Hello everyone. I'm looking through an old book on optics and something there has me really confused. I attached the two pages I'm referring to, or you can find them here: http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Rays/ as pages 14 and 15 in the pdf. Basically, what's going on is that he is talking about rays of light hitting a mirror. He wants to find the equation of a mirror surface that reflects a given system of incoming rays to a single focal point, without yet saying anything about the nature of the incoming rays. The cosines of the angles an incoming ray makes with the x, y, and z-axis are α, β, and γ, while the the cosines for the reflected ray are α', β', and γ'. Equation (E) was already derived as the basic equation for dealing with reflections. For reference here, equation (E) is [tex] (\alpha + \alpha ')dx + (\beta + \beta ')dy + (\gamma + \gamma ')dz = 0[/tex] Next he argues that [tex]\alpha 'dx + \beta 'dy +\gamma 'dz[/tex] is an exact differential with an argument that makes sense. My trouble comes next. He says I understand neither why that expression needs to be an exact differential of two variables, nor why that last equation (equation (F)) represents that condition. Any help would be appreciated!