Strings excited by Plucking and by Resonance. 579 
isolated load whose mass is » times that of the string, at a 
point dividing the string in the ratio z : 1—z lowers the pitch 
of the sth overtone through the interval * 
/1+2usin? sz7. 
The relative displacements of the pitches will of course 
alter with the position of the load. On examination, it appears 
that a greater flattening of the higher partials is the more 
probable result. . For example, if the load is on either of the 
outside thirds of the string, the octave is too flat, relatively to 
the fundamental. But witha proper arrangement of density- 
variations along the string, this cause of disturbance might 
imitate either of the two others. 
Comparison of Observed Results with Theory. 
In the case of metal strings, which may be taken to be 
uniform, we should expect the forward or backward slope to 
appear according as the effect of motion of the ends, or of the 
rigidity of the string, predominates. If we write I+ 
and 1 + Qs? for the expressions occurring under heads (a) and 
(6) above, we may call P and Q the end-coefficient and the 
rigidity-coefficient respectively. Itis obvious that the rigidity 
effect will rapidly increase in importance as we go to overtones 
of high order ; but the form of the curve will be settled in the 
main by the relative pitch-displacements of the lower partials. 
The magnitude — will give a measure of the tendency of the 
forward slope to prevail; e. g., if 6 >4, then the displaced 
octave is flat in reference to the displaced fundamental. 
Putting in the values we get 
P_ end-coefficient — 16 ([T* 
Q  rigidity-coetticient M7’ a?r*g¢ 
It is convenient to use the frequency n and the density p 
of the string instead of T and a. The formula then becomes 
Fwy 64 Pnip? 
Oon Mar, ¢ 
Thus increasing the length of the string and raising the 
pitch would tend to bring on the forward slope. The radius 
of the string does not appear in the final expression ; so the 
form of the vibration-curve should be the same for strings of 
* Rayleigh, ‘Sound,’ vol..i. p. 215. 
