From these comparisons it is obvious, that the mean altitudes, or the values of q, are very nearly reciprocally proportional to the diameters of the tubes ; for, in the ex periments on water, the value of q deduced from this ratio is 15.895, which differs little from 15.9034, the value found from experiment ; and that in accurate experiments, the correction made by the addition of the sixth part of the diameter of the tube is indispensably requisite.
If the section of the pipe in which the fluid ascends is a rectangle, whose greater side is a, and its lesser side d, then the base of the elevated column will be = a d, and its perimeter c = 2 a + 2 d. Hence, the value of the menis a a 8 d _a 2 2 cus will be 4 1 ), that is, q 2 h 2 'r + 4(1 ) . Hence, if in the general equation No. 1. we substitute for c its equal 2 a + 2 d, and for V its equal a d q, we have gDqad=2ge x 2 a+ 2 d, and dividing by a and by g D, we have 2 e e d q = 2 ---- X 1 + and g a g 9 = 2 2 X 1+ g D a d In applying this formula to the elevation of water be tween two glass plates, the side a is very great corn pared with d, and therefore the quantity being almost insensible, may be safely neglected. Hence the formula becomes e e X q = g d By comparing this formula with the formula No. 2. it is obvious, that water will rise to the same height between plates of glass as in a tube, provided the distance d be tween the two plates of glass is equal to r, or hall* the di ameter of the tube. This result was obtained by Newton, and has been confirmed by the experiments of succeeding writers.
' As the constant quantity 22e e --- is the same as already g found for capillary tubes, we may take its value, viz. 15,1311, and substitute it in the preceding equation, we then have 15.1311 1=14.1544 ; and since 1.060 2 ( 4 1 - 4 ), subtracting 1 --)=0.1147, we have 2 which differs very little from 13.574, the ob served 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 appears also, that if the section of the tube is a regular polygon, the altitudes of the fluid will be reciprocally proportional to the homo logous lines of the similar base, a result which, as we have seen, M. Gellert obtained from direct experiment. Hence
in all prismatic tubes, whose sections are polygons inscrib ed in the same circle, 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 : 31, or very nearly as 7 : 8.
M. La Place has remarked, that there may be several states of equilibrium in the same tube, provided its width is not uniform. If we suppose two capillary tubes com municating with one another, so that the smallest is placed above the greatest, we may then conceive their diameters and lengths to be such, that the fluid is at first in equili brium above its level in the widest tube, and that in pour ing 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 di minishes by insensible 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 diminishing, becomes nothing in a state of equilibrium. Beyond this it becomes negative, and consequently the fluid tends to descend. Thus the first equilibrium is sta ble, since the fluid, being a little removed from this state, tends to return to it. In continuing to raise the fluid, 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 re move from this state, which is not stable. In a similar manner it will be seen, that the third state is stable, the fourth instable, and so on.
Although the preceding method of considering the phe nomena of capillary attraction is extremely simple and ac curate, yet it does not indicate the connection which sub sists between the elevation and depression of the fluid, and the concavity or convexity of the surface which every fluid assumes in capillary spaces. The object of Al. La Place's first method, contained in his first supplement, is to deter mine this connection.