for Optical Purposes. 199 
dividing-engine may now and then make a grating which is 
good in one spectrum but not in all. And so we often find 
that one spectrum is better than another. Furthermore Prof. 
Young has observed that he could often improve the definition 
of a grating by slightly bending the plates on which it was 
ruled. 
. From the above theorem, we see that if a plate is ruled in 
circles whose radius is r sin /j, and whose distance apart is 
Ar sin /*, where Ar is constant, then the ruling will be appro- 
priate to bring the spectrum to a focus at a distance r and 
angle of incidence fi. Thus we should need no telescope 
to view the spectrum in that particular position of the grating. 
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 effect of causing the 
dust in the slit and the spectrum to have different foci — a fact 
sometimes observed. 
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 represented by 
\ r sin fi / 
in order to bring the light to a focus at the angle fx and dis- 
tance r, c being a constant and x the distance from some point 
on the plate. If fi changes sign, the 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. Prof. Peirce has measured some of 
Mr. Putherfurd's gratings and found that the spaces increased 
in passing along the grating; and he also found that the foci 
of symmetrical spectra were different. But this is the first 
attempt to connect the two. The definition of a grating may 
thus be very good even when the error of run of the screw is 
considerable, provided it is linear. 
Concave Gratings. 
Let us now take the special case of lines ruled on a spherical 
surface; and let us not confine ourselves to light coming back 
to the same point, but let the light return to another point. 
Let the coordinates of the radiant-point and focal point be 
y=0, x= — a, and y = 0, x— +a, and let the centre of the 
