  An exact expression for the gravitational field strength in a
self-gravitating dust continuum is derived within the Lagrangian picture of
continuum mechanics. From the Euler-Newton system a transport equation for the
gravitational field strength is formulated and then integrated along
trajectories of continuum elements. The resulting integral solves one of the
Lagrangian equations of the corresponding Lagrange-Newton system in general.
Relations to known exact solutions without symmetry in Newtonian gravity are
discussed. The presented integral may be employed to access the
non-perturbative regime of structure formation in Newtonian cosmology, and to
apply iterative Lagrangian schemes to solve the Lagrange-Newton system.
