210 M. F. Lindemann on the Forms of the 



4A(A + B) 3 ^ 4 -4AL(A + B) 2 (4A + B> 3 



+ 12A 2 L 2 (A + B)(2A + B> 2 



- A 2 L [12AL 2 (A + B) + 4A 2 L 2 + * 2 B 2 T 2 > + 4A 4 L 4 



= -4B(A + B) 3 ^ 4 + 12ABL(A + B) V 



- 12A 2 BL 2 (A + B> 2 + A 2 BL(4AL 2 - * 2 BT> 



+ * 2 A 2 B 2 L 2 T 2 . 



This equation has to subsist for all values of x, which can 

 only be the case if 



A , t. A , 2 4A 2 L 2 4L 2 

 A + B=0 and « 2 = -^^ =-7p. 



The second of these conditions shows us the dependence 

 between the vibration-period, on the one hand, and the length, 

 mass, and tension, of the string, on the other (conf. no. 2); it 

 is the same relation which is otherwise obtained from the 

 theory of Fourrier's series for transversal vibrations. Here it 

 was to be demonstrated in another way ; the same could also 

 have been done in the well-known manner with the aid of 

 d'Alembert's solution. 



The first of the relations found expresses B in terms of A. 

 The functions a, b,f g, X, determined in no. 11, now become 



b = -Ax, f=iATx, 



a=A(L-a), g=AT(L-x), 



£=?U* (26) 



Herewith the relations subsisting for the discontinuity-place 

 t=T — ^X are spontaneously fulfilled. • 

 Accordingly equations (21) change into 



^ * -c x 

 y=A(h—x)t for 0f:rp = 2j]> 



v -Ax(--t) A<l<2L-5 f (2?) 



y=-A(L-*)(T-0„ ^<i<l- 



By these the motion of the string is represented completely. 



* In the determination of % we might have started from the relation 

 aMi^j =\ J which, according to Christoffel, must he fulfilled, and by 



virtue of which one of the conditions of discontinuity results from the 

 other. 



