Page:Optics.djvu/90

 In any particular case it is easy to put the proper values of $$m, t, r,$$ and $$r'$$ in the equations, and determine accurately the value of $$\Delta''$$ or $$F,$$ but no simple general expression can be obtained for them.

92.The sphere may be considered as a sort of lens. In fact, it is a particular species of double convex, in which the thickness is twice the radius.

In investigating its focal length, it will be most convenient to refer the distances to the centre, as in Art. 77.

In Fig. 89, if $$EQ=q,\ Eq=q',\ ET=q'',\ ER= r,$$ we have

$$\frac{1}{q'}= -\frac{m-1}{r} + \frac{m}{q},$$

$$EA$$ and $$EQ$$ being in the same direction,

$$\frac{1}{q''} = \frac{\frac{1}{m}-1}{r} + \frac{1}{mq'}$$ (Here $$r$$ is negative.)

$$= - \frac{m-1}{mr} - \frac{m-1}{mr} + \frac{1}{q}$$

$$= -2 . \frac{m-1}{mr} + \frac{1}{q}.$$

The principal focal length is of course $$-\frac{mr}{2 (m-1)},$$ the negative sign meaning that the focus is on the opposite side from that whence the light proceeds.

If the sphere be of glass, and placed in air, $$m=\frac{3}{2},$$ and $$F=\frac{3}{2} r,$$

if of water, $$........................... m=\frac{4}{2},$$ and $$F=2 r.$$

93.There is one case in which a ray will pass through a lens without deviation, that is, the emergent ray will be parallel to the incident: it is when the surfaces at which it enters and emerges, are parallel.