HT. A. Rowland— Concave Gratings for Optical Purposes. 91 
ie MP (sin M+sin v) 
a (b*—a") (y—y') —B*y (e«—2#') = 
~ sinxu+sin vy cos asin € 
2a cos d6=r cos u—R cos v 
2a sin d=r sin u—R sin vy. 
On substituting these values and reducing, we find 
x. 2Rrcos a cos € 
pt=—, — 
r cos’ v+Reos’ 
Whence the focal length is 
pR cos’ 
2R cos a cos €—p cos’ v 
For the transmitted beam, change the sign oe o Supposing 
?, Rand » to remain constant and r and pz to , this equa- 
tion will then give the line on bina all the palin and the 
central image are brought to a 
By far the most interesting case is tobias hy making 
r=pcosu R=pcos vy, 
since these values satisfy the equation. The line of foci is 
then a circle with a radius equal to one-half p. Hence if a 
r= 
* A more simple solution is the following ; must be constant in the direction 
in which the dividing engine rules. If the —- engine rules in the direction 
of the axis, y, the differential of this with respect to y must be zero. But we can 
also bese! e the reciprocal of this quantity and so we can write for the equation of 
con 
Taking a circle as our curve we can write 
(a—a’* +(y—y’P=p* 
and (2—2? + (y—y"”P=R? 
poe toe aa 
a(R = ~y a 
SF Fg yy (SE) So 
d d(R+r) —a!! ¢—2/" —2”")(y—y”) @-7\y-¥") 
ae ee Rat te eee) 
+ (xa— —0)| nits eer} t =o 
aking o=0, y=20, y’=0, 2’ =p 
— =t- a Se =0, 
2 Ae COB M+COSY- 2Rr cos a cos € 
p= Rr Fost y+ Room «roost y+ Roose 
