Exact analytical solutions are given for the three finite disks with surface density \Sigma _ { n } = \sigma _ { 0 } ( 1 - R ^ { 2 } / \alpha ^ { 2 } ) ^ { n - 1 / 2 } \textrm { with } n = 0 , 1 , 2 .
Closed-form solutions in cylindrical co-ordinates are given using only elementary functions for the potential and for the gravitational field of each of the disks .

The n=0 disk is the flattened homeoid for which \Sigma _ { hom } = \sigma _ { 0 } / \sqrt { 1 - R ^ { 2 } / \alpha ^ { 2 } } .
Improved results are presented for this disk .
The n=1 disk is the Maclaurin disk for which \Sigma _ { Mac } = \sigma _ { 0 } \sqrt { 1 - R ^ { 2 } / \alpha ^ { 2 } } .
The Maclaurin disk is a limiting case of the Maclaurin spheroid .
The potential of the Maclaurin disk is found here by integrating the potential of the n=0 disk over \alpha , exploiting the linearity of Poisson ’ s equation .
The n=2 disk has the surface density \Sigma _ { D 2 } = \sigma _ { 0 } \left ( 1 - R ^ { 2 } / \alpha ^ { 2 } \right ) ^ { 3 / 2 } .
The potential is found by integrating the potential of the n=1 disk .