Annular wall (16, 18) of a turbine engine combustion chamber, comprising a cold side (16a, 18a) and a hot side (16b, 18b), said annular wall comprising:. a plurality of primary holes or dilution holes distributed according to a circumferential row to allow circulating air of the cold side (16a, 18a) of said annular wall to enter the hot side (16b, 18b) to respectively create an air/fuel mixture or ensure dilution of the air/fuel mixture; and. a plurality of cooling orifices to allow the circulating air of the cold side (16a, 18a) of said annular wall to enter the hot side (16b, 18b) to form a film of cooling air along said annular wall, said cooling orifices being distributed according to a plurality of circumferential rows spaced axially from one another and the geometric axes of each of said cooling orifices being inclined, in an axial direction D of flow of combustion gases, by an angle of inclination θ1 relative to a normal N to said annular wall;. a plurality of additional cooling orifices arranged directly downstream of said primary holes or dilution holes and distributed according to a plurality of circumferential rows spaced axially from one another, the geometric axes of each of said additional cooling orifices being arranged in a plane perpendicular to said axial direction D and inclined by an angle of inclination θ2 relative to a normal N to said annular wall, characterized in that it further comprises at the level of a transition zone (28B, 30B) formed directly downstream of said plurality of rows of additional orifices and directly upstream said plurality of rows of cooling orifices, exactly two rows of orifices whereof the geometric axes of each of said orifices are inclined of 30° and 60° respectively, relative to a plane perpendicular to said axial direction D.. a plurality of primary holes or dilution holes distributed according to a circumferential row to allow circulating air of the cold side (16a, 18a) of said annular wall to enter the hot side (16b, 18b) to respectively create an air/fuel mixture or ensure dilution of the air/fuel mixture; and. a plurality of cooling orifices to allow the circulating air of the cold side (16a, 18a) of said annular wall to enter the hot side (16b, 18b) to form a film of cooling air along said annular wall, said cooling orifices being distributed according to a plurality of circumferential rows spaced axially from one another and the geometric axes of each of said cooling orifices being inclined, in an axial direction D of flow of combustion gases, by an angle of inclination θ1 relative to a normal N to said annular wall;. a plurality of additional cooling orifices arranged directly downstream of said primary holes or dilution holes and distributed according to a plurality of circumferential rows spaced axially from one another, the geometric axes of each of said additional cooling orifices being arranged in a plane perpendicular to said axial direction D and inclined by an angle of inclination θ2 relative to a normal N to said annular wall, characterized in that it further comprises at the level of a transition zone (28B, 30B) formed directly downstream of said plurality of rows of additional orifices and directly upstream said plurality of rows of cooling orifices, exactly two rows of orifices whereof the geometric axes of each of said orifices are inclined of 30° and 60° respectively, relative to a plane perpendicular to said axial direction D.