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render the surface included between the rings convex. Let us then gradually elevate the upper ring, and we shall produce a cylinder of greater height than
the first. If we repeat the same manipulation a suitable number of times, we
cylinder. Hence it follows that one of the terms of the quantity ** # dis
appears, and that the other is constant; this same quantity is, therefore, con-
to a plane surface. It is, therefore, evident that the bases of our liquid cylinder must necessarily be convex, as is shown to be the case by experiment; for as equilibrium requires that the pressures should be the same throughout the whole extent of the figure, these bases must produce a greater pressure than that which corresponds to a plane surface.
Our plane figure, then, fully satisfies theory; but verification may be urged still further. Theory allows us to determine with facility the radius of those spheres of which the bases form a part. In fact, if we represent this radius by 2, the formula (1) of paragraph 4 will give, for the pressure corresponding to the spheres in question,
Now, as this pressure must be equal to that corresponding to the cylindrical surface, we shall have
from which we may deduce
x=22. Thus the radius of the curvature of the spherical segments constituting the bases is equal to the diameter of the cylinder.
Hence, as we know the diameter, which is the same as that of the solid rings, we may calculate the height of the spherical segments; and if by any process we afterwards measure this height in the liquid figure, we shall thus have a verification of theory even as regards the numbers. We shall now investigate this subject.
40. If we imagine the liquid figure to be intersected by a meridional plane, the section of each of the segments will be an arc belonging to a circle, the radius of which will be equal to 21, according to what we have already stated, and the versed sine of half this arc will be the height of the segment. If we suppose the metallic filaments forming the rings to be infinitely small, so that each of the segments rests upon the exact circumference of the cylinder, the chord of the above arc will also be equal to 21; and if we denote the height of the segments by h, we shall have
h=1(2-V3)=0.268... Now, the exact external diameter of my rings, or the value of 21, corresponding with my experiments, was 71.4 millimeters, which gives h=9.57 millimeters. But as the metallic wires have a certain thickness, and the segments do not rest upon the external circumference of the rings, it follows that the chord of the meridional arc is a little less than 21, and that, consequently, the true theoretical height of the segments is a little less than that given by the preceding formula. To determine it exactly, let us denote the chord by 2c, which will give
h=21–1412 - 07. Now, let us remark that the meridional plane intersects each of the rings in two small circles to which the meridional arc of the spherical segment is tan. gential, and upon each of which the chord of this arc intercepts a small circular segment. The meridional arc being tangential to the sections of the wire, it follows that the above small circular segments are similar to that of the spheri. cal segment; and as the chord of the latter differs but very slightly from the radius of the circle to which the are belongs, the chords of the small circular segments may be considered as equal to the radius of the small sections, which radius we shall denote by r. It is moreover evident that the excess of the ex• ternal radius of the ring over half the chord c is nothing more than the excess
of the radius r over half the chord of the small circular segments, which half chord, in accordance with what we have stated, is equal to . Thence we get 2-c=, -, whence c=i-r, and we have only to substitute this value in the preceding formula to obtain the true theoretical value of h. The thickness of the wire forming my rings is 0.74 millimeters ; hence r=0.18 millimeters, which gives as the true theoretical height of the segments under these circumstances,
h=9.46 millimeters. I may remark that it is difficult to distinguish in the liquid figure the precise limit of the segments, i. e., the circumfcrences of contact of their surfaces with those of the rings. To get rid of this inconvenience, I measured the height of the segments, commencing only at the external planes of the rings; i. e., in the case of each segment, commencing at a plane perpendicular to the axis of revolution, and resting upon the surface of the ring on that side which is opposite the summit of the segment. The quantity thus measured is evidently equal to the total height minus the versed sine of the small circular segments which we have considered above; consequently these small circular segments being similar to that of the spherical segment, we obtain for the determination of this
h f .:, versed sine, which we shall denote by f, the proportion = , which in the
case of our liquid figure gives f=0.05 millimeters, whence
h-f=9.41 millimeters. This, then, is definitively the theoretical value of the quantity which was required to be measured.
