Page:Optics.djvu/92

68

Now, suppose a second lens be placed close to the first, (Fig. 94.) having for its principal focal length $F',$ or ${\frac {m'-1}{\rho '}}.$

In order to find $\phi ',$ the distance of the focus after the second refraction, we must consider $\phi$ and $\phi '$ as representing $\Delta$ and $\Delta ''$ in the formula, so that

${\frac {1}{\phi '}}={\frac {1}{F'}}+{\frac {1}{\phi }},$ or ${\frac {m'-1}{\rho '}}+{\frac {1}{\phi }};$

$\therefore {\frac {1}{\phi '}}={\frac {1}{F}}+{\frac {1}{F'}}+{\frac {1}{\Delta }},$ or ${\frac {m-1}{\rho }}+{\frac {m'-1}{\rho '}}+{\frac {1}{\phi }}+{\frac {1}{\Delta }}.$

And in like manner, if there be any number $n$ of lenses acting together, we shall have

${\frac {1}{\phi ^{(n)}}}={\frac {1}{F}}+{\frac {1}{F'}}+....+{\frac {1}{F^{(n)}}}+{\frac {1}{\Delta }},$

or $={\frac {m-1}{\rho }}+{\frac {m'-1}{\rho '}}+....+{\frac {m^{(n)}-1}{\rho ^{(n)}}}+{\frac {1}{\Delta }};$

so that their joint effect is the same as that of a single lens, having, for its principal focal length unity divided by

${\frac {1}{F}}+{\frac {1}{F'}}+{\frac {1}{F''}}+....+{\frac {1}{F^{(n)}}},$

or $={\frac {m-1}{\rho }}+{\frac {m'-1}{\rho '}}+{\frac {m''-1}{\rho ''}}+....+{\frac {m^{(n)}-1}{\rho ^{(n)}}}.$

95. Mr. Herschel calls the reciprocal quantity ${\frac {1}{F}}$ the power of a lens, and enounces the last result thus:

"The power of any system of lenses is the sum of the powers of the component lenses."

Of course, regard must be had to the signs: the power of a concave-lens must be considered as positive, that of a convex one, negative.

96. The same method by which we found the focal length of a lens may be easily applied to any number of surfaces, having a common axis.