peror by the senate, was defeated and killed, A.D. 276. (Vopiscus, Historia Augusta.) PROCURATOR, a manager or agent, whence the word proctor is formed. A Roman procurator was a person appointed by another to manage or conduct a cause for him. It required no particular words to constitute a man a procurator; nor was it, as in the case of a cognitor, necessary for the opposite party to consent to his appointment. A man might commence a suit as a procurator, without showing his authority; but it seems that he was obliged to produce it before the judges came to a decision, or to enter into security that the plaintiff would abide by his acts. (Gaius, iv. 84.) Under the Empire the governor of a province was, in certain cases, called a Procurator, or Procurator Cæsaris. Sometimes this Procurator had not the government of a province, but only managed affairs of revenue (res fisci). [PROVINCIA.] PRO'CYON. (Astronomy.) [SIRIUS AND PROCYON.] PRO'CYON. (Zoology.) [RACCOON.] PRO'DICUS, a native of Cos, or, as some think, of Chios, flourished B.C. 435. He was a disciple of Protagoras, became a celebrated Sophist, and had among his followers Socrates, Euripides, Isocrates, and Xenophon. Prodicus travelled through Greece from town to town, to deliver his lectures, for which he demanded payment of his hearers, sometimes to an extravagant amount. Several antient writers refer to these lectures, or harangues, as worthy of a philosopher. Prodicus however is reported to have been put to death by the Athenians, because they thought | that he corrupted the youth by his teaching; and it is further remarkable that he is numbered among the atheists by Cicero (De Nat. Deorum, i. 42). None of the writings of Prodicus are extant except a beautiful episode preserved by Xenophon (Mem., ii. 1), usually called 'The Choice of Hercules.' This has been paraphrased in English verse by Shenstone and by Bishop Lowth; and there is a prose translation in the 'Tatler.' Three others of this name are noticed by Fabricius, but very little concerning them is known. (Fabricius, Bibliotheca Græca.) PRODUCT, a term really equivalent to result, but used only when the result is the one obtained by the multiplication of two or more quantities. PROFANENESS. [BLASPHEMY.] PROFESSION, PROFESSED. [MONK.J PROFIT, one of the three parts into which all that is derived from the soil by labour and capital is distributed, the other two being wages and rent: from these three sources arise all the revenues of the community. Profit is therefore the surplus which remains to the capitalist after he has been reimbursed for the wages advanced and the capital consumed during the process of production. To obtain this surplus is the only object for which capital is employed. Profits have a tendency to fall to the same level in all branches of industry; for if the ratio of profit in proportion to the capital employed be greater in one than in another, more capital will be directed to that which affords the highest profit; and the powers of production being inereased, the supply is greater, prices fall, and the equilibrium of profit is restored. A distinction must however be made between real and apparent profit. When the employment of capital is attended with extraordinary risk, profits are nominally high; but after deducting the losses to which it is exposed, the real profits tend to the same level as the ordinary rate. The case is similar in occupations of a disagreeable or agreeable nature, the pleasantness of the latter counterbalancing the low rate of profit. A wholesale merchant and a retail trader both dealing in the same commodities may appear to obtain different rates of profit; but in the latter case wages are confounded with profits, and when they are properly distinguished, the apparent disproportion is diminished. Unless we reduce profits from their apparent to their real value, there is no truth in the maxim that the rate of profit is uniform in the same country at the same time. In different stages of society the rate of profit is subject to certain changes. In all new countries the employment of capital is attended with large returns, and profits are high. As population advances, it becomes necessary to cultivate land of inferior degrees of fertility, or to apply more capital to the land already cultivated. The labourer receives a greater proportional share of the produce, though it may be less absolutely than before this change took place; and more | capital being necessary to obtain the same quantity of pro duce, the proportion of profit to the capital employed is therefore less: that proportion of the produce which is distributed as rent increases. The natural tendency then of profits (whether arising from capital employed in agriculture or in manufactures) is to decline as the necessities of the population render it necessary to have recourse to inferior soils. Happily, improvements in machinery and in the art of agriculture, better combinations of labour and capital, and greater freedom of commerce, are calculated to arrest this retrograde movement; and to such sources of relief every highly advanced country must look as a means for sustaining its prosperity; for whatever diminishes the necessity of raising