520 Prof. Arthur Schuster on 



h n 7 7tx 7 2tt% 

 -%+bx cos -j- + o 2 cos —j- + . . 



where 



= ljo^ 



cos — r— «\. 

 a 



Let the reflecting part cover a portion 2a of the grating, and 

 the opaque part a portion 2c. The function to be expanded 

 has a value one from ^ = to #• = &; the value, will vanish 

 between os=a and x = a + 2c, and regain the unit value from 

 x — cl — a to x~d. 

 We find in this way, 



7 2 ,., x . mTra 



fy» = (1 + cos mir) sin — =— . 



W17T a 



or if, as in the previous investigation, we put d = 27r/q, 



fix) — — + - sin aq cos qx + 5 sin 2«<7 cos 2g^ + -r sin \.aq cos 4g^ + 



The first term represents that part of the grating which 

 acts simply as a reflecting surface. In order that the first- 

 order spectrum should be as bright as possible sinag=l, or 

 d=4:a, in which case the spectra of even orders would all 

 disappear. The factor 2/ir, taken in conjunction with what 

 we have previously proved, shows that a grating constructed 

 in the manner indicated gives a first- order spectrum whose 

 intensity cannot exceed A/7T 2 , where A is the intensity of the 

 reflected image, if there are no lines on the grating. The 

 strongest second-order spectrum has an intensity A/47T 2 . 



The direct image could be much diminished, and therefore 

 the spectra increased in intensity, if a grating could be made 

 on a glass surface — say the largest surface of a right-angled 

 prism, — periodicity being introduced by silvering the glass 

 along parallel lines. If light were then allowed to fall in- 

 ternally on the silvered surface, the reflexions taking place 

 between glass air and glass silver respectively would have 

 nearly opposite phases, which is the condition required for the 

 diminution of intensity of the direct image. 



It appears, then, that although we have introduced an 

 imaginary grating called a simple grating, the effects are 

 exactly the same as those of a real grating, the latter con- 

 sisting of a superposition of simple gratings, as has already 

 been pointed out. 



11. We return now to our more immediate object, and take 

 as a second example of a disturbance analysed by a grating 

 that of a single impulsive velocity v reaching the point 



