which is sensibly equal to 2wv, if v be small compared with the absolute eastward velocity of the earth's surface @R cos A. The sign of v is positive or negative, according as the relative velocity which it denotes is eastward or westward. The excess 2wv may be resolved into 2wv cos λ vertical, and 2wv sin a horizontal, of which the latter must be equal and opposite to the constraining force P. III. The value of P in both cases (and therefore also for all intermediate directions) is thus 2w sin λ. v, which, if the second It is the same as the constrain be the unit of time, is v sin x ing force required for motion in a circle of radius N C 02 02 ... sin a . ß= cos a . ; .. 2 28 = tan a cotan X. The radius of the sphere is here the unit of length. In terms of any unit BC2 2AB of length, we have =R cotan A. as the tangent to a circular arc A C, and But BC may be regarded BC2 2A B is the expression for the radius of curvature. The force which would produce deflection from C B into C A, in the case of a body moving with velocity v, is therefore the same as the constraining force required to keep it on the circumference of a circle of radius R cotan λ. V. Since, for given velocity of circular motion, the constraining force varies inversely as the radius, P is to the force which would produce deflection from CB into CA in the ratio dinary cases the value of this expression ranges from 100 to several hundreds. The tendency to swerve is therefore sensibly the same for motion along a circle of latitude as for motion along a great circle touching it, in the neighbourhood of the point of contact. VI. The constraining force on a body moving along a circle of latitude is to the body's weight in the ratio. Р If the foot 9 Р v g sin λ 220,000 Call and second be units, g is about 32.2, and is m this Then if the air between two parallels of latitude is moving east or west with velocity v, the change of pressure is the same in going m feet along a meridian as in rising 1 foot, viz. 00114 inch of mercury. The change of pressure per degree of latitude (365,000 feet), expressed in inches of mercury, is 365000 m ×·00114=·0019v sin λ, v being in feet per second. The average observed difference per degree is about 01 of an inch. This would require v to be about 5 sin λ feet per second. VII. For a cyclone, if r denote distance from its axis, and v the component velocity perpendicular to r, centrifugal force computed as if the earth were at rest gives a barometric difference. equivalent to rising a height = dr. The earth's rotation adds to this a difference equivalent to dr rising a height m VIII. The following investigation can be employed instead of I., II., III. The carth's rotation o may be resolved into a translation, a rotation about a horizontal axis (at the place considered), and a rotation about a vertical axis. The two former may be neglected. The third is w sin λ, which call ωρ The lateral constraining force is therefore the same for a body moving horizontally in a straight line or in a great circle, as for a body travelling along a radius of a horizontal disk revolving about its centre with angular velocity w Let denote distance from Hence, for horizontal motion in a great circle, the tendency to swerve is rigorously equal for all directions of motion, a result which will be found to agree with the comparison of II. and V. XXV. Account of Experiments upon the Resistance of Air to the Motion of Vortex-rings. By ROBERT STAWELL BALL, A.M., Professor of Applied Mathematics and Mechanism, Royal College of Science for Ireland, Dublin*. THE HE experiments, of which the following is an abstract, were carried out with the aid of a grant from the Royal Irish Academy. A paper containing the results has been laid before the Academy. A brief account of one serics of the experiments and a Table embodying them will be given. Air-rings 9 inches in diameter were projected from a cubical box, each edge of which is 2 feett. The blows were delivered by means of a pendulum called the striker, which, falling from a constant height, ensured that the rings were projected with a constant velocity. In the experiments described in the present series, this velocity was a little over 10 feet per second. The pendulum was arranged so as to open a circuit at the instant of its release. After the ring had traversed a range which varied from 2 inches to 20 feet, it impinged upon a target. The blow upon the target closed the circuit, which had been opened at the release of the striker. A chronoscope measured the interval of time between the release of the striker and the impact upon the target. The target was placed successively at distances of 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 feet from the orifice of the box. Not less than ten observations of the time were taken at each range. The probable error of the mean time at each range is in every case less than 1 per cent. of the whole amount. A special series of experiments, which need not be described, determined the value of the chronoscope readings in seconds. The observations are next represented in a curve, of which the abscissæ are the ranges, and the ordinates the corresponding mean chronoscope readings. By drawing tangents to this curve, Communicated by the Author, being an abstract of a paper read before the Royal Irish Academy. This method was suggested by Professor Tait, See a paper by Sir William Thomson, Phil. Mag. July 1867; also a paper by the author, Phil. Mag. July 1868. the velocity of the ring at its different points is approximately found. A second projection is made in which the abscissæ are the ranges and the ordinates are the velocities; the points thus determined are approximately in a straight line. It follows that the rings are retarded as if acted upon by a force proportional to the velocity, and an approximate value of the numerical coefficient becomes known. A more accurate value having been determined by the method of least squares, the results are embodied in the following Table, of which a description is first given. I. contains a series of numbers for convenience of reference. II. It was found that the motion of the ring in the immediate vicinity of the box was influenced by some disturbing element. The zero of range was therefore taken at a point 4 feet distant from the orifice. This column contains the ranges. III. The interval between the release of the striker and the arrival of the ring at a point 4 feet from the orifice is 6.5 chronoscopic units, or about 0.93 second. This constant must be subtracted from the mean readings of the time in order to reduce the zero epoch to the instant when the ring is 4 feet from the orifice. This column contains the mean readings of the chronoscope corrected by this amount. IV. When the ranges are taken as abscissæ and the corresponding times as ordinates, it is found that a curve can be drawn through or near all the points thus produced. To identify the points with the curve, small corrections are in some cases required. These corrections are shown in column IV. In the case of No. 5 the correction amounts to 0.7. This is about 0-09 second. The magnitude of this error appears to show that some derangement, owing perhaps to a current of air or other source of irregularity, has vitiated this result. For the sake of uniformity, however, the corrected value has been retained. V. This column merely contains the corrected means, as read off upon the curve determined by the points. VI. The value of the chronoscope unit after the first few revolutions is 0.1288 second, with a probable error of 0.0002 second. By means of this factor the corrected means in column V. are evaluated in seconds in column VI. VII. This column contains the time calculated on the hypothesis that the rings are retarded as if acted upon by a force proportional to the velocity, the coefficients being determined by the method of least squares; the formula is t=9016-6.25 log (27.7—s). Phil. Mag. S. 4. Vol. 42. No. 279. Sept. 1871. Р VIII. This column shows that the difference between the corrected mean time and the calculated time in no case exceeds 0.01 second. IX. The approximate velocities deduced by drawing tangents to the curve. X. The true velocities calculated from the formula TABLE of Experiments showing the retardation which a Vortexring of air experiences when moving through air at the same temperature and pressure. The vortex-ring is 9 inches in diameter, and has an initial velocity of 10-2 feet per second. The retarding force is proportional to the velocity; and after 2.34 seconds the ring has moved 16 feet, and its velocity is reduced to 4.3 feet per second. |