and the altitude of its centre of gravity 48 inches: therefore we shall have

"Now the sum of the momenta, divided by the sum of the areas, will

will give

278613.98966016

7528-7808

= 37.006 inches, the altitude.

of G, the centre of gravity of the section ABCD above the bottom. In like manner, the altitude of R, the centre of gravity of the section MMCD, will be found to be equal

123093-98966016

4936,7808

= 24.934 inches; and consequently their difference, or the value of GR 12.072 inches, will be found.

RN

5.6975

"Suppose the vessel to heel 15°, and we shall have the following proportion; namely, As radius: tangent of 15°:: MX= 54 inches: 14.469 inches ME or MF; and consequently' the area of either triangle MXE or MXF = 390-663 square inches. Therefore, as 4936-7808: 390-663 :: 72 = nn=AB: 5.6975 inches RT; and, again, as radius: sine of 15°:: 12·072 = GR: 3.1245 inches RN; consequently RT 3.1245 = 2.573 inches sw, the stability required. "Moreover, as the sine of 15°: radius : : 5-6975 22-013 Rs, to which, if we add 24.934, the altitude of the point R, we shall have 46.947 for the height of the metacentre, which taken from 72, the whole altitude, there remains 25.053; from which, and the half width = 54 inches, the distance BS is found 59.529 inches very nearly, and the angle SBV = 80° 06′ 42′′; from whence sv 58-645 inches.

= =

66

RT:

Again: Let us suppose the mean length of the vessel to be 40 feet, or 480 inches, and we shall have the weight of the whole vessel equal to the area of the section MMCD = 4936-7808 multiplied by 480 2369654.784 cubic inches of water, which weighs exactly 85708 pounds avoirdupoise, allowing the cubic foot to weigh 62.5 pounds.

"And, finally, as sv: sw (i. e.) as 58.645: 2.573 :: 85708:3760+ the weight on the gunwale which will sustain the vessel at the given inclination. Therefore a vessel of the above dimensions, and weighing 38 tons, 5cwts. 28lbs. will require a weight of 1 ton, 13 cwts. 64 lbs. to make her incline 15°.

"In this example, the deflecting power has been supposed

to act perpendicularly on the gunwale at B; but if the vessel is navigated by sails, the centre velique * must be found; with which, and the angle of deflection, the projected distance thereof on the line sv may be obtained; and then the power, calculated as above, necessary to be applied at the projected point, will be that part of the wind's force which causes the vessel to heel, and conversely, if the weight and dimensions of the vessel, the area and altitude of the sails, the direction and velocity of the wind, be given, the angle of direction may be found."

The centre velique (a term first used by M. Bouguer) is the centre of gravity of the equivalent sail; that is, of the sail whose position and magnitude are such as cause it to be acted upon by the wind when the vessel is sailing, in a manner equivalent to the action of the wind upon all the sails together which the vessel actually carries. See also Euler on Vessels, Book III. Chap. ii. § 10. 11., and Bossut Hydrodynamique, part i. ch.

12, 13, 14.

CHAPTER IV.

On the Phenomena of Attraction in Capillary Tubes.

426. THE appellation capillary, in a general sense, is given to any thing on account of its extreme fineness, because it in that respect resembles hair. In physics, capillary tubes are small pipes of glass whose canals are extremely narrow. The internal diameter of these tubes may vary from to of an inch: indeed Dr. Hook affirms that he drew tubes in the flame of a lamp much smaller, and resembling a spider's thread.

If several capillary tubes of different diameters, and open at both ends, be immersed a little way into water, the fluid will be seen to stand higher in the tubes than the surface of the water without, and this is in a greater degree as the tube is smaller; the height of the surface of the fluid in the tube above that of the fluid in the reservoir, being nearly in the reciprocal ratio of the diameter of the tube: so that the diameter of the tube* multiplied into the altitude of the fluid in it (above that in the reservoir) is nearly a constant quantity for the same fluid, This constant quantity is found by experiment to be about .048 of an inch when the fluid is water; 036 of an inch for vinegar or ol. tar. per deliquium; and 024 of an inch for sweet oil. When quicksilver is put into the tube, the contrary to all this takes place; for that fluid stands lower within the tube than its surface in the vessel, and the lower as the tube is smaller.

Pour

Among the various methods of determining the diameter of a uniform capillary tube, the following seems the best and most accurate. into the tube a certain quantity of mercury whose weight in troy grains is w, and measure carefully the length of the tube which it occupies; then is the diameter d= 0192523. For, the specifie gravity of mcrcury being 13568, a cubic inch weighs 3435-16 grains; hence 1: del + 785398 :: 3435-16: w. Whence, by multiplying means and extremes, &c. we find d✔. ='0192523√ T.

