Some properties of mutivariate distributions with pdf's constant on ellipsoids
Malay Ghosh, Edward Pollak · Communications in Statistics · 1975
For random variables X1...., XK+1 with unbounded support, having a joint pdf constant on ellipsoids, the joint distribution of the ratios Yi = Xi / Xk+1(1 ≤ i ≤ k) is obtained. If the Xi's have a spherically symmetric pdf. the Yi's have a multivariate Cauchy distribution. The condition of unbounded support of the Xi's is not necessary, although it cannot always be relaxed.