Page:Elementary Text-book of Physics (Anthony, 1897).djvu/428

surface, $\mu '$ the new index, and $p'$ the new focal distance, we have ${\frac {\mu '}{p'}}-{\frac {1}{s}}={\frac {\mu '-1}{r'}}\cdot$

If we suppose the lens to be very thin, we may put $s=p''.$ If we suppose also that the medium to the right is the same as that to the left of the lens, $\mu '$ is equal to ${\frac {1}{\mu }}\cdot$ On these suppositions ${\frac {\frac {1}{\mu }}{p'}}-{\frac {1}{p''}}={\frac {{\frac {1}{\mu }}-1}{r'}}\cdot$ Multiplying through by $\mu ,$ we have

${\frac {1}{p'}}-{\frac {\mu }{p''}}={\frac {1-\mu }{r'}}=-{\frac {\mu -1}{r'}}\cdot$

Eliminating $p''$ between this equation and equation (110), we obtain

${\frac {1}{p'}}-{\frac {1}{p}}=(\mu -1)\left({\frac {1}{r}}-{\frac {1}{r'}}\right),$

(111)

which expresses the relation between the conjugate foci of the lens. It should be noted that $r$ in the above formulas represents the radius of the surface on which the light is incident, and $r'$ that of the surface from which the light emerges. All the quantities are positive when measured toward the source of light. Fig. 104 shows sections of the different forms of lenses produced by combinations of two spherical surfaces, or of one plane and one spherical surface.

An application of equation (111) will show that for the first three, which are thickest at the centre, light is concentrated, and for the second three diffused. The first three are therefore called converging, and the second three diverging, lenses. Let us consider the first and fourth forms as typical of the two classes. The first