88 
THE HOX. J. W. STEUTT OX THE THEOEY OF EESOXAXCE. 
removing by means of an air-pump the pressure of the atmosphere on the outside of the 
bulb, the liquid fell in the tube, but only to an extent which indicated an increase in 
the capacity of the flask of about a ten-thousandth part. This corresponds in the ordi- 
nary arrangement to a doubled density of the contained air. It is clear that so small a 
yielding could produce no sensible effect on the pitch of the air-vibration.] 
Open Organ-pipes. 
Although the problem of open organ-pipes, whose diameter is very small compared to 
their length and to the wave-length, has been fully considered by Helmholtz, it may not 
be superfluous to show how the question may be attacked from the point of view of the 
present paper, more especially as some important results may be obtained by a compa- 
ratively simple analysis. The principal difficulty consists in finding the connexion 
between the spherical waves which diverge from the open end of the tube into free 
space, and the waves in the tube itself, which at a distance from the mouth, amounting 
to several diameters, are approximately plane. The transition occupies a space which 
is large compared to the diameter, and in order that the present treatment may be 
applicable must be small compared to the wave-length. This condition being fulfilled, 
the compressibility of the air in the space mentioned may be left out of account and the 
difficulty is turned. Imagine a piston (of infinitely small thickness) in the tube at the 
place where the waves cease to be plane. The motion of the air on the free side is 
entirely determined by the motion of the piston, and the vis viva within the space con- 
sidered may be expressed by 
where X denotes the rate of total flow at the place of the piston, and c is, as before, a 
linear quantity depending on the form of the mouth. If Q is the section of the tube 
and T the velocity potential, 
X = Q 
A P 
dx 
The most general expression for the velocity-potential of plane waves is 
sin B cos Jcxj cos 2 7rnt + ( 3 cos Jcx sin 2ffnt, . . 
cos Jcx — BJc sin Jcx) cos 2 imt—filc sin Jcx sin 2 mt. 
where 
Jc= 
2tt 2tt» 
A a 
(13) 
cos 2‘zr?zzf -f- /3 sin 2 ^nt, 
d'b 
dx 
=A cos 2 nut. 
When #=0, 
