420 MR. C. A. BELL ON THE SYMPATHETIC VIBRATIONS OF JETS. 
take place in the direction of the jet, and let the (liquid) jet play into free air. The 
motions of the orifice being rapid, and the liquid mobile, the velocity of the stream 
through the centre may be supposed to remain constant. During a forward motion 
of the orifice we shall then have acceleration, and during a backward motion 
retardation, of the velocities of the outer layers of the jet. The first change 
necessarily implies expansion and diminished vorticity, the second contraction and 
increased vorticity. If we imagine the motions of the orifice to be produced by 
two rapidly succeeding impulses in opposite directions, the result will be the 
formation of two vortex rings of unequal size and strength, which wall accordingly 
begin to act upon each other in a known manner. The foremost ring will go on 
expanding, its velocity at the same time diminishing, while the hindmost ring 
will contract, its velocity at the same time increasing. This action will continue 
until either the constricted portion of the jet gives way, or the vortical motions 
are destroyed by internal friction, and the forces of figure gain the upper hand. 
The growth of an expansion and contraction are thus accounted for, if we suppose 
that the action of neighbouring parts of the jet on the two vortices is small in 
comparison with their mutual action. Moreover, as this mutual action of two rings 
varies inversely as some power of the distance between them higher than the second, 
changes once started at the orifice must increase very rapidly with the time. 
Now it is evident that the portion of the jet which has passed through the orifice 
during the time that this has taken to execute a complex vibration, must present 
swellings and contractions at successive points proportional to the varying velocity of 
the orifice; and in this case, for the sake of simplicity, we may imagine as many pairs 
of unequal vortex rings to be developed in the jet as there are simple vibrations in 
the impressed complex vibration. The larger rings will then grow at the expense 
of all smaller rings ; and the result will be that the changes in the jet will go on 
increasing until disruption occurs at a distance from the orifice. 
Since the expansions and contractions of the rings are attended respectively by 
diminution and increase of their velocites, both rotatory and translatory, there 
must be corresponding variations in the velocities, measured along the jet-path, of 
the particles of fluid involved in them. Leaving out of consideration the motion of 
translation of each ring, the motions of the central and circumferential particles are 
in opposite directions, and the sum of the velocities in one direction must be 
equal to the sum of the velocities in the opposite direction. The changes produced 
by vibration must therefore be apparent at every point of a jet; but since the stream 
lines of a vortex ring are crowded together near its centre, these changes must be 
most intense along the jet axis, and most feeble along its outer portions. Again, 
when the rings are destroyed, either by friction or by directing the jet into a 
trumpet-shaped tube, the opposing external and internal velocities neutralise each 
other in the main. On the other hand, when the jet strikes upon any object, such 
