VOLUME XIV SLICE VI Inscriptions to Ireland, William Henry (continued 9)

(T. Ba.)

1 Introduction to the Principles of Morals and Legislation (Clarendon Press edition of 1879).

2 The Times, July 26, 1887.

3 R. v. Keyn, 2, Ex.D. 63.

4 Address at Saratoga Springs, N.Y., 1896 (Law Quarterly Review, October 1896).

5 Commentaries on the Law of England, 4th ed., iv. 66.

6 Austin’s view, as set out in the Province of Jurisprudence Determined, is that laws proper, or properly so-called, are commands; laws which are not commands are laws improper or improperly so-called. A command implies a definite superior in a position to enforce the command. Where there is no superior to impose obedience there is no law. Rules which “are imposed among nations or sovereigns by opinions current among nations are usually styled the law of nations or international law. Now, a law set or imposed by public opinion is a law improperly so-called” (p. 147). For Sir H. Maine’s views see below.

7 Introduction to the Principles of Morals and Legislation (Oxford, 1879), pp. 24 et seq.

8 Province of Jurisprudence Determined (1861), p. 177; Austin explains his view more fully at p. 127.

9 International Law, p. 50.

10 Droit des gens (1896), i. 22. Compare Savigny: “A community of judicial conscience can be formed among nations like that which positive law creates in the bosom of one people. The foundations of that intellectual community are constituted partly by a community of race, partly and especially by a community of religious convictions. Such is the basis of the law of nations which exists principally among European Christian states, but which was not known to the peoples of antiquity. We are entitled to look upon this law as a positive law, although it is an incomplete judicial formation” (eine unvollendete Rechtsbildung), System des heutigen römischen Rechts (1840), i. § 11.

11 Elements (London, 1885), pp. 22 et seq.

12 “It seems to me,” says Professor L. Oppenheim, “that most writers confound the conception of ‘source’ with that of ‘cause,’ and through this mistake come to a standpoint from which certain factors which influence the growth of International Law appear as sources of rules of the Law of Nations. This mistake can be avoided by going back to the meaning of the term ‘source’ in general. Source means a spring or well, and has to be defined as the rising from the ground of a stream of water; and, wanting to know whence it comes, we follow the stream upwards until we come to the spot where it rises naturally from the ground. On that spot, we say, is the source of the stream of water. We know very well that this source is not the cause of the existence of the stream of water. ‘Source’ signifies only the natural rising of water from a certain spot of the ground, whatever natural causes there may be for that rising. If we apply the conception of source in this meaning to the term ‘source of law’ the confusion of source with cause cannot arise. Just as we see streams of water running over the surface of the earth, so we see, as it were, streams of rules running over the area of law. And if we want to know whence these rules come, we have to follow these streams upwards until we come to their beginning. Where we find that such rules rise into existence there is the source of them. Of course, rules of law do not rise from a spot on the ground as water does; they rise from facts in the historical development of a community. Thus a good many rules of law rise every year from the Acts of Parliament. Source of Law is therefore the name for an historical fact out of which rules of conduct rise into existence and legal force” (International Law, London, 1905, sec. 15.).

13 International Law (London, 1905) sec. 19.

14 Note 8 to Grotius, L., ii. c. iii. § 3.

15 Bishop Percy’s translation (1847), p. 138.

16 We have seen this in the progress made in the three instances given above at the Congress of Paris (1856), the Conference of Berlin (1878) and the Hague Conference of 1907.

INTERNATIONAL LAW (PRIVATE). There is in every territory the law of the land, or territorial law, by which the courts decide all cases that include no circumstances connected with any foreign territory. Often, however, such a circumstance suggests the question whether justice does not require that the law of some other territory shall be applied. Thus the Gretna Green marriages, by which English minors escaped the necessity of banns or the consent of parents or guardians, suggested the question, which was answered in the affirmative, whether even in England their validity ought not to be tried by the law of Scotland, where they were celebrated. Often, again, the question is suggested whether justice does not require that the courts of law should allow some effect to foreign legal proceedings, such as a judgment obtained or litigation pending abroad. Such questions as these are answered by private international law, which, since both laws and legal proceedings are emanations of public authority, may be defined as the department of legal science which is concerned with the effect to be given in the courts of law of any territory to public authority of another territory. The extradition of criminals is also an effect given to foreign public authority, but rather by the government which surrenders the criminal (see Extradition) than by the courts of law, whose only function is to check the surrender so far as the domestic legislation allows them to do so. If private international law were defined as the effect to be given by any mode in one territory to the public authority of another, extradition would be included in it, as is often done; but since the principles governing extradition have little to do with those applicable to other cases, it seems best to treat it as a separate department of law, as is generally done in England.

Comity of Nations.—In the 17th century the Dutch jurists Paul and John Voet and Huber brought forward a view which has since been largely adopted in England and the United States, namely, that the effect given by courts of law to foreign public authority is only due to the comity of nations, but for which every possible question before them would have to be decided by the law of the land. Comity, in that phrase, may only be intended to express the truth that foreign public authority has no inherent effect, without denying that the effect which domestic public authority allows to it is dictated by justice. But the limitations implied in the popular meaning of comity have sometimes been made the ground for deciding questions of private international law in the manner supposed to be most for the interest of litigants belonging to the territory; the phrase is consequently reprobated by most European continental writers, and had better be dropped. The justice on which private international law is founded acknowledges no interest but the general one of intercourse between persons sharing a common civilization in different countries. This interest, as manifesting itself in the domain of law, it seeks to satisfy and it is therefore a true legal justice, rightly classed under law, droit, recht, diritto, derecho and other corresponding terms.

Of the two words which, together with law, make up the title of our subject, private is justified by the fact that its application is between litigants in courts of law, and not between governments except so far as they may be such litigants. International (although interterritorial would be better) is justified by the facts that public authority, which may be internationally foreign, has to be considered, and that governments display a great interest in the question by concluding treaties about it, and occasionally even by suspending diplomatic relations when a court of one country has applied to the subjects of another a rule which the government of the latter deems unjust. But those who think that the primary division of law should be into public and private, and not into international (or interterritorial) and territorial, object to the order in which the three words of the name are usually placed, and call the subject “international private law.”

Conflict of Laws.—This is another name for our subject, and indeed an older one than “private international law,” besides being still much used. But although laws may differ, they cannot properly be said to conflict, unless each can lay a just claim to application in the same circumstances. Now this does not happen. The justice which points out that in certain cases effect ought to be given in one territory to the laws or legal proceedings of another really traces the limits of laws and legal proceedings in space; and the tracing of limits is rather the prevention of conflict than its solution. Savigny has well pointed out that our subject is analogous to the determination of the limits of laws in time, which has to be made when the just application of a new enactment is to be distinguished from the ex post facto application which cannot justly be allowed it. The truth which is aimed at in the phrase “conflict of laws” is that the main problem of our subject is the selection of a law for each given case; but different laws are candidates for selection, not from anything in them as laws, but from differing opinions about the justice of the case. From this selection, again, will be seen the contrast between private international law and attempts at the assimilation of the laws of different countries. To a great extent such assimilation is desirable, especially in mercantile law, but it must always be limited by different views of social order and differences in national habits of thought and action. So far as it is realized, private international law comes to an end with the occasion for selection.

Territory.—This word, as entering into the definition of private international law, does not imply a separate state, whether sovereign or semi-sovereign; it includes every geographical area having a separate legal system, England and Scotland, as well as France or Germany. The case of the Gretna Green marriages illustrates the necessity of rules of private international law between all such, as well as between areas internationally foreign to one another; and indeed the rules are so applied, and in the language of our subject, the area of every separate legal system is foreign to every other such area. Only where a rule contemplates a person as attached more or less permanently to a particular territory, the tie which so attaches him to it may be either nationality or domicile if the territory is a separate state, as France; but it can only be domicile if the territory is combined with others in one state. Nothing but domicile can distinguish British subjects as belonging to England, Scotland or Jamaica, or citizens of the United States as belonging to New York or Pennsylvania.

Legal rules must have relation to the physical and mental characters, and the consequent habits of action, of the populations for which they are intended; they would not satisfy legal justice if they endangered social order as understood and desired by those populations, or if they failed to give due effect to the expectations of parties. This must be true for the rules of private international law as well as for those of any territorial law, and it leads us to ask whether the differences which preclude the universal identity of the latter must not also preclude the existence of the former. The answer is: (1) That where circumstances connected with different territories are concerned, wise rules for the selection of a law will generally give better effect to the expectations of the parties than an exclusive adherence to the territorial law of the court; (2) That the circumstances in which a foreign law is held to apply are exceptional as compared with those in which the domestic law applies, and naturally occur oftenest among the persons and in the affairs having most of a cosmopolitan character, so that the moral shock of applying to them a law founded on a foreign social order is greatly attenuated; (3) That throughout Christendom (to which Japan has now been added for legal purposes) there does exist, though not an identity, yet a considerable similarity in views of social order and prevalent habits of thought and action. Within the same geographical limits there also exists another requisite for the working of a system of private international law, namely, a mutual confidence between countries in the enlightenment and purity of their respective judicatures, to whose proceedings the respect enjoined by the rules of our subject is to be mutually given.

