AGGREGATE EFFECT OF INTERFERENCE. 161 



Also the middle term is 



(hm-i = 2m. 



We thus obtain as the value of the intensity, 

 He 



2 



*He 



4a 



J dx — {cos (6m - 4) x - 2 cos (6m - 6)x + ...+ m\ 



{ c -<«»-«>«_ 2£? - (6 "'- 6)a + ... + m] 



rrHe 



j {1 - 2 + 1 + 2 - 4 + ... to (3m - 2) terms + m\ 



4 a 

 ir He 



{1 . (6m - 4) - 2 (6m - 6) + ... to (3m - 2) terms}. 



Now 1 — 2 + l+...+ m is obviously half the above expansion, when 

 = 1, and is consequently zero. 



Also l.(6m- 4) - 2 (6m - 6)+ 



= 1 (6m - 4) + 2(6m - 10) + + m(6m - 6m - 2) 



- 2 (6m- 6)- 4 (6m - 12)- -2(m - 1) (6m - 6m - 6) 



+ 1 (6m - 8) + 2 (6m - 14) + + (m - 1) (6m - 6m- 4) 



= 1 .(12m - 12) +2 (12m - 24)+...+ (m - 1) (12m - 12m - 12) 



- 2 (6m- 6) — 4 (6m -12)- - 2 (m -1) (6m - 6m - 6) 



+ m (6m — 6m — 2) 

 = 2 m. 



Hence the whole intensity is - Hem = - H x space left uncovered, the 

 same result as before. 



5. Lastly, let us take the most general case, of a grating in which 

 the thickness of the bars bears any proportion whatever to the spaces left 

 uncovered. 



Vol. VII. Part II. X 



