Now, suppose a second lens be placed close to the first, (Fig. 94.) having for its principal focal length
or
In order to find
the distance of the focus after the second refraction, we must consider
and
as representing
and
in the formula, so that
![{\displaystyle {\frac {1}{\phi '}}={\frac {1}{F'}}+{\frac {1}{\phi }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6625c10a30d10c8a5eacf22890a868801c64d314)
or
![{\displaystyle {\frac {m'-1}{\rho '}}+{\frac {1}{\phi }};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/363f38aeff3f3b8681e01f12fdd89d4dc2ab35c5)
![{\displaystyle \therefore {\frac {1}{\phi '}}={\frac {1}{F}}+{\frac {1}{F'}}+{\frac {1}{\Delta }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2ef5646f36f6e10a1ce4cecabfbf44dc5a9c09)
or
![{\displaystyle {\frac {m-1}{\rho }}+{\frac {m'-1}{\rho '}}+{\frac {1}{\phi }}+{\frac {1}{\Delta }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55d156b8ca7c88e2341a64a697d9222c05247a98)
And in like manner, if there be any number
of lenses acting together, we shall have
![{\displaystyle {\frac {1}{\phi ^{(n)}}}={\frac {1}{F}}+{\frac {1}{F'}}+....+{\frac {1}{F^{(n)}}}+{\frac {1}{\Delta }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c93e8b41f36d3b9402aa018a2f90b73b000e8d)
or
![{\displaystyle ={\frac {m-1}{\rho }}+{\frac {m'-1}{\rho '}}+....+{\frac {m^{(n)}-1}{\rho ^{(n)}}}+{\frac {1}{\Delta }};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87478c48e7b98cd2b9740b23372096f0eea3052d)
so that their joint effect is the same as that of a single lens, having, for its principal focal length unity divided by
![{\displaystyle {\frac {1}{F}}+{\frac {1}{F'}}+{\frac {1}{F''}}+....+{\frac {1}{F^{(n)}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ef7e0b33a5e68d95e1789322868c3af0743a450)
or
![{\displaystyle ={\frac {m-1}{\rho }}+{\frac {m'-1}{\rho '}}+{\frac {m''-1}{\rho ''}}+....+{\frac {m^{(n)}-1}{\rho ^{(n)}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f372bd4af35d9c9e64f7bb7d069daa8915f829c5)
95. Mr. Herschel calls the reciprocal quantity
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.