the Secondary Maxima of Grating Spectra, 481 



Our broken line now becomes essentially a smooth curve. 

 We have our first minimum when it forms a complete circle, 

 the phase difference between disturbances from the first 

 and last lines being 360°, or the path difference X. The first 

 secondary maximum occurs when the line has wound up into 

 a circle and a half. The ratio of the intensity of the secondary 

 to that of the principal maximum is obviously the ratio of the 

 square of the diameter of the circle of 1\ turns to the square 

 of the total length of the line. This we easily find by winding 

 up a strip of paper of known length. It will be found to be 

 about 1 : 23. 



This shows us that, no matter how many lines we have in 

 the grating, our spectrum-lines will always be accompanied 

 by close companions, having at least ^ of their brightness. 

 For an eight-line grating the ratio is not very different, being 

 about 2V It occurs for a phase difference approximately 

 such that we have a regular pentagon, three sides of 

 which are made of double lines. For a four-line grating it 

 is about T a g (triangle with one side double)*, and for a 

 three-line grating ^. This last is the maximum value of the 

 ratio. 



It seems quite surprising that even for optical gratings the 

 secondary maxima have a brilliancy very nearly one half of 

 that which obtains in the case of a three-line grating. Their 

 angular distance from the spectrum-lines is such as to make 

 the path difference between disturbances coming from the first 



and last lines of the grating '— - more than the path difference 



at the line. This angle is obviously that subtended by one and 

 one half waves, at a distance equal to the width of the ruled 

 surface. The distance between the secondary maxima is thus 

 seen to depend upon the width of the grating, and not upon 

 the number of lines. At first sight there may appear to be 

 some difficulty about this, since there are n — 2 secondary 

 maxima ; and we might very naturally expect an increase of 

 n to push them nearer together. This is, however, only true 

 when the " grating space " remains constant, i. e. when we 

 add new lines of the same spacing. 



Suppose we have a grating of given width with 20 lines 

 which gives 18 secondary maxima. If we interpolate lines 

 between the lines already present, we double the number of 

 secondary maxima to be sure; but the principal maxima 

 (spectra) of odd order disappear by interference ; in other 



* This, however, is not the centre or brightest point of the maximum,, 

 which is for a P.D. of 135°, the figure having the form shown in fig. 2, 

 At this point the intensity is about T " T that of the principal maxima. 



