  We find the joint generalized singular value distribution and largest
generalized singular value distributions of the $\beta$-MANOVA ensemble with
positive diagonal covariance, which is general. This has been done for the
continuous $\beta > 0$ case for identity covariance (in eigenvalue form), and
by setting the covariance to $I$ in our model we get another version. For the
diagonal covariance case, it has only been done for $\beta = 1,2,4$ cases
(real, complex, and quaternion matrix entries). This is in a way the first
second-order $\beta$-ensemble, since the sampler for the generalized singular
values of the $\beta$-MANOVA with diagonal covariance calls the sampler for the
eigenvalues of the $\beta$-Wishart with diagonal covariance of Forrester and
Dubbs-Edelman-Koev-Venkataramana. We use a conjecture of MacDonald proven by
Baker and Forrester concerning an integral of a hypergeometric function and a
theorem of Kaneko concerning an integral of Jack polynomials to derive our
generalized singular value distributions. In addition we use many identities
from Forrester's {\it Log-Gases and Random Matrices}. We supply numerical
evidence that our theorems are correct.
