Dispersion is splitting of white light into its constituent colours. This band of Colours of light is called its spectrum.

(i) Let us consider an object OO' of height h placed on the principal axis with its height perpendicular to the principal axis, The inverted real image II' is formed which has a height h' as shown in Figure.
(ii) The lateral (or) transverse magnification m is defined as the ratio of the height of the image to height of the object.
\(m=\cfrac { II' }{ OO' } \)  ...(i) 
(iii) From the two similar triangles ∆POO' and ΔPII', we can write,
\(\cfrac { II' }{ OO' } =\cfrac { PI }{ PO } \)  ...(ii) 
On applying sign convention,

\(\cfrac { -{ h' } }{ { h } } =\cfrac { v }{ -u } \) 
(iv) Substituting this in the equation (ii) for magnification,
\(m=\cfrac { -{ h }' }{ { h }} =\cfrac { v }{ -u } \) 
After rearranging,
\(m=\cfrac { { h }'}{ h } =\cfrac { v }{ u } \)   ...(iii) 
(v) The magnification is negative for real image and positive for virtual image. In the case of a concave lens, the magnification is always positive and less than one.
We can also have the other forms of equations for magnification by combining the lens equation as,
\(m=\cfrac { { h }' }{ { h }} =\cfrac { f }{ f+u } (or)m=\cfrac { { h }' }{ { h }} =\cfrac { f-v }{ f } \)  ...(iv)  

Sign conventions for lens
(i) The sign of focal length is not decided on the direction of measurement of the focal length from the pole of the lens as they have two focal lengths, one to the left and another to the right.
(ii) The focal length of the thin lens is taken as positive for a converging lens and negative for a diverging lens.
(iii) The other sign conventions for object. distance, image distance, radius of curvature, object height and image height remain the same for thin lenses as that of spherical mirrors.

An endoscope is an instrument used by doctors which has a bundle of optical fibres that are used to see inside a patient's body. Endoscopes work on the phenomenon of total internal reflection. The optical fibres are inserted in to the body through mouth, nose or a special hole made in the body.

Consider two lenses 1 and 2 of focal length f₁ and f₂ are placed coaxially in contact with each other so that they have a common principal axis.
O be the object which is placed beyond the focus of the first lens on the principal axis. I' is the image of object O which is formed beyond the lens 2. Then, I' acts as an object for the lens 'P' is the common optical centre of the two lenses.
From the figure, PO =u, PI' = v' for lens I
PI' =  v' (object distance) PI = v (image distance) for lens 2

For lens 1,
\(\cfrac { 1 }{ v' } -\cfrac { 1 }{ u } =\cfrac { 1 }{ { f }_{ 1 } } \)   .........(1) 
For lens 2,
\(\cfrac { 1 }{ v } -\cfrac { 1 }{ u' } =\cfrac { 1 }{ { f }_{ 2 } } \)    .........(2)
Adding (1) of (2)
 \(\cfrac { 1 }{ v } -\cfrac { 1 }{ u } =\cfrac { 1 }{ f_1 }+\cfrac{1}{f_2} \)   .........(3)
(vi) If the combination acts as a single lens of focal length f so that for an object at the position O it forms the image at I,
Then,
\(\cfrac { 1 }{ v } -\cfrac { 1 }{ u } =\cfrac { 1 }{ f } \)   .........(4) 
Comparing equations (3) and (4) we can write, 
\(\cfrac { 1 }{ F } =\cfrac { 1 }{ { f }_{ 1 } } +\cfrac { 1 }{ { f }_{ 2 } } \) .........(5)
The above equation can be extended for any number of lenses in contact as, 
\(\cfrac { 1 }{ f } =\cfrac { 1 }{ { f }_{ 1 } } +\cfrac { 1 }{ { f }_{ 2 } } +\cfrac { 1 }{ { f }_{ 3 } }+\cfrac { 1 }{ { f }_{ 4 } } +..........\)