food from the poorer soils, tends to maintain the rate of profit. Two other causes have great influence upon the rate of profit, namely, wages and taxation. A rise in wages will diminish profits, unless industry becomes more productive; but if the latter take place, both may rise at the same time, either in the same or in different proportions according te circumstances. Taxation will diminish profits, unless wages fall or industry become more productive. Taxes on profits, when they fall alike upon all capital engaged in productive industry. are paid by the owners of capital, who have not the power of charging the tax upon consumers. Consumption is checked and the power to accumulate diminished.” Wher the profits of only certain classes of traders are taxed, they would betake themselves to other occupations not taxed, unless they could charge the consumers with the tax: the tax therefore falls upon the consumers. The effect of the competition of capitalists in reducing the rate of profit has not been much discussed by writers on political economy. Mr. M'Culloch says:- Competition cannot affect the productiveness of industry, and therefore has nothing to do with the average rate of profit.' In reply to this assertion it has been remarked (Edin. Rev., No. 142, p. 443) that although the inferior fertility of newly cultivated soils be the immediate cause of the diminution of the rate of profit, yet it is nothing but the competition of capitalists which drives capital to seek the inferior soil, and induces its owners to be content with a lower rate of profit. The capitalists who had accumulated at the old rate of profit are content with a new investment producing a lower rate, instead of consuming their savings unproductively. (Ricardo, Principles of Political Economy and Taxation, chaps. v. and xiii.; Mill, Elements of Political Economy, c. ii., sec. 3; and c. iv. sec. 6; M'Culloch's ed. of the Wealth of Nations, note vii.; The Laws of Wages, Profits, and Rent investigated, by Professor Tucker, Philadelphia, 1837.) PROGNO'SIS (in Medicine) is the opinion formed re specting the probable future events of any disease, as, for example, whether it will terminate in recovery or in death how long it is likely to continue, what other diseases may be expected to arise in its course, what are the chances of relapse, and what those of some permanent injury of structure or function being produced by the morbid processes. PROGRESSION. A series of numbers following any law should be called a progression, but the word is usuall restricted to two sorts of progression, which are called, but by no means correctly, arithmetical and geometrical: the analogies pointed out in RECTANGLE give the origin of these terms. An arithmetical progression is one in which the terms continually increase or diminish equally, including, as an extreme case, that in which they do not increase nor diminish at all. Thus 7 7, 7, 7, &c. 7, 8, 9, 10, &c. 10, 94, 9, 84, &c. 2, 31, 41, 51, &c. are sets of terms in arithmetical progression. The following proposition contains the principal part of their theory:— If a be the first term of an arithmetical progression, and Aa the difference between any two terms (negative, if the terms diminish); and if a be the nth term from and after an a exclusive, and S" the sum of n terms, we have a 1 -r = a + ar+ +ar" + ... ar"+1 any three of these being given, the other two can be found, I series be carried ad infinitum. The general equation, made subject however to this restriction, that the problem is un- absolutely true, after stopping at ar" in the series, is meaning when n is not a whole number, whether it be given or found. These theorems are only the simplest case of a more general pair, in which, taking any series, and supposing neither the differences nor the differences of the differences, &c. to be equal, an expression is given for any term of a series, or for the sum of n terms, which frequently gives finite forms in the place of indefinite ones. Calculate, as in the article DIFFERENCE, the value of Aa, ▲2a, &c., from a, a, a,, &c., and let Then 2 a2 = a + n, ▲ a + n2 ▲3 a + n ̧ 43a +, &c. S2 = n1a + n,▲a + n ▲2a +n, A3 a +, &c. Thus in the series 1 + 5 + 17 +43 + 89 +161+&c., the law of whose terms is undiscoverable at first sight, we shall, by what the beginner may, till he knows better, call an accidental circumstance, discover both the law of the terms and that of their sum, as follows— An = 1 + 4n, + 8ng + 6n, = n3 + (n + 1) S2 = n + 4n, + 8n + 6n, =(2 n (n + 1) (2 n + 1) + 6 2 Thus the seventh term (the sixth after 1, n = 6) is 63 + 73, or 265, and the sum of 6 terms (make n = 6 in the second formula, in which remember that S, is the sum of n terms, not of n terms after a) is (}.6.5)2 + 1.6.7.13, or 316, which may easily be verified. [SUM.] The apparently accidental circumstance above alluded to, is the vanishing of all the differences of a from and after the fourth. But it is to be observed, that the series was originally constructed so as to make all differences vanish after the fourth, and that the preceding theorem will never change indefinite into definite formula, except when all differences after a certain one vanish. The rule is, when a, is an