one of the scalar fields in terms of the others (see Appendix A for details). The resulting N independent equations are where H ≡ a $_{a}$, $^{(4)}$□ ≡ γ $^{µν}$∂$_{µ}$ ∂$_{ν}$ , and in equation (51), as throughout, we have assumed that there are no direct couplings between the 5D scalar fields in the scalar potential (in Appendix A we derive the general form of these equations when couplings between the fields are included). These dynamical equations can be written more compactly as where ξ has suppressed discrete indices which run over the N − 1 fundamental scalar fields, Ψ is the graviscalar, the D y i are y -dependent differential operators (i.e., linear differential oper- ators having y -dependent coefficients and acting only on functions of y ) also with suppressed discrete indices, and □ ≡ γ $^{µν}$∂$_{µ}$ ∂$_{ν}$ is the 4D wave operator. The boundary conditions are determined by integrating the equations of motion across each brane. Integrating equations (50) and (51), these are found to be These boundary conditions can be put in the form 22 $^{(4)}$□ Ψ − Ψ ′′ − ( 9 H − 2 a 2 1 ¯ χ ′ N ∂W ∂χ$_{N}$ ∣ ¯ χ ) Ψ ′ − ( 12 H 2 + 4 H ′ − 1 2$^{(4)}$ R − 4 a $^{2}$H 1 ¯ χ ′ N ∂W ∂χ$_{N}$ ∣ ¯ χ ) Ψ = − 4 a 2 3 M 3 ∗ N − 1 ∑ a =1 ( ∂W ∂χ$_{a}$ ∣ ¯ χ − ¯ χ ′ a ¯ χ ′ N ∂W ∂χ$_{N}$ ∣ ¯ χ ) ξ$_{a}$ (50) $^{(4)}$□ ξ$_{a}$ − ξ ′′ a − 3 H ξ ′ a − a 2 ∂ $^{2}$W ∂χ 2 a ∣ ¯ χ ξ$_{a}$ = − 3¯ χ ′ $_{a}$Ψ ′ − 2 a 2 ∂W ∂χ$_{a}$ ∣ ¯ χ Ψ , (51) □ Ψ + D y $_{1}$Ψ = D y $_{2}$ξ (52) □ ξ + D y $_{3}$ξ = D y $_{4}$Ψ , (53) [Ψ $^{′}$]$_{y}$$_{i}$ − 2 a 2 1 ¯ χ ′ N ∂W ∂χ$_{N}$ ∣ ¯ χ Ψ ′ ∣ y i − 4 a $^{2}$H 1 ¯ χ ′ N ∂W ∂χ$_{N}$ ∣ ¯ χ Ψ ∣ y i = 4 a 2 3 M 3 ∗ N − 1 ∑ a =1 ( ∂W ∂χ$_{a}$ ∣ ¯ χ − ¯ χ ′ a ¯ χ ′ N ∂W ∂χ$_{N}$ ∣ ¯ χ ) ξ$_{a}$ ∣ y i (54) [ ξ ′ $_{a}$]$_{y}$$_{i}$ + a 2 ∂ $^{2}$W ∂χ 2 a ∣ ¯ χ ξ$_{a}$ ∣ y i = 2 a 2 ∂W ∂χ$_{a}$ ∣ ¯ χ Ψ ∣ y i . (55) Ψ $^{′}$( x, y$_{i}$ ) = A$_{1}$ ( y$_{i}$ )Ψ( x, y$_{i}$ ) + A$_{2}$ ( y$_{i}$ ) ξ ( x, y$_{i}$ ) (56) ξ $^{′}$( x, y$_{i}$ ) = B$_{1}$ ( y$_{i}$ )Ψ( x, y$_{i}$ ) + B$_{2}$ ( y$_{i}$ ) ξ ( x, y$_{i}$ ) , (57)