282 
PHYSICS: A. G. WEBSTER 
These results give a fair agreement considering that we have used for the 
conductivity of the mouth the simple formula c = 0.6 R which is true only for 
cross-sections infinitesimal compared with the wave-length, whereas in the 
case of the wooden phonograph horn, the actual radius is nearly one-fourth of 
the wave-length. 
A paper on the subject of the impedance of such an end will shortly appear. 
In the case of an exponential section we have 
<j — (Toe 
+ m -f + p = 0, -w +X = 0, 
dx dx dx dx 
p = e-V^-k^x [ji cos kx -{- B sin kx}, 
X = g-Vi^A; [c COS kx-\-D sin kx}. 
and it is noticeable that the pressure vanishes at the same cross-section as for 
a straight tube. 
Finally, in the case 
we may solve the equation by means of the confluent hyper-geometric func- 
tion. 
It is to be noticed that in none of these cases, except the straight tube, are 
the different overtones harmonic. Thus, the characteristic tone of the "brass" 
is not due to the substance, but is entirely a matter of geometry as is shown 
by the heavy casting in plaster of Paris of a trombone bell used by the writer, 
the tone of which cannot be distinguished from that of the brass bell. I 
believe this phenomenon is well known. 
Inasmuch as all musical instruments are composed either of resonators 
combined with strings, bars, plates, and horns, I feel that the above theory, 
while merely an approximation as to accuracy, will go far toward enabling us 
to complete the theory of musical instruments. Of course, the actual tones 
•emitted by a brass instrument will depend upon the dynamics of the Hps 
which is reserved for a future paper. 