algebraically rational and integral function of n of the p order, that is, of the form kn? +In+, &c., all differences after the pth vanish, and then only. Geometrical progression is when the terms of a series increase or diminish by the use of the same multiplier, whole or fractional, including, as an extreme case, that in which the multiplier is unity. Thus, the multipliers being 1,,, and 2, the four following sets of terms are in geometrical progression: | Other points connected with this equation will be mentioned in the article SERIES. There is no doubt that every whole can be subdivided into parts without limit, or, in common language, can be divided into an infinite number of parts. An old fallacy, mentioned in MOTION, receives its explanation from the preceding. If we make a= 1−r, the equation carried ad infinitum becomes 1 = (1 − r) + (1 − r)r + (1 − r) r2 +, &c., ad. inf. By giving different values to r, we have therefore an infinite number of ways of subdividing unity into an infinite antecedent is followed by a consequent; and if dividing the number of parts. If then we take a problem in which an antecedent into an infinite number of parts, we consider separately the parts of the consequent which belong to those of the antecedent, we shall of course divide the consequent into an infinite number of parts. It would be a gross fallacy to infer that the consequent must be infinitely great, because it is produced in a never-ending succession of parts, since that never-ending succession was produced by dividing the avowedly finite antecedent into an infinite number of parts. No one could fail to detect the following: 'Let M be divided into an infinite number of parts, a, b, c, &c.; let each of these parts be doubled; then the result is made up of 2a, 2b, 2c, &c., ad infinitum; consequently 2a+ 26+2c+, &c., being made up of an infinite number of quantities, is infinite.' Nevertheless this fallacy was not only produced in an ingenious form, as a sophism [MOTION], but has even reappeared in modern times as a serious argument. The sophism is known by the name of Achilles and the Tortoise.' The swiftest of men runs after the slowest of beasts, without (says the sophism) the possibility of ever overtaking it. For if, when they set out, Achilles be at A and the tortoise at T, then by the time Achilles has run over AT, how fast soever he may run, the tortoise will have gone over some length, say TB, while the hero goes spite of the sophism) goes over BC, and so on ad inover TB, his dinner (for dinner he may have out of it, in finitum. How then, asks the objector, is it possible that &c. a A T B C DE, Achilles can ever come up with the tortoise, since it is unquestionable (and this is perfectly correct), that let him go as far as he may, he must always come up to where the tortoise was before he can reach the point at which he is; so that it requires an infinite number of parts of time (but here the sophism quietly introduces an infinite time) to catch the tortoise? The answer is, that Achilles will certainly overtake the tortoise at a finite distance from A, say at a· any contrivance which subdivides Aa into an infinite number of parts, does the same with the time in which Achilles runs over Aa; and there is no more reason to say that the time is therefore infinitely great, than to say that Aa is made infinitely great by the subdivision. This would be a sufficient answer, since it would throw upon the sophist the onus of showing that the infinite number of parts of time makes an infinite time; but a more complete answer consists in positive proof that it is not so, as follows: Let AT be called a, and let Achilles move m times as fast as the tortoise; then TB is necessarily the mth part of AT, BC of TB, CD of BC, &c. Hence, if t be the time in which Achilles moves over AT, this time, added to his times of going over TB, BC, CD, &c., or t, t÷m, t÷m3, &c., make up 1 In the work of a celebrated political economist there is the following argument, to show that a tax on wages must fall on the labourers; for if it did not so fall, wages would rise, whence the price of goods would rise, which would again cause a rise of wages, and this again a rise in goods, and so on ad infinitum, which is inferred to be absurd. This is of course precisely a repetition of the preceding case; and granting all the premises, the conclusion by no means follows. For that conclusion is that the rise would go on without limit, which need not be the case. PROHIBITION, a writ to prohibit a court and parties to a cause then depending before it from further proceeding in the cause. It will be convenient to define,-1, out of what courts it may issue; 2, to what courts it may be addressed; 3, under what circumstances it is grantable; 4, at whose instance it may be obtained; 5, at what time it may be obtained; 6, the form and incidents of the proceeding. 1. A writ of prohibition may issue from any of the three superior courts of common law at Westminster, and also from each of the common-law courts of Chester and Lancaster. It is generally stated that a writ of prohibition may issue from the Court of Chancery; but the Court of Chancery acts by injunction addressed only to the parties, and does not interfere with the court. [INJUNCTION.] 