ON THE SPECTRUM OF HYDROGEN. 
381 
inspection of the ends of the lines (cf. Nicholson and Merton, ‘ Phil. Trans.,’ A, 
vol. 216, p. 459, 1916).) The edges of the lines at a, b, e, f, c and d denote a certain 
critical intensity, I c , which is represented by a dotted line in the upper part of the 
figure. It will be seen that we assume only that a constant degree of blackening of 
the plate is produced by a light of constant but entirely unspecified intensity. 
Let x x and x 2 denote the distances of a and b, or c and d , respectively from the true 
maximum of intensity of the line in the double-order position (x — 0, a = tt/2), and « 0 the 
distance of e and/from the maximum in the single-order position (x — 0, a = 0). Then 
expressing a 0 and x in circular measure we have for the single-order position 
e /,a " 2 sin 2 rx, 
0/ a o > 
(i.) 
and for the double-order position 
L/In 
— e 
' kx sin 2 
a. a 
1 3 
I,/Io = 
,-kxX 
sim 
a 2 /a 2 5 
(ii.), (iii.) 
where rx 1 and a 2 are the angles corresponding to the points a and b , or d and c, 
respectively. 
Putting 
sin 2 « 0 / a o 2 — 1/ sin 2 a x /a x 2 = P, and sin 2 « 2 / a 2 2 = Q> 
we have 
log (R/P) = kot 0 2 - kx x 2 , log (R/Q) = ka 0 2 - kx 2 2 . . . (iv.), (v.) 
Now the plates are measured with a photo-measuring micrometer with which readings 
of the positions of the points a, b, c, d, e and / are obtained on an arbitrary scale, and 
since x x is not exactly equal to x 2 , the number of micrometer divisions between a and c, 
or b and d, does not correspond to an angle n. 
Since it is not possible to measure directly the number of micrometer divisions which 
are equal to the separation of successive orders, both equations (iv.) and (v.) are required 
to solve for k. 
The measurements give 2a 0 and (aq -j- x i) in micrometer divisions, and it is 
necessary to find a value of (x x — x 2 ) such that equations (iv.) and (v.) give the same 
value for k, from which the value of <U, the half-width, at once follows, since the 
difference in wave-length corresponding to the separation of successive orders is known 
from the optical constants of the grating. ( x x — x 2 ) is very small, and can readily be 
found by trial of a series of values, which can be plotted against the resulting values of 
[^fromOv.))—■&<from(v.))] o n squared paper. 
(9) Experimental Results. 
We have measured, in the manner described in the preceding section, the half-widths 
of three lines in the secondary spectrum, A X 6018, 6028 and 6225 A, and also the half- 
