On the Apollonius Solutions to the GPS Equations

(PREVIOUS TITLE: ON THE SOLUTIONS TO THE GPS EQUATIONS) It has been shown previously that the Problem of Apollonius can be used to obtain closed form solution to the GPS pseudo-range equations. While the Problem of Apollonius can have up to 16 solutions, it has been demonstrated that the GPS problem can have at most 2 solutions when only direct signal transmission from transmitters to receiver are considered. This correspondence shows that if reflected signals are taken into account, it is theoretically possible to obtain 16 solutions for the GPS problem. Furthermore, non-physical solutions to GPS equation can assume physical significance.