Page:Optics.djvu/81

 $$\mathrm {Then} \frac{RF}{EF}=\frac{\sin REF}{\sin ERF}=\frac{\sin ERQ}{\sin ERF}=m, \quad \mathrm {or } \frac{1}{m},$$

and putting $$AF$$ for $$RF$$ as before,

$$AF=m.EF, \quad \mathrm {or} \quad \frac{1}{m} \cdot EF,$$

whence$$AF = \pm \frac{m}{m-1} AE,\quad \mathrm {or} \quad \mp \frac{1}{m-1} AE,$$ as above.

It is important to observe, that in all cases, the distance (AF) of the principal focus from the surface is to its distance (EF) from the centre as the sine of incidence to the sine of refraction.

81. If we introduce the distance $$f$$ into the formulæ, we shall have in

Cases 1 and 2, $$\frac{1}{\Delta'} = \frac{1}{f} + \frac{1}{m \Delta'},$$

3 and 4, $$\frac{1}{\Delta'} = \frac{1}{f} + \frac{m}{\Delta}$$.

82. A spherical refracting surface may, in fact, be said to have two principal foci, one for rays proceeding, parallel to the axis, from the rarer into the denser medium, the other for parallel rays proceeding in the contrary direction. They are on opposite sides of the surface, and at different distances from it, as may easily be seen from the formulæ, for in Cases 1 and 4, $$f$$ is positive, that is, $$F$$ lies on the side whence the light proceeds; in cases 2 and 3, $$f$$ is negative.

In Fig. 75 and 76, $$F$$ and $$f$$ are the two principal foci above described, $$F$$ for parallel rays entering the denser medium, $$f$$ for those proceeding out of it into the rarer one.

83. We will now proceed to examine the varieties of position that $$Q$$ and $$q$$, the conjugate foci, are capable of.

Case 1. In the first place, when $$Q$$ is at an infinite distance, the place of $$q$$ is $$F$$, (Fig. 71.)

When $$Q$$ is at $$E,\ q$$ is likewise at $$E.$$

In all intermediate cases, that is, when $$Q$$ is beyond $$E,\ q$$ lies between $$E$$ and $$F,$$ (Fig. 66.)