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We may, therefore, I think, legitimately generalize the above law in accordance with the whole of the remarks made in the preceding section, and deduce the following conclusions: 1. If we conceive a cylinder of mercury formed in vacuo or in air, of sufficient length to furnish several spheres, its convex surface being entirely free, and its length such that the divisions assume exactly their normal length, the time which will elapse from the origin of the transformation to the instant of the rupture of the lines will be exactly or apparently proportional to the diameter of this cylinder. 2. The same very probably applies to a cylinder formed of any other very slightly viscid liquid, as water, alcohol, &c., and supposed to exist under the same circumstances. 3. It is possible that this law is completely general. i. e., applicable to a cylinder formed, always under the same circumstances, of any kind of liquid whatever; but our experiments leave us in doubt on this point. 66. Let us now enter upon the consideration of the absolute value of the time in question for a given diameter, the cylinder always being supposed to be produced in vacuo or in air, of sufficient length to furnish several spheres, its entire convex surface free, and its length such that its divisions assume their normal length. It is clear that this absolute value must vary according to the nature of the liquid; for it evidently depends upon the density of the latter, upon the intensity of its configuring forces, and, lastly, upon its viscidity. The experiments which we have detailed give with regard to oil a very remote superior limit; this results, first, from the two causes which we have mentioned in § 64, and which are due to the presence of the alcoholic liquid; but with these two causes is connected a third, which we must make known. If we imagine a cylinder of oil formed under the above conditions, the sum of the lengths of a constriction and a dilatation will necessarily be much greater in regard to this cylinder than in regard to one of our short cylinders of oil of the same diameter; for in the former this sum is equivalent to the length of a division; and in consequence of the great viscidity of the oil, this latter quantity must greatly exceed the length corresponding to the limit of stability. Now, it may be laid down as a principle, that, all other things being equal, an increase in the sum of the lengths of a constriction and a dilatation tends to render the transformation more rapid, and consequently to abbreviate the total and partial durations of the phenomenon. In fact, for a given diameter, the more the sum in question differs from the length corresponding to the limit of stability, the more the forces which produce the transformation must act with .#. moreover, as the transformation ceases to take place immediately above the limit of stability, the duration of the phenomenon may then be considered as infinite, whence it follows that when this limit is exceeded, the duration passes from an infinite to a finite value, consequently it must decrease rapidly as it deviates from this limit; lastly, this is also confirmed by the results of observation, as we shall show hereafter. Thus, even if it were possible to form in vacuo or in air one of our very short cylinders of oil, consequently to eliminate the two causes of retardation due to the presence of the alcoholic liquid, the duration relative to the cylinder would still exceed that which would relate to a cylinder of oil of the same diameter formed under the conditions we have supposed. I have said that the principle above established is confirmed by experiment, i. e., for the same diameter, the same liquid, and the same external actions, if any exist; when, from any cause, the sum of the lengths of a constriction and a dilatation augments, the total and partial durations of the transformation become less. We shall proceed to make this evident. In the experiments of the preceding section, the partial duration relating to the cylinder, the diameter of which was 15 millimeters, was, for instance, about 30 seconds, the mean, as shown by the table. Consequently, if we were to form in the alcoholic liquid a similar
cylinder of oil, the diameter of which is 4 millimeters, the partial duration of this, in virtue of the law which we have found, would be nearly equal to 30" + 4
= 8". Now, in the nearly cylindrical figure of oil of § 47, which figure is also formed in the alcoholic liquid, the mean diameter was ($ 56) about 4 millimeters. In this and the preceding figure, the diameter, the liquid, and the external actions then are the same; but in the former, the sum of the lengths of the constriction and the dilatation would only be equal to 4 millimeters,
+ 3.6 = 14.4 millimeters, whilst in the second, this sum, which is equivalent to the length of a division, was ($ 56) approximatively 66.7 millimeters. Now, on observing this latter figure, we recognize easily that the duration of its transformation is much less than 8". In truth, from the nature of the experiment, it is impossible with regard to this same figure to fix upon the commencement of the formation of a given constriction or dilatation, so that the complete duration should considerably exceed that which would be deduced by the simple inspection of the phenomenon; but the latter does not amount to one second, and there cannot be any doubt that it would be going too far to extend the complete duration, and à fortiori, the portion which terminates at the rupture of the lines, to two seconds. Thus in the case we have just considered, the sum of the length of a constriction and a dilatation becoming about four and a half times greater, the partial duration becomes at least four times less.
