H. A. Rowland—Concave Gratings for Optical Purposes. $9 
CONCAVE GRATINGS. 
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 codrdinates of the radiant point and focal point be y=0, 
=—a and y=0, z=-+a, and let the center of the sphere 
whose radius is p be at 2’, y’. Let r be the distance from the 
radiant point to the point a, y, and let R be that from the focal 
point to z, y. Let us then write 
2b=R+re. 
Where c is equal to +1 according as the reflected or transmit- 
ted ray is used. Should we increase } by equal quantities and 
draw the ellipsoids 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 Huyghen’s zones. By actually drawing 
these zones on the surface, we form a grating which will dif- 
fract 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 z, y. Then our preseat problem is to 
nd the width of Huyghen’s zones on the surface, that is ds in 
terms of db, 
The equation of the circle is 
(e—2')’+ (y—y')"="" 
and of the ellipse or hyperbola 
R+re=26 
or a’) a +0'y'=8' (BP —a’) 
in which ¢ has disappeared | 
ns a 3. de _y—y 
ds=/ dx +dy’; ay sree 
dz | (Ba!) e—¥ ZS | bia (ty +a) }db 
dy | — (a) 22 4 by =) {2b'—(a*+y' +a’) }db 
_ oo 2b°— (a? +4" +a") . 
| ab Pa) (yy) 2—F ey 
This equation gives us the proper distance of the rulings on the 
surface, and if we could get a dividing engine to rule according 
to this formula the problem of bringing the spectrum to a 
