  We consider the Bianchi identity as a field equation for the distortion of
the shapes of images produced by weak gravitational lensing. Using the spin
coefficient formalism of Newman and Penrose [1962], we show that certain
complex components of the Weyl and Ricci curvature tensors are directly related
to fundamental observables in weak gravitational lensing. In the case of weak
gravitational fields, we then show that the Bianchi identity provides a field
equation for the Ricci tensor assuming a known Weyl tensor. From the Bianchi
identity, we derive the integral equation for weak lensing presented by
Miralda-Escude [1996], thus making the Bianchi identity a first principles
equation of weak gravitational lensing. This equation is integrated in the
important case of an axially symmetric lens and explicitly demonstrated in the
case of a point lens and a SIS model.
