HYDRODYNAMICS. 



477 



15.1311 , . 



9= -/= 14.1. jit; and since 



1 .\J\jAJ 



^ 

 i m 



tn the 



-'' 



fe ' 

 r *ur- 



h = f f 1 -^-Y subtracting 



^Y 1 ) = 0.1 147, we have 



h = 14.0397, which differs very little from 13.57*, the 

 observed altitude. 



It will be seen from the formula No. 2. that of all 

 tubes that have a prismatic form, the hollow cylinder 

 is the one in which the volume of fluid raised is the 

 least possible, as it has the smallest perimeter. It ap- 

 pear* also, that if the section of the tube is a regular 

 polygon, the altitudes of the fluid will be reciprocally 

 proportional to the homologous lines of the similar base, 

 a molt which, as we have seen, M. Gellert obtained 

 from direct experiment. Hence in all prismatic tulx.-s 

 whose sections are polygons inscribed in the same cir- 

 cle, the fluid will rise to the same mean height. If one 

 of the two bases is, for example, a square, and the other 

 an equilateral triangle, the altitudes will be as 2 : 3J, 

 or very nearly as 7 : 8, 



M. La Place has remarked, that there may be sevc- 

 ral states of equilibrium in the same tube, provided its 

 width is not uniform. If we suppose two capillary 

 tubes communicating with one another, so that the 

 smallest is placed above the greatest, we may then con- 

 ceive their diameters and lengths to be such, that the 

 fluid i* at first in equilibrium above its level in the 

 widest tube, and that in pouring in some of the same 

 fluid, so as to reach the smaller tube, and fill part of it, 

 the fluid will still maintain itself in equilibrium. When 

 the diameter of a capillary tube diminishes by insensi* 

 ble gradations, the different states of equilibrium are 

 alternately stable and instable. At first the fluid tends 

 to raise itself in the tube, and this tendency diminish- 

 ing, becomes nothing in a state of equilibrium. Be- 

 yond this it become* negative, and consequently the 

 fluid tends to descend. Thus the first equilibrium is 

 stable, since the fluid, being a little removed from this 

 state, tends to return to it. In continuing to raise the 

 iukl, its tendency to descend diminishes, and becomes 

 nothing in the second state of equilibrium. Beyond 

 this it becomes positive, and the fluid tends to rise, and 

 consequently to remove from this state which is not 

 stable. In a similar manner it will be seen, that the 

 third state is stable, the fourth iiutable, and so on. 



Although the preceding method of considering the 

 phenomena of capillary attraction is extremely simple 

 and accurate, yet it does not indicate the connection 

 which subsists between the elevation and depression of 

 the tlu "l. ""I tnc concavity or convexity of the surface 

 which every fluid assumes in capillary spaces. Toe ob- 

 ject of M. La Place's irst method, contained in his first 

 supplement, is to determine this connection. 



By mean* of the methods for calculating the attrac- 

 tion of spheroids, he determine* the action of a mas* of 

 fluid terminated by a spherical surface, concave or con* 

 vex, upon a column of fluid contained in an infinitely 

 narrow canal, directed towards the centre of this surface. 

 By this action La Place means the pressure which the 

 fluid contained in the canal would exercise, in virtue of 



the attraction of its entire mass upon a plane base si- On Capillary 

 tuated in the interior of the canal, and perpendicular Attraction 

 to its sides, at any sensible distance from the surface, , *" . 

 this base being taken for unity. He then shews that "fluid*. 

 this action is smaller when the surface is concave than -_- _' 

 when it is plane, and greater when the surface is con- 

 vex. The analytical expression of tins action is compo- 

 sed of two terms : the first of these terms, which is 

 much greater than the second, expresses the action of 

 the mass terminated by a plane surface,* and the se- 

 cond term expresses the part of the action due to the 

 sphericity of the surface, or, in other word?, the action 

 of the meniscus comprehended between this surface and 

 the plane which touches it This action is either ad- 

 ditive to the preceding, or subtractive from it, accor- 

 ding as the surface is convex or concave. It is reci- 

 procally proportional to the radius of the spherical sur- 

 face ; for the smaller that this radius is, the meniscus i* 

 the nearer to the point of contact. 



From these results relative to bodies terminated by 

 sensible segments of a spherical surface, La Place de- 

 duces this general theorem. " In all the laws which 

 render the attraction insensible at sensible distances, 

 the action of a body terminated by a curve surface up. 

 on an interior canal infinitely narrow, perpendicular to 

 this surface in any point, is equal to the half sum of 

 the actions upon the same canal of two spheres, which 

 have for their racjii the greatest mid the small i-M of the 

 radii of the osculating circle of the surface at this 

 point" 



By means of this theorem, and the laws of hydro - 

 statics, La Place has determined the figure which a 

 mass of fluid ought to take when acted upon by gra- 

 vity, or contained in a vessel of a given figure. The 

 nature of the surface is expressed by an equation of 

 partial difference* of the second order, which cannot be 

 integrated by any known method. If the figure of the 

 surface i* one of revolution, the equation is reduced to one 

 of ordinary differences, and i* capable -of being integra- 

 ted by approximation, when the surface is very small. 

 La Place next shews, that a very narrow tube ap- 

 proaches the more to that of a spherical segment as the 

 diameter of the tube becomes smaller. If these seg- 

 ment* are similar in different tubes of the same sub- 

 stance, the radii of their surfaces will be inversely as 

 the diameter of the tubes. This similarity of the sphe- 

 rical segments will appear evident, if we consider that 

 toe distance at which the action of the tube cease* to be 

 sensible, is imperceptible ; so that if, by means- of a very 

 powerful microscope, this distance should be found 

 equal to a millimetre, it is probable that the same mag- 

 nifying power would give to the diameter of the tube 

 an apparent diameter of several metre*. The surface 

 of the tube may therefore be considered a* very nearly 

 plane, in a radius equal to that of the sphere of sensi- 

 ble activity ; the fluid in this interval will therefore de- 

 scend, or rise from this surface, very nearly as if it were 

 plane. Beyond this the fluid being subjected only to 

 the action of gravity, and the mutual action of it* own 

 particles, the surface will be very nearly that of a sphe- 

 rical segment, of which the extreme planes being those 

 of the fluid surface, at the limits of the sphere of the 

 sensible activity of the tube, will be very nearly in 

 different tubes equally inclined to their sides. Hence 

 it follow* that all the segment* will be similar. 



M. La Plae* \ of opinion, that th nuptmtoa of mercury to a barometer tube, at a height two or three titnei greater than that 

 wfckb ! due U lW prtMur* of tb* almwptwr*, depend* on this term. He cMKtiv** too, that the refracting fore* of tract parent bo- 

 *c, cobofeo, and in general chemical aflttty, ttftmt also upon iu 



