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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 approximation. Thus a continued fraction equivalent to is

of which the successive convergents are

the fourth of which is accurate to the sixth decimal place, since the error lies between 1/q4q5 or .0000002673 and a6/q4q6 or .0000002665.

Similarly the continued fraction given by Euler as equivalent to 1/2 , viz.

may be used to approximate very rapidly to the value of e.

The value of such a fraction is the positive root of a quadratic equation whose coefficients are real and of which one root is negative. Since the fraction is infinite it cannot be commensurable and therefore its value is a quadratic surd number. Conversely every positive quadratic surd number, when expressed as a simple continued fraction, will give rise to a recurring fraction. Thus

Here we have

whence

and we readily find that

whence the value of the fraction taken to infinity is 1/4 .

It is always possible to find the value of the n^ convergent to a recurring continued fraction. If r be the number of quotients in the recurring cycle, we can by writing down the relations connecting the successive p's and q's obtain a linear relation connecting

in which the coefficients are all constants. Or we may proceed as follows. . Let the fraction be

right-hand side is not necessarily zero.

The tests for convergency are as follows:

Let the continued fraction of the first class be reduced to the form

series d3 + d5 + d7 + ..., d2 + d4 + d6 + ... diverges, and oscillates if both these series converge.

For the convergence of the continued fraction of the second class there is no complete criterion. The following theorem covers a large number of important cases.

Two cases have been given by Legendre as follows:--

and it is evident that we have

The continuant

is also equal to the determinant

from which point of view continuants have been treated by W. Spottiswoode, J. J. Sylvester and T. Muir. Most of the theorems concerning continued fractions can be thus proved simply from the properties of determinants .

Perhaps the earliest appearance in analysis of a continuant in its determinant form occurs in Lagrange's investigation of the vibrations of a stretched string .

and the first two partial quotients are given by

which we can transform into

a result given by Euler.

Here the fraction converges to the sum to infinity of the series. Its n^ convergent is not equal to the sum to n terms of the series. Expressions for 0, 1, 2, ... by means of determinants have been given by T. Muir .

A method was given by J. H. Lambert for expressing as a continued fraction of the preceding type the quotient of two convergent power series. It is practically identical with that of finding the greatest common measure of two polynomials. As an instance leading to results of some importance consider the series

We have

whence we obtain

which may also be written

These results were given by Lambert, and used by him to prove that and ? incommensurable, and also any commensurable power of e.

Gauss in his famous memoir on the hypergeometric series

There is another type of continued fraction called the ascending continued fraction, the type so far discussed being called the descending continued fraction. It is of no interest or importance, though both Lambert and Lagrange devoted some attention to it. The notation for this type of fraction is

It is obviously equal to the series

to have deduced, no one knows how, from Wallis' formula for

CONTRABAND , a term given generally to illegal traffic; and particularly, as "contraband of war," to goods, &c., which subjects of neutral states are forbidden by international law to supply to a belligerent.

According to current practice contraband of war is of two kinds: absolute or unconditional contraband, i.e. materials of direct application in naval or military armaments; and conditional contraband, consisting of articles which are fit for, but not necessarily of direct application to, hostile uses. There is much difference of opinion among international jurists and states, however, as to the specific materials and articles which may rightfully be declared by belligerents to belong to either class. There is also disagreement as to the belligerent right where the immediate destination is a neutral but the ultimate an enemy port.

An attempt was made at the Second Hague Conference to come to an agreement on the chief points of difference. The British delegates were instructed even to abandon the principle of contraband of war altogether, subject only to the exclusion by blockade of neutral trade from enemy ports. In the alternative they were to do their utmost to restrict the definition of contraband within the narrowest possible limits, and to obtain exemption of food-stuffs destined for places other than beleaguered fortresses and of raw materials required for peaceful industry. Though the discussions at the conference did not result in any convention, except on the subject of mails, it was agreed among the leading maritime states that an early attempt should be made to codify the law of naval war generally, in connexion with the establishment of an international prize court .

Mails.

Meanwhile, on the subject of mails, important articles were adopted which figure in the "Convention on restrictions in the right of capture" . They are as follows:--

The provisions of the preceding paragraph do not apply, in case of violation of blockade, to correspondence destined for, or proceeding from, a blockaded port.

