computed from the solenoid geometry. The spheres in Fig. 5 show the computed magnitude of Γ = −∇ U at the surface of three droplets, radius a = 5 . 0 mm, 7 . 5 mm and 10 . 0 mm. Radiating lines indicate the direction and magnitude of Γ , which is directed toward the interior of the droplet. The variation in | Γ | over the surface of the droplet is due to the octopole component of the potential trap. We now consider the effect on the droplet’s eigenfrequencies of adding a harmonic com- ponent c ′ $_{j}$, j ≥ 2 to the potential trap. The analysis proceeds as above, however, we must now include higher order harmonics in the eigenfunction of the shape oscillation since l is not, in general, a good eigennumber in a non-spherical potential. In principle, we should decompose the shape into spherical harmonics Y m l , since the degeneracy in m is also lifted in a non-spherical potential (see Fig. 6). However, our method of inducing shape oscillations in the droplet tends to excite only the axisymmetric shapes (i.e. with m = 0), since the air jet is aligned along the solenoid axis. For this reason, we derive here the frequencies of the m = 0 oscillations only (Eqn. 14), which are sufficient to interpret the experimental results. We summarize the treatment of the general case | m | ≤ l in the Appendix. The velocity potential is The magnetogravitational potential U at the surface of the drop is (see Eqn. 9a) where − Γ$_{r}$( a, θ ) = c $^{′}$$_{0}$( a )+ c ′ $_{j}$( a ) P$_{j}$ (cos θ ) in this case (see Eqn. 4). Inserting Eqn. 16, Eqn. 14 and Eqn. 15 into Eqn. 6, and equating the time-varying terms, we obtain The product P$_{l}$P$_{j}$ appearing on the RHS of this equation can be expanded as a sum of Legendre polynomials [15], which, for our purposes, is most conveniently written P$_{l}$P$_{j}$ = ∑ j + l p = | j − l | Q ( l, j, [ j + l − p ] / 2) P$_{p}$ , in which [15] 12 Q ( l, j, s ) = A ( l − s ) A ( s ) A ( j − s ) A ( j + l − s ) $^{(}$2 j + 2 l − 4 s + 1 2 j + 2 l 2 s + 1$^{)}$ (18) aω 2 ∑ l ≥ 1 b$_{l}$P$_{l}$ l = ∑ l ≥ 1 [ ( c $^{′}$$_{0}$( a ) + c ′ $_{j}$( a ) P$_{j}$ ) + T ρa $_{2}$( l − 1)( l + 2)$^{]}$ b$_{l}$P$_{l}$. (17) U ( R ) = U ( a ) − ϵ Γ$_{r}$( a, θ ) sin( ωt ) ∑ l ≥ 1 b$_{l}$P$_{l}$ (cos θ ) , (16) φ ( r, θ ) = − ϵω cos ωt ∑ l ≥ 1 b$_{l}$r $^{l}$l − $^{1}$a − l $^{+1}$P$_{l}$ (cos θ ) . (15) r = R ( θ, t ) = a + ϵ sin ωt ∑ l ≥ 1 b$_{l}$P$_{l}$ (cos θ ) , (14)