by Curved Diffraction-gratings. 185 



In the last equation put 180 + 0' for 6', and —r 1 for /. 

 The equation then becomes identical with the corresponding 

 equation. Hence curves whose equation is 



cos 2 6 _ cos 1 

 r c d 



have the property that, if the source of light is at any point 

 on one of these curves, the whole of the reflected spectra pro- 

 duced by the grating lie on the same curve. 



If we start with the equation for transmitted light, 



sinfl' — sin 0=n- > 



we shall arrive at the same result. Hence we see that each 

 of the curves whose equation has just been found is the locus 

 of the foci of all the diffracted rays, when the source of light 

 is at any point on the curve. 



These curves I will call " diffraction-curves." It is obvious 

 from the equation that they are independent of the distance 

 between the lines of the grating. 



A table is given at the end of the paper showing the values 

 of r for every five degrees in the value of 6, d having the 



/? 

 values o? c, 3c, c being taken as 1000 ; and the forms of the 



o 



curves are shown in fig. 3. 



When d is infinite, the diffraction-curve is a circle having 

 the radius of curvature of the grating as diameter, and a 

 straight line through D tangential to the grating. In every 

 other case the curve is formed of two loops, one lying inside 

 the circle and the other outside, touching one another at D. 

 The inner loop is always an oval, which is infinitely small 

 when d is zero, and increases as d increases, until d becomes 

 infinite, when the inner loop coincides with the diffraction- 

 circle. The outer loop is finite when d is less than c ; and 

 increases as d increases, until d equals c, when the outer loop 

 becomes infinite, and resembles a parabola. When d is greater 

 than c the outer loop takes somewhat the form of an hyperbola, 

 with the asymptotes inclined to the axis at an angle whose 



cosine is -^, and intersecting one another at a distance from the 



Co 



C 3 



grating = — 2 r 2 . One of the two branches into which the 



outer loop is now divided passes through D, always retaining 

 the resemblance to a branch of an hyperbola, and ultimately, 

 when d is infinite, becomes a straight line tangential to the gra- 

 Phil. Mag. S. 5. Yol. 15. No. 93. March 1883. 



