It is evident that the different values of the length of a division, with a single exception, are all obviously greater than all those which relate to a cylinder of the same diameter, the surface of which only touches the glass by a single line, (§ 54.) We must thence conclude that, all other things being the same, the length of the divisions increases with the external resistance; consequently, under the action of the same resistance, this length is necessarily greater than it would be if the convex surface of the cylinder had been perfectly free.

In the above series neither of the results appears to be very regular; but we can readily understand that the mean of the values of the third column will approach the normal length of the divisions. This is, moreover, confirmed by the tables in §§ 54 and 55. If we take in the former the respective means of the values of the two series, we find for one 6.77 millimeters, and for the other 7.17 millimeters, quantities, the first of which is nearly equal to the length 6.67 millimeters, which may be considered as the normal length, and from which the second does not differ much; and if likewise we take the relative mean in the following table, we find 13.15 millimeters, a quantity very near the length 13.33 millimeters, which in the case of the second table may also be regarded as the normal length. Now, the corresponding mean in the above table is 10.81 millimeters; consequently, in the case of two lines of contact we have

10.81

1.05

10.29 as the approximate value of the proportion of the normal length of the divisions to the diameter of the cylinder, whilst in the case of a single line of contact we found only 6.35. Hence the proportion between the normal length of the divisions and the diameter of the cylinder increases by the effect of an external resistance.

59. Let us proceed to the influence of the nature of the liquid. All liquids are more or less viscid; i. e., their molecules do not enjoy perfect mobility with regard to each other. Now, this gives rise to an internal resistance, which must also render the transformation less easy; and as external resistances increase the length of the divisions, we can understand that the viscidity will act in the same manner; consequently, all other things being equal, the proportion now under consideration will increase with this viscidity. But, on the other hand, with equal curvatures, the intensities of the forces which produce the transformation vary with the nature of the liquid; for these intensities depend, in the case of each liquid, upon that of the mutual attraction of the molecules. Now, it is clear that the viscidity will exert so much more influence upon the length of the divisions as the intensities of the forces in question are less. Thus, leaving external resistances out of the question, the proportions of the normal length of the divisions to the diameters will be greater in proportion to the viscidity of the liquid and the feebleness of the configuring forces.

The intensities of the configuring forces corresponding to different liquids may be compared numerically for the same curvatures. In fact, let us first bear in mind that the pressure corresponding to one element of the superficial layer, and reduced to unity of the surface, is expressed by (§ 4,)

Now, the value of the part P of this pressure being the same for all the elements of the superficial layer, and the pressures being transmitted throughout the mass, this part P will always be destroyed, whether equilibrium exists in the liquid figure or not; so that the active part of the pressure (that which consti

tutes the configuring force) will have for its measure simply(+r)

Hence it is evident that when the curvatures are equal, the intensity of the configuring force arising from one element of the superficial layer is proportional to the coefficient A. Now, this coefficient is the same as that which enters into the known expression of the elevation or depression of a liquid in a capillary tube: consequently the measures relating to this elevation or depression will give us, in the case of each liquid, the value of the coefficient in question. Hence we may also say that the proportion of the normal length of the divisions to the diameter of the cylinder will be greater as the liquid is more viscid and as the value of A which corresponds to the latter diminishes. For instance, oil is much more viscid than mercury; on the other hand, it would be easy to show that the value of A is much less for the first than for the second of these two liquids; lastly, this value must be much diminished in regard to our figure of oil by the presence of the surrounding alcoholic liquid, the mutual attraction of the molecules of the two liquids in contact diminishing the intensities of the pressures, (§ 8.) This is why the proportion belonging to a cylinder of oil formed in the alcoholic mixture considerably exceeds that belonging to a cylinder of mercury resting upon a plate of glass, notwithstanding the slight external resistance to which the latter is subjected.

60. It follows from this discussion concerning the resistances that the smallest value which the proportion of the normal length of the divisions to the diameter of the cylinder could be supposed to have corresponds to that case in which there is simultaneously complete absence of external resistance and of viscidity; and, after the demonstration given in § 57, this least value will be at least equal to the limit of stability. Now, as all liquids are more or less viscid, it follows that, even on the hypothesis of the annihilation of all external resistance, the proportion in question will always exceed the limit of stability; and since this is more than 3, this proportion will, à fortiori, be always more than 3.

