The method of mixtures, explained in Section 2, is applied to derive the distribution functions of a positive quadratic form in normal variates and of the ratio of two independent forms of this type.
"Application of the Method of Mixtures to Quadratic Forms in Normal Variates." Ann. Math. Statist. 20 (4) 552 - 560, December, 1949. https://doi.org/10.1214/aoms/1177729947