Page:Optics.djvu/75

 Also $\phi'+\psi'=\iota;\quad\therefore d\phi'+d\psi'=0$. Eliminating therefore $$d\phi'$$ and $$d\psi'$$, we find $\cos\psi'.\cos\phi.d\phi+\cos\phi'.\cos\psi.d\psi=0;$ $\therefore d\psi=-\frac{\cos\phi}{\cos\phi'}.\frac{\cos\psi'}{\cos\psi}d\phi,$ and $d\delta=d\phi\left\{1-\frac{\cos\phi}{\cos\phi'}.\frac{\cos\psi'}{\cos\psi}\right\}$. Now this will clearly be equal to nothing, when $$\phi$$ and $$\psi$$ are equal, as their cosines and those of $$\phi'$$ and $$\psi'$$ will also be equal, and therefore $$\frac{\cos\phi}{\cos\psi}.\frac{\cos\psi'}{\cos\phi'}=1$$.

It appears then, that the deviation is a minimum, when the incident and emergent ray make equal angles with the sides of the prism.

72.A thin conical pencil of rays pass nearly perpendicularly through both sides of a very thin prism; required the focus of the emergent rays.

Let $$Q$$, (Fig. 64.) be the focus of the incident rays, $$QO$$ a perpendicular to the surface. The focus of the rays in their passage through the prism will be $$x$$, a point in $$OQ$$, such that $Ox=m\times OQ,$ (Art. 63.).

From $$x$$ draw $$xy$$ perpendicular to the second side of the prism, then, as $$x$$ is the focus of rays incident on that surface at $$S$$, their focus after the second refraction will be a point $$q$$, such that $yx=m\times yq$.

73. find experimentally the refracting power of any given substance, we may form a piece of it, if solid, into a prism, and observe an object through it. See Fig. 65, where $$P$$ is the place of the eye, $$Q$$ the object observed. Let the angles at $$P, \ Q$$ be measured; also the angle $$PAH$$ or $$PAI,$$ in order to have the angle of incidence at $$A$$.