It seems a lifetime away now, but I began my scientific career in astrophysics, with a doctorate on galactic dynamics supervised by Wyn Evans. For my thesis work, we modelled a disk galaxy as an infinitesimally thin disk, whose density (mass per unit area) varied as an inverse power law of radius. We calculated the stability of this disk to gravitational perturbations within the plane of the disk. The stars in the disk have a tendency to clump together because of their mutual gravitational attraction, so you might think the disk would just collapse into its centre. However, if the disk is spinning fast enough, the tangential motion of the stars will counteract this tendency, and the disk may remain stable. We examined the circumstances under which stable modes are possible, for a variety of different assumptions about the density profile of the disk. This paper is basically the mathematical methods we used to do the analysis.