In this paper we have solved certain distribution problems connected with the simultaneous analysis of variance test  and have proved that the power function of this test has the monotonicity property.
K. V. Ramachandran. "On the Simultaneous Analysis of Variance Test." Ann. Math. Statist. 27 (2) 521 - 528, June, 1956. https://doi.org/10.1214/aoms/1177728275