3435 16X 785398 × /

w

Ifl be the whole length of the tube in inches, and w the difference in troy grains between its weight when empty and when full of mercury, the same theorem will obviously ascertain the diameter.

Another phenomenon of these tubes is, that such of them as would naturally only discharge water by drops, when electrified yield a continued and accelerated stream; and the acceleration is proportional to the minuteness of the tube. Nay, the effect of electricity is so considerable that it produces a continued stream from a very small tube, out of which the water would not drop at all previous to the excitation by electricity. But we shall not attempt here any explanation of this phenomenon of the continued stream: our present object being merely to state the most approved method of accounting for the ascent and suspension of fluids in these tubes, according to the principle of attraction or of adhesion.

In accounting for these phenomena of capillary tubes we must deviate from the method we have generally followed hitherto, of demonstrating a series of connected propositions : for the mode of elucidation we adopt has, at most, only probability on its side; and we would not willingly delude the student with an appearance of strict demonstration, when that kind of proof is incompatible with our present knowledge of the subject.

427. It will be necessary first to premise that the attraction between the particles of glass and water is greater than the cohesive attraction between the particles of water itself. For if this were not the case, the least quantity or drop of water applied to the underside of a glass tube placed horizontally would not adhere to it, but fall down immediately in the direction of gravity; whereas this does not happen till the bulk and weight of the drop be so far increased as to exceed the attractive power of the glass, and then it falls off.

Since, then, we find such a strong attractive power exerted at the surface of glass, it will be natural to conceive that such power must act sensibly on the surface of a fluid that is not viscid (water, for instance) contained within the small cavity of a glass tube, and that it will be proportionally stronger as the internal diameter of the tube is the smaller: for that the efficacy of the power to hold up the fluid in the tube will follow the inverse proportion of the diameter will be highly probable, if we consider that only such particles as are in contact with the fluid, and those immediately above the surface, can affect it.

428. Hence most philosophers assert that the suspension of the fluid in capillary tubes is owing to the attraction of the narrow ring of glass contiguous to the upper surface of the fluid. The reasoning adduced is of this kind: Every ring of glass below the surface attracts the water above it as much downwards as it attracts the water below it upwards, and con

sequently can contribute nothing towards the support of the column: and the action of the lowest ring upon all the fluid of the tube within its surface of attraction, must either concur with the force of gravity to bring the fluid downwards, or acting upon it at right angles can have no effect in suspending it within the tube. The fluid, therefore, can only be supported by the ring of glass contiguous to its upper surface, which, attracting upwards, opposes the action of gravitation by which the fluid is solicited downwards. And the same kind of reasoning may be applied to the fluid raised between parallel planes of glass.

429. The preceding reasoning being admitted, it will follow, in conformity with the experiments, that in capillary tubes the heights to which the fluid rises by virtue of the attraction are inversely as the internal diameters. For the fluid being suspended by the action of the annulus of glass contiguous to the upper surface, and the distance to which the attraction of glass upon any one fluid reaches being unvaried, the force which sustains the fluid will be as the number of attracting particles, that is, as the circumference, or as the diameter of the upper ring, or of the tube. Let q, q, then, represent the quantities of fluid to be raised in two tubes of different bores; D, d, the respective internal diameters; and H, h, the heights to which fluids rise in the tubes: then, because 0,9, represent two cylinders of the fluid, we have a:q:: D'H: d'h; and, from the nature of this attraction, which is as the diameters of the tubes, D:d :: Q: q ; consequently D'H: d'h;: D: d, or DH: dh:: 1:1, or, finally, D:d::h: H.

Dr. Jurin, who first offered this solution of the phenomenon, says, the effect is the same in vacuo as in the air: but in his time the air-pumps would not exhaust sufficiently to determine this point: the pumps which are now made may perhaps shew that the water will not be supported after a very great degree of exhaustion.

Mr. Martin says, the power by which the fluid is raised will keep it in the tube for any time without exhaling or evaporation; as he tried by hanging several capillary tubes thus charged with their fluids for months together in the summer sun, whose heat did not appear to diminish the fluids in the least sensible. degree.

430. Another curious circumstance ascribed to the same cause is the following: Between two glass plates, meeting on one side, and kept open at a small distance on the other, water will rise unequally; and its upper surface will form a hyperbolic curve, in which the altitudes of the several points above the surface of the fluid in the reservoir will be to one another