  We present three new identities in law for quadratic functionals of
conditioned bivariate Gaussian processes. In particular, our results provide a
two-parameter generalization of a celebrated identity in law, involving the
path variance of a Brownian bridge, due to Watson (1961). The proof is based on
ideas from a recent note by J. R. Pycke (2005) and on the stochastic Fubini
theorem for general Gaussian measures proved in Deheuvels et al. (2004).
