278 PHYSICS: A. G. WEBSTER 
and may be termed the sensitiveness of the phonometer. Where 
7 = 5% = S^pa^/V, a = Kft, /3 = 5W/27ra, 
« = / - + 5 V/F, u = S^pa'/V - pu^/c] =S% (l - k^V/c) (14) 
As described in my recent article the back of the piston is exposed to the 
sound, figure 4. Then 
P — p = ZiXi = Z2X2 _\ 
p^Zo(Xx + X,) ^^^^ 
from which 
Xi= ^ (16) 
ZqZi 4" Z1Z2 H" Z2ZQ 
Tubes and Horns. — Beside the above described phone and phonometer, the 
theory of which assumed a resonator so small that the pressure is supposed to 
be the same at every internal point, I have made use of many arrangements 
employing tubes or cones, in which we must take account of wave-motion. 
The famihar theory of cylindrical pipes may be included in the following gen- 
eralized theory, which I have found experimentally to serve well. 
Let us consider a tube of infinitesimal cross section <r varying as a function 
of the distance x from the end of the tube. Then if q is the displacement of 
the air, p the pressure, 5 the compression, we have the fundamental equations 
p = es = pa^s = — e div ^ = — - ^ (ig) 
a dx 
^ = a'Ap = a' div grad p ^ a' \ ^- ^ ( a^-£)\ (19) 
dt^ ^ iadxK dx/) 
icr dx ^1 
3=_l^ = ,^l^i^(,,)|. (20) 
at p dx dx 
For a simple periodic motion we put />, q proportional to e*"', and obtain 
^ + Ii2i5^ + *V = 0, ^ + ^l? + ^-J2|f 5 + A^g = o. (21) 
dx dx dx dt dx dx dx 
Both these linear equations may be solved by means of series, and if we call 
u{kx)^ v{koc) two independent solutions we have 
p = Au-\- Bv, ^q = Au' + Bv', (3 = pa^k, 
where the accents signify differentiation according to kx. If we denote values 
at one end x = Xi and at the other end x = X2hy suffixes 1, 2, respectively, 
and form the determinants 
