198 Prof. H. A. Rowland on Concave Gratings 
lines and the form of the surface on the definition of the gra- 
ting. Mr. C. S. Peirce has recently shown, in the 'American 
Journal of Mathematics,' that a periodic error in the ruling 
produces what have heen called " ghosts " in the spectrum. At 
first I attempted to calculate the effect of other irregularities 
by the ordinary method of integration; but the results obtained 
were not commensurate with the labour. I then sought for a 
simpler method. Guided by the fact that inverse methods in 
electrical distribution are simpler than direct methods, I soon 
found an inverse method for use in this problem. 
In the use of the grating in most ordinary spectroscopes the 
telescopes are fixed as nearly parallel as possible, and the gra- 
ting turned around a vertical axis to bring the different spectra 
into the field of view. The rays striking on the grating are 
nearly parallel : but for the sake of generality I shall a-ume 
that they radiate from a point in space, and shall investigate 
the proper ruling of the gracing to bring the rays back to the 
point from which they started. The wave-fronts will be a 
series of spherical shells at equal- distances apart : if these 
waves strike on a reflecting surface they will be reflected 
back, provided they can do so all in the same phase. A 
sphere around the radiant-points satisfies the condition for 
waves of all lengths; and this gives the case of ordinary reflec- 
tion. Let any surface cut the wave-surface in any manner, 
and let us remove those portions of the surface which are cut 
by the wave-surface. The light of that particular wave-length 
can then be reflected back along the same path and in the 
same phase; and thus, by the above principle, a portion will be 
thus sent back. But the solution only holds for one wave- 
length; and so white light will be drawn out into a spectrum. 
Hence we have the important conclusion that a theoretically 
perfect grating for one position of the slit and eyepiece can 
be ruled on any surface, flat or otherwise. This is an ex- 
tremely important 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 counter- 
balanced by errors in the flatness of the plate ; so that a bad 
tions, for which I have not yet had time, I did not publish it at the time. 
I was much surprised soon after to see an article on this subject, which 
had been presented to the Physical Society and was published in the 
Philosophical Magazine. The article contains nothing more than an ex- 
tension of my remarks at the Physical Society, and formulae similar to 
those in this paper. As I have not before this published any thing except 
a preliminary notice of the concaye grating, I expected a little time to 
work up the subject, seeing that the practical work of photographing the 
spectrum has recently absorbed all my time. But probably Ihave waited 
too lonor. 
