Intersection Volume of N dimensional ball and square

Question

I want to calculate the intersection volume of an 3N dimensional ball and "kind of square". $$ B=[(p_1,p_2,...p_{3N})\in\mathbb{R}^{3N}:p_1^2+...+p_{3N}^2<R^2] $$ $$ S=[(p_1,p_2,...p_{3N})\in\mathbb{R}^{3N}:p_1^2+p_2^2+p_3^2<a^2,...,p_{3N-2}^2+p_{3N-1}^2+p_{3N}^2<a^2] $$ I first transformed N times into 3D spherical coordinates: $$ (4\pi)^N\int...\int_{r_1^2,...,r_N^2<a^2\quad and \quad r_1^2+...+r_N^2<R^2}r_1^2 r_N^2dr_1...dr_N $$ But now I can't transform into N dimensional spherical coordinates because in the first inequality wouldn't transform well. Is there maybe another way how to deal with this ?