Page:Optics.djvu/93

 Let $$r, r', r'' ....$$ be the successive radii, each having its own proper sign as well as magnitude.

$$m, m' ,m''.... $$ the indices of refraction at the several surfaces.

$$\Delta$$ the original focal distance.

$$\Delta', \Delta, \Delta' $$ those after one, two, three, refractions.

Then, if only we neglect the distances between the surfaces along the axis, we shall have

$$\frac{1}{\Delta'} = \frac{m-1}{mr} + \frac{1}{m \Delta},$$

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

$$\frac{1}{\Delta'} = \frac{m-1}{mr} + \frac{1}{m \Delta}$$

$$= \frac{m-1}{mm'mr} + \frac{m'-1}{m'mr'}+\frac{m-1}{mr}+\frac{1}{mm'm\Delta},$$

and so on.

When the surfaces are those of lenses, $$m'=\frac{1}{m}, \ m'=\frac{1}{m},$$ and the equations are reducible to those we have already seen.