Even within the geographical limits just mentioned there are certain differences on points of social order, especially on marriage or divorce, which have hitherto prevented a complete agreement being attained in the rules of private international law. But no attempt has ever been made to establish any system of the kind as between Christian communities and Mahommedan or other polygamous ones, or between countries enjoying a Christian standard of civilization and those, of which China may be taken as an example, which, whether polygamous or not, do not inspire the necessary confidence in their judicatures. In Turkey and other Eastern countries (in which designation Japan is no longer included for purposes of law) Christians are placed by treaty under the jurisdiction in civil matters of their respective consuls. When in the courts of Christian countries Eastern persons or circumstances connected with Eastern laws have to be dealt with, the peculiar institutions of those countries are not enforced; and while in other respects the judges may be assisted by some of the rules of private international law, especially such as have for their object to carry into effect the reasonable intentions of parties, yet those rules are not applied as parts of an authoritative system.

Rules for the selection of the territorial law to be applied in the different classes of cases, or for the recognition of foreign legal proceedings, have sometimes been made the subject of international treaties, and have often been enacted by territorial legislatures. England possesses a few such enactments, as in the Bills of Exchange Act 1882, and many other countries possess them to a much larger extent in their codes. Where such enactments exist, or where treaty stipulations have been entered into, and the territorial law makes such stipulations binding on the judges, the courts of law must obey and apply them as they must obey and apply any other part of the law of the land. If, as in England, judicial precedents are held to be binding, so that the law of the land consists in part of judge-made law, a similar result is produced; an English court must follow English precedents on the application of foreign law or the refusal to apply it, to the same extent to which it would be bound to follow them on any other point. So far as our matter remains open for a judge, he has, to assist him towards a just decision, the treaties, written laws and judicial precedents of other countries as examples, and a vast literature which has grown up in all Christian countries. That this apparatus is far from having furnished concordant results is due, not only to the divergences on points of social order referred to, but also to the different bases of the legal systems with which the respective governments and writers have been familiar. The legal systems of different countries have been founded on Roman law, feudal law, English common law and still other bases. The arguments of lawyers are affected by the prepossessions thence arising, and they have consequently failed to arrive by their unaided efforts at so much agreement on the rules of private international law as would have been compatible with the conditions and modes of life and action surrounding them. But the general acceptance of a complete body of rules on private international law is a goal which for other countries than England is well within sight by the road of international treaties concluded under the joint direction of professional and non-professional minds.

The most remarkable steps taken in or towards the conclusion of such treaties are those initiated, to its high credit, by the government of the Netherlands. That government first moved in the matter in 1874, and has succeeded in assembling at the Hague the official representatives of nearly all European powers in conferences held in 1893, 1894, 1900 and 1904. At these conferences rules on many branches of private international law were agreed on for submission to the respective governments, which has led to conventions, one of the 14th of November 1896, three of the 12th of June 1902, and four of the 19th of July 1905, regulating the selection of the laws for determining the validity of marriage and of contracts made on the occasion of marriage, their effects on property and on the status of the wife and children, divorce and judicial separation, the guardianship of minors and of interdicted persons, the validity of testamentary dispositions and the rules of intestate succession, and many points of judicial procedure. These conventions may be found at length in the Revue de droit international et de legislation comparée, t. 28, pp. 574-579; 2e série, t. 4, pp. 485-500; and 2e série, t. 7, pp. 646-678. A draft relating to bankruptcy was also prepared at the conference of 1904, but was intended to serve, not as a general convention, but as the base of separate conventions to be concluded between particular states. The extent to which the continent has become united with regard to private international law appears from the fact that France, Germany, Italy, the Netherlands, Portugal, Rumania and Sweden are parties to all the conventions—that Luxemburg, Russia and Spain are parties to those relating to judicial procedure—and that all the ten except Russia, but with the addition of Austria, Belgium and Switzerland, are parties to those on the validity of marriage, divorce and judicial separation, and the guardianship of minors; while all remain open to adhesion by other powers. It is much to be regretted that the British government has declined all invitations to take part in this great international work. The fact must in part be ascribed to the hindrance which the difference between the English common law and the Roman law places, even for lawyers, in the way of joint action with the continent, and in part to the necessity that the rules laid down in any convention should be enacted for the United Kingdom by parliament, the leaders of which belonging to either party take no interest in any such matters.

Next in importance among combined official efforts should be mentioned the congress of seven South American states at Montevideo in 1888-1889, which on many branches of private international law drew up rules intended for adoption by treaty on that continent.

Nationality: Domicile.—Coming now to the particular rules of private international law which are received in England, or have been most widely received elsewhere, the most obvious cases which present themselves for admitting foreign circumstances to influence the decision of a judge are those in which rights are so connected with the person of an individual that the justice of deciding on them by a law having relation to his person speaks almost for itself. Hence arises the notion of a personal law, which must be that either of the person’s political nationality or of his domicile, these being the only circumstances that for the time being are fixed for the individual, irrespectively of the spot where he may happen to be, and of the transaction in which he may happen to engage. We have seen in the article on Domicile what is the legal meaning of that term, how its existence is ascertained, that in and long after the middle ages it was the usual criterion of the personal law, and that in modern times political nationality has largely replaced it as such criterion on the continent of Europe. Thus as well by the conventions mentioned as by the codes of many states—France, Italy and Germany among the number—the capacity and status of persons is now governed by the law of their political nationality. In Latin America the criterion of the personal law is still generally held to be domicile, which is among the reasons why the South American states prefer to pursue the codification of private international law independently of European conferences and conventions.

The English courts were slow to recognize a personal law at all and as late as Lord Eldon’s time they held that the competency of a person to contract depended on the law of the place where the contract was made. Their decisions have since come into line with the continental decisions so far as to make capacity and status depend on a personal law, but not so far as to make nationality its criterion. Hence in England, and in a minority of European continental countries, of which Denmark is an example, the capacity of a party to enter into a contract, whether it be disputed on the ground of his age, or, in the case of the contract of marriage, on the ground of his consanguinity or affinity with the other party, will be decided by the law of his domicile. Guardians, curators and committees of foreign minors or lunatics, deriving their authority from the law or jurisdiction of the latter’s domicile or nationality, can sue and give receipts for their personal property. A court will not decree the divorce of persons not domiciled within its jurisdiction, and it will recognize foreign divorces if, and only if, they have been decreed by a jurisdiction to which the parties were subject by domicile or nationality. And the legitimation of a child by the subsequent marriage of its parents will be held to depend on the law of its father’s domicile or nationality. But the reference to the place of contract, carried to North America with the rest of the English jurisprudence of that date, still maintains in the courts of the United States a struggle with the doctrine of personal law as governing capacity and status.

Here must be noticed a difficulty which arises about the application of any foreign law to the capacity for contracting. It will be understood by the German provision intended to meet it, namely, that “if a foreigner enters in Germany into a transaction for which he is incapable or has only a restricted capacity, he is to be treated for that transaction as being so far capable as he would be by the German legislation. This, however, does not apply to transactions with regard to rights of family or of succession, or to those disposing of foreign immovable property” (Art. 7 of the statute enacting the code). In a spirit similar to that which dictated the German enactment, the French courts have not generally allowed a Frenchman to suffer from the incapacity, by his personal law, of a foreigner who contracts in France, when the foreigner would have been capable by French law, and the Frenchman was in good faith and without great imprudence ignorant of his incapacity. Lately a disposition has been shown to limit this protection of nationals to the case in which the foreigner has been guilty of fraud. English courts usually hold themselves to be more stringently bound by rules, whether those enacted by parliament or those adopted for themselves; and if they should continue to profess the doctrine that capacity depends on the law of the domicile, it is not probable that they will deem themselves entitled to make exceptions for the protection of persons contracting in England with foreigners not enjoying such capacity. The point furnishes an illustration of the fact that to deal satisfactorily with so complex a subject as private international law requires the assistance of the legislature, which again cannot be given with full utility unless uniform provisions, to be enacted in different countries, are settled by international convention.

Another ground for the application of a personal law is furnished by the cases in which masses of property and rights have to be dealt with collectively, by reason of their being grouped around persons. The principal instances of that kind are when it is necessary to determine the validity and operation of a marriage settlement or contract, or the effect of marriage on the property of the husband and wife in the absence of any express settlement or contract, and when property passes on death, either by a will or by intestate succession.

These matters, at least when the property concerned is movable, are generally referred to the personal law of the husband at the time of the marriage, or to that of the deceased respectively; but about them, besides the question between domicile and nationality, there arises the question whether immovable property is to be included in the mass governed by the personal law, or is to follow the territorial law of its own situation (lex situs). Here we touch the distinction between real and personal statutes which arose in the middle ages, when the local legislation of the free cities was contrasted, under the name of statutes, with the general Roman law. That distinction did not bear the same character at all times, but in the 16th century, under d’Argentré, it acquired its most developed form, absorbing all laws into one or other of the two classes, and giving a vast extension to the real class, for which was claimed exclusive application to immovables situate in the territory of the law. In accordance with this system, the highly feudal character of which was very sympathetic to English jurisprudence, English practice has refused to include English immovables in the mass to be dealt with as a unit on marriage or death. But it refers the validity and operation of a marriage settlement, at least as to movables, and the effect of marriage, in the absence of express contract, on the movable property of the husband and wife, to the law of the husband’s domicile at the time of the marriage, called the matrimonial domicile. And with regard to the succession to movables on death, it adopts the principle of massing them irrespectively of their situation, so far as is permitted by the peculiar system under which the property in movables situate in England does not pass directly to the legatees or next of kin, but to the executors or administrators, who are charged with the duty of paying the debts of the deceased and distributing the beneficial surplus. The validity of a will of movables, otherwise than in respect of form (about which more hereafter), and the rights, whether under a will or under an intestacy, in the beneficial surplus arising from them, are determined in England by the law of the testator’s last domicile. On the points glanced at in this paragraph the decisions in the United States generally agree with those in England, only allowing the pecuniary relations of a married couple, in the absence of express contract, to be varied by a change of domicile, notwithstanding that such change is in the husband’s exclusive power, instead of maintaining them as fixed by the matrimonial domicile. On the continent of Europe partisans of a variation after the marriage are scarcely to be found; but as between the nationality and the domicile of the husband or of the deceased, and on the question whether the mass to be governed either by nationality or domicile, on marriage or on death, includes immovables situate under a different law, the division of opinion, legislation and practice is considerable and intricate.

