  We study a model of $n$ one-dimensional non-intersecting Brownian motions
with two prescribed starting points at time $t=0$ and two prescribed ending
points at time $t=1$ in a critical regime where the paths fill two tangent
ellipses in the time-space plane as $n \to \infty$. The limiting mean density
for the positions of the Brownian paths at the time of tangency consists of two
touching semicircles, possibly of different sizes. We show that in an
appropriate double scaling limit, there is a new familiy of limiting
determinantal point processes with integrable correlation kernels that are
expressed in terms of a new Riemann-Hilbert problem of size $4\times 4$. We
prove solvability of the Riemann-Hilbert problem and establish a remarkable
connection with the Hastings-McLeod solution of the Painlev\'e II equation. We
show that this Painlev\'e II transcendent also appears in the critical limits
of the recurrence coefficients of the multiple Hermite polynomials that are
associated with the non-intersecting Brownian motions. Universality suggests
that the new limiting kernels apply to more general situations whenever a
limiting mean density vanishes according to two touching square roots, which
represents a new universality class.
