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...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!
http://www.maths.tcd.ie/pub/HistMath...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
Quote:
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Quote by William Rowan Hamilton
If therefore the equation (E) be integrable, that is, if it can be satisfied by any unknown relation between x,y,z it is necessary that in establishing this unknown relation between those three variables, the part [itex](\alpha dx + \beta dy + \gamma dz)[/itex] should also be an exact differential of a function of the two variables which remain independent; the condition of this circumstance is here
[tex](\alpha + \alpha ')(\frac{\partial \beta}{\partial z} - \frac{\partial \gamma}{\partial y}) + (\beta + \beta ')(\frac{\partial \gamma}{\partial x} - \frac{\partial \alpha}{\partial z}) + (\gamma + \gamma ')(\frac{\partial \alpha}{\partial y} - \frac{\partial \beta}{\partial x})[/tex] |