( 522 ) 
If a string is bowed the fundamental of which has a period T, 
the note will be accompanied by harmonics of periods 7 t T, 1 /, T, 
7 *T etc. respectively. 
The parallel motion of the bridge will cause a periodical change 
of pressure of its left foot on the roof of the violin. When the 
bridge moves to the left the pressure increases and vice versa. The 
change of pressure may be represented by the following series: 
a, nn 2* L + a , *» 2* jij, + 2* Hn 2jr * 
The transverse motion of the bridge will also cause a change in 
the pressure between the left foot and the roof. When the bridge 
is pulled forward the front of the left foot will exert a greater 
pressure on the roof; when the bridge moves back the pressure 
diminishes. This change of pressure may be represented by a series 
of the form 
' *?+**“ 
As the foot of the bridge has only a small area compared to the 
large surface of the violin which is set in motion, we may assume 
that the pressure changes which are due to the parallel and the 
transverse motions of the bridge respectively, occur at the same point 
of the roof. In order to find the total change of pressure produced 
by both motions together we must therefore add the two above 
series. If we assume that the excursion of the roof at the point 
where the left foot is attached to it is proportional to the change 
of pressure, the sum of the two series multiplied by a constant will 
give us the type of motion Of the roof at that point. 
It is well known that in general a sound becomes mellower according 
as the partial overtones become weaker and that the intensification 
of the even overtones especially renders the sound sharper. Many 
instances of this are to be found in Helmholtz’s work already repeat¬ 
edly quoted (p. 129—133 and p. J 51—152). As an illustration 
of the influence of the overtones on a mixed sound we may also 
mention the sound of a piano when octaves are played. When an 
octave is struck on the piano the two notes cannot easily be heard 
separate, as they can be e. g. with thirds. But only very slight 
musical training is required to hear in a musical recital that running 
octaves are played: the sound is then sharper and rougher. The same 
holds for running octaves on the violin. 
When in the above series we diminish the coefficients a lf a ti a, 
etc. while leaving the b t , b 4 , b t unchanged as far as possible, the 
