Interference Phenomena. 519 



It follows that, whatever the direction of the telescope, the 

 disturbance passing through its focus is of the same period as 

 that of the incident light, that is to say, whatever the period- 

 icity of the grating it has no power of altering the periodicity 

 of the disturbance. The amplitude of the disturbance is 



Q ^ a sin kI. The manner in which this varies according to 

 q 2 — /c 2 6 



the direction of the telescope is shown by expressing I in 

 terms of N, when it is seen that we may write for the ampli- 

 tude ^pql sin [2ttN(a: + ?)/?] 

 K±q 2irN (/c+q)/g 



The second factor is of the form (sin a)/«, which reaches 

 its maximum value when a = 0. The successive maxima are 

 very close together if N is large, so that the whole light is 

 near the points for which k= + q. The direction of the optic 

 axis in that case is given by +qa=p<y. The wave-length of 

 the original light being fjL=2ira/p and the distance between 

 the lines d=27r/q = 27r//c, the relation becomes 



fj, = dy= + d(sm a, — sin ft), 



which is the well-known condition that a wave-length fi shall 

 have a first-order maximum in the direction defined by /9. 

 The other maxima of (sin a) /a which our equation gives are 

 those commonly ascribed to diffraction from the edge of the 

 aperture of the telescope. 



The principal maximum has an amplitude which, restoring 

 the constant factors, is plh cos /3/27raF, or, introducing the 

 wave-length ja of the incident light, the amplitude becomes 

 hi cos /3/F/^. The numerator of this fraction is half the 

 effective aperture of the grating, and the intensity of the 

 spectrum is therefore one quarter of what it would be if the 

 grating were replaced by a reflecting surface, and the colli- 

 mator moved until the direct image coincided with the focus 

 of the telescope. A surface which acts like a grating must 

 also, as pointed out by Rayleigh, act as an absorbing surface, 

 so that the sum of the intensities in the diffracted spectra 

 will be smaller than that of the original light. 



10. Ordinary gratings are generally taken to contain 

 parts which are alternately completely reflecting or trans- 

 parent and opaque. Calling d the distance from the centre 

 of one line to the centre of the next, and taking the origin at 

 the centre of one of the lines, we may express the reflecting 

 properties of the grating (confining ourselves to reflecting 

 gratings) as a periodic function of a 9 say /(#), which may 

 be expressed in a series of the form 



