ON CONCAVE GRATINGS FOR OPTICAL PURPOSES 495 



dzl (b z a 2 ) xPy ^^ } = b\W 



- (a? 



x x 



. - ,b 

 " 



(V + y* + a 2 ) 



This equation gives us the proper distance of the rulings on the sur- 

 face, and if we could get a dividing engine to rule according to this 

 formula the problem of bringing the spectrum to a focus without tele- 

 scopes would be solved. But an ordinary dividing engine rules equal 

 spaces and so we shall further investigate the question whether there 

 is any part of the circle where the spaces are equal. We can then write 



ds __ n 

 db~ 



And the differential of this with regard to an arc of the circle must 

 be zero. Differentiating and reducing by the equations 



dx _ _y y' . db _ p 

 ~dy ~ x=2' ~dy ~ G (x a/)' 

 we have 



P { 2xb (y y'} - 2yb (x x'}- - [6i a - (a? + y 1 + a 1 )] } 



It is more simple to express this result in terms of E, r, p and the 

 angles between them. 



Let fi. be the angle between p and r, and v that between p and R. Let 

 us also put 



Let /?, f and 3 also represent the angles made by r, R and p respec- 

 tively with the line joining the source of light and focus, and let 



Then we have 



_ R cos f + r cos ,5 _ R sin f + r sin p _r cos /3 R cos y 



-I 2/ 9 9. " 



