688] 
GEOMETRICAL CONSIDERATIONS ON A SOLAR ECLIPSE. 
315 
and also twice touching the limit. If we consider, as before, the intersections of KV 
with the corresponding penumbral curve, this will be a curve extending from K x so 
as to touch the limit, and thence onward to K 2 , the portion from K x to the contact 
with the limit being the line “ eclipse begins at transit,” and the portion from the 
limit to K 2 the line “ eclipse ends at transit.” I say “ transit ” instead of midday, 
since for a circumpolar place the phenomenon may happen at one or the other transit 
of the sun over the meridian. It is to be remarked, that the node of the figure of 
eight is a point, such that the eclipse there begins at sunrise and ends at sunset; 
this point does not appear to be an important one in the geometrical theory. 
The two loops of the critic line may be of very unequal magnitudes, and in 
particular one of them may actually vanish; viz. the points K x and K„ then coincide 
together, and the critic curve is a closed cuspidal curve touching the horizon-envelope 
at the cusp; moreover, instead of two contacts with the limit there is one proper 
contact, and an improper contact at the cusp, that is, the limit simply passes through 
the cusp. And through this special separating case, we pass to the case where, 
instead of the figure of eight, we have a single oval, not touching the horizon-envelope 
(viz. the points K x , K., have become imaginary), but still touching the limit twice ; 
this is a distinct type for an eclipse of the second class. 
And, similarly, in an eclipse of the first class, where the points K x , do not 
in general exist (viz. geometrically they are imaginary), these points may present 
themselves in the first instance as two coincident points, viz. instead of the sunrise 
oval or the sunset oval (as the case may be), we have then a cuspidal curve; or 
they may be two real points, viz. instead of the same oval, we have then a figure 
of eight touching the horizon-envelope twice, and also touching each of the two limits. 
These are thus the several cases. 
When the Earth traverses the penumbral cone, the critic curve is 
1. A pair of ovals: 
2. An oval and a cuspidate oval: 
3. An oval and a figure of eight. 
And when the Earth does not traverse the penumbral cone, the critic curve is 
4. A figure of eight: 
5. A cuspidate oval: 
6. An oval. 
To which may be added the transition case which separates 1 and 4, viz. here the 
Earth just has an internal contact with the penumbral cone, and the critic curve is 
7. Two ovals touching each other. 
But of course 2, 5, and 7 are so special that they may be disregarded altogether; 
and 3 and 6 are of rare occurrence. I have not sufficiently examined the conditions 
for the occurrence of these forms 3 and 6; my attention was called to them, and 
indeed to the whole theory, by a question proposed by Prof. Adams in the Cambridge 
Smith’s Prize Examination for 1869. 
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