Lex situs, lex loci actus, lex loci contractus, lex fori.—The law of the territory in which they are situate (lex situs) is generally applied to the property in particular things, whether movable or immovable, so far as they are not included in any mass grouped round a person; in England, therefore, always to immovables. In drawing up documents and conducting ceremonies public functionaries must necessarily follow the law from which they derive their authority, wherefore the law of the place where any public document is entered into, or any public ceremony performed (lex loci actus), is the only one that can be followed in its external form. This maxim applies to the forms of notarial acts, and to that of marriage celebrated with the official concurrence of clergymen, registrars and so forth. And since documents and ceremonies entered into without official concurrence are rarer on the continent of Europe than in England, the inevitableness of the form of the lex actus, when such concurrence is had, has generally led to that form being also held sufficient whenever the affair comes to be inquired into later. Nor in England has the sufficiency of the form of the lex loci actus for the celebration of marriage ever been doubted, but a will made by a notarial act in accordance with that law was not admitted. Disregarding the distinction between external form and internal validity and operation, a will of English land could not take effect unless made in English form (that is, since the Wills Act of 1837, with two witnesses), and a will of personal estate could not be admitted in England to probate unless made in the form of the law of the testator’s last domicile. But now, by Lord Kingsdown’s Act, passed in 1861, there are given for wills of personal property made by British subjects, besides the form of their last domicile, three alternative forms, namely, the form of the place of making the will, that of the testator’s domicile at the time when it was made, and that of the part of the British dominions where he had his domicile of origin—only the first of the three, however, being offered when the will is made in the United Kingdom; and no will is to be revoked or invalidated by a change of the testator’s domicile after making it.

The law of the place of contract lex loci contractus, is distinguished into that of the place where the contract is entered into, lex loci contractus celebrati, and that of the place where it is to be performed, which, from the particular case in which the performance consists only in a payment, is called lex loci solutionis. To the first of these is generally referred the formal validity of a contract, so far as entered into without the intervention of a functionary, and therefore not covered by the principle of the lex loci actus, and so far also as the performance is not tied to any particular place. For example, the form for contracting marriage, whether with official intervention as in England, or by private and even oral contract as in Scotland, depends, both as to necessity and as to sufficiency, on the law of the place of contracting it. But as to the internal validity, interpretation and operation of a contract, there has been and still remains much difference of opinion between the laws of the place of contracting and of that of stipulated performance; the former being supported, among other grounds, on some texts of Roman law which Savigny has shown to have been misunderstood, while the latter agrees much oftener with the intention of the parties. The English decisions do not adhere closely to either of those laws, but while repeating much of the traditional language about the lex loci contractus, they aim at doing substantial justice by referring a contract to that place with which its matter has the closest connexion, or which the intention of the parties points out.

In matters of legal procedure every court follows its own practice exclusively (lex fori), as, for instance, whether the remedy on a contract shall be damages or specific performance, and whether a judgment may be executed against the person or only against the property of a party. A point much disputed under this head is whether the time of limitation of actions shall, as held in the United Kingdom, be decided by the lex fori, as an incident to the procedure, or by the lex loci contractus in one of its varieties, as an essential modality of the obligation.

Renvoi.—We will now suppose that the rules of private international law, as practised in any country (A), refer a case arising in its courts to the law of another country (B), as being that of the domicile or nationality of a person, and that those rules as practised in (B) in turn refer (renvoient) the same case to the law of (A), as being that of the nationality or domicile or perhaps of the locus actus: what are the courts of (A) to decide? This question, which involves nothing less than that of the meaning in which the reference to a law is to be understood in our subject, has during recent years excited great discussion both among the jurists and in the courts of all nations. It is answered by the English courts to the effect that (B) by its reference back (renvoi) has disclaimed the control of the case, which must therefore be decided without regard to (B)’s particular laws. See In re Trufort, 36 Ch. D. 600, and In re Johnson, 1903, 1 Ch. 821. This principle practically gives efficacy to the renvoi, and coincides with the express provisions both of the above-mentioned convention of the 12th of June 1902, Art. 1, as to the right of contracting marriage, and of the statute enacting the German code, Art. 27, as to capacity generally. The English law agrees in opinion, and is supported by a numerical preponderance of the judicial precedents in France and Belgium; but it must be admitted that a numerical preponderance of the jurists who have declared themselves hold that the courts of (A) ought to apply the particular laws of (B).

Public Order.—It must not be supposed that the law of the land, the proper territorial law of the court which has to deal with a case in which foreign circumstances arise, always gives way to the foreign law pointed out by the general maxims which even that particular court accepts. All rules for the application of foreign laws are subject to an exception commonly called that of public order, i.e. where such application would interfere with essential principles of morality or policy received in the territory. This reservation is usually made in general terms where legislation on private international law is attempted, as in Article 6 of the Code Napoleon, and preliminary Article 12 of the Italian code; but the courts have to administer it, as they have also in England and other countries where it rests only on judicial practice, and the greater or less extent given to it is one of the causes of the uncertainty and want of uniformity in our subject. One example often quoted is the refusal of the courts in all Christian countries to give effect to polygamous marriage, but this case goes deeper still, for none of the countries in which polygamous marriage exists is allowed to enter at all into the communion of private international law. All, so far as Great Britain has settled legal relations with them, are among those in which British subjects live under consular protection and jurisdiction, or (in Egypt) under that of the Mixed Courts. A better instance is afforded by the refusal of courts, normally within the pale of European legal communion, to recognize divorce as dissolving a marriage, notwithstanding that it has been decreed under the personal law. As another instance, there can be little doubt that an incapacity to marry imposed by the personal law in virtue of religious vows or orders would be disregarded by the English courts in the case of a person marrying in England. Again, it is established in England that damages cannot be recovered for a tort unless the act complained of was a wrong both by the law of the country where it was done and by the law of England; and Article 12 of the statute enacting the German code is in accordance with that doctrine. Now the law of the country where the act is done would naturally give the standard for measuring its legal consequences, and it seems to be due to the connexion which laws qualifying acts as wrongs have with public order that respect for that law is tempered by respect for the law of the countries in which it is invoked; but Article 8 of the Belgian code refers the liability for torts to the former law without any restriction.

Foreign Judgments.—In the rules which have passed before us in the foregoing general review it is easy to perceive a leading motive—that of securing, so far as public order allows, the certainty and stability both of personal and of business relations in the international or interterritorial intercourse which has always accompanied civilization, but is now especially frequent and extensive. It has been attempted to erect this motive into a guiding principle of law, laying down that rights once accrued in any territory, or sometimes, it is said, by virtue of any territorial law, are to be recognized and enforced, subject to the requirements of public order, in any other territory in which they may be invoked before a court of justice. From this, which may be called the principle of the acceptance of foreign rights, it is claimed that the rules of private international law are to be deduced, and that by their consonance with it any such rules are to be tested when proposed. The difficulties of the subject, however, do not admit of being unlocked by so simple a key. They meet us again when we inquire in what territory, or by virtue of what territorial law, a particular alleged right has accrued. Persons belonging by domicile or nationality to A enter in B into a contract to be performed in C; where and by virtue of what law does either acquire a right against the other? Is it to be in or by the law of their homes, where they are normally, though not always necessarily, to be sued? Or of the country where they contract, which for various purposes, as those of police, but not for all purposes, has the control of them when they contract? Or of the country where their contract is to be performed, under a similar control by which, perhaps extending to the very acts of performance, they or their agents may be brought by the operation of their contract? Evidently we cannot apply the principle to guide us in our choice of a law till the very problem which that choice presents has first been solved. There is, however, one case in which the principle of the acceptance of foreign rights leads to a conclusion, namely, where the right has been declared by the judgment of a competent court, which may have been given in an ordinary case, presenting no question of private international law, but in which, if such a question arose, it has been solved by choosing the law and basing the judgment on it. The rule in England and in many other countries as to foreign judgments is that the judgments of competent courts in other territories (foreign in the sense of civil law, whether politically foreign or not) are to be enforced without reopening the merits of the questions disposed of by them. In some countries, however, a foreign judgment is examinable on its merits before being enforced. This was formerly the unquestioned rule in France, though the practice there seems to be now turning the other way. In the system adopted in England everything turns on the competence. For judgments in rem, declaring or disposing of the property in a thing, the test of competence is that the thing, whether movable or immovable, was within the territory of the court. Judgments which declare the status of a person, as with regard to marriage or majority, are competent if the person was subject to the jurisdiction by nationality or domicile. The property or the status is treated as being what has been so declared or decreed. For judgments in personam, decreeing the payment of a certain sum, the test of competence for the present purpose is again that the person against whom it was pronounced was subject to the jurisdiction by nationality or domicile; the judgment may then be sued on as giving of itself a good title to the sum decreed by it to be paid. For domestic purposes the competence may exist on quite other grounds. By its own territorial law a court may be authorized to entertain a suit in personam because the plaintiff possesses its nationality, as by Article 14 of the code Napoleon, or because the contract sued on was made or was to be performed in the territory, and so forth. But judgments based on these grounds will not be enforceable outside the territory. Here we touch the root principles of our subject. The distinction between domestic and international grounds of competence can only be explained by the history of law, and we come in sight of the fact that the rules of private international law rest finally on conventions which could not have existed if the civilization of different countries had not so much that was common in its origin and in the course which it has followed, but which suit the life of those countries just because that life is itself another outcome of those common antecedents.

