998 THE FIGURES OF EQUILIBRIUM OF A LIQUID MASS 
corresponding value of @. The charge #', as we have seen, should be such 
that V2gh’ may be very considerable in regard to gé’, or, in other words, that 
aieh! 
the proportion alt may be very great. Let us now take an orifice of less 
0 
diameter, and let 4” denote the charge which fulfils in regard to this second 
orifice the same condition as /’ with regard to the former; let, also, 6’ denote 
what @ becomes in the case of the new orifice. If we wish, in the movement 
of the liquid in the continuous portion of the vein which flows from the latter, 
to have the same degree of uniformity as ia the continuous portion of the 
preceding one, we must evidently make 
V 2gh' — V2gh!'! 
gé go"! 
which gives 
Vii or 
Val ? 
consequently 
h! g!2 
hl! oie 
But the time 0, at least in the case of mercury, is proportionate to the diameter 
of the contracted section, consequently to that of the orifice, (¢ 74;) hence, in 
12 
the case of this liquid, we may substitute for ny that of the squares of the 
diameters of the two orifices; whence it follows that, in passing from any deter- 
minate orifice to one which is less, the charge under consideration will decrease 
as the square of the diameter of the orifice. Now it must be considered as 
very probable that the weakest charge at which Savart’s law begins to be 
realized will decrease in an analogous manner, 2. e., in a much greater propor- 
tion than that of the diameters. As we have several times stated, we are not 
aware whether the considerations relative to mercury are applicable or not to 
all other liquids; but we know at least that they are very probably so to all 
those the viscosity of which is very slight; consequently the above conclusion 
is very probably also true in regard to any of the latter liquids—such, for 
instance, as water. 
79. Let us provisionally admit the preceding conclusions as perfectly demon- 
strated, and let us pass to the other law, ¢. e., that which governs the length 
of the continuous portion when the diameter of the orifice is made to vary. I 
say, in the first place, that, in the case of mercury, this law will coincide with 
the second of those of Savart, when we give to the common charge the value 
at which the vein escaping from the largest of the orifices employed would 
begin in reality to satisfy the first of these laws. In fact, let us remark, first, 
that with the charge in question, aad which we shall denote by 4, the veins 
escaping from all the lesser orifices will exist a fortiori in the effective con- 
ditions of the first law. Consequently, if for a moment we substitute for this 
charge h, a sufficiently considerable charge to render the velocity of the liquid 
sensibly uniform throughout all the continuous parts, and if we again pass from 
this second charge to the preceding, the respective lengths of the continuous 
parts will all decrease in the same proportion, 2. e., in that of the square roots 
of the two charges. Now, with the largest of the latter, the lengths in ques- 
tion were to each other as the diameters of the corresponding orifices, (§ 74;) 
it will also be the same with the charge 4,; consequently with this charge the 
second of Savart’s laws will be satisfied. 
In the second place, [ say that with a lower charge than 2; the same will 
not hold good. ‘To show this, let h, be this new charge; and let us denote by 
