(2) 
418 ASTRONOMY: SHAPLEY AND NICHOLSON 
coefficient. Then the total light of a star is given by- 
sir 1 
L^'Joj j (l-x + x Vl - r2) rdrdd 
0 0 
Introducing (1) and integrating once 
L = 27rJo ^jv (1 -x)dv + jv^xdi^ 
Suppose that from any given element of surface the width of the 
absorption lines is very small compared with the displacement due to 
radial motion of the element; and let the amount by which the con- 
tinuous spectrum is decreased by a given absorption hne be Z' = L 
times a constant. The equation of the intensity curve of a Hne in the 
/ dL' 
spectrum of the whole surface would then be 7- = — which may be 
dL' 
written because of the essentially linear relation (throughout very 
short intervals of the spectrum) between wave-length, X, and velocity; 
and hence, from (2), 
~ =2ir[v{\- x) + vH] (3) 
'0 
where /o is the intensity for maximum absorption {v = 1) divided 
by 27r and is independent of the degree of darkening. 
For a uniform disk a; = 0 and 
- = Itv (4) 
while for one darkened to zero at the limb, x = 1, and 
- ^lir'U' (5) 
These intensity curves are plotted in figure 1. For a star of the spectral 
character of the sun, x is f in the neighborhood of X = 4400 A. There- 
fore 
/o 2 ' 
and the intensity curve Hes between those for uniform and completely 
darkened disks, but much nearer the latter, as shown by the broken 
curve in figure 1. 
It is obvious from the above discussion that darkening at the limb 
will aid in concealing whatever asymmetry there may be in the spectral 