Authorities.—The best authority on the history of private international law to the end of the 18th century is Lainé, Introduction au droit international privé (2 vols., Paris, 1888). For modern progress the most copious materials are to be found in the Revue de droit international et de législation comparée (Brussels, from 1869); the Journal du droit international privé et de la jurisprudence comparée (Paris, from 1874); and the Annuaire de l’institut de droit international (Paris, from 1877). The most comprehensive general treatise is that of von Bar, of which the 2nd edition appeared at Göttingen in 1889, and has been translated: The Theory and Practice of Private International Law, by L. v. Bar, 2nd ed., translated, by Gillespie (Edinburgh, 1892). Other works, many of great merit, are numerous in all languages; but in this, as in every department of law, the first place for England and the United States must be given to the different Law Reports, since in those countries it is not in the study but on the bench that the highest legal intellect is usually displayed, and the judgments delivered are often essays on the points involved. The following works, however, among others, treat the subject from the English or United States point of view: Story, Commentaries on the Conflict of Laws, Foreign and Domestic, 8th ed., by Bigelow (Boston, 1883); Wharton, A Treatise on the Conflict of Laws or Private International Law (2nd ed., Philadelphia, 1881); J. Westlake, A Treatise on Private International Law, with Principal Reference to its Practice in England (4th ed., London, 1905); Foote, A Concise Treatise on Private International Jurisprudence, based on the Decisions in the English Courts (3rd ed., London, 1904); A. V. Dicey, A Digest of the Law of England with Reference to the Conflict of Laws (2nd ed., London, 1908); Beale, A Selection of Cases on the Conflict of Laws, with Notes and Summary (Cambridge, Mass., 1900-1903); Bate, Notes on the Doctrine of Renvoi (1904). (Jno. W.)

INTERPELLATION (from Lat. interpellare, to interrupt), a term meaning, in general, an interruption, more particularly used of a method of procedure adopted in some of the legislative chambers of continental Europe, especially those of France and Italy, and somewhat similar to that of a motion to adjourn the House in the British parliament. It was originally confined to the asking of a question, after due notice, on some affair of state. It is now, however, the chief means by which the policy or action of the ministry of the day is challenged. An interpellation can be brought on without the consent of the minister to be attacked; it is usually made the subject of a general debate, and generally ends with a vote of confidence or want of confidence in the ministry. The right of permitting or vetoing an interpellation rests with the chamber. In France a tendency has been growing among deputies to use the interpellation as a method of attack on or accusation against individual colleagues.

INTERPLEADER, in English law, the form of action by which a person who is sued at law by two or more parties claiming adversely to each other for the recovery of money or goods wherein he has no interest, obtains relief by procuring the rival claimants to try their rights between or among themselves only. Originally the only relief available to the possessor against such adverse claims was by means of a bill of interpleader in equity. The Interpleader Act 1831 enabled the defendant in such cases, on application to the court, to have the original action stayed and converted into a trial between the two claimants. The Common Law Procedure Act of 1860 further extended the power of the common law courts in interpleader; and the Judicature Act 1875 enacted that the practice and procedure under these two statutes should apply to all divisions of the High Court of Justice. The Judicature Act also extended the remedy of interpleader to a debtor or other person liable in respect of a debt alleged to be assigned, when the assignment was disputed. In 1883 the acts of 1831 and 1860 were embodied in the form of rules by the Rules of the Supreme Courts (1883), O. lvii. by reference to which all questions of interpleader in the High Court of Justice are now determined. The acts themselves were repealed by the Statute Law Revision Act of the same year. Interpleader is the equivalent of multiplepoinding in Scots law.

INTERPOLATION (from Lat. interpolare, to alter, or insert something fresh, connected with polire, a polish), in mathematics, the process of obtaining intermediate terms of a series of which particular terms only are given. The cubes, for instance, shown in the second column of the accompanying table, may be regarded as terms of a series, and the cube of a fractional number, not exceeding the last number in the first column, may be found by interpolation. The process of obtaining the cube of a number exceeding the last number in the first column would be extrapolation; the formulae which apply to interpolation apply in theory to extrapolation, but in practice special precautions as to accuracy are necessary. The present article deals only with interpolation.

Number. Cube of Number.

0 0

1 1

2 8

3 27

4 64

5 125

6 216

. .

. .

. .

The term is usually limited to those cases in which there are two quantities, x and u, which are so related that when x has any arbitrary value, lying perhaps between certain limits, the value of u is determinate. There is a given series of associated values of u and of x, and interpolation consists in determining the value of u for any arbitrary value of x, or the value of x for any arbitrary value of u, lying between two of the values in the series. Either of the two quantities may be regarded as a function of the other; it is convenient to treat one, x, as the “independent variable,” the other, u, being treated as the “dependent variable,” i.e. as a function of x. If, as is usually the case, the successive values of one of the quantities proceed by a constant increment, this quantity is to be regarded as the independent variable. The two series of values may be tabulated, those of x being placed in a column (or row), and those of u in a parallel column (or row); u is then said to be tabulated in terms of x. The independent variable x is called the argument, and the dependent variable u is called the entry. Interpolation, in the ordinary sense, consists in determining the value of u for a value of x intermediate between two values appearing in the table. This may be described as direct interpolation, to distinguish it from inverse interpolation, which consists in determining the value of x for a value of u intermediate between two in the table. The methods employed can be extended to cases in which the value of u depends on the values of two or more independent quantities x, y,...

In the ordinary case we may regard the values of x as measured along a straight line OX from a fixed point O, so that to any value of x there corresponds a point on the line. If we represent the corresponding value of u by an ordinate drawn from the line, the extremities of all such ordinates will lie on a curve which will be the graph of u with regard to x. Interpolation therefore consists in determining the length of the ordinate of a curve occupying a particular position, when the lengths of ordinates occupying certain specified positions are known. If u is a function of two variables, x and y, we may similarly represent it by the ordinate of a surface, the position of the ordinate being determined by the values of x and of y jointly.

The series or tables to which interpolation has to be applied may for convenience be regarded as falling into two main groups. The first group comprises mathematical tables, i.e. tables of mathematical functions; in the case of such a table the value of the function u for each tabulated value of x is calculated to a known degree of accuracy, and the degree of accuracy of an interpolated value of u can be estimated. The second group comprises tables of values which are found experimentally, e.g. values of a physical quantity or of a statistical ratio; these values are usually subject to certain “errors” of observation or of random selection (see Probability). The methods of interpolation are usually the same in the two groups of cases, but special considerations have to be taken into account in the second group. The line of demarcation of the two groups is not absolutely fixed; the tables used by actuaries, for instance, which are of great importance in practical life, are based on statistical observations, but the tables formed directly from the observations have been “smoothed” so as to obtain series which correspond in form to the series of values of mathematical functions.

It must be assumed, at any rate in the case of a mathematical function, that the “entry” u varies continuously with the “argument” x, i.e. that there are no sudden breaks, changes of direction, &c., in the curve which is the graph of u.

Various methods of interpolation are described below. The simplest is that which uses the principle of proportional parts; and mathematical tables are usually arranged so as to enable this method to be employed. Where this is not possible, the methods are based either on the use of Taylor’s Theorem, which gives a formula involving differential coefficients (see Infinitesimal Calculus), or on the properties of finite differences (see Differences, Calculus of). Taylor’s Theorem can only be applied directly to a known mathematical function; but it can be applied indirectly, by means of finite differences, in various cases where the form of the function expressing u in terms of x is unknown; and even where the form of this function is known it is sometimes more convenient to determine the differential coefficients by means of the differences than to calculate them directly from their mathematical expressions. Finally, there are cases where we cannot even employ finite-difference formulae directly. In these cases we must adopt some special method; e.g. we may instead of u tabulate some function of u, such as its logarithm, which is found to be amenable to ordinary processes, then determine the value of this function corresponding to the particular value of x, and thence determine the corresponding value of u itself.

In considering methods of interpolation, it will be assumed, unless the contrary is stated, that the values of x proceed by a constant increment, which will be denoted by h.

In order to see what method is to be employed, it is usually necessary to arrange the given series of values of u in the form of a table, as explained above, and then to take the successive differences of u. The differences of the successive values of u are called its first differences; these form a new series, the first differences of which are the second differences of u; and so on. The systems of notation of the differences are explained briefly below. For the fuller discussion, reference should be made to Differences, Calculus of.