41. Before pointing out the process which I employed for this purpose, and communicating the result of the operation, I must preface a few important remarks. If the densities of the alcoholic mixture and of the oil are not rigorously equal, the mass has a slight tendency to rise or descend, and the height of one of the segments is then a little too great, whilst that of the other is a little too small; but we can understand that if their difference is very small, an exact result may still be obtained by taking the mean of these two heights. We thus avoid part of those preliminary experiments which the establishment of perfect equality between the two densities requires. But one circuinstance which requires the greatest attention is the perfect homogeneity of each of the two liquids. If this condition be not fulfilled with regard to the alcoholic mixture, i. e., if the upper part of this mixture be left containing a slightly greater proportion of alcohol than the lower portion, the liquid figure may appear regular and present equal segments; all that is required for this is, that the mean density of that part of the mixture, which is at the same level as the mass, must be equal to the density of the oil; but under these circumstances the level of the two segments is too low. In fact, the oil forming the upper segment is then in contact with a less dense liquid than itself, and, consequently, has a tendency to descend, whilst the opposite applies to the oil forming the inferior segment.* Heterogeneity of the liquid produces an opposite effect, i. e., it renders the height of the segments too great. In fact, the least dense portions rising to the upper part of the mass tend to lift it up, whilst the most dense portions descend to the lower part, and tend to depress it. Now,
* By intentionally producing very great heterogeneity in the alcoholic mixture, (0 9 of the preceding memoir,) and employing suitable precautions, a perfectly regular cylinder may be formed, the bases of which are absolutely plane.
the quantities of pure alcohol, and that at 16° added to the alcoholic mixture to balance the mass, necessarily produce an alteration in the homogeneity of the oil; for, in the first place, the oil during these operations being in contact with mixtures which are sometimes more, sometimes less charged with alcohol, must absorb or lose some of this by its surface; in the second place, these same additions of alcohol to the mixture diminish the saturation of the latter with the oil, so that it removes some of it from the mass; and this action is undoubtedly not equally exerted upon the two principles of which the oil is composed. Hence, before taking the measures, the different parts of the oil must be intimately mixed together, which may be effected by introducing an iron spatula into the mass, moving it about in it in all directions, and this for a long time, because the mixture of the oil can only be perfectly effected with great difficulty on account of its viscidity.
#. avoid the influence of the reactions which render the oil heterogeneous, the operations must be conducted in the following manner: The mass being introduced into the vessel and attached to the two rings, and the equality of. the densities being perfectly established, allow the mass to remain in the alcoholic liquid for two or three days, re-establishing from time to time the equilibrium of the densities altered by the chemical reactions and the variations of temperature. Afterwards remove the two rings from the vessel, so that the mass remains free; remove almost the whole of this, by means of a siphon, into a bottle, which is to be carefully corked; withdraw with the syringe the small portion of oil which is left in the vessel, and reject this portion. Next replace the two lings, and mix the alcoholic liquid perfectly; then again introduce the oil into the vessel, taking the precaution of enveloping the bottle containing it with a cloth several times folded, so that the temperature may not be sensibly altered by the heat of the hand.* Then attach the mass to the lower ring only, the upper ring being raised as much as possible; mix the oil intimately, as we have said above; then depress the upper ring, cause the mass to adhere to it, elevate it so as to form an exact cylinder, and proceed immediately to the Imeasurement.
* The following is the reason why the oil must be removed from the vessel before employing it for the experiment. After having remained a considerable time in the alcoholic liquid, the oil becomes enveloped by a kind of thin pellicle; or, more strictly speaking, the superficial layer of the mass has lost part of its liquidity, an effect which undoubtedly arises from the unequal action of the alcohol, upon the principles of which the oil is composed. The necessary result of this is, that the mass loses at the same time part of its tendency to assume a determinate figure of equilibrium, which tendency must, therefore, be completely restored to it. This is why the oil is withdrawn by the siphon. In fact, the pellicle does not trate the interior of the latter, and during its contraction continues to envelop the small portion remaining; so that after the latter has been removed by the syringe, which ultimately absorbs the pellicle itself, we get completely rid of the latter.
Before using the siphon, the thickness and consistence of the pellicle are too slight to enable us distinctly to perceive its presence; but when the operation of the siphon is nearly terminated, and the mass is thus considerably reduced, we find that the surface of the latter forms folds, hence implying the existence of an envelope. Moreover, when the siphon is removed, the small residuary mass, which then remains freely suspended in the alcoholic liquid, no longer assumes a spherical form, but retains an irregular aspect, appearing to have no tendency to assume any regular form.
