96 H. A. Rowland—Concave Gratings for Optical Purposes. 
Developing the value of cos 0’ in terms of 6, we have 
ae Ay, P08 My | na 
cos 6 =cos 6 } 143] 1+ Pr | _ 
p-sin wf 20) 14. 0} 
oR, be +A(1 +) Jo + &e. 
As the cases we are to consider are those where A is small, 
it will be sufficient to write 
»_ PCOS ft, 
tan 6’= R, 6. 
Whence we have 
sin u=sin pz, cos 6 | 1+ cot 4, A645 1 + ogre Ie 
ras pcos He) _ psin M, 2P\ | os t 
Se cot j., (142s oor (1+4 (1450) é + &e. 
We can write the value of sin» from symmetry. But we have 
9 _. : 
Fe = Sin MH +sin v. 
In this formula, dd can be considered as a constant depending 
on the wave-length of light, ete., and ds as the width apart of 
the lines on the grating. The dividing engine rules lines on 
the curved surface according to the formula 
2 # 00s 6 (sin uw, +sin v,). 
But this is the second approximation to the true theoretical 
ruling. And this ruling will not only be approximately cor- 
rect, but exact when all the terms of the series except the first 
vanish. In the case where the slit and focus are on the circle 
of radius 4p, as in the automatic arrangement described above, 
we have A=O and the second and third terms of the series - 
disappear, and we can write since we have : 
R r 
—2=cos #, and -=cos v, 
p p 
1 sin utanu,+siny, ofan gs +e.) 
| 
2 Pes 6(sin MM 9 tsin La ‘' 
sin “@,+sin v 
ds 
But in the partir arrangement we also have y,=0, and so 
the formula becom 
db : ; 1 
2 5 mae 6 (sin 4, +8in v,) | we tan pu, 0° + &e. t 
To find the greatest departure from theoretical perfection, @ 
must refer to the edge of the grating. In the gratings which 