I. Interpolation from Mathematical Tables

A. Direct Interpolation.

1. Interpolation by First Differences.—The simplest cases are those in which the first difference in u is constant, or nearly so. For example:—

Example 1.—(u = log10x). Example 2.—(u = log10x).

x. u. 1st Diff. x. u. 1st Diff.

+ +

4.341 .6375898 7.40 .86923

1000 59

4.342 .6376898 7.41 .86982

1000 58

4.343 .6377898 7.42 .87040

1000 59

4.344 .6378898 7.43 .87099

1000 58

4.345 .6379898 7.44 .87157

In Example 1 the first difference of u corresponding to a difference of h ≡ .001 in x is .0001000; but, since we are working throughout to seven places of decimals, it is more convenient to write it 1000. This system of ignoring the decimal point in dealing with differences will be adopted throughout this article. To find u for an intermediate value of x we assume the principle of proportional parts, i.e. we assume that the difference in u is proportional to the difference in x. Thus for x = 4.342945 the difference in u is .945 of 1000 = 945, so that u is .6376898 + .0000945 = .6377843. For x = 4.34294482 the difference in u would be 944.82, so that the value of u would apparently be .6376898 + .000094482 = .637784282. This, however, would be incorrect. It must be remembered that the values of u are only given “correct to seven places of decimals,” i.e. each tabulated value differs from the corresponding true value by a tabular error which may have any value up to ± ½ of .0000001; and we cannot therefore by interpolation obtain a result which is correct to nine places. If the interpolated value of u has to be used in calculations for which it is important that this value should be as accurate as possible, it may be convenient to retain it temporarily in the form .6376898 + 944 82 = .6377842 82 or .6376898 + 94482 = .637784282; but we must ultimately return to the seven-place arrangement and write it as .6377843. The result of interpolation by first difference is thus usually subject to two inaccuracies, the first being the tabular error of u itself, and the second being due to the necessity of adjusting the final figure of the added (proportional) difference. If the tabulated values are correct to seven places of decimals, the interpolated value, with the final figure adjusted, will be within .0000001 of its true value.

In Example 2 the differences do not at first sight appear to run regularly, but this is only due to the fact that the final figure in each value of u represents, as explained in the last paragraph, an approximation to the true value. The general principle on which we proceed is the same; but we use the actual difference corresponding to the interval in which the value of x lies. Thus for x = 7.41373 we should have u = .86982 + (.373 of 58) = .87004; this result being correct within .00001.

2. Interpolation by Second Differences.—If the consecutive first differences of u are not approximately equal, we must take account of the next order of differences. For example:—

Example 3.—(u = log 10x).

x. u. 1st Diff. 2nd Diff.

6.0 .77815

+718

6.1 .78533 −12

+706

6.2 .79239 −11

+695

6.3 .79934 −11

+684

6.4 .80618 −11

+673

6.5 .81291

In such a case the advancing-difference formula is generally used. The notation is as follows. The series of values of x and of u are respectively x0, x1, x2, ... and u0, u1, u2, ... ; and the successive differences of u are denoted by Δu, Δ2u, ... Thus Δu0 denotes u1 − u0, and Δ2u0 denotes Δu1 - Δu0 = u2 − 2u1 + u0. The value of x for which u is sought is supposed to lie between x0 and x1. If we write it equal to x0 + θ(x1 − x0) = x0 + θh, so that θ lies between 0 and 1, we may denote it by xθ, and the corresponding value of u by uθ. We have then

uθ = u0 + θΔu0 − θ (1 − θ) Δ2 u0 + θ (1 − θ) (2 − θ) Δ3 u0 − ...

2! 3!

(1).

Tables of the values of the coefficients of Δ2u0 and Δ3u0 to three places of decimals for various values of θ from 0 to 1 are given in the ordinary collections of mathematical tables; but the formula is not really convenient if we have to go beyond Δ2u0, or if Δ2u0 itself contains more than two significant figures.

To apply the formula to Example 3 for x = 6.277, we have θ = .77, so that uθ = .79239 + (.77 of 695) − (.089 of −11) = .79239 + 535 15 + 0 98 = .79775.

Here, as elsewhere, we use two extra figures in the intermediate calculations, for the purpose of adjusting the final figure in the ultimate result.

3. Taylor’s Theorem.—Where differences beyond the second are involved, Taylor’s Theorem is useful. This theorem (see Infinitesimal Calculus) gives the formula

uθ = u0 + c1θ + c2 θ2 + θ3 + ...

2! 3!

(2),

where, c1, c2, c3, ... are the values for x = x0 of the first, second, third, ... differential coefficients of u with regard to x. The values of c1, c2, ... can occasionally be calculated from the analytical expressions for the differential coefficients of u; but more generally they have to be calculated from the tabulated differences. For this purpose central-difference formulae are the best. If we write

μδu0 = ½ (Δu0 + Δu−1)

δ2u0 = Δ2u−1

μδ3u0 = ½ (Δ3u−1 + Δ3u−2)

&c.

(3),

so that, if (as in §§ 1 and 2) each difference is placed opposite the space between the two quantities of which it is the difference, the expressions δ2u0, δ4u0, ... denote the differences of even order in a horizontal line with u0, and μδu0, μδ3u0, ... denote the means of the differences of odd order immediately below and above this line, then (see Differences, Calculus of) the values of c1, c2, ... are given by

c1 = μδu0 − 1⁄6μδ3u0 + 1⁄30μδ5u0 − 1⁄140μδ7u0 + ... c2 = δ2u0 − 1⁄12δ4u0 + 1⁄90δ6u0 − 1⁄560δ8u0 + ... c3 = μδ3u0 − 1⁄4μδ5u0 + 7⁄120μδ7u0 − ... c4 = δ4u0 − 1⁄6δ6u0 + 7⁄240δ8u0 − ... c5 = μδ5u0 − 1⁄3μδ7u0 + ... c6 = δ6u0 − 1⁄4δ8u0 + ... . . . . . .

(4).

If a calculating machine is used, the formula (2) is most conveniently written

uθ = u0 + P1θ P1 = c1 + 1⁄2P2θ P2 = c2 + 1⁄3P3θ . . . . . .

(5).

Using θ as the multiplicand in each case, the successive expressions ... P3, P2, P1, uθ are easily calculated.

As an example, take u = tan x to five places of decimals, the values of x proceeding by a difference of 1°. It will be found that the following is part of the table:—

Example 4.—(u = tan x).

x. u. 1st Diff. 2nd Diff. 3rd Diff. 4th Diff.

+ + + +

65° 2.14451 732 16

10153 96

66° 2.24604 828 19

10981 115

67° 2.35585 943 18

To find u for x = 66° 23′, we have θ = 23/60 = .3833333. The following shows the full working: in actual practice it would be abbreviated. The operations commence on the right-hand side. It will be noticed that two extra figures are retained throughout.

u0. μδu0. δ2u0. μδ3u0. δ4u0.

2.24604 +1056700 +82800 +10550 +1900

− 1758 − 158

——— ——— ——— ———

c1 = +1054942 c2 = +82642 c3 = +10550 c4 = +1900

P1θ = +410567 ½P2θ = + 16102 1⁄3P3θ = + 1371 1⁄8c4θ = + 182

——— ——— ——— ———

uθ = 2.28710 P1 = +1071044 P2 = +84013 P3 = +10732

The value 2.2870967, obtained by retaining the extra figures, is correct within .7 of .00001 (§ 8), so that 2.28710 is correct within .00001 1.

In applying this method to mathematical tables, it is desirable, on account of the tabular error, that the differences taken into account in (4) should end with a difference of even order. If, e.g. we use μδ3u0 in calculating c1 and c3, we ought also to use δ4u0 for calculating c2 and c4, even though the term due to δ4u0 would be negligible if δ4u0 were known exactly.

4. Geometrical and Algebraical Interpretation.—In applying the principle of proportional parts, in such a case as that of Example 1, we in effect treat the graph of u as a straight line. We see that the extremities of a number of consecutive ordinates lie approximately in a straight line: i.e. that, if the values are correct within ±½ρ, a straight line passes through points which are within a corresponding distance of the actual extremities of the ordinates; and we assume that this is true for intermediate ordinates. Algebraically we treat u as being of the form A + Bx, where A and B are constants determined by the values of u at the extremities of the interval through which we interpolate. In using first and second differences we treat u as being of the form A + Bx + Cx2; i.e. we pass a parabola (with axis vertical) through the extremities of three consecutive ordinates, and consider that this is the graph of u, to the degree of accuracy given by the data. Similarly in using differences of a higher order we replace the graph by a curve whose equation is of the form u = A + Bx + Cx2 + Dx3 + ... The various forms that interpolation-formulae take are due to the various principles on which ordinates are selected for determining the values of A, B, C ...

B. Inverse Interpolation.

5. To find the value of x when u is given, i.e. to find the value of θ when uθ is given, we use the same formula as for direct interpolation, but proceed (if differences beyond the first are involved) by successive approximation. Taylor’s Theorem, for instance, gives

θ = (uθ − u0) ÷ (c1 + c2 θ + ...)

2!

= (uθ − u0) ÷ P1

(6),

We first find an approximate value for θ: then calculate P1, and find by (6) a more accurate value of θ; then, if necessary, recalculate P1, and thence θ, and so on.

II. Construction of Tables by Subdivision of Intervals

6. When the values of u have been tabulated for values of x proceeding by a difference h, it is often desirable to deduce a table in which the differences of x are h/n, where n is an integer.

