288 THE FIGURES OF EQUILIBRIUM OF A LIQUID MASS 
This admitted, let us consider the infinitely thin section which constitutes 
the neck of a constriction at the moment at which it quits the contracted sec- 
tion. This section will descend with a constant velocity, and at the same time 
its diameter will continually diminish until the constriction to which it belongs 
becomes transformed into a line, and then the section in question will occupy 
the middle of this line; the line will become disunited, to be converted into 
spherules. As we have shown above, the time employed in the accomplish- 
ment of these phenomena, and during which the liquid section we have con- 
sidered has traversed the distance comprised between the contracted section 
and the place which the middle of the line occupies at the precise instant of 
rupture, is independent of the velocity of transference ; consequently, if the 
diameter of the orifice does not change, this time will be constant, whatever 
may be the charge. Now, when the movement is uniform, the space traversed 
during a determinate time being in proportion to the velocity, the above dis- 
tance will be in proportion to V¥2gh, consequently to Vi. As we shall fre- 
quently have oceasion to make use of this distance, we shall represent it, for 
the sake of brevity, by D. 
Now, it is easily understood that in our vein the length of the continuous 
part does not differ sensibly from the distance D. In fact, the continuous part 
terminates at the exact place at which, in each line, the most elevated of the 
points of rupture of the latter is produced; for at the instant at which the rup- 
ture takes place, the phases of transformation of all that portion which is above 
the unit in question are less advanced, (§ 69,) and therefore it still possesses 
continuity, whilst all that below this point is necessarily already discontinuous. 
Thus, on the one hand, the continuous part of the vem commences at the ori- 
fice and terminates at the place at which the most elevated point of rupture 
of each filament is produced; and, on the other hand, the distance D com- 
mences at the contracted section, and terminates at the point corresponding to 
the middle of the length of each of the lines at the instant. of their rupture. 
The continuous part then takes its origin rather higher up, but also terminates 
a little above the distance D ; the difference in the origins of these two magni- 
tudes and that of their terminations must, consequently, partially compensate 
each other ; and as these differences are both very minute, the excess of one 
over the other will, a fortior?, be very small, so that the two magnitudes to 
which they refer may, as I have stated, be regarded, without any sensible 
error, as equal to each other.* In virtue of this equalicy, the length of the 
continuous part of the vein which we are considering will then apparently 
follow the same law as the distance D, 7. e., it will be very nearly proportional 
to Vi. 
Thus, in the imaginary ease of uniform velocity of transference, we again 
recognize the first of the laws given by Savart. Now, it is clear that in a real 
vein the velocity will deviate from uniformity so much the less as the charge 
is greater; whence we may infer that, for sufficiently great charges, the length 
of the continuous part of the real vein must still exactly follow this law. We 
rhall, moreover, demonstrate this in a rigorous manner. 
73. Let us, then, take the real case, 7. e., let us consider a vein submitted to 
the action of gravity, in which, consequently, the movement of transference is 
accelerated. ‘Then the velocity possessed, after any time ¢, by a horizontal 
section of the liquid conveyed by the movement of transference, will have for 
its value ¥2gs+-g¢, the first term representing the portion of the velocity due 
to the charge, the second the portion due to the action of gravity upon the 
vein, and # being reckoned from the moment at which the liquid section passes 
the contracted section. It must be borne in mind that, in virtue of the accele- 
* We shall recur to this point, and shall then establish it more clearly. 