This indifference to assume figures of equilibrium, arising from a diminution in the liquidity of the superficial layer, constitutes a new and curious proof of the fundamental principle relating to this layer, (§§ 6 bis and 10 to 16.) M. Hagen (Mémoire sur la Surface des Liquides, in the Memoirs of the Academy of Berlin, 1845) has observed a remarkable fact, to which the preceding appears to be related. It consists in this, that the surface of water, left to itself for some time, undergoes a peculiar modification, in consequence of which the water then rises in capillary spaces to elevations which are very distinctly less than is the case when its surface is exempt or freed from this alteration. This fact might, perhaps, be explained by admitting that the water dissolves a small proportion of the substance of the solid with which it is in contact, and that the external air acts chemically at the surface of the liquid upon the substance dissolved, thus giving rise to the formation of a slight pellicle which modifies the effects of the molecular forces.
42. The instrument best suited for effecting the latter operations in an exact manner is undoubtedly that which has received the name of cathctometer, and which, as is well known, consists of a horizontal telescope moving along a vertical divided rule. The distance comprised between the summits of the two segments is first measured by the aid of this instrument; the distance included between the external planes of the two rings (§ 40) is then measured by the same means. The difference between the first and the second result evidently gives the sum of the two heights, the mean of which must be taken ; and, consequently, this mean, or the quantity sought, h - f, is equal to half the difference in question.
The determination of the distance between the external planes of the rings requires peculiar precautions. First, as the points of the rings at which we must look are not exactly at the external surface of the figure, the oil interposed between these points and the eye must produce some effects of refraction, which would introduce a slight error into the value obtained. To avoid this inconvenience, we need only expose the rings by allowing the liquids to escape from the vessel by the stop-cock, (note 2 to $ 9,) then remove the minute portions of the liquid which remain adherent to the rings by passing lightly over their surface a small strip of paper, which must be introduced into the vessel through the second aperture. The drops of alcoholic liquid remaining attached to the inner surface of the interior side of the vessel must also be absorbed in the same manner. In the second place, as it would be difficult for the rings to be rigorously parallel, their distance must be measured from two opposite sides of the system, and the mean of the two valves thus found taken. The following are the results whicb I obtained: The mensuration of the distance between the summits gave first, in four successive operations, the values 76.77, 76.80, 76.85, and 76.75 milli neters, the mean of which is 76.79 millimeters. But after the alcoholic liquid had been again agitated for some time, to render its homogeneity more certain, two new measurements taken immediately afterwards gave 77.05 and 77.00 millimeters, or a mean of 77.02 millimeters. The distance between the external planes of the rings was found, on the one hand, by two observations, which agreed exactly, to be 57.73 millimeters; on the other hand, two observations furnished the values 57.87 and 57.85 millimeters, or as the mean 57.86 millimeters. Taking, then, the mean of these two results, we get 57.79 millimeters as the value of the distance between the centres of the external planes. Hence, if we assume the first of the two values obtained for the distance of the summits, 76.79 millimeters, we find
76.79 — 57.79 N j =
=9.50 millimeters ; and if from the second result, 77.02 millimeters, we find
2 =9.61 millimeters. These two elevations evidently differ but little from 9.41 millimeters, the altitude deduced from theory, (§ 40;) in the first case the difference does not amount to the tooth part of this theoretical value, and in the second it hardly exceeds 18 óths. These differences undoubtedly arise from slight remains of heterogeneity in the liquids ; it is probable that in the first case neither of the two liquids was absolutely homogeneous, and that the two contrary effects which thence resulted (§ 41) partly neutralized each other, whilst in the second case, the alcoholic liquid being rendered perfectly homogeneous, the effect of the slight heterogeneity of the oil exerted its full influence. Ilowever this may be, these differences in each case are so small that we may consider experiment as in accordance with theory, of which it evidently presents a very remarkable confirmation.
43. Mathematically considered, a cylindrical surface extends indefinitely in the direction of the axis of revolution. Hence it follows that the cylinder