It is conceivable that the least value considered above, i. e., that which the proportion would have in the case of complete absence of resistance, both internal as well as external, would be equal to the limit of stability itself, or would very slightly exceed it. In fact, on the one hand, the proportion approximates to this limit as the resistances diminish, and on the other hand, if it exceeds it, the transformation becomes possible, (§ 57;) hence we see no reason why it should differ sensibly from it if the resistances were absolutely null. The results of our experiments, moreover, tend to confirm this view. First,. since the proportion belonging to our cylinder of mercury descends from 10.29 to 6.35, passing from that case in which the cylinder touches the glass at two lines to that where it touches it at a single one only, (§ 58,) it is clear that if this latter contact itself could be suppressed, which would leave the influence of the viscidity alone remaining, the proportion would become much less than

6.35; and as, on the other hand, it must exceed 3, we might admit that it would at least lie between the latter number and 4, so that it would closely approximate the limit of stability. If, then, it were possible to exclude the viscidity also, the new decrease which the proportion would then experience, would very probably bring the latter to the very limit in question, or at least to a value differing but exceedingly little from it. Thus, on the one hand, the least value of the proportion, that corresponding to the complete absence of resistances, would not differ, or scarcely so, from the limit of stability; and on the other hand, under the influence of viscidity alone, the proportion appertaining to the mercury would be but little removed from this least value. Hence it is evident that the influence of the viscidity of mercury is small, which is moreover explained by the well-known feebleness of this same viscidity.

We can now understand in the case of other but very slightly viscid liquids, such as water, alcohol, &c., where the viscidity is not able to form more than a minimum resistance, that this viscidity, notwithstanding the differences in the intensities of the configuring forces, will also exert only a feeble influence upon the proportion in question. Hence it results that, in the absence of all external resistance, the values of this proportion respectively corresponding to the various very slightly viscid liquids cannot be very far removed from the limit of stability; and as the smallest whole number above this is 4, we may in regard to these liquids adopt this number as representing the mean approximative probable value of the proportion in question.

Starting from this value, calculation gives us the number 1.82 as the proportion of the diameter of the isolated spheres which result from the transformation to the diameter of the cylinder, and the number 2.18 for the proportion between the distance of two adjacent spheres and this diameter.

61. There is another consequence arising from our discussion. For the sake of simplicity let the diameter of the cylinder be taken as unity. The propor tion of the normal length of the division to the diameter will then express this normal length itself, and the proportion constituting the limit of stability will express the length corresponding to this limit. This admitted, let us resume the conclusion at which we arrived at the commencement of the preceding sec-tion, which conclusion we shall consequently express here by stating that in the case of all liquids the normal length of the divisions always exceeds the limit of stability; we must recollect, in the second place, that the sum of the lengths of one constriction and one dilatation is equal to that of a division, (§ 57;) and, thirdly, at the first moment of the transformation the length of one constriction is equal to that of a dilatation, (§ 46.) Now, it follows from all. these propositions, that when the transformation of a cylinder begins to take place, the length of a single portion, whether constricted or dilated, is necessarily greater than half the limit of stability; consequently the sum of the lengths of three contiguous portions, for instance two dilatations and the intermediate constriction, is once and a half greater than this same limit. Hence, lastly, if the distance of the solid bases is comprised between once and once and a half the limit of stability, it is impossible for the limit of stability to give rise to three: portions, and it will consequently only be able to produce a single dilatation in juxtaposition with a single constriction. This, in fact, as we have seen, always took place in regard to the cylinder, in § 46, which was evidently in the above condition, and the want of symmetry in its transformation now becomes explicable.. 62. As stated at the conclusion of § 48, we have yet to describe a remarkable fact which always accompanies the end of the phenomenon of the transformation of a liquid cylinder into isolated masses.

In the transformation of large cylinders of oil, whether imperfect or exact, (§ 44 to 46,) when the constricted part is considerably narrowed, and the separation seems on the point of occurring, the two masses are seen to flow back rapidly towards the rings or the disks; but they leave between them a cylindri

18 s

cal line which still establishes, for a very short time, the continuity of the one with the other, (Fig. 28;) this line then resolves itself into partial masses. It

generally divides into three parts, the two extreme ones of which become lost in the two large masses, the intermediate one forming a spherule, some millimeters in diameter, which remains isolated in the middle of the interval which separates the large masses; moreover, in each of the intervals between this spherule and the two large masses, another very much smaller spherule is seen, which indicates that the separation of the parts of the above line is also effected by attenuated lines. Fig. 29 (Pl. VIII) represents this ultimate state of the liquid system. The same effects are produced when the resolution of the thin and elongated cylinder of oil of § 47 into spheres occurs, only there is in one or the other of the intervals between the spheres frequently a larger number of spherules, and, besides, the formation of the principal line is less easily observed, in consequence of the more rapid progress of the phenomena. Lastly, in the case of our cylinders of mercury, the resolution into spheres takes place also in too short a time to allow of our perceiving the formation of the lines; but we always find, in several of the intervals between the spheres, one or two very minute spherules, whence we may conclude that the separation is effected in the same manner.

