300 THE FIGURES OF EQUILIBRIUM OF A LIQUID MASS 
This bemg established, let us now begin by examining the table relating to 
the orifice of 6 millimetres. It is evident that the proportion of the length of 
the continuous portion to the square root of the charge diminishes considerably 
from the first charge to the last; whence it follows that, in the case of a vein 
of water escaping from an orifice 6 millimetres in diameter, if the charge be 
not made to exceed 47 centimetres, Savart’s first law is far from being satisfied. 
Thus the first conclusion of § 78 is conformable with experiment. Moreover, 
the diminution of the proportion determines the direction in which the true law 
differs from that of Savart within the limit at which this begins to be sufli- 
ciently approximative ; it is evident that the length of the continuous portion 
then augments less rapidly than the square root of the charge. In the second 
place, as the proportion in question increases, we find that the latter converges 
towards a certain limit, which must be a little less than 23, 2. e., the value cor- 
responding to the charge of 47 centimetres. In fact, whilst the charge receives 
successive augmentations of 7.5, 15 and 20 centimetres, the proportion dimin- 
ishes successively by 14, 8.9 and 4.5 units, and the latter difference is still 
tolerably slight in regard to the value of the latter proportion; whence we 
may presume that, if the charge were still farther increased, the further dimi- 
nution of the proportion would be very small, and that a sensibly constant 
limit would soon be attained, at which limit Savart’s first law would be 
satisfied. 
Let us now find the proportion of the velocity of transference of the liquid 
at the extremity of the continuous part to that at the contracted section, in the 
case of the vein escaping under a charge of 47 centimetres. We shall disregard 
here the small alternate variations which have been treated of in § 77, and shall 
therefore consider the velocity of transference of a horizontal section of the 
liquid of the vein as being also that which this section would have if it had 
fallen freely and in a state of isolation from the height of the level of the liquid 
in the vessel. Then, on neglecting the small interval comprised between the 
orifice and the contracted section, we shall have for the velocity in question, 
at any distance Z of this section, the value Y2g.(4+/); if, then, 7 denotes the 
length of the continuous portion, the proportion of the velocity at the end of 
this length to that at the contracted section will be expressed generally by 
2 
BEAM more simply by af 
V 2h 
sion for 2 and / the value relative to the vein in question, 7. e., 47 and 158, we 
find for the relation between the extreme velocities the value 2.1. Thus, al- 
though, under a charge of 47 centimetres, the vein escaping from an orifice of 
6 millimetres may probably nearly exist under the effective conditions of Savart’s 
first law, the velocity at the end of the continuous portion is even more than 
double the velocity at the contracted section, so that the movement of transfer- 
ence of the liquid is still more considerably accelerated. The second conclusion 
of § 78, therefore, appears so far to agree, like the first, with the results of ex- 
periments. : 
Let us pass to the table relating to the orifice of 3 millimetres. Here it is 
evident that the proportion of the length of the continuous portion to the 
square root of the charge is very nearly the same for all the charges; whence 
it follows that, with this orifice, the vein already begins to come within the 
effective conditions of Savart’s first law under a charge of 4.5 centimetres. 
But, according to what we have stated, the orifice being 6 millimetres, the vein 
does not satisfy these conditions except under a charge at least equal to 47 
centimetres; the charge at which Savart’s first law begins to be realized, then, 
augments and diminishes with the diameter of the orifice, and much more 
se than this diameter. Now, this is the substance of the conclusion 
of § 78. 
h 
l Pe ON 
as . On now substituting in this expres- 
7 
