ON CONCAVE GRATINGS FOR OPTICAL PURPOSES 501 



centre. Let fi and v be the general values of the angles and //<, and V Q 

 the angles referred to the centre of the mirror. The condition is that 



o 



-^ = sin // + sin v 

 L/ 



shall be a constant for all parts of the surface of the grating. Let us 

 then develope sin // and sin v in terms of /* , v and the angle d between 

 the radii drawn to the centre of the grating and to the point under con- 

 sideration. Let d' be the angle between R and R . Then we can write 

 immediately 



/> sin fi = p sin /./ cos 8' + R sin d' p cos // sin 8', 



sin /j. = sin // cos d' \ 1 + r J l A tan 8 f i , 



iOsmjH, y 



where * _ -. _ p cos ,u 



Developing the value of cos d' in terms of d, we have 

 cos " = cos S { 1 + A [l + '' 8 "'"| ' 



As the cases we are to consider are those where A is small, it will be 

 sufficient to write 



tan * : = 



Whence we have 



sin <,. = sin // cos d 







+ ,5 3 + &c. 



\ - 

 We can write the value of sin v from symmetry. But we have 



2 -7- = sin fj. + sin v . 

 as 



In this formula, db can be considered as a constant depending on 

 wave-length of light, etc., and ds as the width apart of the lines on the 

 grating. The dividing engine rules lines on the curved surface accord- 

 ing to the formula 



2 -7- = cos 8 (sin //,+ sin v ). 



CL8 



But this is the second approximation to the true theoretical ruling. 

 And this ruling will not only be approximately correct, but exact when 