If n is even it may be advisable to form an intermediate table in which the intervals are ½h. For this purpose we have

u1/2 = ½ (U0 + U1)

(7),

where

U = u − 1⁄8δ2u + 3⁄128δ4u − 5⁄1024δ6u + ... = u − 1⁄8[δ2u − 3⁄16 {δ4u − 5⁄24 (δ6u − ...) } ]

(8).

The following is an example; the data are the values of tan x to five places of decimals, the interval in x being 1°. The differences of odd order are omitted for convenience of printing.

Example 5.

x. u ≡ tan x. δ2u. δ4u. δ6u. U. u = mean of values of U. x.

+ + +

73° 3.27085 2339 100 5 3.26794 95

3.37594 73½°

74° 3.48741 2808 132 23 3.48392 98

3.60588 74½°

75° 3.73205 3409 187 18 3.72783 17

3.86671 75½°

76° 4.01078 4197 260 51 4.00559 22

4.16530 76½°

77° 4.33148 5245 384 64 4.32501 07

If a new table is formed from these values, the intervals being ½°, it will be found that differences beyond the fourth are negligible.

To subdivide h into smaller intervals than ½h, various methods may be used. One is to calculate the sets of quantities which in the new table will be the successive differences, corresponding to u0, u1, ... and to find the intermediate terms by successive additions. A better method is to use a formula due to J. D. Everett. If we write φ = 1 − θ, Everett’s formula is, in its most symmetrical form,

uθ = θu1 + (θ + 1) θ (θ − 1) δ2u1 + (θ + 2) (θ + 1) θ (θ − 1) (θ − 2) δ4u1 + ...

3! 5!

+ φu0 + (φ + 1) φ (φ − 1) δ2u0 + (φ + 2) (φ + 1) φ (φ − 1) (φ − 2) δ4u0 + ...

3! 5!

(9).

For actual calculations a less symmetrical form may be used. Denoting

(θ + 1) θ (θ − 1) δ2u1 + (θ + 2) (θ + 1) θ (θ − 1) (θ − 2) δ4u1 + ...

3! 5!

(10)

by θV1, we have, for interpolation between u0 and u1,

uθ = u0 + θΔu0 + θV1 + 1−θV0

(11),

the successive values of θ being 1/n, 2/n, ... (n − 1)/n. For interpolation between u1 and u2 we have, with the same succession of values of θ,

u1+θ = u1 + θV1, V2 + 1−θV1

(12).

The values of 1−θV1 in (12) are exactly the same as those of θV1 in (11), but in the reverse order. The process is therefore that (i.) we find the successive values of u0 + θΔu0, &c., i.e. we construct a table, with the required intervals of x, as if we had only to take first differences into account; (ii.) we construct, in a parallel column, a table giving the values of θV1, &c.; (iii.) we repeat these latter values, placing the set belonging to each interval h in the interval next following it, and writing the values in the reverse order; and (iv.) by adding horizontally we get the final values for the new table.

As an example, take the values of tan x by intervals of ½° in x, as found above (Ex. 5). The first diagram below is a portion of this table, with the differences, and the second shows the calculation of the terms of (11) so as to get a table in which the intervals are 0.1 of 1°. The last column but one in the second diagram is introduced for convenience of calculation.

Example 6.

x. u = tan x. δu. δ2u. δ3u. δ4u.

+ + + +

11147 62

74°.0 3.48741 700 8

11847 70

74°.5 3.60588 770 9

12617 79

x u0 + θΔu0. θV1. 1−θV0. θV1 + 1−θV0. u.

73°.6 · −22 35 · · ·

73°.7 · −39 11 · · ·

73°.8 · −44 71 · · ·

73°.9 · −33 54 · · ·

74°.0 3.48741 00 3.48741

74°.1 3.51110 40 −24 58 −33 54 −58 12 3.51052

74°.2 3.53479 80 −43 02 −44 71 −87 73 3.53392

74°.3 3.55849 20 −49 18 −39 11 −88 29 3.55761

74°.4 3.58218 60 −36 89 −22 35 −59 24 3.58159

74°.5 3.60588 00 3.60588

The following are the values of the coefficients of u1, δ2u1, δ4u1, and δ6u1 in (9) for certain values of n. For calculating the four terms due to δ2u1 in the case of n = 5 it should be noticed that the third term is twice the first, the fourth is the mean of the first and the third, and the second is the mean of the third and the fourth. In table 3, and in the last column of table 2, the coefficients are corrected in the last figure.

Table 1.—n = 5.

co. u. co. δ2u. co. δ4u. co. δ6u.

+ − + −

.2 .032 .006336 .00135168 = 1/740 approx.

.4 .056 .010752 .00226304 = 1/442 ”

.6 .064 .011648 .00239616 = 1/417 ”

.8 .048 .008064 .00160512 = 1/623 ”

Table 2.—n = 10.

co. u. co. δ2u. co. δ4u. co. δ6u.

+ − + −

.1 .0165 .00329175 .000704591

.2 .0320 .00633600 .001351680

.3 .0455 .00889525 .001887064

.4 .0560 .01075200 .002263040

.5 .0625 .01171875 .002441406

.6 .0640 .01164800 .002396160

.7 .0595 .01044225 .002115799

.8 .0480 .00806400 .001605120

.9 .0285 .00454575 .000886421

Table 3.—n = 12.

co. u. co. δ2u. co. δ4u. co. δ6u.

+ − + −

1/12 .013792438 .002753699 .000589623

2/12 .027006173 .005363726 .001145822

3/12 .039062500 .007690430 .001636505

4/12 .049382716 .009602195 .002032211

5/12 .057388117 .010979463 .002307357

6/12 .062500000 .011718750 .002441406

7/12 .064139660 .011736667 .002419911

8/12 .061728395 .010973937 .002235432

9/12 .054687500 .009399414 .001888275

10/12 .042438272 .007014103 .001387048

11/12 .024402006 .003855178 .000748981

III. General Observations

7. Derivation of Formulae.—The advancing-difference formula (1) may be written, in the symbolical notation of finite differences,

uθ = (1 + Δ)θ u0 = Eθ u0

(13);

and it is an extension of the theorem that if n is a positive integer

un = u0 + nΔu0 + n (n − 1) Δ2 u0 + ...

2!

(14),

the series being continued until the terms vanish. The formula (14) is identically true: the formula (13) or (1) is only formally true, but its applicability to concrete cases is due to the fact that the series in (1), when taken for a definite number of terms, differs from the true value of uθ by a “remainder” which in most cases is very small when this definite number of terms is properly chosen.

Everett’s formula (9), and the central-difference formula obtained by substituting from (4) in (2), are modifications of a standard formula

uθ = u0 + θδu1/2 + θ (θ − 1) δ2 u0 + (θ + 1) θ (θ − 1) δ3 u1/2 + (θ + 1) θ (θ − 1) (θ − 2) δ4 u0 + ...

2! 3! 4!

(15),

which may similarly be regarded as an extension of the theorem that, if n is a positive integer,

un = u0 + nδu1/2 + n (n − 1) δ2 u0 + (n + 1) n (n − 1) δ3 u1/2 + ...

2! 3!

(16).

There are other central-difference formulae besides those mentioned above; the general symbolical expression is

uθ = (cosh θhD + sinh θhD) u0

(17),

where

cosh ½hD = μ, sinh ½hD = ½δ

(18).

8. Comparative Accuracy.—Central-difference formulae are usually more accurate than advancing-difference formulae, whether we consider the inaccuracy due to omission of the “remainder” mentioned in the last paragraph or the error due to the approximative character of the tabulated values. The latter is the more important. If each tabulated value of u is within ±½ρ of the corresponding true value, and if the differences used in the formulae are the tabular differences, i.e. the actual successive differences of the tabulated values of u, then the ratio of the limit of error of uθ, as calculated from the first r terms of the series in (1), to ½ρ is the sum of the first r terms of the series

1 + o + θ (1 − θ) + θ (1 − θ) (2 − θ) + 7⁄12θ (1 − θ) (2 − θ) (3 − θ) + ¼θ (1 − θ) (2 − θ) (3 − θ) (4 − θ) + 31⁄360θ (1 − θ) ... (5 − θ) + ...,

while the corresponding ratio for the use of differences up to δ2pu0 inclusive in (4) or up to δ2p u1 and o2p u0 in (9) (i.e. in effect, up to δ2p+1 u1/2) is the sum of the first p + 1 terms of the series

1 + θ (1 − θ) + (1 + θ) θ (1 − θ) (2 − θ) + (2 + θ) (1 + θ) θ (1 − θ) (2 − θ) (3 − θ) + ...,

1.1 (2!)2 (3!)2

it being supposed in each case that θ lies between 0 and 1. The following table gives a comparison of the respective limits of error; the lines I. and II. give the errors due to the advancing-difference and the central-difference formulae, and the coefficient ρ is omitted throughout.

Table 4.

Error due to use of Differences up to and including

1st. 2nd. 3rd. 4th. 5th. 6th. 7th.

