Sums of the triple divisor function over values of some quadratic forms

Let ₃ (n) be the triple divisor function. It is proved that ₁ ₍䃑, ₍䃒, ₍䃓 ₗ₃ (n₁²+n₂²+n₃²) =c₁x^3{2} (x) ²+ c₂x^3{2} x +c₃x^3{2} +O_ (x^13{10+}) for some constants c₁, c₂ and c₃, updating a result of the second author and Zhang. Moreover, we show that ₁ ₍䃑, ₍䃒, ₍䃓, ₍䃔 ₗ₃ (n₁²+n₂²+n₃²+n₄²) =c₄x^2 (x) ²+c₅x^2 x+c₆x^2 +O_ (x^5{3+}) for some constants c₄, c₅ and c₆, which improves the results in 6.