Page:Elementary Text-book of Physics (Anthony, 1897).djvu/449

the parallel lines $ab,cd,$ etc., at such an angle that the distance from the centre of $a$ to the foot of the perpendicular let fall from the centre of the adjacent opening on $ab$ shall be equal to some definite wave length of light. It is evident that $an$ will contain an exact whole number of wave lengths, $co$ one wave length less, etc. The line $mn$ is, therefore, tangent to the fronts of a series of elementary waves which are in the same phase, and may be considered as a plane wave, which, if it were received on a converging lens, would be concentrated to a focus. If the obliquity of the lines be increased until $ae$ equals $2\lambda ,3\lambda ,$ etc., the result will be the same. Let us, however, suppose that $ae$ is not an exact multiple of a wave length, but some fractional part of a wave length, ${\tfrac {99}{100}}\lambda$ for example. Let $m$ be the fifty-first opening counting from $a;$ then $an$ will be ${\tfrac {99}{100}}\lambda \times 50=49.5\lambda .$ Hence the wave from the first opening will be in the opposite phase to that from the fifty-first. So the wave from the second opening will be in the opposite phase to that from the fifty-second, etc. If there were one hundred openings in the screen, the second fifty would exactly neutralize the effect of the first fifty in the direction assumed. Light is found, therefore, only in directions given by

$\sin \theta ={\frac {n\lambda }{d}},$

(114)

where $n$ is a whole number, $d$ the angle between the direction of the light and the normal to the grating, and $d$ the distance from centre to centre of the openings, usually called an element ot the grating. Gratings are made by ruling lines on glass at the rate of

some thousands to the centimetre. The rulings may also be made on the polished surface of speculum metal, and the same effects as described above are produced by reflection from its surface. Since the number of lines on one of these gratings is several thousands, it is seen that the direction of the light is closely confined to the