Page:Harold Dennis Taylor - A System of Applied Optics.djvu/29

 The radius of curvature A..C is to be considered as an intrinsically positive quantity whether the surface be convex or concave; and then—

For instance, in the case of Fig.2d we have $$\textstyle{ \frac{1}{A..q}=\frac{1}{F}-\frac{1}{A..Q} }$$, but by convention A..Q is a negative quantity, therefore the formula is $$\textstyle{ \frac{1}{A..q}=\frac{1}{F}-\frac{1}{-A..Q} }$$ or $$\textstyle{ \frac{1}{F}+\frac{1}{A..Q} }$$, therefore A..Q comes out divergent and positive.

Should Q..A or A..Q be infinite or the rays of the incident pencil be parallel, then of course $$\textstyle{ \frac{1}{A..Q} }$$becomes zero, and $$\textstyle{ \frac{1}{A..q} }$$ becomes $$\textstyle{ \frac{2}{A..C} }$$ or $$\textstyle{ \frac{1}{F} }$$, and the rays converge to or diverge from the principal focus of the mirror.

The dotted lines in the figures indicate negative distances, and the full lines the positive distances.

Plane Refracting Surfaces

In the case of normal or perpendicular incidence of small pencils at a plane refracting surface bounding a transparent substance whose refractive index = μ, while that of the left-hand medium = o, the simple relationship A..q=μ(A..Q) holds good. See Figs.3a and 3b.

Spherical Refracting Surfaces

In the case of direct refraction of normal pencils by spherical surfaces, as in Figs. 4a, b, c, d, e, f, g, and h, the formula