* We cannot avoid recognizing an analogy between the phenomenon of the formation of liquid lines and that of the formation of lamina. In fact, in the experiment in § 23, for instance, the plane layer begins to be formed when the two opposite concave surfaces are almost in contact with each other at their summits; and in the resolution of a cylinder into spheres, the formation of the lines commences when all the meridional sections of the figures almost touch each other by the summits of their concave portions.

When treating of the layers, we have considered their formation as indicating a kind of tendency towards a particular state of equilibrium, which results from the circumstance that in the case of the thin part of the liquid system the ordinary law of pressure is modified. For the analogy between the two orders of phenomena to be complete, it would, therefore, be necessary that excessively delicate liquid lines should connect thick masses, and should thus form with these masses a system in equilibrio, notwithstanding the incompatibility of this equilibrium with the ordinary law of pressures. Now, we shall show that this equilibrium is in reality possible, at least theoretically. Let us always take as example the resolution of our unstable cylinder into partial masses. When the cylindrical lines form, their diameter is even then very small in comparison with the dimensions of the thick masses; consequently their curvature in the direction perpendicular to the axis is very great in comparison with the curvature of these masses. The pressure corresponding to the lines is then originally much greater than those corresponding to the thick masses, whence it follows that the liquid must be driven from the interior of the lines towards these same masses, and that the lines, like the layers, ought to continue diminishing. Moreover, their curvatures, and conse quently their pressure augmenting in proportion as they become more attenuated, their tendency to diminish in thickness will go on increasing, and consequently if we disregard the instability of the cylindrical form, we see that they must become of an excessive tenuity. But I say that the augmentation of the pressure will have a limit, beyond which this pressure will progressively diminish, so that it may become equal to that which corresponds to the thick parts of the liquid system.

In fact, without having recourse to theoretical developments, it is readily seen that if the diameter of the line becomes less than that of the sphere of the sensible activity of the mole cular attraction, the law of the pressure must become modified, and, the diameter continuing to decrease, the pressure must finish by also progressively diminishing, notwithstanding the increase of the curvatures, in consequence of the diminution in the number of attracting molecules. Hence the pressure may diminish indefinitely; for it is clear that it would entirely vanish if the diameter of the line became reduced to the thickness of a single molecule. Those geometricians who study the theory of capillary action know that the formule

As we are now acquainted with the entire course which the transformation of a liquid cylinder into isolated spheres must take, we can represent it graphi

Fig. 30.

cally. Fig. 30 represents the successive forms through which the liquid figure passes, commencing with the cylinder up to the system of isolated spheres and of this theory cease to be applicable in the case of very great curvatures, or those the ra lii of which are comparable to that of molecular attraction. Now, it follows from what has been stated, that we may always suppose the thinness of the line to be such that the corresponding pressure may be equal to that existing in thick masses which have attained a state of equilibrium. In this case, adinitting that the lines are mathematically regular, so that the pressure there may be everywhere rigorously the same, consequently that they have no tendency to resolve themselves into small partial masses, equilibrium will necessarily exist in the system. In this case the form of the thick masses will not be mathematically spherical; for their surface must become slightly raised at the junctures with the lines by presenting concave curvatures in the meridional direction. This form will be the same as that of an isolated mass, traversed diametrically by an excessively minute solid line, (§ 10.) This system, like those into the composition of which layers enter, is composed of surfaces of a different nature; but this heterogeneity of form becomes possible here, as in the case of the layers, in consequence of the change which the law of pressures undergoes in passing from one to another kind of surface.

We can, moreover, understand that the equilibrium in question, although possible theoretically, as we have shown, can never be realized, in consequence of the cylindrical form of the lines. The same does not apply to the case of the plane layers; for, as we shall show in the following series, the plane surfaces are always surfaces of stable equilibrium, whatever may be their extent.