2. It may be addressed by any of the three superior courts to any other temporal court; such as to the Admiralty courts, to courts-martial, a court baron, any other inferior court in a city or borough, to the Cinque-Ports courts, the duchy or county palatine courts, the chancery of Chester, the Stannary courts, the Court of Honour of the EarlMarshal, to the Commissioners of Appeals of Excise, to any court by usurpation without lawful authority, or to a court whose authority has expired. When any one has a citation to a court out of the realm, a prohibition lies to prevent his answering. It seems also that it might issue to the Court of Exchequer and to the Court of Common Pleas; but not to the Court of Chancery, nor is there any instance of a prohibition to the King's Bench. It may be granted by any of the three superior common-law courts to any spiritual court, and by the common-law courts of Chester and Lancaster to the spiritual courts within the county palatine and duchy. 3. The writ is grantable in all cases where a court enter tains matter not within its jurisdiction; or where, though the matter is within its jurisdiction, it attempts to try by rules other than those recognised by the law of England. Matter may be said to be not within the jurisdiction of a court in two senses: 1, when the subject-matter entertained is in its nature not cognizable by the court; 2, where the subject-matter is in its nature cognizable by the court, but lies out of the local district where only that court has jurisdiction; or, in the case of a court whose jurisdiction is general, when the subject-matter lies in a local district exempt from the general jurisdiction of the court or where the subject-matter of the cause relates to persons over whom the court has no jurisdiction. In order to ascertain those cases which fall under the first head, we must consider the nature and character of the subject-matters over which the jurisdiction of the court extends. It is obvious, that if we have those clearly defined, we shall see whether the subject-matter in question is or is not within that jurisdiction. This general rule may be useful, because the cases that may occur in which a prohibition will lie, are endless. The examples of cases which have occurred will assist in the application of it. To begin with those relating to temporal courts. A prohibition will lie if one sue another in a court-baron, or other court not a court of record, for charters concerning inheritance or freehold; or in the county court for trespass vi et armis; or in the county courts or courts baron for a matter of 408. or upwards, -and the plaintiff cannot evade the prohibition by dividing his demand into smaller sums; to the courts of Admiralty, if they entertain questions of a contract made or to be executed within the kingdom; to an inferior court, if an action be brought in it on a judgment in one of the superior courts; to the court of honour of the earl-marshal, if it holds plea of things determinable by the common law. To a spiritual court a prohibition will lie, if it takes cognizance of any plea concerning a title to lands or tenements, or an advowson of a church, or an office, or goods, money, or chattels; and this applies even in the case of goods or ornaments given to a church, or matters of a criminal nature punishable only temporally: in short, as it has been said, anything for which a remedy exists at common law. Yet it has been held that a pension which commenced by the grant of the patron or ordinary may be sued for either in the temporal or spiritual court. Lord Coke however goes further, and says that prohibition lies to a spiritual court in any case, though of a spiritual nature, where a remedy is given by statute in a temporal court, unless the jurisdiction of the spiritual court is saved by the same statute. Perhaps this assertion cannot be maintained to the full extent. In some cases it has been so held, as where a suit in an ecclesiastical court was instituted for preaching without licence or marrying without banns, these having been created offences by act of parliament. Still a prohibition does not lie in a suit for small tithes or for contribution to the repairs of the church, though other remedies are given in these cases by statute (7 and 8 Wm. III., c. 6, and c. 34), and perhaps in some other cases. In cases where no remedy exists elsewhere, a spiritual court may still be restrained from entertaining questions as to matters not within its jurisdiction. With regard however to the spiritual courts, various exceptions and restrictions must be applied to what has been said. A spiritual court can hold plea for goods, money, and chattels to which a spiritual character attaches; as for instance, tithes, provided they are under a fourth part of the value of the church, or oblations, mortuaries, &c., that is, payments by communicants, or payments for marriages, christenings, churchings, and burials, or pensions; or a sum promised to be given as a marriage portion. It can also hold plea for matters testamentary, such as a legacy, even of a chattel real. Though a will disposes of land as well as personalty, the granting of probate belongs to