WITHDRAWN FROM THE ACTION OF GRAVITY. 289 
ration of the velocity, the vein, if it did not become divided, would continue to 
become indefinitely thinner from above downwards, (§ 69.) 
This admitted, let us imagine that another vein of the same liquid, placed 
under the imaginary condition of the preceding paragraph, flows off with the 
same charge from another orifice of the same diameter, in the same time as the 
true vein in question. Let @ denote the time occupied by this second vein in 
traversing the distance which we have denoted by D, ¢. ¢., that which is com- 
prised between the instant at which the liquid section that constitutes the neck 
of a constriction passes to the contracted section, and the instant of the rupture 
of the line into which this constriction becomes transformed. In the expression 
of the velocity relative to the first vein, let 0, which gives for this velocity, 
after the time 0, the value V2eh+g0; in other words, let us consider the ve- 
locity of a liquid section belonging to the true vein, after the time necessary 
for a-.section belonging to the imaginary vein to have traversed the distance D. 
According to what we have seen in the preceding section, if the orifice remains 
the same, this time is constant, whatever the charge may be; so that in the 
above expression the term g0 remains invariable when 4 is made to vary. 
Hence, whatever may be the value of 0, we may suppose the charge h to be 
sufficiently large for the term 2h to be very great in proportion to the term 
g, and that the latter consequently may be neglected without any sensible 
error. In the case of a value of 4 which will realize this condition, and, a Sfor- 
tior2, in the case of all still greater values, the velocity of a section of the true 
vein during the time 0 may be regarded as constant, and equal to that of a sec- 
tion of the imaginary vein ; so that throughout the entire space traversed by 
the first during this time, commencing at the contracted section, the real vein, 
if it did not become divided, would preserve exactly the same diameter, and 
might be regarded as identical with the imaginary vein, also assumed to be free 
from divisions. 
Now, it necessarily follows, from this approximative identity, that during the 
time 0 the same will apparently occur in like manner in both veins; conse- 
quently the time @ will be very nearly that which, in the true vein, the liquid 
section, corresponding to the neck of a constriction, would employ in accom- 
. plishing the modifications which we have considered, and the space which it 
will traverse during these modifications may be regarded as equal to the dis- 
tance D relative to the imaginary vein. 
Now, as the continuous part of the true vein terminates a little below this 
space, and is consequently included in the same portion of the vein, it follows, 
. from the above approximative identity, that this continuous part will be exactly 
equal in length to that of the imaginary vein, and therefore, commencing with 
the least of the charges considered above, the lengths of the continuous parts 
of both veins must be very nearly governed by the same law. 
We arrive then, lastly, at this conclusion, that for the same orifice, and com- 
mencing, with a low but sufficient charge, the length of the continuous part of 
the true vein must be in proportion to the square root of the charge. 
In accordance with the preceding demonstration, the low charge in question 
is that at which the movement of transference of the liquid begins to remain 
apparently uniform in all that portion of the true vein which is comprised be- 
tween the contracted section and the point occupied by the middle of each line 
at the instant of rupture; but as the extremity of the continuous part is very 
little distant from this point, (§ 72,) we may neglect the small difference, and 
say simply that the low charge in question is that which begins to render the 
movement of transference of the liquid exactly uniform as far as the extremity 
of the continuous part of the vein. 
Thus, under the condition of a low charge sufficient to produce this approxi- 
mative uniformity, which condition is always realizable, the law indicated by 
Savart as establishing the relation between the length of the continuous part 
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