110 
THE HOH. J. W. STEUTT OH THE THEOET OF EESOHANCE. 
Ihe last formulas, (55) and (56), are perhaps more elaborate than is required in the 
present state of acoustical science, and it is rather in the theory of electricity that their 
interest would lie ; but they present themselves so readily as generalizations of previous 
results that I hope that they are not altogether out of place in the present paper. In 
all these cases we have the advantage that the quantity sought is determined by a mini- 
mum property, and is therefore subject to a much smaller error than exists in the condi- 
tions which determine it. 
PAET III. 
Experimental. 
The object of this Part is to detail some experiments on resonators instituted with a 
view of comparing some of the formulse of Parts I. and II. with observation. Helm- 
holtz in his paper on organ-pipes has compared his own theory with the experiments of 
Sondliauss and Wertheim for the case of resonators whose communication with the 
external atmosphere is by simple holes in their sides. The theoretical result is embodied 
in (5) and (23), or for circular holes (24) and runs, 
a 
(57) 
or when the area of the opening is approximately circular and of magnitude er, 
n: 
CUT* 
2MS* 
(58) 
On calculation Helmholtz finds 
^=56174 4*, 
the unit of length being metrical. 
The empirical formula found by Sondhauss is 
?a=52400 4 , 
which agrees completely with theory as regards its form, but not so well in the value it 
assigns to the constant multiplier. The difference corresponds to more than a semitone, 
and is in the direction that the observed notes are all too low. I can only think of two 
explanations for the discordance, neither of which seem completely satisfactory. In the 
first place, Sondhauss determined his resonant notes by the pitch of the sound produced 
when he blew obliquely across the opening through a piece of pipe with a flattened end. 
It is possible that the proximity of the pipe to the opening was such as to cause an 
obstruction in the air-passage which might sensibly lower the pitch. Secondly, no 
account is taken of the thickness of the side of the vessel, the effect of which must be 
* The velocity of sound is taken at the freezing-point ; other-wise the discordance -would be greater. 
