John's equation is an ultrahyperbolic partial differential equation satisfied by the X-ray transform of a function. It is named after German-American mathematician Fritz John. Given a function with compact support the X-ray transform is the integral over all lines in \R^n. We will parameterise the lines by pairs of points x \ne y on each line and define u as the ray transform where Such functions u are characterized by John's equations which is proved by Fritz John for dimension three and by Kurusa for higher dimensions. In three-dimensional x-ray computerized tomography John's equation can be solved to fill in missing data, for example where the data is obtained from a point source traversing a curve, typically a helix. More generally an ultrahyperbolic partial differential equation (a term coined by Richard Courant) is a second order partial differential equation of the form where n \ge 2, such that the quadratic form can be reduced by a linear change of variables to the form It is not possible to arbitrarily specify the value of the solution on a non-characteristic hypersurface. John's paper however does give examples of manifolds on which an arbitrary specification of u can be extended to a solution.