the spiritual court; but this grant in no way determines the validity of the will so far as relates to the land. A spiritual court has also jurisdiction as to offices merely spiritual, but not for lay offices, even though they are in the spiritual court. It has also jurisdiction over offences committed within the spiritual court itself, as perjury or extortion in all officers of the court, or for brawling and committing a nuisance within a churchyard, or for defamation where no damages are demanded, or where a crime either merely or in part spiritual is imputed, or the words spoken are mere words of passion. Where violence has been done to a spiritual person, he may maintain a suit in the ecclesiastical court, to punish the party by ecclesiastical censures. The spiritual court also has cognizance of a suit for maintaining a way to the church, or where the question is not of a right of way generally, but solely as to the right of way to a church, or a way by which a parson carries off his tithes. Prohibition lies equally both where the matter of the suit is not cognizable by the court, and where, though the substance is cognizable, matter arises during the progress of it, and is clearly about to be tried, over which the court has no cognizance. As to this, perhaps some confusion and contradiction will be found in the authorities. It has been said that where a suit is brought in the spiritual court for a thing within the cognizance of that court, and temporal matter becomes incident, it shall be determined there, and there can be no prohibition. (12 Co., 65.) However this must always have been understood with the condition, that as to such things the ecclesiastical court was bound to try according to course of common law. Perhaps it was only applicable to cases where parties neglected to apply for prohibition till after sentence, and where no want of jurisdiction appeared on the face of the proceedings. More recent cases have clearly established that if in a cause properly cognizable by a spiritual court a question arises and is necessarily about to be tried as to the existence of a custom, or a prescription, or the limits of a parish, or where in a suit for tithes there is plea of a modus, or that the lands are discharged by statute, or tithes are claimed of things for which no tithes are due, or the defendant makes title by lease, &c., a prohibition will lie immediately. It has been laid down broadly that the ecclesiastical courts cannot try any matter triable at common law. It is otherwise however where the construction of a statute may come in question: prohibition will not lie on the mere suggestion that the spiritual court is not competent to construe it. A prohibition is in all cases grantable where a court allows | illegal or disallows legal evidence, as where the commissioners of appeals for the excise determine by the minutes of evidence taken by a justice of the peace, instead of examining the witnesses vivâ voce, or a spiritual court disallows proof of payment, &c. because proof of it is made only by a single witness, or where it has misconstrued an act of parliament, or disallows an award when it is good by Where a suit is for matter within the cognizance of the court, which is combined with things over which the court has no cognizance, prohibition will issue as to that over which the court has no cognizance. But in those cases where both parties to the suit are spiritual persons, as where the question is whether the tithes belong to the rector or the vicar, no prohibition lies. law. Where the matter is cognizable by the court, but lies out of its local jurisdiction, the question is merely one of boundary; as in cases where an inferior court holds plea of matter out of its limits, the duchy courts or courts palatine of land out of the duchy, &c. This is also the case where one spiritual court trespasses on the district of another, as if a man resident in one diocese or peculiar be cited to appear in another; in this case it is however to be observed that no prohibition will lie if the proper ordinary refuses or neglects to act in the case, or is party to the suit, or, under certain circumstances provided for by the canons, refers the matter to his immediate superior. A prohibition will also lie where a court attempts to extend its jurisdiction to parties over whom it has none, as where a court-martial inquires into the conduct of a person not a soldier or sailor; the Stannary courts, where neither parties are tinners, nor the matter in question respecting tin, &c. 4. A prohibition may be obtained at the instance of either party to an ecclesiatical suit. In the case of a suit for tithes against a lessee, it may be obtained by the reversioner. Where a court has no jurisdiction over the matter of the suit, a prohibition is grantable at the request of a mere stranger. 