that the total length of the cylinder, although limited, is equal to the product of the normal length of the divisions by a whole number, or rather a whole number plus a half, nothing will prevent the divisions from exactly assuming this normal length. If, on the other hand, which is generally the case, the total length of the cylinder fulfils neither of the preceding conditions, we should think that the divisions would assume the nearest possible to the normal length; and then, all other things being equal, the difference will evidently be as much less as the divisions are more numerous, or, in other words, as the cylinder is longer. We should also believe that the transformation would adopt that of the two methods which is best adapted to diminish the difference in question, and this is also confirmed by experiment, as we shall see presently. Hence, although, as I have already stated, the transformation of the cylinder of mer. cury almost always ensues in one of the two normal methods, the result is rarely very regular; we must, therefore, admit that slight accidental disturbing causes in general render the divisions formed in any one experiment unequal in length; but then the expressions of a obtained above evidently give in each experiment the mean length of these divisions, or, in other words, the common length which the divisions would have taken if the transformation had occurred in a perfectly regular manner, giving rise to the same number of isolated spheres and to the same state of the terminal masses. Lastly, since the third method of transformation presents itself, i. e., since it sometimes happens that each of the bases is occupied by a mass of the small kind, if we would leave out of consideration the particular cause of irregularity inherent in this method, (the preceding paragraph,) and find the corresponding expression of 2, it need only be remarked that each of the terminal masses then proceeds from a semi-constriction or the fourth of a division, which will 2 54. I shall now relate the results of the experiments. The diameter of the copper wires, consequently of the cylinder, was 1.05 millimeter. I first gave the cylinder a length of 90 millimeters, and repeated the experiment ten times, noting after each the number of isolated spheres produced, and the state of the masses adherent to the bases; I then calculated for each result the corresponding value of the length of a division, by means of that of the three formulæ of the preceding paragraph which refers to this same result, I afterwards made ten more experiments, giving the cylinder a length of 100 millimeters, and alzo calculated the corresponding values of the length of a division. The table contains the results furnished by these cylinders, and the values deduced for the length of a division. I only obtained a perfectly regular result in one case in each series; I have placed an * opposite the corresponding number of isolated spheres. This table shows, in the first place, that the different values obtained for the length of a division are not so far removed from each other as to prevent our perceiving a constant value, the uniformity of which is only altered by the influence of slight accidental causes. In the second place, out of twenty exriments, it happened once only that the masses adherent to the bases were oth of the small kind. In the third place, both the perfectly regular results have given identically the same value for the length of a division; this value, expressed approximatively to two decimal places, is 6.67 millimeters; but its exact expression is 63 millimeters; for the operation to be effected consists, in the case of the first series, in the division of 90 millimeters by 13.5, and, in the case of the second series, in the division of 100 millimeters by 15. As the two lengths given to the cylinder are considerable in proportion to the diameter, and consequently the numbers of division are tolerably large, this value, 63 millimeters, ought very nearly, if not exactly, to constitute that of the normal length of the divisions. It is seen, moreover, that to give the divisions this closely approximative or exact value of the normal length, the transformation has chosen, in one case the first, in the other case the second method. 55. Let us pursue our inquiry into the laws of the phenomenon with which . we are engaged; we shall soon make an important application of them, and it will then be understood why so extensive a development is given to this part of our work. It might be regarded as evident a priori that two cylinders formed of the same liquid and placed in the same circumstances, but differing in diameter, would tend to become divided in the same manner, i.e., that the respective normal lengths of the divisions would be to each other in the proportion of the diameters of these cylinders. In order to verify this law by experiment, I procured some copper wires, the diameter of which was exactly double that of the first, therefore equal to 2.1 millimeters, and I made with them a new series of ten experiments, giving the cylinder a length of 100 millimeters. This series also furnished me with only a single perfectly regular result, which I have denoted as before by an * placed opposite the corresponding number of isolated spheres. The following is the table relating to this series: | Number of Masses adherent to the bases. Longth isolated of a spheres. | division. millims. 