Page:EB1911 - Volume 07.djvu/45

Rh CONTINENTAL SHELF, the term in physical geography for the submerged platform upon which a continent or island stands in relief. If a coin or medal be partly sunk under water the image and superscription will stand above water and represent a continent with adjacent islands; the sunken part just submerged will represent the continental shelf and the edge of the coin the boundary between it and the surrounding deep, called by Professor H. K. H. Wagner the continental slope. If the lithosphere surface be divided into three parts, namely, the continent heights, the ocean depths, and the transitional area separating them, it will be found that this transitional area is almost bisected by the coast-line, that nearly one-half of it (10,000,000 sq. m.) lies under water less than 100 fathoms deep, and the remainder 12,000,000 sq. m. is under 600 ft. in elevation. There are thus two continuous plain systems, one above water and one under water, and the second of these is called the continental shelf. It represents the area which would be added to the land surface if the sea fell 600 ft. This shelf varies in width. Round Africa—except to the south—and off the western coasts of America it scarcely exists. It is wide under the British Islands and extends as a continuous platform under the North Sea, down the English Channel to the south of France; it unites Australia to New Guinea on the north and to Tasmania on the south, connects the Malay Archipelago along the broad shelf east of China with Japan, unites north-western America with Asia, sweeps in a symmetrical curve outwards from north-eastern America towards Greenland, curving downwards outside Newfoundland and holding Hudson Bay in the centre of a shallow dish. In many places it represents the land planed down by wave action to a plain of marine denudation, where the waves have battered down the cliffs and dragged the material under water. If there were no compensating action in the differential movement of land and sea in the transitional area, 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

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

Continental writers frequently use the notation

The terminating continued fractions

reduced to the forms

are called the successive convergents to the general continued fraction.

Their numerators are denoted by p1, p2, p3, p4; their denominators by q1, q2, q3, q4

We have the relations

In the case of the fraction

we have the relations pn = anpn−1 − bnpn−2,    qn= anqn−1 − bnqn−2.

Taking the quantities a1, b2 to be all positive, a continued fraction of the form

is called a continued fraction of the first class; a continued fraction of the form

is called a continued fraction of the second class.

A continued fraction of the form

where a1, a2, a3, a4 are all positive integers, is called a simple continued fraction. In the case of this fraction a1, a2, a3, a4 are called the successive partial quotients. It is evident that, in this case,

p1, p2, p3, q1, q2, q3 ,

are two series of positive integers increasing without limit if the fraction does not terminate.

The general continued fraction

is evidently equal, convergent by convergent, to the continued fraction

where 2, 3, 4, are any quantities whatever, so that by choosing 2b2 = 1,   23b3 = 1, &c., it can be reduced to any equivalent continued fraction of the form

 Simple Continued Fractions.

1. The simple continued fraction is both the most interesting and important kind of continued fraction.

Any quantity, commensurable or incommensurable, can be expressed uniquely as a simple continued fraction, terminating in the case of a commensurable quantity, non-terminating in the case of an incommensurable quantity. A non-terminating simple continued fraction must be incommensurable.

In the case of a terminating simple continued fraction the number of partial quotients may be odd or even as we please by writing the last partial quotient, an as an − 1 + $1⁄1$.

The numerators and denominators of the successive convergents obey the law pnqn−1 − pn−1qn = (−1)n, from which it follows at once that every convergent is in its lowest terms. The other principal properties of the convergents are:—

The odd convergents form an increasing series of rational fractions continually approaching to the value of the whole continued fraction; the even convergents form a decreasing series having the same property.

Every even convergent is greater than every odd convergent; every odd convergent is less than, and every even convergent greater than, any following convergent.

Every convergent is nearer to the value of the whole fraction than any preceding convergent.

Every convergent is a nearer approximation to the value of the whole fraction than any fraction whose denominator is less than that of the convergent.

The difference between the continued fraction and the nth convergent is less than and greater than. These limits may be replaced by the following, which, though not so close, are simpler, viz. and

Every simple continued fraction must converge to a definite limit; for its value lies between that of the first and second convergents and, since

so that its value cannot oscillate.

The chief practical use of the simple continued fraction is that by means of it we can obtain rational fractions which approximate to any quantity, and we can also estimate the error of our