In this thesis, balanced function are made a detailed study coming to following aspects: 1) If a elementary function g(t) satisfies g′(t)≥0 and g″(t)≤0 then B 1(x 1,…,x m)=mj=2g(x j) is a balanced function 2) If a elementary function h(t) satisfies ( lnh(t))″≤0 then B 2(x 1,…,x m)=mj=1h(x j) is a balanced function. 3) If B (x 1,…,x m) is a balanced function and the elementary function φ(t) satisfies φ′(t)≥0 on the range of B, then (φB)(x 1,…,x m)+c(c is a constant) is a balanced function and induces the same model of variable weight that B(x 1,…,x m) does. 4) As deduction of the above , we give the balanced functions as ∑ 1(x 1,…,x m)=mj=1x j, ∏ 1(x 1,…,x m)=∏mj=2x α j(α>0) and ∑ α(x 1,…,x m)=mj=2x α j (0≤α≤1), and corresponding weight fomulas. At last, a example is given to note the variable weight principle