, the whole of the land
would be gradually planed down to a submarine platform, and all the
globe would be covered with water. There are, however, periodical
warpings of this transitional area by which fresh areas of land are
raised above sea-level, and fresh continental coast-lines produced,
while the sea tends to sink more deeply into the great ocean basins, so
that the continents slowly increase in size. "In many cases it is
possible that the continental shelf is the end of a low plain submerged
by subsidence; in others a low plain may be an upheaved continental
shelf, and probably wave action is only one of the factors at work" (H.
R. Mill, _Realm of Nature_, 1897).

CONTINUED FRACTIONS. In mathematics, an expression of the form

b2
a1 +- -----------
b3
a2 +- -----------
b4
a3 +- ----------
b5
a4 +- ----
a5 +- ...,

where a1, a2, a3, ... and b2, b3, b4, ... are any quantities whatever,
positive or negative, is called a "continued fraction." The quantities
a1 ..., b2 ... may follow any law whatsoever. If the continued fraction
terminates, it is said to be a terminating continued fraction; if the
number of the quantities a1 ..., b2 ... is infinite it is said to be a
_non-terminating_ or _infinite_ continued fraction. If b2/a2, b3/a3 ...,
the _component fractions_, as they are called, recur, either from the
commencement or from some fixed term, the continued fraction is said to
be _recurring_ or _periodic_. It is obvious that every terminating
continued fraction reduces to a commensurable number.

The notation employed by English writers for the general continued
fraction is

b2 b3 b4
a1 +- -- -- -- ...
a2 +- a3 +- a4 +-

Continental writers frequently use the notation

b2 b3 b4 b2 | b3 | b4 |
a1 +- -- +- -- +- -- +- ..., or a1 +- |----| +- |----| +- |----| +- ...
a2 a3 a4 | a2 | a3 | a4

The terminating continued fractions

b2 b2 b3 b2 b3 b4
a1, a1 + --, a1 + -- --, a1 + -- -- --, ...
a2 a2 + a3 a2 + a3 + a4

reduced to the forms

a1 a1a2 + b2 a1a2a3 + b2a3 + b2a1
--, ---------, --------------------,
1 a2 a2a3 + b3

a1a2a3a4 + b2a3a4