ASTRONOMY: SHAPLEY AND NICHOLSON 
421 
not too small, however, and the widening due to rotation is not too 
large compared with the normal width of the line, these three factors 
will not appreciably affect the form of the line given by (6). The 
narrowest lines observed in stellar spectra are in width rarely less 
than 0.2A, which is of the same order as the semi-amplitude of the 
shift for a t3^pical Cepheid variable. 
The true form of a line in the spectrum of a pulsating star is there- 
fore not as given in figure 1, but may be approximated by integrating 
(6) over the whole surface of the star. Any given incremental annulus 
on the stellar disk has a velocity and contributes an incremental 
spectral line whose intensity curve is given by (6) : 
jf _ J/ ^-KKv-vny- 
The intensity of this line for maximum absorption, Iq, occurs for 
V = Vn] and hence from (3), 
/J = /, = 27rZoK(l -x)+vlx] (9) 
Substituting (9) in (6) 
r = 27r lo [vn (1 - ^) + vl x] e-""'^'-'^^' 
and summing up these incremental lines for all chosen values of v„ 
from 0 to 1, we obtain, as closely as we care to compute, the intensity 
curve from the whole surface : 
I 
r = F{v,x) = 
Writing c\ = v from (6), and /q = 27r/o, we have, in the limit, for 
the intensity of the built-up absorption line at any point X, 
J// 3 
=f{\,x) ={[v„a-x)+ vl x] e-^'^'"--'^^' dv, (10) 
Equation (10) has been integrated mechanically for dift'erent values of 
X and different ratios of line width to displacements; figures 3a and 3b 
show the resulting intensity curves for line widths of about 0.4A and 
0.2A, respectively, adopting in both cases a radial velocity that would 
yield a measured displacement of about 0.3 A. From these curves we 
obtain the results of the following table: 
WIDTH 
OF UNDIS- 
PLACED 
LINE 
WIDTH OF DISPLACED LINE 
MEASURED DISPLACEMENT 
ACTUAL DIS- 
PLACEMENT 
AT CENTER 
OF DISK 
*=l 
x = h 
x=l 
From figure 3a 
From figure 3b 
0.34A 
0.17A 
0.40 A 
0.23 A 
0.38A 
0.22A 
0.30A 
0.32 A 
0.32 A 
0.33A 
0.40 A 
0.40 A 