67. But if, in reckoning the absolute duration in the case of one of our short cylinders of oil, we only obtain with regard to this liquid one upper limit, and this much too high, the cylinder of mercury in § 55 (which cylinder is formed ia the air, and the length of which in proportion to the diameter is sufficient for the divisions to have assumed exactly, or very nearly, their normal length) will furnish us, on the contrary, in regard to this latter liquid, with a limit which is probably more approximative and which will be very useful to us.
First, in the case of this cylinder, the diameter of which, as we have said, was 2.1 millimeters, the transformation does not take place in a sufficiently short time for us to estimate with any exactitude the total duration of the phenomenon; I say the total duration, because in so rapid a transformation it would be very difficult to determine the instant at which the rupture of the lines occurs. To approximate as closely as possible to the value of this total duration, I have had recourse to the following process.
By successive trials, I regulated the beats of a metronome in such a manner, that on rapidly raising, at the exact instant at which a beat occurs, the system of glass strips belonging to the apparatus serving to form the cylinder, (s 50 and 51,) the succeeding beat appeared to me to coincide with the termination of the transformation; then having satisfied myself several times that this coincidence appeared very exact, I determined the duration of the interval between two beats, by counting the oscillations made by the instrument during two minutes, and dividing this time by the number of oscillations. I thus found the value 09.39 for the interval in question. The total duration of the transformation of our cylinder of mercury may therefore be valued approximatively at 0".39, or more simply, at 0.4.
But the entire convex surface of this cylinder is not free, and its contact with the plate of glass must exert an influence upon its duration, both directly as well as by the increase which it produces in the length of the divisions. Let us examine the influence in question under this double point of view.
The direct action of the contact with the plate is undoubtedly very slight; for as soon as the transformation commences, the liquid must detach itself from the glass at all the intervals between the dilated parts, so as only to touch the solid plane by a series of very minute surfaces belonging to these dilated parts; consequently, if the direct action of the contact of the plate were alone eliminated, i. e., if we could manage so that the entire convex surface of the cylinder should
be free, but that the divisions formed in it should acquire the same length as beforo, the total duration would scarcely be at all diminished.
There still remains the effect of the elongation of the divisions. The length of the divisions of our cylinder is equal to 6.35 times the diameter, (§ 56,) whilst, according to the hypothesis of the complete freedom of the convex surface, this length would very probably be less than four times the diameter, ($ 60.) Now, in virtue of the principle established in the preceding section, this increase in the length of the divisions necessarily entails a diminution in the duration, which diminution is more considerable in proportion as it occurs in the vicinity of the limit of stability; consequently, if it could be managed so that the elongation in question should not exist, the total duration would be very considerably increased. Thus the suppression of the direct action of the contact of the plate would only produce a very slight diminution of the total duration ; and the annihilation of the elongation of the divisions would produce, on the other hand, a very considerable increase in this same duration. If, then, these two influences were simultaneously eliminated, or, in other words, if the entire convex surface of our cylinder were free, the total duration of our transformation would be very considerably greater than the direct result of observation.
Now, the quantity which we have to consider is the partial, and not the total duration ; but, under the same circumstances, the first must be but little less than the second; for when the lines are about to break, the masses between which they extend even then approximate to the spherical form ; consequently, in accordance with the conclusion obtained above, we must admit that the partial duration under our present consideration, 2. e., that referring to the case of the complete freedom of the convex surface of the cylinder, would still exceed considerably the total duration observed, i, e., 0".4.
In starting from this value 0.4 as constituting the lower limit corresponding to a diameter of 2.1 millimeters, the law of the proportionality of the partial duration to the diameter will immediately give the lower limit corresponding to any other diameter; we shall find, e. g., that for 6 millimeters this limit would be 0".4 + 10
-=1".9, or more simply 2". If, then, we imagine a cylinder of mercury a centimeter in diameter, formed in vacuo or in air, of sufficient length to furnish several spheres, entirely free at its convex surface, and of such a length that its divisions assume their normal length, the time which will elapse from the origin of the transformation of this cylinder to the instant of the rupture of the lines will considerably exceed two seconds.
68. It will not be superfluous to present here a resumé of the facts and laws which the experiments we have described have led us to establish with respect to unstable liquid cylinders.
1. When a liquid cylinder is formed between two solid bases, if the proportion of its length to its diameter exceeds a certain limit, the exact value of which is comprised between 3 and 3.6, the cylinder constitutes an unstable figure of equilibrium.
The exact value in question is that which we denominate the limit of stability of the cylinders.
2. If the length of the cylinder is considerable in proportion to its diameter, it becomes spontaneously converted, by the rupture of equilibrium, into a series of isolated spheres, of equal diameter, equally distant, having their centres upon the right line forming the axis of the cylinder, and in the intervals of which, in the direction of this axis, spherules of different diameters are placed ; except that each of the solid bases retains a portion of a sphere adherent to its surface.
