for integer s and we define Q ( l, j, s ) = 0 for half-integer s ; A ( n ) = 1 × 3 × 5 × . . . × (2 n − 1) /n !. Equating the coefficients of P$_{l}$ , we obtain This equation has the form of an eigenvalue problem ω $^{2}$b$_{l}$ = H$_{lλ}$b$_{λ}$ = ( H (0) lλ + V$_{lλ}$ ) b$_{λ}$ (using the summation convention, l ≥ 1), where H $^{(0)}$, representing the first two terms on the RHS of Eqn. 19 is a diagonal matrix and V , representing the third term on the RHS is not, in general, diagonal. Treating V as a perturbation, the first order correction ( σ 2 $_{j}$) (1) = b (0) l V$_{lλ}$b (0) λ due to a harmonic component j can be computed analytically to obtain the eigenfrequencies ω 2 = σ 2 T + σ 2 0 + ( σ 2 $_{j}$) (1) + . . . . For a quadrupole harmonic j = 2 we obtain For an octopole harmonic j = 3, and all odd j , ( σ 2 $_{j}$) (1) = 0. This explains why our spherical- well approximation works so well in predicting the eigenfrequency spectrum of the droplet: we have minimized c ′ 2 by careful adjustment of B$_{0}$ and we expect, from the above analysis, that the effect of the octopole harmonic c ′ 3 (which is comparable to the spherically symmetric component c ′ $_{0}$) on the measured eigenfrequencies in our experiment, to be minimal. Fig. 6 shows the eigenfrequencies determined by using a computer to solve the eigenvalue problem numerically (i.e. beyond first order). The eigenfrequencies of an a = 7 . 5 mm droplet, for various c ′ 2 and c ′ $_{3}$, are shown (thick lines), along with the first-order result Eqn. 20 for comparison (thin lines). In addition, we plot the eigenfrequencies of modes with m = 0 (broken lines); the calculation of the frequency of these modes is outlined in the Appendix. The lower plot of Fig. 6 shows that the effect of the octopole harmonic c ′ 3 on the measured eigenfrequencies is minimal for all m for | c ′ 3 | ≲ 1. For a quadrupole harmonic c ′ $_{2}$, the eigenfrequencies depend strongly on m , as shown in the upper plot of Fig. 6. However, since we have reduced | c ′ 2 | to smaller than ≈ 0 . 05 as described above, and since our method of exciting the oscillations tends to excite only the m = 0 modes, this does not alter our explanation of why the spherical-well approximation works so effectively. 13 ( σ 2 $_{j}$) (1) = c ′ $_{2}$( a ) a l $^{2}$( l + 1) (2 l + 3)(2 l − 1) . (20) ω $^{2}$b$_{l}$ = c ′ $_{0}$( a ) l a b$_{l}$ + T ρa $_{3}$l ( l − 1)( l + 2) b$_{l}$ + c ′ $_{j}$( a ) l a j + l ∑ p = | j − l | b$_{p}$Q ( p, j, [ j + p − l ] / 2) . (19)