314 THE FIGURES OF EQUILIBRIUM OF A LIQUID MASS 
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§ 5. Let us suppose now that, through the means indicated by Savart, we 
have withdrawn the vein from the influence of the vibrations caused by the 
fall of the liquid into the vessel which receives it, and from that of external 
noises; and that then, the vein being left to the sole action of the contigura- 
tive forces, we transmit to the vessel from which it escapes, and consequently 
to the liquid contained therein, a sound exactly in unison with that which 
would be rendered by the impact of the discontinuous part against a membrane. 
The liquid which flows from the interior of the vase towards.the orifice, passes 
through it under the action of the resulting vibrations; and if these are com- 
municated in a vertical direction, each portion of the vein which escapes at the 
contracted section, under the influence of a descending vibration, will be pro- 
pelled by the velocity V2gi increased by that of the vibration, and will, con- 
sequently, contain more liquid than the portion which would have passed in 
the same time in the absence of the vibrations. ‘The excess of velocity will 
tend, it is true, to be communicated to the part of the vein situated below that 
which we are considering; but, putting out of view for a moment the config- 
urative forces, we must at least admit that this inferior part will oppose a cer- 
tain resistance in virtue of its inertia, and that, therefore, the excess of liquid, 
superinduced by the excess of velocity, will tend to disperse itself in a hori- 
zontal direction, or, in other words, to dilate the portion to which it pertains. 
This being premised, if the nearly cylindrical figure which the vein would 
assume under the sole effects of the movement of translation of the liquid and 
of the circular form of the orifice was a stable figure of equilibrium, the portion 
which, under the action of the descending vibration, dilates while it passes at 
the contracted section, would at the same time exert an effort to return. to its 
first form; whence it necessarily follows, upon the hypothesis in question, that 
in proportion as the dilatation is formed, it would be propagated to the subjacent 
sections, and would constitute on the surface of the vein a dilated wave of a 
certain length, which would advance with a velocity equal to the sum of the 
velocity of its own propagation and of that of the liquid. ‘Then also the por- 
tion of the vein which would afterwards pass at the* contracted section under 
the action of an ascending vibration, and which” would consequently traverse 
that section with the velocity V2 gh diminished by that of the vibration, would 
produce, for the opposite reason, a constricted “wave of the same length with 
the dilated wave, and which would advance behind the latter with the same 
velocity ; there would then come a new dilated wave followed by a new con- 
stricted one, and so on, as long as the communication of vibratory movements 
was continued. But, by reason of the instability of the cylindrical figure and 
of the tendency of the vein to transformation into isolated spheres, things will 
pass in quite another manner. Let us imagine that the lower extremity of one 
of the dilatations, which would be formed by the sole action of the configura- 
tive forces due to the instability, should traverse the contracted section at the 
precise moment when a descending vibration commences in the liquid. Now, 
since the configurative forees impel in a continuous manner into this portion of 
the vein an excess of liquid which dilates it, without any tendency in the vein 
to return upon itself, we see that the quantity of liquid superinduced at the 
same time by the additional velocity due to the descending vibration may be 
distributed in the horizontal direction and contribute to the formation of the 
dilatation, without having to surmount a contrary tendency. Moreover, since 
the duration of the vibration is equal to the time which the portion of the vein, 
whose configurative forces would themselves alone form an incipient dilatation, 
occupies in passing at the contracted section, the upper extremity of that por- 
tion will traverse the contracted section at the precise moment when the vibra- 
tion terminates, so that the immediate action of this vibration will be exerted 
on the whole portion in question, and only on that portion. In fine, since the 
dilatation produced by the combined actions of which we have just spoken has 
