which is the time of passage of the axis of the projectile from the outer circle to the inner and back to the outer. K is assumed to be constant throughout this small interval. With regard to the particular integral, it is quite negli gible till & becomes large, but it then becomes important for both s (the stability factor) and are then large. It has been suggested that some means should be adopted to decrease the spin more rapidly and thus reduce s, so as to lessen this disturbing factor at the apex of a trajectory. The For initial elevations up to as much as 45° or 50° the effect of the trajectory on the yaw is negligible. explanation of this important fact has only recently been given and is worthy of a simple explanation in words. We have the well-known results in the theory of the spinning-top: if the precession is hurried the top rises; if it is retarded the top sinks. Now the projectile, neglecting the motion of its centre of gravity, is like a top spinning upon its point. The centre of gravity of the projectile corresponds to the point of the top. The air-pressure acting through the centre of pressure corresponds to gravity acting upon the top. In the case of the top, if the gravity couple were increased, the precession would be hurried and the top would rise. Now the curvature of the path, when the projectile's axis is pointing above the tangent, tends to increase the pressure-couple. Hence the precession is hurried and the nose of the projectile is lowered. When its axis is pointing below the tangent, the curvature of the path tends to decrease the pressure-couple. Hence the precession is retarded and the nose of the projectile is lowered still further. It will be noted that the oscillations of the axis of the projectile are referred to the direction of motion of the centre of gravity of the projectile. There should, therefore, be added the helical oscillations of the latter in order to obtain the resultant oscillations relatively to the plane trajectory. It has, however, been shown that the angular helical oscillations of the centre of gravity are, so far as their most important periodic term is concerned, only a very small fraction of 4, the yaw of the shell referred to the direction of motion of the centre of gravity. It is, accordingly, quite unnecessary to take them into account in questions connected with the stability of the projectile. CXI. An Experimental Determination of the Inertia of a Sphere Vibrating in a Liquid. By N. C. KRISHNAIYAR, M.A., Lecturer in Physics, University College, Rangoon *. SIR IR G. G. STOKES † has shown analytically that in the case of a sphere attached to a fine wire of which the effect is neglected, and swung in an unconfined mass of incompressible liquid, the force exerted on the sphere by the motion of the surrounding liquid is given by the relation dy F= -Ad2y -B dt2 dt' where y is the displacement and t is the time and the coefficients A and B have the following values, A=(†+9/4X»)M';_B=(9/4Xr) (1+ 1,) M'a, where M' is the mass of the fluid displaced by the sphere, r is the radius, a=2π/T where T is the period of oscillation, and λ=(a/2)12 and coefficient of viscosity density. The first term in the expression for the force F has the same effect as increasing the inertia of the sphere. To take this term into account it will be sufficient to conceive a mass M collected at the centre of the sphere, adding to its inertia without adding to its weight. The second term gives a frictional force varying as the velocity, and the effect is mainly to produce diminution in the are of oscillation. The assumptions made were that the centre of the sphere performed small periodic oscillations along a straight line, that the sphere had a motion of translation only, that the velocities were so small that their squares could be neglected, and that there was no slipping of the fluid along the surface of the solid in contact with it. The application of the theory to the pendulum observations of Bessel and Baily was very satisfactory in the case of pendulums swinging in the air; but in the case of pendulums swinging in water or a more viscous liquid, conclusive results were not possible on account of the rapid diminution of the are of oscillation. G. F. McEwen measured the force acting on a sphere swinging in water and in a very viscous oil by employing a forced vibration method in which the sphere was suspended *Communicated by the Author. + Cambridge Philosophical Transactions,' 1850; 'Collected Papers,' vol. iii. p. 34. The same result has been obtained by a different method. Lamb's Hydrodynamics,' 3rd ed., p. 584. | Phys. Rev., 5th ser. vol. xxxiii. pp. 492–511 (1911). by a fine wire, and had a slow motion of translation only along a vertical line. The oscillations were small, and either diminished very slowly or were maintained constant for any desired length of time. The forcing was done by a mechanical arrangement and a satisfactory measurement of the force was obtained. H. T. Barnes*, by an arrangement in which a spherical pendulum bob moving under water was received by a ballistic pendulum, found that its effective inertia comes from the addition to its own mass of half the mass of the liquid that would fill the space occupied by the sphere. Gilbert Cook † determined the inertia of a large spherical mine-case falling freely through water under the influence of gravity. The theoretical increase of inertia in such cases is the mass of the displaced liquid, and Gilbert Cook obtained the value 0:46. In