the position of equilibrium let it ex- 

 tend along the positive X axis from ^=0 to « = L. Then L 

 is its length; let M denote its mass, and P the force stretching- 

 it ; and the form of the string at the time t is determined by 

 the differential equation (2), if M . a 2 = L . P. 



In order to treat the motion of the a twitched string," let 

 us assume that the given string is drawn on one side from its 

 position of rest y=0 by means of a pointed peg, while the 

 point of contact of the peg with the string, x—a^ has moved 

 in the positive direction to the distance b from its resting-place. 



* Schlomilch, in his Compendium der hoheren Analysis, lays down con- 

 ditions for the possibility of differentiating geometrical series term by term 

 (2nd edition, vol. ii. p. 139). In the present case those conditions are 

 fulfilled but not applicable ; for they imply the presupposition that the 

 differential quotient of the series in question canlikewise be represented 

 by a trigonometrical series. But the second differential quotient of (1) 

 is ; and can only be represented by a trigonometric series whose co- 

 efficients are collectively =0. 



■\Annali di Mateniatica, serie 2, t. viii. p. 81. 