3. The course of the phenomenon is as follows: The cylinder at first gradually swells at those portions of its length which are situated at equal distances from
each other, whilst it becomes thinner at the intermediate portions, and the length of the dilatations thus formed is equal, or nearly so, to that of the constrictions; these modifications become gradually more marked, ensuing with accelerated rapidity, until the middle of the constrictions has become verythin; then, commencing at the middle, the liquid rapidly retires in both directions, still, however, leaving the masses united two and two by an apparently cylindrical line; the latter then experiences the same modifications as the cylinder, except that there are in general only two constrictions formed, which consequently include a dilatation between them; each of these little constrictions becomes in its turn converted into a thinner line, which breaks at two points and gives rise to a very minute isolated spherule, whilst the above dilatation becomes transformed into a larger spherule; lastly, after the rupture of the latter lines, the large masses assume completely the spherical form. All these phenomena occur symmetrically as regards the axis, so that, throughout their duration, the figure is always a figure of revolution. 4. We denominate dirisions of a liquid cylinder, those portions of the cylinder, each of which must furnish a sphere, whether we conceive these portions to exist in the cylinder itself, before they have begun to be apparent, or whether we take them during the transformation, i. e., whilst each of them is becoming modified so as to arrive at the spherical form. The length of a division consequently measures the constant distance which, during the transformation, is included between the necks of two adjacent constrictions. Moreover, by normal length of the divisions, we denominate that which the divisions would assume, if the length of the cylinder to which they belong were infinite. In the case of a cylinder which is limited by solid bases, the divisions also assume the normal length when the length of the cylinder is equal to the product of this normal length by a whole number, or rather a whole number and a half. Then, if the second factor is a whole number, the transformation becomes disposed in such a manner that during its accomplishment the figure terminates on one side with a constriction, and on the other with a dilatation; if the second factor is composed of a whole number and a half, the figure terminates on each side in a dilatation. When the length of the cylinder fulfils neither of these conditions, the divisions assume that length which approximates the most closely possible to the normal length, and the transformation adopts that of the two above dispositions which is most suitable for the attainment of this end. 5. In the case of a cylinder of a given diameter, the normal length of the divisions varies with the nature of the liquid, and with certain external circumstances, such as the presence of a surrounding liquid, or the contact of the convex surface of the cylinder with a solid plane. ''. all the subsequent statements we shall take the simplest case, i.e., that of the absence of external circumstances; in other words, we shall always suppose that the cylinders are produced in cacuo or in air, and that they are free as regards their entire convex surface. 6. Two cylinders of different diameters, but formed in the same liquid, and the lengths of which are such that the divisions assume in each of them their normal length, become subdivided in the same manner, i. e., the respective normal lengths of the divisions are to each other as the diameters of these cylinders. In other words, when the nature of the liquid does not change, the normal length of the divisions of a cylinder is proportional to the diameter of the latter. The same consequently applies to the diameter of the isolated spheres into which the normal divisions become converted, and to the length of the intervals which separate these spheres. 7. The proportion of the normal length of the divisions to the diameter of the cylinder always exceeds the limit of stability.
8. This proportion is greater as the liquid is more viscid and as the configuring forces in it are weaker.
9. In the case of a cylinder of mercury, this proportion is much less than 6, and we may admit that it is less than 4.
In the case of a cylinder composed of any other very slightly viscid liquid, such as water, alcohol, &c., it is very probable that the proportion in question is very nearly 4. Hence, in the case of the latter liquids, we have for the probable approximative value of the proportion of the diameter of the isolated spheres resulting from the transformation and the diameter of the cylinder, the number 1.82; and for that of the proportion of the distance of two adjacent spheres to this same diameter, the number 2.18.
10. If mercury is the liquid, and the divisions have their normal length, the time which elapses between the origin of the transformation and the instant of the rupture of the lines, is exactly or apparently proportional to the diameter of the cylinder.
This law very probably applies also to each of the other very slightly viscid liquids.
This same law may possibly be general, i. e., it may be applicable to all liquids; but our experiments leave this point uncertain.
11. For the same diameter, and when the divisions are always of their normal length, the absolute value of the time in question varies with the nature of the liquid.
12. In the case of mercury, and with a diameter of a centimeter, this absolute value is considerably more than two seconds.
13. When a cylinder is formed between two solid bases sufficiently approximated for the proportion of the normal length of the cylinder to the diameter to be comprised between once and once and a half the limit of stability, the transformation gives only a single constriction and a single dilatation ; we then obtain for the final result only two portions of a sphere which are unequal in volume and curvature, respectively adherent to solid bases, besides interposed spherules.
(TO BE CONTINUED IN THE NEXT REPORT.)