88 H. A. Rowland— Concave Gratings for Optical Purposes. 
the solution holds for ony one wave-length and so white light 
will be drawn out into a spectrum. Hence we have the im- 
Bortant conclusion that a  rectoneally: perfect grating for one 
position of the slit and eyepiece can be ruled on any surface, 
at or otherwise. This is an extremely celal practical 
conclusion and explains many facts which have been observed 
in the use of gratings. For we see that errors of the dividing 
engine can be counterbalanced by errors in the flatness of the 
plate, so that a bad dividing engine may now and then gme 6 a 
grating which is good in one spectrum but not in all. An 
we often find that one spectrum is better than another. Far: 
thermore Professor Young has observed that he could often 
improve the definition of a grating by slightly bending the 
plate on which it was ruled. 
From the above theorem we see that if a plate is ruled in 
circles whose radius is r sin and whose distance apart is 
view the spectrum in that particular position of the ate: 
Had the wave surfaces been cylindrical instead of spherical the 
lines would have been straight instead of circular, but at the 
above distances apart. In this case the spectrum would have 
been brought to a focus, but would have been diffused in the 
direction of the lines. In the same way we can conclude that 
in flat gratings any departure from a straight line has the effeet 
of causing the dust in the a Rate the spectrum to have differ- 
ent foci, a fact sometimes o 
We also see that, if the departure from equal spaces is small, 
or, in other words, the distance r is great, the lines must be 
ruled at distances apart ei sheres by 
(1-2 
in order to bring the light to a focus at the angle # and distance 
r, c being a constant and « the distance from some point on 
the plate. If » changes sign, then r must change in sign. 
Hence we see that the effect of a linear error in the spacing is 
to make the focus on one Side shorter and the other side longer 
than the normal amount. Professor Peirce has measured some 
of Mr. Rutherfurd’s gratings and found that the fe incfeased 
in passing along the grating, and he also found that the foci of 
symmetrical spectra were different. But this is the ee attempt 
to connect the two, The definition of a grating may thus be 
very good even elas the error of run of the screw is consider- 
able, provided it is linear. 
ae + ke.) 
