PROFESSOR AT HEIDELBERG 197 



carried by a tuning-fork which vibrated under the influence of 

 an electro-magnet once to four vibrations of the string. In 

 this way he discovered that the starch granules described a 

 shining curve, the horizontal abscissae of which corresponded 

 with the displacements of the tuning-fork, and the vertical 

 ordinates with the displacements of the string. This motion 

 may therefore be imagined as consisting of two different kinds 

 of vibration, the first of which greatly preponderates in regard 

 to amplitude, while its period corresponds with the period of 

 the fundamental tone of the strings, independent of the point 

 to which the bow is applied ; the second weaker motion, on the 

 contrary, makes very minute deflexions in the curve, since its 

 vibration-period corresponds with one of the higher partials of 

 the string ; at all nodes of the partial the principal motion alone 

 appears. Experiment showed in regard to the principal motion 

 that every point of it first advances with constant velocity in 

 one direction, and then returns to its first position with another 

 constant velocity, from which, in view of the fact that the 

 vibrations of a string occur in a plane, the analytic expression 

 of the displacement of any point may be stated with the aid of 

 Fourier's series as a function of the distance of the point 

 from one end of the string, and of the time. Along with this 

 principal form of vibration, other lesser vibrations, which 

 may be expressed in precisely the same way, are produced 

 if the bow touches any point of which the distance from the 

 nearest end of the string is the reciprocal value of a whole 

 number of lengths of the string ; in such a case the 

 over-tones (on the analogy of Young's investigations for the 

 strings of a harp) of which the multiples correspond to that 

 whole number will not be heard, although the ear can plainly 

 distinguish all other over-tones. Helmholtz concludes from the 

 simple analytical consideration of this combination, that during 

 the motion of the string the base of the abscissa of its point 

 of greatest displacement moves to and fro along the line of 

 equilibrium with constant velocity, while the apex itself 

 describes two parabolic curves that run above and below the 

 position of equilibrium and through the ends of the string, and 

 the actual form of the string at any instant is given by the two 

 straight lines that join a point on the parabolic curves with the 

 ends of the string. 



