Let R(n) denote the number of representation of the natural number n by quaternary quadratic form f(x1; x2; x3; x4) = x21 + x22 + 5(x23 + x24 ), where x1; x2; x3, and x4 are integers. In this paper we establish the asymptotic formulae of ∑n ⩽ x R(n) and ∑ n⩽x R2(n).