The author obtains some special relations between the orders of entire functions of \(n\) complex variables represented by multiple Dirichlet series by taking asymptotic behaviour in their coefficients. The following theorem is typical (for brevity we set \(n=2)\). Theorem 1. Let \[ f_ j(s_ 1,s_ 2)=\sum^ \infty_{p,q=0} a^{(j)}_{p,q} \exp (s_ 1 \lambda_ p^{(j)}+s_ 2 \mu_ q^{(j)}), \] \(j=0,1,\dots,q\), where \(00, \dots, \alpha_ q>0\), \(\alpha_ 1+\cdots+\alpha_ q=1\), the conditions \(\lambda_ p^{(j)} \sim \lambda_ p^{(0)}\), \(\mu_ q^{(j)} \sim \mu_ q^{(0)}\) \((j=1,\dots,q)\) and \[ \sum^ q_{j=1} \alpha_ j \ln {1 \over a^{(j)}_{p,q}} \sim \ln {1 \over a^{(0)}_{p,q}} \] fulfilled, then \(\rho_ 0 \leq \sum^ q_{j=1} \alpha_ j \rho_ j\).