5. If a court has no cognizance of the matter of a suit, prohibition will lie immediately after appearance, and it may be obtained by either plaintiff or defendant at any future time, even after sentence, appeal, and affirmation; or after judgment and execution, provided it appears by the libel, or by the libel and the proceedings, that the court had no jurisdiction. Where the court has cognizance of a cause, prohibition will not lie until the matter out of its jurisdiction has not only arisen, but is also clearly in progress of being tried. If that matter is then admitted by the litigant parties, the court is still entitled to entertain cognizance of the suit. If not admitted, and these circumstances, though not appearing on the face of the proceedings, are duly brought forward before sentence, a prohibition will then lie. If how ever a prohibition is not then applied for, but the party submit to the trial in the court where the suit has been commenced and sentence is pronounced, no prohibition will lie unless it appear on the libel, or the libel and the proceedings, not only that matter out of the jurisdiction of the court has arisen, but also that the matter has been wrongly decided. (Gould v. Gapper, 5 East, 345; Byerly v. Windus, 5 B. & C. 1.) If a spiritual court has cognizance of part of the charge and not of the rest, the court will not grant a prohibition after sentence. In cases where the suit is determined, it would appear that these observations can at all events only apply to permanent_courts, and where something still remains to be done. In the case of an occasional court, as a court-martial, it would be impossible to carry the principle into execution. 6. A writ of prohibition is applied for by motion in court, which sets out the proceedings in the suit. If the proceedings are not sufficient to show the want of jurisdiction in the court against which prohibition is prayed, suggestions must be added, verified by affidavit, showing such want of jurisdiction. cient cause appears for a prohibition, or he may plead such matters as he thinks proper to show that the writ ought not to issue, and conclude by praying that it may not issue. If matters of fact are put in issue, they are tried by a jury. Judgment is given either on the demurrer or after nonsuit or verdict. The party succeeding is entitled to the costs of these proceedings, and, if a trial takes place, the jury may assess damages. If the court decide in favour of the party applying, the writ issues and forbids the court and other party from further proceeding. In such case, if the ground of application was that the court had no jurisdiction at all in the suit, the writ of prohibition is final. But, if the ground is that something had arisen not cognizable by the court, during the progress of a suit, concerning a matter properly within its jurisdiction, the prohibition is not final. In such case the question is referred to the proper tribunal for trial, and if found against the applicant, the suit may be then resumed. In either case, where the court decides in favour of the party against whom prohibition is prayed, or the verdict has been afterwards in his favour, the court awards a consultation, as it is called, by which the cause is again remitted to the original court. If parties proceed after a writ of prohibition has been obtained and served, they are liable to an attachment for contempt. No prohibition for the same matter lies after a consultation has been awarded upon the merits. (Comyns's Digest; Bacon's Abridgment; Viner's Abridgment; tit. Prohibition,' 2 Inst., 599; 3 Bl. Com., c. 7.) The right of the common-law courts to issue writs of prohibition, and the mode in which they exercised that right, have often been the subject of great dispute between the common-law judges and the ecclesiastics. The latter have several times exhibited many articles of grievance before the parliament and privy council against the former. The most famous of these are the 'Articuli cleri,' exhibited by Archbishop Bancroft, in the name of the whole clergy, in the third year of the reign of James I. They are given at length by Lord Coke (2 Inst., 599), with a full view of the nature of the controversy between the parties, and the unanimous answers of the judges. PROITHERA. [NIGHT-JARS, vol. xvi., p. 229.] PROJECTILES, THEORY OF. This subject usually comprehends the investigation of the relations between the space described, the time of motion, and the velocity acquired by a body when impelled in any direction by some motive force. The circumstances of a body descending from a high place towards the earth by the action of gravity, and those of a body projected vertically upwards from the earth, on the supposition, in both cases, that the body moves in a nonresisting medium, have been noticed in the article FALL OF BODIES; and the circumstances attending the motion, both in a resisting and a non-resisting medium, of a body impelled by fired gunpowder, when the impulse is in a direction parallel or oblique to the horizon, have been investigated in the article GUNNERY. It is intended therefore in this place only to consider the laws of the vertical ascent and descent of bodies in resisting media, the force of gravity, or of terrestrial attraction, being supposed to be constant; and in non-resisting media, under the condition that the force of gravity is variable. Let a spherical body descend vertically from a state of rest in a resisting medium (air, water, &c.) supposed to be of uniform density; and let it be admitted, agreeably to the Newtonian hypothesis (Princip., lib. ii., sec. I; Schol.), that