WITHDRAWN FROM THE ACTION OF GRAVITY. 303 
number of impulses produced in a given time is equal then to that of the divi- 
sions which pass in the same time to the contracted section, and is consequently 
proportionate to the square root of the charge. 
In the second place, as the normal length of the divisions of a cylinder, sup- 
posed to exist under the conditions of our laws and composed of a given liquid, 
is proportionate to the diameter of this cylinder, it follows that, for any liquid, 
the length of the divisions of the imaginary vein is proportionate to the diame- 
ter of the contracted section, and therefore exactly proportionate to that of the 
orifice. Now, for a given velocity of escape, the number of divisions which 
pass in a given time to the contracted section is evidently in inverse ratio to 
the length of these divisions; if, then, the liquid remains the same, this number 
is exactly in inverse ratio to the diameter of the orifice. 
Thus the two laws which, according to Savart, regulate the sounds produced 
by the veins, would necessarily be satisfied with regard to our imaginary 
veins. Now, I say that the sound produced by a true vein will not differ from 
that which the corresponding imaginary vein would produce, if the charge is 
sufficient relatively to the diameter of the orifice for the velocity of transfer- 
ence of the liquid to augment very slightly from the contracted section to a 
distance equal to the length of the divisions of the imaginary vein. Then, in 
fact, within this extent, the two causes which tend to modify the length of the 
divisions, (§ 76,) z.e., the acceleration of the velocity of the liquid and the 
resulting diminution in the diameter of the vein, will both be very small; and 
as they act in opposite directions, their resulting action will be insensible, so 
that the divisions will freely acquire at their origin the length corresponding 
to that of the corresponding imaginary vein. Now, it is clear that in this case 
the number of divisions which will pass in a given time to the contracted sec- 
tion will be the same in the real and the imaginary vein; consequently the 
sounds produced by both the veins will also be identical. 
But in confining ourselves to very slightly viscid liquids, as water, we know 
that the relation between the normal length of the divisions of a cylinder 
imagined to exist under the conditions of our laws and the diameter of this 
cylinder must very probably differ but little from 4; consequently the same 
applies to the relation between the length of the divisions of an imaginary 
vein formed of one of these liquids and the diameter of the contracted section 
of this vein. If, then, ina true vein formed of one of these liquids the in- 
crease in the velocity of transference is very slight at a distance from the con- 
tracted section equal to 4 times the diameter of this section, the condition laid 
down above will very probably be satisfied: however, to avoid any chance of 
being deceived, we will take, for instance, 6 times this diameter. 
It is, moreover, clear that, if the condition thus rendered precise is fulfilled 
with regard to a given charge and orifice, it will be so, a fortiori, for the same 
orifice and greater charges, and for the same charge and smaller orifices. We 
arrive, then, at the following conclusions: 
1. When a series of veins, formed of a very slightly viscid liquid, flow suc- 
cessively from the same orifice and under different charges, if the least of them 
is sufficient for the velocity of transference of the liquid to augment very 
slightly, as far as a distance from the contracted section equal to about 6 
times the diameter of this section, the number of vibrations corresponding re- 
spectively to the sounds produced by each of the veins of the series will neces- 
sarily satisfy the first of the two laws discovered by Savart. 
2. When a series of veins, formed of a very slightly viscid liquid, escapes 
under a common charge and from orifices of different diameters, if the common 
charge is sufficient for the same condition to be fulfilled with regard to the vein 
which escapes from the larger orifice, the number of vibrations corresponding 
respectively to the sounds produced by each of the veins of the series will 
necessarily satisfy the second law. It now remains for us to show that the 