7 | Two small------------------------------------------------------- 13.33 6 Two large-------------------------------------------------------- 13.33 6 One large and one small----------------------------- -------------- 14.28 7 | One large and one small ------------------------------------------ 12.50 *6 Two large-------------------------------------------------------- 13.33 6 Two large-------------------------------------------------------- 13.33 6 One large and one small------------------------------------------- 14.28 8 || One large and one small------------------------------------------- 11.11 8 Two small ---------------------------- --------------------------- 11.76 6 || One large and one small --------------------------- ---------------- 14.28 By stopping at the second decimal place, we have, as is evident, 13.33 millimeters for the value of the length of a division corresponding to the perfectly regular result; but as the operation which yields it consists in the division of 100 by 7.5, the value when perfectly expressed is 13% millimeters. This then is very nearly, if not exactly, the normal length of the divisions of this new cylinder; now this length, 134 millimeters, is exactly twice the length.6% millimeters, which belongs to the divisions of the cylinder of the preceding paragraph; these two lengths arc therefore, in fact, in the proportion to each other of the diameters of the two cylinders. As the perfectly regular result of the above table has given a mass of the larger kind to each base, it follows, that to enable the divisions of the cylinder itself to assume their normal length, or the nearest possible length to this, the transformation has necessarily ensued according to the former method; whilst in regard to a cylinder the diameter of which is a half less, and the total length of which is the same, 100 millimeters, the transformation ensued according to the second method, ($ 54.) Here, also, the case in which there are two masses of the small kind to the solid bases is the least frequent, although it occurred twice. Lastly, the different values of the length of a division are more concordant than in the second series relating to the first diameter, and consequently show the tendency towards a constant value better; we also see that the normal length is that which is most frequently reproduced. 56. According to the law which we have just established, when the nature of the liquid and external circumstances do not change, the normal length of the divisions is proportional to the diameter of the cylinder; or, in other words, the proportion of the normal length of the divisions to the diameter of the cylinder is constant. As we have seen, the diameter of the cylinder in paragraph 54 was 1.05 millimeters, and the normal length of its divisions was very little less than 6.67 millimeters; consequently, when the liquid used is mercury and the cylinder rests upon a plate of glass, the value of the constant proportion in question is 6.67 105 = 6.35, which approximates closely. To ascertain whether the nature of the liquid and external circumstances exert any influence upon this proportion, we shall now determine the value of the latter in the case of a cylinder of oil formed in the alcoholic mixture, which may be effected, at least approximatively, with the aid of the result of the experiment in paragraph 47. To simplify the considerations, we shall suppose that the transformation does not commence until the rapidity of transference has entirely ceased. The point of the funnel, on the one hand, and the section by which the imperfect liquid cylinder is in contact with the mass which collects at the bottom of the vessel, on the other hand, may then be regarded as playing the part of the two bases of the figure. Now it is evident that, as regards the second of these bases, the last portion of the figure which is transformed should be a constriction; for if it constituted a dilatation, there would be discontinuity of the curvature at the junction of the respective surfaces of the latter and the large mass, which is inadmissible. But the same reason does not apply to the other base; and experiment shows that in this case a dilatation is formed, because after the termination of the phenomenon we always find at the point of the funnel a mass comparable to the isolated spheres. Hence in this experiment the transformation ensues according to the second method. Therefore, as the whole length of the figure is about 200 millimeters, and as the transformation constantly yields two isolated spheres, the mean length of the divisions has 200 (s 53) for its approximative value millimeters = 66.7 millimeters; I say the mcan length, because, as the diameter of the figure increases slightly from the summit towards the base, the divisions are probably not exactly equal in length. It must be added here, that the transformation ensues under circumstances which are always identical, and consequently, in the absence of acci. dental disturbing canses, the above quantity ought to represent the normal length of the divisions, or the nearest possible length to the latter. Now, I 200 estimate the mean diameter of the figure before the transformation at about 4 millimeters; we should consequently have = 16.7 as the approximative value of the proportion between the normal length of the divisions and the diameter of the cylinder. This is, therefore, approximatively the constant proportion sought in the case of a cylinder of oil formed in the alcoholic mixture; now this proportion, as is evident, is much greater than that which belongs to the case of a cylinder of mercury resting upon a plate of glass. In fact, the length 66.7 millimeters may differ somewhat materially from the normal length; for if, on the one hand, the whole length of the figure of oil is considerable in regard to its diameter, on the other hand, the number of divisions which form there is very small. Let us then see, for instance, what is the least value which the normal length of these divisions may have. We must in the first place remark, that in this case, notwithstanding the absence of disturbing causes, the third method of transfor:nation is possible; in fact, as the lower constriction is adherent to a liquid base, nothing can prevent the oil which it loses from traversing this base to reach the large mass, so that in the third method, also, the direction of the movements of transport may be the same in regard to all the constrictions, (§ 52.) This granted, as the denominator of the expression which gives the length of one division can only vary by half units, (53,) and as the length which we have found resulted from the division of 200 millimeters by 3, it follows that the length immediately below would be millimeters = 57.1 millimeters, which would correspond to three isolated spheres and a transformation disposed according to