200 Prof. H. A. Rowland on Concave Gratings 
sphere whose radius is p be at x' ', 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 + rc, 
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 ellipsoids or hyperboloid so indicated, we could use 
the surfaces in the same way as the wave-surface above. The 
intersections of these surfaces with any other surface form 
what are known as Huyghens's zones. By actually drawing 
these zones on the surface we form a grating which will 
reflect or refract 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's zones on the surface — that is, 
ds in terms of db. 
The equation of the circle is 
and of the ellipse or hyperbola, 
R + rc = 2b, 
or 
(6 2 -aV + % 2 = Zr(& 2 -a 2 ), 
in which c has disappeared. 
. dx y—y 1 
ds = s/da? + dy\ fy = —^r, 
dx{ (h 2 -a^-lPy^~j =b{2b*-(x* + if + a*)\db, 
dy{ -(V-a')x^ + Vy}=b{2V-(x*+f + a*)\ 
<h _ 2b 2 -(x 2 +y 2 + a 2 ) 
" db P {b' i -d i ){y-y')x-b\x-x')y 
This equation gives us the proper distance of the rulin o-s on 
the surface; and if we could get a dividing-engine to rule 
according to this formula, the problem of bringing the spec- 
trum to a focus without telescopes 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~ > 
db 
