494 HENKY A. ROWLAND 



effect of a linear error in the spacing is to make the focus on one side 

 shorter and the other side longer than the normal amount. Professor 

 Peirce has measured some of Mr. Eutherfurd's gratings and found that 

 the spaces increased in passing along the grating, and he also found 

 that the foci of symmetrical spectra were different. But this is the 

 first attempt to connect the two. The definition of a grating may 

 thus be very good even when the error of run of the screw is consider- 

 able, provided it is linear. 



CONCAVE G-KATINGR 



Let us now take the special case of lines ruled on a spherical surface; 

 and let us not confine ourselves to light coming back to the same point, 

 but let the light return to another point. Let the co-ordinates of the 

 radiant point and focal point be y<=0, x = a and y = 0, x*+- a, and 

 let the centre of the sphere whose radius is p be at x r , y'. Let r be the 

 distance from the radiant point to the point x, y, and let R be that from 

 the focal point to x, y. Let us then write 



2b = R -f re, 



where c is equal to 1 according as the reflected or transmitted ray is 

 used. Should we increase b by equal quantities and draw the ellip- 

 soids or hyperboloids so indicated, we could use these surfaces in the 

 same way as the wave surfaces above. The intersections of these 

 surfaces with any other surface form what are known as Huyghens' 

 zones. By actually drawing these zones on the surface, we form a 

 grating which will diffract the light of a certain wave-length to the 

 given focal point. For the particular problem in hand, we need only 

 work in the plane x, y for the present. 



Let s be an element of the curve of intersection of the given surface 

 with the plane x, y. Then our present problem is to find the width of 

 Huyghens' zones on the surface, that is ds in terms of db. 



The equation of the circle is 



(x-xy + (y-y'? = f>* 

 and of the ellipse or hyperbola 



R + re = 2* 



or (i 2 a 2 ) x 3 + fry 2 = tf(V a' i ) 



in which c has disappeared. 



dx y y' 



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