204 Prof. H. A. Rowland on Concave Gratings 
tional to the line A C. Or, for any given spectrum, the wave- 
length is proportional to that line. 
If a micrometer is fixed at C, we can consider the case as 
follows: 
1 JtN . , . 
C = 2^ = 2 (sm/i+Sm,/) ' 
dk w 
— = == cos a. 
du N ^ 
If D is the distance the cross-hairs of the micrometer move 
forward for one division of the head, we can write for the 
point C, 
, D 
and for the same point fi is zero. Hence 
AX——. 
But this is independent of v ; and we thus arrive at the 
important fact that the value of a division of the micrometer 
is always the same for the same spectrum and can always 
be determined with sufficient accuracy from the dimensions 
of the apparatus and number of lines on the grating, as well 
as by observations of the spectrum. 
Furthermore, this proves that the spectrum is normal at 
this point and to the same scale in the same spectrum. Hence 
we have only to photograph the spectrum to obtain the nor- 
mal spectrum, and a centimetre for any of the photographs 
always represents the same increase of wave-length. 
It is to be specially noted that this theorem is rigidly true 
whether the adjustments are correct or not, provided only 
that the micrometer is on the line drawn perpendicularly 
from the centre of the grating, even if it is not at the centre 
of curvature. 
As the radius of curvature of concave gratings is usually 
great, the distance through which the spectrum remains practi- 
cally normal is very great. In the instrument which I prin- 
cipally use, the radius of curvature, p, is about 21 ft. 4 inches, 
the width of the ruling being about 5*5 inches. In such an 
instrument the spectrum thrown on a flat plate is normal 
within about 1 part in 1,000,000 for six inches, and less than 
1 in 35,000 for eighteen inches. In photographing the 
spectrum on a flat plate, the definition is excellent for twelve 
inches ; and by use of a plate bent to 11 ft. radius, a plate of 
twenty inches length is in perfect focus, and the spectrum 
