TUB GREAT ECLIPSE OF AUGUST 17, 1868. 
269 
In fig. 4, c is the point to which the cone of shadow would 
reach in case 2, were it not for the interposition of the earth. 
u'u' bound the umbral, pj9, p'p' the penumbral cone. In 
this figure, as in the former, the angle ucu' is greatly exagge- 
rated, but EC bears its just proportion to the radius of the 
earth e. 
If we consider that the lines cit, pp, cw', p'p', in both figures 
meet (in pairs) upon the moon’s globe, we shall see that the 
penumbral section through c has a radius exactly equal to 
twice the moon’s diameter. Hence clearly pp' is greater in fig. 3 
and less in fig. 4 than twice the moon’s diameter. We see then, 
that in an annular eclipse the penumbra extends {cceteris 
paribus) over a wider range of country than the penumbra in 
a total eclipse.* 
Now the greatest possible value of u u' (fig. 4) can easily be 
determined. For the sun should be at its smallest — that is, 
should subtend 3F 31"*8, and this, therefore, is the value of the 
angle itcu' ; also it is known that ec is equal to 4| radii of the 
earth (when the moon is in perigee and the sun in apogee). 
Therefore it is readily seen that the arc uu' is equal to 31' 
31"*8 increased in the proportion of 4J to 1, or as nearly as 
possible to 2-^°. Hence the greatest radius of the circular 
umbra thrown on the earth, as in fig. 4, is 2^°, or about 173v 
miles. 
But it is clear that an eclipse of this extent cannot happen 
once in many thousands of years, nor can one happen often 
which approaches even pretty closely to the conditions here 
required. 
It is obvious that if the moon is removed by any considerable 
arc from her perigee, or the sun from his apogee, there will be 
a much smaller umbra. And there is another consideration to 
be noticed. The shadow does not necessarily fall, as shown in 
fig. 4, directly towards the centre of the earth. It might fall 
near p or p' for example ; in which case the point p and p' 
being further from the moon than u and u' the shadow would; 
clearly be less than u u' (for ucuf is a cone decreasing towards 
c, and p is farther from the moon than u is). In point of fact, 
most of the noted total eclipses have fallen far away from the 
equator, and thus have been less considerable than those which 
can take place in equatorial or sub-tropical regions. 
But in the great eclipse of the present year, nearly all the 
conditions which tend to increase the moon’s shadow are pretty 
closely fulfilled. 
First as respects the sun’s apparent diameter, which should 
* It mny easily be shown that pp' (in fig. 3), diminished by a a', is equal 
to pp' (in fig. 4) increased by uu'. 
VOL. VII. — JSO. XXVIII. U 
