Consider earth's atmosphere to be spherically symmetric above the surface, with index of refraction \(n=n(r)\), where \(r\) is measured from the center of the earth. Using polar coordinates \(r, \theta\) to describe the trajectory of a light ray entering the atmosphere from high altitudes,

(a) find a first-order differential equation in the variables \(r\) and \(\theta\) that governs the ray trajectory;

(b) show that \(n(r) r \sin \varphi=\) constant along the ray, where \(\varphi\) is the angle between the ray and a radial line extending outward from the center of the earth. This is the analog of the equation \(n(y) \sin \theta=\) constant for a flat atmosphere.