202 Prof. H. A. Rowland on Concave Gratings 
sm fju + sin v cos a sin e 
2a cos S=r cos /*— R cos v, 
2a sin S = r sin /u,— R sin v. 
On substituting these values and reducing, we find 
* „-,-. cos « cos e 
*p = 2Rr 2 — ro 2~ > 
r cos 2 v + R cos z /r 
whence the focal length is 
:pR ; 
2R cos a cos e — /o cos 2 v 
For the transmitted beam, change the sign of R. 
Supposing p, R, and v to remain constant and r and p, to 
vary, this equation will then give the line on which all the 
spectra and the central image are brought to a focus. 
By far the most interesting case is obtained by making 
r = p cos fi, R=pcosv, 
ds 
* A more simple solution is the following. — must be constant in the 
direction in which the dividing-engine rules. If the clividing-engine 
rules in the direction of the axis, the differential of this with respect to y 
must be zero. But we can also take the reciprocal of this quantity ; and 
so we can write for the equation of condition, 
d d(R+r) _ Q 
dy ds 
Taking a circle as our curve, we can write 
(x-x'f+iy-y-y =p> 
and 
(x-x"f+(y-y"f = R 2 , 
(z-a?y+w-t/"y=r*, 
d d(R+r) If x-af' x -*"' _ , r(x-x")jy-y") (x-x"')(y-y'"y 
dy ds "pi R + r U y) L W + ~~ T 3 
+ (,-,")[^V-^]}=0. 
Making x=0, y=0, y'=0. x' = p, we have 
x" . x'" ix" 2 x" n \ n 
R+--p( W + -)=0, 
or 
P cosp-t-cosi/ _op . cos « cose 
rcos 2 v+Rcos 2 n ~ rcos 2 i>+Rcos a M' 
