480 Prof. R. W. Wood on a Simple Treatment of 



difference in this case being 144°. At the centre (180°) we 

 have intensity one, as in the case of the three-line grating. 



It will be noticed that we have a zero value only when the 

 starting and terminal points of our broken line of vectors 

 coincide. With a phase difference of 180° this will happen 

 when the number of lines in the grating is even. It may 

 also happen when a closed figure such as a triangle, square, 

 or regular polygon is formed. 



For a six-line grating we must show the existence of 4 

 secondary maxima. The illumination w T ill be zero for the 

 180° phase difference, also for that of 60° when we have a 

 hexagon, and for 120° when we have two superposed triangles. 

 A star figure cannot be formed of 6 lines compounded as 

 described. 



In the case of a seven-line grating we have the zero minima 

 for the regular polygon (P.D. 51°*6) and for two star-shaped 

 figures (P.D. 102° and 154°), giving us five secondary maxima 

 between the two principal maxima. The eight-line grating- 

 gives zero when the amplitude lines form an octagon, two 

 superposed squares, an eight-pointed star (P.D. = 135°) and 

 also at the centre of symmetry (P.D. = 180°). 



In the case of the twelve-line grating the lines form in 

 succession a twelve-sided polygon, two superposed hexagons, 

 three squares, four triangles, and a twelve-pointed star, the 

 phase difference in the latter case being 150°. 



The following method of ascertaining the number of 

 possible figures will be found useful. 



Arrange around a circle as many equidistant dots as there 

 are lines in the grating, and join the dots by straight lines, 

 first shifting one dot, then two, three, four, &c. With twelve 

 dots we get the twelve-pointed star when we skip four "dots 

 each time. For a grating of thirteen lines we find it possible 

 to form thirteen-pointed stars in five different ways, between 

 0° and 180° phase difference. Each of these gives zero 

 illumination, also the polygon of thirteen sides; consequently 

 we have six minima between 0° and 180°, or eleven secondary 

 maxima. In the case of twelve dots we get but one star, 

 skipping 1, 2, and 3 dots giving the hexagons, squares, and 



triangles. 



We thus see that, in the case of a grating of n lines, we 

 have (n — 2) secondary maxima between the principal maxima, 

 the intensity of which can be easily calculated from diagrams 

 similar to those given. 



We will now consider the case of the optical grating with 

 many thousand lines, and see what part the secondary maxima, 

 which accompany the spectrum-lines, play. 



