for Optical Purposes. 207 
psm/ji = p sin /x Q cos 8' + B sin S'—p cos //. sin S', 
sin //,= sin u, cose)' ^ lH r-^ — AtanS' V, 
I psmyLfco J 
where 
A _ 1 _pcos/* ?f 
E 
Developing the value of cos 8' in terms of 8, we have 
As the cases we are to consider are those where A is small, it 
will be sufficient to write 
whence we have 
sin jm= sin jjl cos 8 -j 1 + cot /* AS 4-^1 + -p ^ 2 
+ [a.^(i + e-»)-eg*( 1+A ( 1+ g)], + & ,} 
We can write the value of sin v from symmetry. But we 
have 
a db . 
2 -^ = sinu+ sin v. 
In this formula db can be considered as a constant depending 
on the wave-length of light &c, 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 -r- = cos 8 (sin p, + 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 \p, as in the automatic arrangement described above, 
we have A=0, and the second and third terms of the series 
disappear, and we can write, since we have — = cos fi and 
