and g = 9 . 8 ms − $^{2}$. The center of the levitating droplet coincides with the position of the stable levitation point L$_{0}$ at the center of the trap, as shown. (There is a second point of unstable levitation L$_{1}$ , at the saddle point in U , as shown in Fig. 1. However, this point is not useful for these experiments, and so we do not discuss it further here.) The net force on a unit mass of water, Γ = −∇ U , is zero at the levitation point. The equilibrium shape of a liquid with no surface tension follows the contours of U , as demonstrated recently, using liquid H$_{2}$ close to the critical point [7]. For water droplets with a ∼ 1 cm, however, the surface tension dominates the magnetic and gravitational forces on the drop, so that its equilibrium shape is nearly spherical. The shape of the magnetogravitational potential trap can be altered by adjusting the current in the magnet solenoid coils. A convenient measure of the current is the magnetic field at the geometric center of the solenoid, B$_{0}$ = B (0 , 0). The field B ( x, z ) is everywhere proportional to B$_{0}$ . To levitate water requires B$_{0}$ ≈ 16 − 17 T using our magnet. Fig. 1 shows the U ( x, z ) for B$_{0}$ = 16 . 5 T. The field profile B ( x, z ) of the magnet, used to construct this plot, was computed by numerical integration of the Biot Savart integral, using a thin-shell approximation for the current density in the solenoid. We expand the potential U in a multipole expansion about L$_{0}$ : where r and θ are spherical coordinates with origin at L$_{0}$ ; θ is the polar angle (i.e. r sin θ = x ). Only the derivative of U normal to the droplet’s surface (i.e. the radial component of the force) influences the eigenfrequencies of the normal modes, where c ′ j = ∂c$_{j}$/∂r . By adjusting B$_{0}$ , we reduce the quadrupole component until it is small compared to the spherically-symmetric component ( c ′ 2 ≪ c ′ $_{0}$). The octopole harmonic ( c ′ $_{3}$), which cannot be reduced this way, remains comparable to c ′ $_{0}$. All other harmonics are small compared to c ′ $_{0}$. We shall return to discuss these points further in Sec. V. Fig. 1, top, shows a schematic diagram of the water droplet levitating in the vertical magnet bore. The droplet at rest is close to spherical; we measure the ratio of the equatorial (horizontal) diameter to the polar (vertical) diameter to be 1 . 00 ± 0 . 02. An optical fiber directs light from a HeNe laser at the droplet. The drop focuses the light onto the aperture 4 Γ$_{r}$( r, θ ) = − ∂U ( r, θ ) ∂r = ∑ j ≥ 0 c ′ $_{j}$( r ) P$_{j}$ (cos θ ) , (4) U ( r, θ ) = ∑ j ≥ 0 c$_{j}$ ( r ) P$_{j}$ (cos θ ) , (3)