Foodstuffs and pre-emption.

As regards food-stuffs Great Britain has long and consistently held that provisions and liquors fit for the consumption of the enemy's naval or military forces are contraband. Her Prize Act, however, provides a palliative, in the case of "naval or victualling stores," for the penalty attaching to absolute contraband, the lords of the admiralty being entitled to exercise a right of pre-emption over such stores, i.e. to purchase them without condemnation in a prize court. In practice, purchases are made at the market value of the goods, with an additional 10% for loss of profit.

On the continent of Europe no such palliative has yet been adopted; but moved by the same desire to distinguish unmistakable from, so to speak, constructive contraband, and to protect trade against the vexation of uncertainty, many continental jurists have come to argue conditional contraband away altogether. This change of opinion has especially manifested itself in the discussions on the subject in the Institute of International Law, a body composed exclusively of recognized international jurists. The rules this body adopted in 1896, though they do not represent the unanimous feeling of its members, may be taken as the view of a large proportion of them. The majority comprised German, Danish, Italian, Dutch and French specialists. The rules adopted contain a clause, which, after declaring conditional contraband abolished, states that: "Nevertheless the belligerent has, at his option and on condition of paying an equitable indemnity, a right of sequestration or pre-emption as to articles which, on their way to a port of the enemy, may serve equally in war or in peace." This rule, it is seen, is of wider application than the above-mentioned provision of the British Prize Act. To become binding in its existing form, either an alteration of the text of the Declaration of Paris or a modification in the wording of the clause would be necessary, seeing that under the Declaration of Paris "the neutral flag covers enemy goods, except contraband of war." It may be said that, in so far as the continent is concerned, expert opinion is, on the whole, favourable to the recognition of conditional contraband in the form of a right of sequestration or pre-emption and within the limits Great Britain has shown a disposition to set to it as against herself.

Coal.

Controversy with Russia in Russo-Japanese War.

The Japanese regulations gave rise to no serious difficulties. Those issued by Russia, on the other hand, led to much controversy between the British government and that of Russia, in connexion with the latter's pretension to class coal, rice, provisions, forage, horses and cotton with arms, ammunition, explosives, &c., as absolute contraband. On June 1, 1904, Lord Lansdowne expressed the surprise with which the British government learnt that rice and provisions were to be treated as unconditionally contraband--"a step which they regarded as inconsistent with the law and practice of nations." They furthermore "felt themselves bound to reserve their rights by also protesting against the doctrine that it is for the belligerent to decide what articles are as a matter of course, and without reference to other considerations, to be dealt with as contraband of war, regardless of the well-established rights of neutrals"; nor would the British government consider itself bound to recognize as valid the decision of any prize court which violated those rights. It did not dispute the right of a belligerent to take adequate precautions for the purpose of preventing contraband of war, in the hitherto accepted sense of the words, from reaching the enemy; but it objected to the introduction of a new doctrine under which "the well-understood distinction between conditional and unconditional contraband was altogether ignored, and under which, moreover, on the discovery of articles alleged to be contraband, the ship carrying them was, without trial and in spite of her neutrality, subjected to penalties which are reluctantly enforced even against an enemy's ship." In particular circumstances provisions might acquire a contraband character, as, for instance, if they should be consigned direct to the army or fleet of a belligerent, or to a port where such fleet might be lying, and if facts should exist raising the presumption that they were about to be employed in victualling the fleet of the enemy. In such cases it was not denied that the other belligerent would be entitled to seize the provisions as contraband of war, on the ground that they would afford material assistance towards the carrying on of warlike operations. But it could not be admitted that if such provisions were consigned to the port of a belligerent they should therefore be necessarily regarded as contraband of war. The test was whether there were circumstances relating to any particular cargo to show that it was destined for military or naval use.

The Russian government replied that they could not admit that articles of dual use when addressed to private individuals in the enemy's country should be necessarily free from seizure and condemnation, since provisions and such articles of dual use, though intended for the military or naval forces of the enemy, would obviously, under such circumstances, be addressed to private individuals, possibly agents or contractors for the naval or military authorities.

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