the resistance of the medium is proportional to the square of the velocity, v, acquired at any moment in the descent; then, if we suppose U to be the velocity which a body falling towards the earth in the resisting medium would acquire when that resistance becomes equal to the accelerative force of gravity, the latter being, as usual, represented by g 2,2 (=322 feet), we shall have U2: g :: v2 : term represents the resistance of the medium at the instant when the velocity is v; hence the accelerative force by which the falling body is urged at such moment is expressed by gU£E. v and the last If the court grants a rule, the other party is heard in answer. The court may then decide, either to refuse the prohibition, or, if they incline to grant it, direct the party applying to declare in prohibition. The mode of doing this is regulated by 1 Will. IV., c. 21. The declaration must Now s being the space descended by the body in the time contain a concise statement of the grounds of the applica-t, and v being the velocity as before, an accelerative force dv dt tion, and conclude by praying that the writ may issue. To is represented by this the other party may demur on the ground that no suffi dv dt d's dt and by [FORCE.] Therefore the relation between the space descended and the time of ; and integrating this descent may be thus investigated. Let be the radius of the earth, and let the force of gravity at the surface be represented by g; then a being any distance from the centre, the attracting force acting on gx Therefore since the the body at that distance will be distance of the body from the centre diminishes, while the time reckoned from the moment of departure increases, we d2x gx shall have =- This equation will be found to be df U2 sing from logarithras to numbers, 2gt verified by assuming a=a cos. t√+b sin. t; which 2g being differentiated once, gives =-Ɑ These equations for s and v give the space descended and the velocity acquired at the end of any given time t from the moment when the motion commenced. For tables of the values of U (the terminal velocities) for iron balls, see Dr. Hutton's 'Tracts,' tract 37. Next, let a body be projected vertically upwards in a uniformly resisting medium with an initial velocity V; and let the body be of a spherical form so that U may be the same as before then, the force of gravity and the resistance of the medium acting in a direction opposite to that of the gvi U dv dt ; whence dx dt Now, in the equation for x, making = dt, making dt (any given distance from the centre) when t=0, we have dx dx a=r'; and in the equation for (the velocity) =0 when t=0, we have b=0. Consequently x=r' cos. t√ whence x is found when t is given: but when x=0, we g time of falling from the surface to the centre of the earth. Let it now be required to investigate the relations between the times, the spaces described, and the acquired velocities when a body falls in vacuo from a point at such a distance from the earth that the attraction of gravity upon it may be considered as variable; and when, agreeably to This equation, being integrated, gives the law of nature, its intensity is inversely proportional to projectile force, we have now = d. -gdt=U 음 ا لارج v the square of the distance. (Princip., lib. i., prop. 74.) = 1 : C-gt dt; Multiplying both mem- + const. The In order to integrate this equation, multiply both sides of dx2 it by 2dx; and then the first integral will be dt 2g72 + const. The constant may be found on conp-x dx sidering that dt (the velocity)=0 when t = 0, when also 2gr dx2 x = 0; therefore const. - and " p t=0; whence const. =—— hyp. log. cos.- : therefore x = U COS. go C-gt U. hyp. log. equation may be put in the form √ px-x2 Making v 0 in the above equation for v, we get the value of when the body has attained its greatest height; by the rules of integration we have t and substituting the value of t so found in the last equation for x, we have that greatest height. When arrived at the greatest height, the body would begin to return towards the earth; and it may be shown that the velocity acquired by the body on arriving at the place from whence it was projected would be less than the initial velocity; also that the time of the descent would differ from the time of ascent. If we imagine the earth to be perforated in the direction of a diameter; and if a body be allowed to descend towards the centre in a non-resisting medium from any point in the line of perforation: the law of attraction being in such a case directly proportional to the distance of the body at any ume from the centre of the earth (Newton. lib. i, prop. 73), 1 = dt; and p-2x +p arc cos. = there is no constant to be added, 2 because x=0 when t=0. p From this equation t may easily be found when x is given: likewise from the equation for dr dt we have the velocity when x is given. And if x be made equal to p-r, the whole distance of the body from the surface of the earth, we shall obtain the whole time of the descent and the velocity acquired at the end of that time. Again: let it be supposed that a body may be projected vertically upwards in vacuo from the surface of the earth, and be subject to a variably accelerative force of attraction |