the third method. But as matters do not take place in this manner, since there are never more than two isolated spheres formed, and the transformation always ensues according to the second method, we must conclude that the normal length of the divisions approximates more closely to the length found, 66.7 millimeters, than the length 57.1 millimeters. If, then, the normal length is greater than the first of these two quantities, it must at least be more than their mean, i. e., 61.9 millimeters; consequently the relation of the normal length of the divisions and the diameter i 61.9 of the cylinder is necessarily greater than = 15.5; now this latter number considerably exceeds the number 6.35, which corresponds to the mercurial cylinder. Thus, the proportion of the normal length of the divisions to the diameter of the cylinder varies, sometimes according to the nature of the liquid, sometimes according to external circumstances, at others according to both these elements. 57. But I say that there is a limit below which this proportion cannot descend, and that this is exactly the limit of stability. Let us imagine a liquid cylinder of sufficient length in proportion to its diameter, comprised between two solid bases, and the transformation of which is taking place with perfect regularity. Suppose, for the sake of clearness, that the phenomenon ensues according to the second method, or, in other words, that the terminal portions of the figure consist one of a constriction, the other of a dilatation; then, as we have seen, ($ 52,) the regularity of the transformation will extend to these latter portions ; i. C., the terminal constriction and the dilatation will be respectively identical with the portions of the same kind of the rest of the figure. Let us then take the figure at that period of the phenomenon at which it still presents constrictions and dilatations, and let us again consider the sections, the diameter of which is equal to that of the cylinder, ($ 52.) Let us start from the terminal constricted portion; the solid base upon which this rests, and which constitutes the first of the sections in question, will occupy, as we have shown, the origin of the constriction itself; we shall then have a second section at the origin of the first dilatation; a third at the origin of the second constriction; a fourth at the origin of the second dilatation, and so on; so that all the sections of the even series will occupy the origins of the dilatations, all those of the odd series the origins of the constrictions. The interval comprised between two consecutive sections of the odd series will therefore include a constriction and a dilatation; and as the figure begins with a constriction and terminates with a dilatation, it is clear that its entire length will be divided into a whole number of similar intervals. In consequence of the exact regularity which we have supposed to exist in the transformation, all the intervals in question will be equal in length; and as the moment at which we enter upon the consideration of the figure may be taken arbitrarily from the origin of the phenomenon to the maximum of the depth of the constrictions, it follows that the equality of length of the intervals subsists during the whole of this period, and that, consequently, the sections which terminate these intervals preserve during this period perfectly fixed positions. Besides the parts of the figure respectively contained in each of the intervals undergoing identically and simultaneously the same modifications, the volumes of all these parts remain equal to each other; and as their sum is always equal to the total volume of the liquid, it follows that, from the origin of the transformation to the maximum of depth of the constrictions, each of these partial volumes remains invariable, or, in other words, no portion of liquid passes from any one interval into the adjacent ones. Thus, at the instant at which we consider the figure, on the one hand, the two sections which terminate any one interval will have preserved their primitive positions and their diameters; and on the other hand these sections will not have been traversed by any portion of liquid. Matters will then have occurred in each interval in the same manner as if the two sections by which it is terminated had been solid disks. But the transformation cannot ensue between two solid disks, if the proportion of the distance which separates the disks to the diameter of the cylinder is less than the limit of stability; the proportion of the length of our intervals and the diameter of the cylinder cannot then be less than this limit. Now, the length of an interval is evidently equal to that of a division; for the first, in accordance with what we have seen above, is the sum of the lengths of a dilatation and a constriction ; and the second is the sum of the lengths of a dilatation and two semi-constrictions, (§ 53 ;) the proportion of the length of a division to the diameter of the cylinder cannot then be less than the limit of stability; and we may remark here that this conclusion is equally true, whether the divisions are able or not to assume exactly their normal length. 58. Let us now attempt to ascertain the influence of the nature of the liquid and that of external circumstances, commencing with the latter. Our liquid cylinder of mercury, along the whole of the line at which it touches the plate of glass, must contract a stight adherence to this plate, which adherence must more or less impede the transformation. To discover whether this resistance exerted any influence upon the normal length of the divisions, consequently upon the proportion of the latter to the diameter of the cylinder, a simple means presented itself, viz., to augment this resistance. To arrive at this result, I arranged the apparatus in such a manner as to remove only one of the strips of glass, so that the liquid figure then remained simultaneously in contact with the plate and the other strip. I again repeated the experiment ten times, using copper wires 1.05 millimeters in diameter, and giving the cylinder a length of 100 millimeters. The following were the results: |