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quadrupole component of the radial force | c ′ 2 | is smallest (for our magnet) at B$_{0}$ = 16 . 2 T in air, and at B$_{0}$ = 16 . 5 T in nitrogen gas; at this field, | c ′ 2 | ≪ | c ′ 0 | and only weakly dependent on r . Increasing B$_{0}$ beyond this value gives a positive c ′ $_{2}$, increasing with r . Decreasing B$_{0}$ below this value gives a negative c ′ $_{2}$, becoming more negative with increasing r . The gradient of the octopole component c ′ $_{3}$, however, is of the same order as c ′ 0 and increases as r $^{2}$. This component is a feature of the magnetogravitational potential trap generated by a solenoid. It can be reduced slightly, by increasing B$_{0}$ , but only at the expense of significantly increasing | c ′ 2 | . Note that the dipole component of the trap c$_{1}$ is necessarily zero (hence c ′ 1 = 0 too), since the net vertical force on the droplet F$_{z}$ must be zero for levitation. Since F$_{z}$ is proportional to the difference between the mean surface potentials of its upper and lower hemispheres, i.e. F$_{z}$ ∝ ∫ 1 0 | w | U ( a, w ) dw − ∫ 0 − 1 | w | U ( a, w ) dw ∝ $^{∑}$$_{j}$c$_{j}$ ∫ 1 − $_{1}$wP$_{j}$ ( w ) dw ∝ c$_{1}$ , it follows that c$_{1}$ = 0, where w = cos θ (= P$_{1}$ ). The harmonic components j > 3, are small compared to both c ′ 0 and c ′ $_{3}$, in our system. Fig. 4 shows the values of the j = 0 and j = 2 , 3 (quadrupole and octopole) coefficients c ′ j at B$_{0}$ = 16 . 5 T (for nitrogen atmosphere) 

