206 Prof. H. A. Rowland on Concave Gratings 
one part in at least five thousand by a simple reading. By 
having a variety of scales, one for each spectrum, we can im- 
mediately see what lines are superimposed on each other, 
and identify them accordingly when Ave are measuring their 
relative wave-length. On now replacing the eyepiece by a 
camera, we are in position to photograph the spectrum with 
the greatest ease. We put in the sensitive plate, either wet 
or dry, and move to the part we wish to photograph. Hav- 
ing exposed for that part, we move to another position and 
expose once more. We have no thought for the focus, for 
that remains perfect, but simply refer to the table giving 
the proper exposure for that portion of the spectrum, and so 
have a perfect plate. Thus we can photograph the whole 
spectrum on one plate in a few minutes from the F line to 
the extreme violet, in several strips each 20 inches long. And 
we may photograph to the red rays by prolonged exposure. 
Thus the work of days with any other apparatus becomes the 
work of hours with this. Furthermore each plate is to scale, 
an inch on any one of the strips representing exactly so 
much difference of wave-length. The scales of the different 
orders of spectra are exactly proportional to the order. Of 
course the superposition of the spectra gives the relative 
wave-lengths. To get the superposition, of course photo- 
graphy is the best. 
Having so far obtained only the first approximation to the 
theory of the concave grating, let us now proceed to a second 
one. The dividing-engine rules equal spaces along the chord 
of the circular arc of the grating ; the cmestion is whether 
any other kind of ruling would be better. For the dividing- 
engine is so constructed that one might readily change it to 
rule slightly different from ecmal spaces. 
The condition for theoretical perfection is that C shall re- 
main constant for all portions of the mirror. I shall therefore 
investigate how nearly this is true. Let p be the radius of 
curvature, and let R and r be the true distances to any point 
of the grating, R and r being the distances to the centre. 
Let /a and v be the general values of the angles, and fi and v 
the angles referred to the centre of the mirror. The condition 
is that 2 
~-= sin/u,-(- sin v 
shall be a constant for all parts of the surface of the grating. 
Let us then develop sin p and sin v in terms of fi , v , and 
the angle 8 between radii drawn to the centre of the grating 
the point under consideration. Let h f be the angle 
Then we can write immediately 