the present paper is described an arrangement in which the inertia of bodies vibrating with constant frequency inside a liquid can be determined under conditions which approximate, to a large extent, to the assumptions made in the hydrodynamical theory. The author has found that thin wires can be maintained in steady transverse vibration if they carry a small direct current and are placed between the poles of an electromagnet actuated by a single-phase alternating current, if the length and tension of the wires are adjusted to give the same natural frequency as the number of cycles of the alternating current. With the same current and the same length of wire the range of tension for the vibration to be maintained is limited, and the tension for resonance can be found from that which gives the maximum amplitude. If a small mass m be at the middle point of a wire of length l and mass per unit length σ, and if m be large in comparison with ol, the tension required to give the system a natural frequency n is given by the relation If the tension is applied by hanging a mass M from the free end of the wire passed over a light pulley, T=Mg. If T1 and T be the values of the tension for best resonance, determined experimentally, when the system is in air and when it is immersed in a large mass of a liquid, and if my *Transactions of the Royal Society of Canada,' vol. xi. June-Sept. 1917. Phil. Mag. vol. xxxix. (1920). and my + km, be the inertia of the body in air and in the liquid, (ol) = T where m is the mass of the liquid displaced by the body and kis the inertia factor of the body in the liquid. The wire also increases in inertia when in the liquid. In the experiments the mass of the wire ol is small in comparison with the mass of the body, and the increase is allowed for by making use of Stokes's table for cylinders vibrating transversely. Moreover, the quantities of and my were so chosen that when Stokes's theory was applied the quantity within the bracket was very nearly equal to unity. The frequency n of the maintained vibration does not enter the final formula, and all that is necessary is the frequency of the alternating current should be steady during the experiment. A silver-plated steel wire of radius 0·009 cm., fixed at one end, passed horizontally under two sharp wedges 30 cm. apart, and then over a fixed light aluminium pulley with ball bearings. Weights could be added to a pan attached to to the free end of the wire. The fixed point and the top of the pulley were slightly higher than the level of the edges, so that the wire was well pressed against the edges. Spheres were fixed with gelatine-glycerine cement in the middle of the wire between the wedges. Near the two edges, on either side of the wire parallel to its length, were the pole-pieces, 10 cm. long, 3 cm. wide, 1 cm. high, of two similar electromagnets. Hence the first one-third and the last one-third parts of the wire were in a periodic field of force, while the middle one-third part containing the sphere was unrestricted in space. The apparatus was placed inside a tank with glass sides, 3 ft. long, 14 ft. broad, and 2 ft. high. A small direct current about one-third of an ampere in the wire and an alternating current of about two amperes through the magnets maintained a steady vibration of the wire in air in the vertical plane if its tension was adjusted for resonance. Kerosene oil (viscosity μ=167 × 0.0088) was poured into the tank, the currents were increased to about half an ampere and four amperes respectively, and the tension to produce resonant vibration was determined. Amplitude of the vibration was observed with a telemicroscope having a glass scale *Collected Papers,' vol. iii. p. 52. in the eyepiece. Correct tension could be determined within one per cent. in air and two per cent. in oil. A preliminary experiment with the wire alone without the sphere showed general agreement with Stokes's tabulated value for thin cylinders. Three determinations were made: first with a pith ball coated with lead foil and then with gelatine-glycerine cement to make it impervious to kerosene oil, second with another pith ball covered with aluminium paint and then with the gelatine-glycerine cement, and third with a polished hollow silver ball. Gelatine melted with a little glycerine produced a cement insoluble in kerosene and was therefore used for fixing the balls on the wire. The excess of the experimental results over the theoretical ones, though small, is larger than the possible errors of experiment. In all the three cases the quantity {1+ol/6(m,+ km2)÷1+ol/5m1}? was equal to 1 correct to two places of decimals, and therefore the approximations used therein could not sensibly affect the calculation. The alternating current was taken from the town supply, and in answer to a query the Chief Electrical Engineer of the Electric Supply Company wrote: “The frequency is kept constant at all times at 50 periods. There is a remote possibility of a very slight variation for a few seconds while turbines are being changed over. This would not amount to more than 1 per cent. at the outside. As the turbines are changed over not more than two or three times in the 24 hours, any slight variation would be only momentary." As has been pointed out before, the actual value of the frequency does not matter if it is constant during a determination. The wire when inside the liquid was not sensibly heated. As is required by the hydrodynamical theory, the centre of the sphere performed smal |