Annular wall of a turbine engine combustion chamber , comprising a cold side and a hot side , 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 of said annular wall to enter the hot side 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 of said annular wall to enter the hot side 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 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 of said annular wall to enter the hot side 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 of said annular wall to enter the hot side 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 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. <EOS>