86 
THE HOH. J. W. STEUTT OX THE TTIEOET OF EESOXAXCE. 
For the other component part, X, — X 3 — 0, so that 
X„=-A., 
n 
m 
12 a 2 (m + 2 )c 
• ( 12 ) 
/i'll 2 I O • 
Thus —E-- , which shows that the second note 
rr ' m 
is the higher. It consists of vibrations in the two reservoirs opposed in phase and mo- 
dified by the connecting passage, which acts in part as a second opening to both, and so 
raises the pitch. If the passage is small, so also is the difference of pitch between 
the two notes. A particular case worth notice is obtained by putting in the general 
equation c 3 =0, which amounts to suppressing one of the communications with the ex- 
ternal air. We thus obtain 
4 i o 2 ( Cc> . Cc> \ i @ r\ 
v + a T + sy 1 ss* = 0 ; 
or if S = S', c l =mc 2 =7nc, 
-\-a~ir g(m+2)-l--g2 m= 0, 
^ t m + 2 ± + 4 * • 
If we further suppose m = 1 or c 2 =c lf 
>r=^(3±^5). 
If N be the number of vibrations for a simple resonator (S, c), 
X 2 
a c 
’4w 2 S 
.-. n\ + N 2 = , 3 ^^= 2 - 61 S, 
X 2 -f 521=5-^=2-618. 
2 3— vo 
It appears therefore that the interval from n 1 to N is the same as from N to n 2 , namely, 
■s/AhTS =1-618, or rather more than a fifth. It will be found that whatever the value 
of rn may be, the interval between the resonant notes cannot be less than 2-414, which 
is about an octave and a minor third. The corresponding value of m is 2. 
A similar method is applicable to any combination of reservoirs and connecting pas- 
sages, no matter how complicated, under the single restriction as to the comparative 
magnitudes of the reservoirs and wave-lengths; but the example just given is sufficient 
to illustrate the theory of multiple resonance. In Part III. a resonator of this sort will 
be described, which was constructed for the sake of a comparison between the theory 
and experiment. In applying the formulae (6) or (12) to an actual measurement, the 
question will arise whether the volume of the necks, especially when they are rather 
large, is to be included or not in S. At the moment of rest the air in the neck is com- 