.5 I. .500 .625 .813 1.086 1.497 2.132 3.147

II. .500 .625 .625 .696 .696 .745 .745

.2 I. .500 .580 .724 .960 1.343 1.976 3.042

II. .500 .580 .580 .624 .624 .653 .653

.4 I. .500 .620 .812 1.104 1.553 2.265 3.422

II. .500 .620 .620 .688 .688 .734 .734

.6 I. .500 .620 .788 1.024 1.366 1.886 2.700

II. .500 .620 .620 .688 .688 .734 .734

.8 I. .500 .580 .676 .800 .969 1.213 1.582

II. .500 .580 .580 .624 .624 .653 .653

In some cases the differences tabulated are not the tabular differences, but the corrected differences; i.e. each difference, like each value of u, is correct within ±½ρ. It does not follow that these differences should be used for interpolation. Whatever formula is employed, the first difference should always be the tabular first difference, not the corrected first difference; and, further, if a central-difference formula is used, each difference of odd order should be the tabular difference of the corrected differences of the next lower order. (This last result is indirectly achieved if Everett’s formula is used.) With these precautions (i.) the central-difference formula is slightly improved by using corrected instead of tabular differences, and (ii.) the advancing-difference formula is greatly improved, being better than the central-difference formula with tabular differences, but still not so good as the latter with corrected differences. For θ = .5, for instance, supposing we have to go to fifth differences, the limits ±1.497 and ±.696, as given above, become ±.627 and ±.575 respectively.

9. Completion of Table of Differences.—If no values of u outside the range within which we have to interpolate are given, the series of differences will be incomplete at both ends. It may be continued in each direction by treating as constant the extreme difference of the highest order involved; and central-difference formulae can then be employed uniformly throughout the whole range.

Suppose, for instance, that the values of tan x in § 6 extended only from x = 60° to x = 80°, we could then complete the table of differences by making the entries shown in italics below.

Example 7.

x. u = tan x. δu. δ2u. δ3u. δ4u. δ5u. δ6u.

+ + + + + +

6775 34

60° 1.73205 425 9

7200 43

61° 1.80405 468 9

7668 52

62° 1.88073 520 9

8188 61

63° 1.96261 581 10

8769 71

64° 2.05030 · 652 · 9

· · · · · · · ·

· · · · · · · ·

· · · · · · · ·

75° 3.73205 · 3409 · 187 · 18

27873 788 73

76° 4.01078 4197 260 51

32070 1048 124

77° 4.33148 5245 384 64

37315 1432 188

78° 4.70463 6677 572 64

43992 2004 252

79° 5.14455 8681 824 64

52673 2828 316

80° 5.67128 11509 1140 64

64182 3968 380

For interpolating between x = 60° and x = 61° we should obtain the same result by applying Everett’s formula to this table as by using the advancing-difference formula; and similarly at the other end for the receding differences.

Interpolation by Substituted Tabulation.

10. The relation of u to x may be such that the successive differences of u increase rapidly, so that interpolation-formulae cannot be employed directly. Other methods have then to be used. The best method is to replace u by some expression v which is a function of u such that (i.) the value of v or of u can be determined for any given value of u or of v, and (ii.) when v is tabulated in terms of x the differences decrease rapidly. We can then calculate v, and thence u, for any intermediate value of x.

If, for instance, we require tan x for a value of x which is nearly 90°, it will be found that the table of tangents is not suitable for interpolation. We can, however, convert it into a table of cotangents to about the same number of significant figures; from this we can easily calculate cot x, and thence tan x.

11. This method is specially suitable for statistical data, where the successive values of u represent the area of a figure of frequency up to successive ordinates. We have first to determine, by inspection, a curve which bears a general similarity to the unknown curve of frequency, and whose area and abscissa are so related that either can be readily calculated when the other is known. This may be called the auxiliary curve. Denoting by ξ the abscissa of this curve which corresponds to area u, we find the value of ξ corresponding to each of the given values of u. Then, tabulating ξ in terms of x, we have a table in which, if the auxiliary curve has been well chosen, differences of ξ after the first or second are negligible. We can therefore find ξ, and thence u, for any intermediate value of x.

Extensions.

12. Construction of Formulae.—Any difference of u of the rth order involves r + 1 consecutive values of u, and it might be expressed by the suffixes which indicate these values. Thus we might write the table of differences

x. u. 1st Diff. 2nd Diff. 3rd Diff. 4th Diff.

· · · · · ·

· · · · · ·

· · · · · ·

· · (−1, 0) · (−2, −1, 0, 1) ·

x0 u0 (−1, 0, 1) (−2, −1, 0, 1, 2)

(0, 1) (−1, 0, 1, 2)

x1 u1 (0, 1, 2) (−1, 0, 1, 2, 3)

(1, 2) (0, 1, 2, 3)

x2 u2 (1, 2, 3) (0, 1, 2, 3, 4)

· · (2, 3) · (1, 2, 3, 4) ·

· · · · · ·

· · · · · ·

· · · · · ·

The formulae (1) and (15) might then be written

u = u0 + x − x0 (0, 1) + x − x0 · x − x1 (0, 1, 2) + x − x0 · x − x1 · x − x2 (0, 1, 2, 3) + ...

h h 2h h 2h 3h

(19),

u = u0 x − x0 (0, 1) + x − x0 · x − x1 (−1, 0, 1) + x − x0 · x − x1 · x − x−1 (-1, 0, 1, 2) + ...

h h 2h h 2h 3h

(20).

The general principle on which these formulae are constructed, and which may be used to construct other formulae, is that (i.) we start with any tabulated value of u, (ii.) we pass to the successive differences by steps, each of which may be either downwards or upwards, and (iii.) the new suffix which is introduced at each step determines the new factor (involving x) for use in the next term. For any particular value of x, however, all formulae which end with the same difference of the rth order give the same result, provided tabular differences are used. If, for instance, we go only to first differences, we have

u0 + x − x0 (0, 1) = u1 + x − x1 (0, 1)

h h

identically.

13. Ordinates not Equidistant.—When the successive ordinates in the graph of u are not equidistant, i.e. when the differences of successive values of x are not equal, the above principle still applies, provided the differences are adjusted in a particular way. Let the values of x for which u is tabulated be a = x0 + αh, b = x0 + βh, c = x0 + γh,... Then the table becomes

x. u. Adjusted Differences

1st Diff. 2nd Diff. &c.

· · · ·

· · · ·

· · · ·

a = xα uα · ·

(α, β)

b = xβ uβ (α, β, γ)

(β, γ)

c = xγ uγ · ·

· · · ·

· · · ·

· · · ·

In this table, however, (α, β) does not mean uβ − uα, but uβ − uα ÷ (β − α); (α, β, γ) means {(β, γ) − (α, β)} ÷ ½(γ − α); and, generally any quantity (η, ... φ) in the column headed “rth diff.” is obtained by dividing the difference of the adjoining quantities in the preceding column by (φ − η)/r. If the table is formed in this way, we may apply the principle of § 12 so as to obtain formulae such as

u = uα + x − c · (α, β) + x − a · x − b · (α, β, γ) + ...

h h 2h

(21),

u = uγ + x − a · (β, γ) + x − c · x − b · (α, β, γ) + ...

h h 2h

(22).

The following example illustrates the method, h being taken to be 1°:—

Example 8.

x. u = sin x. 1st Diff. (adjusted). 2nd Diff. (adjusted). 3rd Diff. (adjusted).

+ − −

20° .3420201

162932 50

22° .3746066 1125 00

161245 00 48 75

23° .3907311 1222 50

158800 00 48 30

26° .4383711 1303 00

156194 00 47 49

27° .4539905 1445 47

151857 60 46 00

32° .5299193 1583 48

145523 67

35° .5735764

To find u for x = 31°, we use the values for 26°, 27°, 32° and 35°, and obtain

u = .4383711 00 + 5 (156194 00) + 5 · 4 (−1445 47) + 5 · 4 · −1 (−46 00) = .5150380,

1 1 2 1 2 3

which is only wrong in the last figure.

If the values of u occurring in (21) or (22) are ualpha, ubeta, ugamma, ... ulambda, corresponding to values a, b, c, ... l of x, the formula may be more symmetrically written

u = (x − b) (x − c) ... (x − l) uα + (x − a) (x − c) ... (x − l) uβ + ...

(a − b) (a − c) ... (a − l) (b − a) (b − c) ... (b − l)

... + (x − a) (x − b) (x − c) ... uλ

(l − a) (l − b) (l − c) ...

(23).

This is known as Lagrange’s formula, but it is said to be due to Euler. It is not convenient for practical use, since it does not show how many terms have to be taken in any particular case.

14. Interpolation from Tables of Double Entry.—When u is a function of x and y, and is tabulated in terms of x and of y jointly, its calculation for a pair of values not given in the table may be effected either directly or by first forming a table of values of u in terms of y for the particular value of x and then determining u from this table for the particular value of y. For direct interpolation, consider that Δ represents differencing by changing x into x + 1, and Δ′ differencing by changing y into y + 1. Then the formula is

ux, y = (l + Delta)x (1 + Δ′)y u0,0;

and the right-hand side can be developed in whatever form is most convenient for the particular case.

References.—For general formulae, with particular applications, see the Text-book of the Institute of Actuaries, part ii. (1st ed. 1887, 2nd ed. 1902), p. 434; H. L. Rice, Theory and Practice of Interpolation (1899). Some historical references are given by C. W. Merrifield, “On Quadratures and Interpolation,” Brit. Assoc. Report (1880), p. 321; see also Encycl. der math. Wiss. vol. i. pt. 2, pp. 800-819. For J. D. Everett’s formula, see Quar. Jour. Pure and Applied Maths., No. 128 (1901), and Jour. Inst. Actuaries, vol. xxxv. (1901), p. 452. As to relative accuracy of different formulae, see Proc. Lon. Math. Soc. (2) vol. iv. p. 320. Examples of interpolation by means of auxiliary curves will be found in Jour. Royal Stat. Soc. vol. lxiii. pp. 433, 637. See also Differences, Calculus of.

