186 On the Spectra formed by Curved Diffraction-gratings. 



ting. The other branch has points of inflection, if d is greater 



3 



than 5C, in the positions for which 3c cos 6=d— \/l8c 2 + d 2 ; 



and when d is greater than 2c, this branch has points which 



are at a minimum distance from D. At these points the dis- 



4c 2 2 c 



tance from D is —r f and cos Q— — -= . Consequently the locus 



of these points is the circle of curvature of the grating. When 

 d becomes infinite, this branch coincides with the tangent to 

 the grating at D, and with the diffraction-circle. 



The diffraction-curve has been shown to consist of two loops, 

 one of which passes through the source of light. This loop is 

 the locus of the spectra of transmitted light; and the wave- 

 length at any point is given by the equation 



n\=a (sin 6 f — sin 6). 



The other loop is the locus of the spectra of reflected light ; 

 and the wave-length at any point is given by the equation 



n\ = cr (sin 6' + sin 0). 



As both loops coincide in the diffraction-circle, this circle is 

 the locus both of the spectra of transmitted and of reflected 

 light when the source of light is on the circle. 



As an example of the determination of the wave-length, 

 suppose the grating to have 25,000 lines to the inch ; then 

 each division of the grating is 40 millionth s of an inch. 

 Divide the diameter of the diffraction-circle into 40 parts, 

 and with the centre of curvature as centre describe circles 

 through these divisions, and number the points in which they 

 cut the diffraction-circle, beginning with the centre of curva- 

 ture as zero, and counting the readings as positive on one side 

 of the zero and negative on the other (see fig, 4, in which only 

 every tenth reading is given). If the source of light be at 

 the centre of curvature, the readings of the diffraction-circle 

 will give the wave-lengths, or multiples of them, in millionths 

 of an inch. Now with the centre of the grating as centre of 

 projection, project the readings of the diffraction-circle on 

 both branches of any diffraction-curve, and place the source 

 of light at the point in which one of the loops cuts the perpen- 

 dicular from the centre of the grating. The readings give the 

 wave-lengths, or multiples of them, as before. 



Let the source of light be now placed at any point of a gra- 

 duated diffraction-curve. Take the reading of the point, and 

 subtract it from the readings of all other points on the same 

 loop ; the new readings will give the wave-lengths, or mul- 

 tiples of them, for transmitted light. Add the reading of the 



