The conditions for minimum and maximum bow force can tell us something interesting about the difficulty of playing the violin. When a simple analysis is done of these two conditions, it turns out that they both depend, among other things, on the position of the bow on the string. Suppose the length of the string is
, and that the bow is applied a distance
from the bridge, where
is usually a rather small number for normal violin playing. Then it can be shown that the maximum bow force is proportional to
, while the minimum bow force is proportional to
. These two conditions can be combined in a graphical form first suggested by John Schelleng in the 1960s. It is most convenient to plot the bow force
and the bow position
