STRINGS 71 



and physiologically. Its occurrence in vibrating systems is of 

 course quite exceptional. Even in the present case, if the 

 string deviate appreciably from uniformity or from perfect 

 flexibility, the above scale of frequencies is at once departed 

 from *. 



We were led in 16 to the conclusion, on physical grounds, 

 that in any system of finite extent the effect of the most 

 general initial conditions consistent with its constitution may 

 be obtained by superposition of the several normal modes, with 

 suitable amplitudes and phase-constants. We infer that the 

 most general motion of a finite string can be represented by 

 the formula 



/?x 



............ (6) 



provided the constants C 8 , e g be properly determined, the 

 summation S extending over all integral values of s. An 

 equivalent form is 



STTCt n . S7TCt\ . STTX /h _ x 



,cos-j- +4am-j-J sm-p, ...... (7) 



where A 8 = C 8 cose 8) B 8 =-C 8 s'm s ............. (8) 



If the string start from rest in a given position at the 

 instant = the coefficients B 8 will vanish; if it be started 

 with given velocities from the equilibrium position (y 0) 

 the coefficients A 8 will vanish. 



Since the value of every term in (6) or (7) recurs whenever 

 t is increased by 2/c, the vibration is essentially periodic, as 

 already proved in | 24. In all other respects the motion of the 

 string when started in an arbitrary manner is, from the present 

 point of view, of a complex character, being made up of an 

 endless series of simple-harmonic vibrations. The resulting 

 note is accordingly made up of a series of pure tones, consisting 

 (in general) of a fundamental, its octave, twelfth, double octave, 

 and so on. 



It is not altogether easy to excite a string in such a way 



* The fact that a particular sequence of notes, musically related to one 

 another, is associated with lengths of string proportional to the quantities 

 1 i> i> i> was known to the Greeks, and was the origin of the name 

 "harmonic" as applied to the numerical series. 



