( 274 ) 
radius, and therefore being the weaker ones; then, as we approach 
the outer limits of the region abed, the rays that come gradually 
straighter from the source. 
A disturbance of this “normal” arrangement of density gradients 
round a vortex may incidentally cause the photospheric light to 
follow a shorter path towards the observer, even through the central 
parts ot the spot. The brilliant tongues and bridges, often observed 
in spots, find a natural explanation in this way. 
Within a streaming mass of gas the velocity usually changes 
periodically (think of gusts of wind!); so there are shells of greater 
and of smaller density following each other alternately, and cutting 
the stream-lines at right angles. In the circulating matter round a 
vortex such shells must assume almost radial directions, but at the 
same time be slightly curved in a spiral shape. Refraction in an 
optical system thus constituted, brings about a distribution of brightness, 
known to us as the radial-filamentary structure of the penumbra l ). 
We do not want, in our spot theory, to make any special suppo¬ 
sition as to the form of the region in which the density is a minimum. 
If it 'resembles a cylinder with its axis pointing towards us, the 
refraction effects will indeed be somewhat greater than if it were, 
for instance; spherical; but in the latter case too they may be strong 
enough and will bear the same principal character. Our explanation 
of the umbra and the penumbra rests on the only supposition, that 
in a certain region of the sun’s atmosphere the density gradients are 
sufficiently great 2 ), and gradually decrease from a central point 
outwards. 
Anomalous dispersion in the region of a sun-spot. 
So far we only dealt with the effects of refraction in general, 
without noticing the disparity of its values with the various waves 
of the spectrum, that is, the dispersion. 
In general the refraction-constant of transparent media is positive; 
in the visible spectrum it increases with decreasing wave-length. 
Substances, however, showing absorption lines, may have refraction 
constants of such extreme variety of magnitude with the waves 
The spiral-shaped flocculi visible 
Journ. XXVIH, Sept. 1908), and cov 
similar origin. 
on Hale’s spectroheliograms with H* (Astroph. 
rering much greater regions, probably have a 
) Astronomers who do not even hesitate to admit violent eruptions in sun-spots, 
and irregular velocities of many kilometres per second as a rule all over the sun’s 
sur ace, will certainly not think the density gradients, required in our argument, 
unreasonably great. 
