Let \(R_1(n), R_2(n)\) denote the numbers of representations of a positive integer n by the quaternary quadratic forms \(g_1(x_1,x_2,x_3,x_4)\) = \(2( x_{1}^{2}+x_1 x_2+ x_{2}^{2})+2x_1x_3 +x_1x_4+ x_2x_3+2x_2x_4+2(x_{3}^{2}+x_3 x_4+x_{4}^{2}), g_2(x_{1},x_2,x_3,x_4)=8( x_{1}^{2}+x_{2}^{2})+x_{3}^{2}+x_{4}^{2}\), respectively, where \(x_1\), \(x_2\), \(x_3\) and \(x_4\) are integers. In this paper, we establish the asymptotic formulae for the sums \(\sum \limits _{n\le x}R_i(n)\) and \(\sum \limits _{n\le x}R_i^{2}(n)\) for \(i=1,2\).

令\(R_1(n), R_2(n)\)表示正整数n的四元二次型表示数\(g_1(x_1,x_2,x_3,x_4)\) = \(2( x_{ 1}^{2}+x_1 x_2+ x_{2}^{2})+2x_1x_3 +x_1x_4+ x_2x_3+2x_2x_4+2(x_{3}^{2}+x_3 x_4+x_{4}^{2}), g_2(x_{1},x_2,x_3,x_4)=8( x_{1}^{2}+x_{2}^{2})+x_{3}^{2}+x_{4}^{ 2}\)，其中\(x_1\)、\(x_2\)、\(x_3\)和\(x_4\)是整数。在本文中，我们建立了和\(\sum \limits _{n\le x}R_i(n)\)和\(\sum \limits _{n\le x}R_i^{2} (n)\)为\(i=1,2\)。