GLYPH<s48> GLYPH<s50> GLYPH<s52> GLYPH<s54> GLYPH<s56> GLYPH<s54> GLYPH<s51>GLYPH<s46>GLYPH<s54> GLYPH<s49>GLYPH<s46>GLYPH<s53> GLYPH<s32> FIG. 2: The dispersion law for a dot array in the FM state along some symmetric directions at different λ = ω$_{0}$/ω$_{int}$ (shown near the curves). Here and below we use the notations Γ, X and M for symmetric points of the Brillouin zone (0,0), (0,1) and (1,1), respectively. of the sums Σ$_{(}$$_{±}$$_{)}$( ⃗ k ) vanishes as k → 0, and the stability of the FM state against the linear long wave excitations is broken if H eff a + H$_{0}$ < 0, or H$_{0}$ < 3 σ (0) / 2 − H$_{a}$ . After the replace- ment 3 σ (0) / 2 → 4 π and M → M$_{s}$ this criterion coincides with the result obtained from the continuum theory for a thin magnetic film. Nevertheless the dependence ω ( ⃗ k ) is essentially different than for thin films. The minimal value of the function ω $^{2}$( ⃗ k ) is attained at the boundary of the Brillouin zone for the quasimomentum, ⃗ k parallel to the (1,0) axis. At this point, the maximal value of is equal to 0 . 5859 = 4 π · 0 . 0466. Because this instability condition is first realized for a non- small ⃗ k ‖ (1,0), it is stricter than the one found in the long-wave approximation. Finally, the ferromagnetic state of the dot array loses stability at H$_{0}$ < H$_{c}$ , where This value of the critical field differs from that obtained by the continuum approximation, 4 πM by more than 12 %. Moreover, the character of instability for the array of dots coupled by the dipole interaction is principally different that for the continuous film. For the dot array − (1 / 2)Σ$_{(}$$_{−}$$_{)}$( ⃗ k ) = 0 . 5[3 ∣ σ$_{c}$ ( ⃗ k ) ∣ − σ (0) + σ ( ⃗ k )] H$_{c}$ = M 2 max [ 3 ∣ σ$_{c}$ ( ⃗ k ) ∣ − σ (0) + σ ( ⃗ k ) ] − H$_{a}$ ≃ ≃ 1 . 125 · 4 πM − H$_{a}$. (12) 10