Forms of Vibrations of twitched and stroked Strings. 197 



A simplification worth noticing in this process arises from 

 the fact that the digits on the collecting slides, the number of 

 which is always odd, are equal at equal distances from the 

 middle. 



12. Since writing the above I have found analogies to the 

 processes described. Lieut.-Col. Oakes, in the Journal of the 

 Institute of Actuaries, Jan. 1863, gives " A method of multi- 

 plication which may be practised mentally ," the knowledge of 

 which from some other source led me to construct the appa- 

 ratus. The other processes came, as it were, by the sponta- 

 neous action of the machine. Methods having a partial cor- 

 respondence with them are given by Dr. Peacock in his 

 "Arithmetic" in the Encyc. Met. as Arabian or Persian 

 methods ; also in his 'Algebra,' vol. i. pp. 100-105. 



Napier's Promptuarium, described in the Appendix to his 

 Rabdologia, multiplies by a process arithmetically identical 

 with that given ; but the construction and manipulation of his 

 apparatus are very troublesome. 



Hampstead, Jan. 2, 1880. 



XXVII. On the Forms of the Vibrations of twitched and 

 stroked Strings. By F. Lindemann*. 



1. "|~\URING my lectures on Fourier's series and their 

 -L' applications it struck me that various objections 

 might be raised against the theories of twitched and of stroked 

 strings as they are given by Helmholtzf. This occasioned 

 the following investigations, which are intended to establish 

 those theories more rigorously. 



In regard to the twitched string, the series 



2bJj 2 ^ 1 . mra . nirx . 2nirt ,+ . 



7/= o T c-2, -nSin -^-sin -^^cosm m , . (1) 



d 7r 2 a(L— a) n 2 L L T ? v J 



derived in the well-known manner from the general theory, 

 cannot be regarded as the integral of the fundamental differen- 

 tial equation 



^= a ^, where a = T , .... (2) 



merely because the latter is satisfied by every single term of 

 the series. In the present case the admissibility of this view 



* Translated from a separate impression, communicated by the Author, 

 from the BericMe uber die Verhandlungen der naturforschenden Gesellschaft 

 zu Freiburg i. B. vol. vii. no. 4, pp. 500-532. 



t Die Lehre von den Tonempjindungen^ Brunswick, 1863, Beilage II. 

 and V. These supplements remain unaltered in the subsequent editions. 