(W. F. Sh.)

INTERPRETATION (from Lat. interpretari, to expound, explain, interpres, an agent, go-between, interpreter; inter, between, and the root pret-, possibly connected with that seen either in Greek φράζειν, to speak, or πράττειν, to do), in general, the action of explaining, or rendering the sense of an obscure form of words or an unknown tongue into a language comprehended by the person addressed. In legal use the word “interpretation” is employed in the sense of ascertaining the meaning of the language of a document, as well as its relation to facts. It is also applied to acts of parliament, as pointing out the sense in which particular words used therein are to be understood. The interpretation of documents and statutes is subject to definite legal rules, the more important of which will be found in the articles Contract, Statute, Will, &c.

INTERREGNUM (Lat. inter, between, and regnum, reign), strictly a period during which the normal constituted authority is in abeyance, and government is carried on by a temporary authority specially appointed. Though originally and specifically confined to the sphere of sovereign authority, the term is commonly used by analogy in other connexions for any suspension of authority, during which affairs are carried on by specially appointed persons. The term originated in Rome during the regal period when an interrex was appointed (traditionally by the senate) to carry on the government between the death of one king and the election of his successor (see Rome: History, ad init.). It was subsequently used in Republican times of an officer appointed to hold the comitia for the election of the consuls when for some reason the retiring consuls had not done so. In the regal period when the senate, instead of appointing a king, decided to appoint interreges, it divided itself into ten decuries from each of which one senator was selected. Each of these ten acted as king for five days, and if, at the end of fifty days, no king had been elected, the rotation was renewed. It was their duty to nominate a king, whose appointment was then ratified or refused by the curiae. Under the Republic similarly interreges acted for five days each. When the first consuls were elected (according to Dionysius iv. 84 and Livy i. 60), Spurius Lucretius held the comitia as interrex, and from that time down to the Second Punic War such officers were from time to time appointed. Thenceforward there is no record of the office till 82 B.C., when the senate appointed an interrex to hold the comitia which made Sulla dictator (Appian, Bell. civ. i. 98). In 55, 53 and 52 interreges are again found, the last-mentioned being on the occasion when Pompey was elected sole consul.

The most noteworthy use of the term “Interregnum” in post-classical times is that of the Great Interregnum in German history between the death of Conrad IV. (1254) and the election of Rudolf of Habsburg (1273). See Germany: History.

INTERSTATE COMMERCE. The phrase “interstate commerce,” as used in the United States, denotes commerce between the citizens of different states of the Union. The words “interstate” and “intrastate” are not found in the constitution nor, until comparatively recently, in decisions of the courts or in legislative acts (probably being first used officially in 1887 in the Interstate Commerce Act). The constitution of 1789 uses the phrase “commerce among the states,” and the first official decision interpreting the phrase says that “it may very properly be restricted to that commerce which concerns more states than one” (Chief Justice Marshall in Gibbons v. Ogden, 9 Wheaton 194). Commerce among the states is there distinguished from “commerce which is completely internal, which is carried on between man and man in a state, or between parts of the same state, and which does not extend to or affect other states.” It was declared (Lehigh case, 145 U.S. 192) that commerce between two persons in the same state is not interstate even when there is a temporary deviation to the soil of another state; but later (Hanley case, 187 U.S. 617, distinguishing the Lehigh case) it was declared that as to transportation, such commerce is interstate. The courts have interpreted commerce to denote not merely a mutual selling or traffic, but as “a term of the largest import,” including intercourse for the purposes of trade in any and all its forms (Gibbons v. Ogden, 9 Wheaton 194, and Welton v. Missouri, 91 U.S. 280). Thus have been included not only the actions of trading, navigation, transportation, and communication, but also the instruments and agents employed, including even telegraph messages and, in the extremest cases, lottery tickets.1

The decision of the question where federal control of interstate traffic ends and state control begins has been one of great practical difficulty. In general it has been held that whenever a commodity begins to move as an article of trade from one state to another, commerce in that commodity between the states has begun. Mere intention to ship goods does not make them subjects of interstate commerce, but they must actually be put in motion or committed to the carrier for that purpose (Coe v. Errol, 116 U.S. 517). As a practical guide in deciding when state control should be resumed, the court as early as 1827 (Brown v. Maryland) laid down the “original package rule,” that the taxing power of the state should begin when the original package in which the goods had been imported into the state had been broken up or sold. The injustice of allowing goods to be held thus, for long periods escaping local taxation, led to a modification of the rule in 1868 (Woodruff v. Parkham, 8 Wall. 123), and such goods after reaching their destination may be taxed as property in common with other property in the state.2

Reason for Federal Control of Interstate Commerce.—Immediately after the close of the War of American Independence in 1783 appeared the separatist tendencies and local jealousies usual in a confederation. The Congress of the Confederation had no power to levy tariff duties or to regulate commerce between the states, and the separate states freely and recklessly exercised their rights in this matter. Though commerce at that time was comparatively unimportant, the results of this restrictive policy were most unfortunate. The Annapolis Convention of 1786 was called by the Virginia legislature to take into consideration the trade of the United States and to consider how far a uniform system in their commercial relations might be necessary to the common interests and their permanent harmony. This conference resulted in the call of the Philadelphia Convention of 1787, which framed the present Constitution. Chief Justice Marshall, in one of the early cases on this subject (Brown v. Maryland, 12 Wheaton 419, in 1827), said in words often since quoted: “It may be doubted whether any of the evils proceeding from the feebleness of the federal government contributed more to that great revolution which introduced the present system than the deep and general conviction that commerce ought to be regulated by Congress.”

Every year has increased the importance of the congressional power of regulating commerce. At the time of the adoption of the Constitution, each neighbourhood supplied nearly all its needs by its own industry, but improving means of transportation and communication have multiplied the commercial ties between the citizens of the various states. This change went on slowly until 1830, more rapidly between 1830 and 1860, and at an ever-hastening pace after the Civil War. Until 1824 no case involving directly the consideration of this power reached the United States Supreme Court. From 1824 to 1840 the Supreme Court decided an average of one-third of a case a year; from 1841 to 1860, an average of three-fourths of a case; from 1861 to 1870, an average of one case; from 1871 to 1880, an average of nearly six cases; from 1881 to 1890, an average of more than seven cases; and from 1891 to 1900, an average of more than ten cases. The decisions have not been entirely uniform, and there were some decisions too contradictory to be explained by any ingenuity. The Supreme Court itself has said (Fargo v. Michigan, 121 U.S. 230) that “it may be admitted that the court has not always employed the same language, and that all of the judges of the court who have written opinions for it may not have meant precisely the same thing.” Though in the period just preceding the Civil War the doctrine of states’ rights tended to weaken somewhat the federal power, the broad outlines of the interpretation by Chief Justice Marshall laid down in 1824 in Gibbons v. Ogden remain to-day almost undimmed.

Interstate Commerce in the Federal Constitution.—Freedom of trade, without discrimination, between the citizens of all the states was in the main ensured by one brief sentence, usually called the “commerce clause” of the federal constitution:—“The Congress shall have power ... to regulate commerce with foreign nations, and among the several states, and with the Indian tribes” (Art. 1, sec. 8, clause 3). Hardly less important is the power “to make all laws which shall be necessary and proper for carrying into execution the foregoing powers, and all other powers vested by this Constitution in the government of the United States, or in any department or officer thereof” (Art. 1, sec. 8, clause 18). To the same end of freedom of commerce, Congress is limited in that “no tax or duty shall be laid on articles exported from any state,” and “no preference shall be given by any regulation of commerce or revenue to the ports of one state over those of another; nor shall vessels bound to or from one state be obliged to enter, clear, or pay duties in another” (Art. 1, sec. 9, clauses 5 and 6). Directly and by implication, Congress was granted a number of other powers over commerce, in that it may coin money, establish uniform laws of bankruptcy, establish post-offices and post roads, regulate weights and measures, exercise admiralty jurisdiction (now interpreted to extend to all public waterways accessible to the traffic of more than one state), grant patents and copyrights, and use the power of taxation to protect, repress or even destroy the agencies of commerce (e.g. state bank notes). But these powers can be exercised only in ways which favour and make free the intercourse among all parts of the nation.

Even if the commerce clause had been omitted from the Constitution, a large part of its object would have been attained by certain prohibitions upon the states as follows: “The citizens of each state shall be entitled to all privileges and immunities of citizens in the several states” (Art. 4, sec. 2). “No state shall, without the consent of the Congress, lay any impost or duties on imports or exports, except what may be absolutely necessary for executing its inspection laws; and the net produce of all duties and impost, laid by any state on imports or exports, shall be for the use of the treasury of the United States, and all such laws shall be subject to the revision and control of the Congress” (Art. 1, sec. 10, clause 2). “No state shall, without the consent of Congress, lay any duty of tonnage” (Art. 1, sec. 10, clause 3). Thus by threefold measures of precaution was ensured domestic freedom of trade from every point in the land to its farthest frontiers.