Chapter 14

(W. R. S.)

1A variant of the name Makkah is Bakkah (Sur.iii. 90; Bakrī, 155 seq.). For other names and honorific epithets of the city see Bakrī,ut supra, Azraqī, p. 197, Yāqūt iv. 617 seq. The lists are in part corrupt, and some of the names (Kūthā and ‘Arsh or ‘Ursh, “the huts”) are not properly names of the town as a whole.2Mecca, says one of its citizens, in Wāqidī (Kremer’s ed., p. 196, orMuh. in Med.p. 100), is a settlement formed for trade with Syria in summer and Abyssinia in winter, and cannot continue to exist if the trade is interrupted.3The details are variously related. See Bīrūnī, p. 328 (E. T., p. 324); Asma‘i in Yāqūt, iii. 705, iv. 416, 421; Azraqī, p. 129 seq.; Bakrī, p. 661. Jebel Kabkab is a great mountain occupying the angle between W. Namān and the plain of Arafa. The peak is due north of Sheddād, the hamlet which Burckhardt (i. 115) calls Shedad. According to Azraqī, p. 80, the last shrine visited was that of the three trees of Uzzā in W. Nakhla.4So we are told by Bīrūnī, p. 62 (E. T., 73).5Wāqidī, ed. Kremer, pp. 20, 21;Muh. in Med.p. 39.6The older fairs were not entirely deserted till the troubles of the last days of the Omayyads (Azraqī, p. 131).7This is the cross-road traversed by Burckhardt (i. 109), and described by him as cut through the rocks with much labour.8Iṣṭakhrī gives the length of the city proper from north to south as 2 m., and the greatest breadth from the Jiyād quarter east of the great mosque across the valley and up the western slopes as two-thirds of the length.9For details as to the ancient quarters of Mecca, where the several families or septs lived apart, see Azraqī, 455 pp. seq., and compare Ya‘qūbī, ed. Juynboll, p. 100. The minor sacred places are described at length by Azraqī and Ibn Jubair. They are either connected with genuine memories of the Prophet and his times, or have spurious legends to conceal the fact that they were originally holy stones, wells, or the like, of heathen sanctity.10Balādhurī, in his chapter on the floods of Mecca (pp. 53 seq.), says that ‘Omar built two dams.11The aqueduct is the successor of an older one associated with the names of Zobaida, wife of Harūn al-Rashīd, and other benefactors. But the old aqueduct was frequently out of repair, and seems to have played but a secondary part in the medieval water supply. Even the new aqueduct gave no adequate supply in Burckhardt’s time.12In Ibn Jubair’s time large supplies were brought from the Yemen mountains.13The corruption of manners in Mecca is no new thing. See the letter of the caliph Mahdi on the subject; Wüstenfeld,Chron. Mek., iv. 168.14The exact measurements (which, however, vary according to different authorities) are stated to be: sides 37 ft. 2 in. and 38 ft. 4 in.; ends 31 ft. 7 in. and 29 ft.; height 35 ft.15The Ka‘ba of Mahomet’s time was the successor of an older building, said to have been destroyed by fire. It was constructed in the still usual rude style of Arabic masonry, with string courses of timber between the stones (like Solomon’s Temple). The roof rested on six pillars; the door was raised above the ground and approached by a stair (probably on account of the floods which often swept the valley); and worshippers left their shoes under the stair before entering. During the first siege of Mecca (A.D.683), the building was burned down, the Ibn Zubair reconstructed it on an enlarged scale and in better style of solid ashlar-work. After his death his most glaring innovations (the introduction of two doors on a level with the ground, and the extension of the building lengthwise to include the Ḥijr) were corrected by Ḥajjāj, under orders from the caliph, but the building retained its more solid structure. The roof now rested on three pillars, and the height was raised one-half. The Ka‘ba was again entirely rebuilt after the flood ofA.D.1626, but since Ḥajjāj there seem to have been no structural changes.16Hobal was set up within the Temple over the pit that contained the sacred treasures. His chief function was connected with the sacred lot to which the Meccans were accustomed to betake themselves in all matters of difficulty.17See Ibn Hishām i. 54, Azraḳī p. 80 (‘Uzzā in Baṭn Marr); Yāḳūt iii. 705 (Otheydā); Bar Hebraeus on Psalm xii. 9. Stones worshipped by circling round them bore the namedawārorduwār(Krehl,Rel. d. Araber, p. 69). The later Arabs not unnaturally viewed such cultus as imitated from that of Mecca (Yāqūt iv. 622, cf. Dozy,Israeliten te Mekka, p. 125, who draws very perverse inferences).18The oldkiswais removed on the 25th day of the month before the pilgrimage, and fragments of it are bought by the pilgrims as charms. Till the 10th day of the pilgrimage month the Ka‘ba is bare.19Before Islām the Ka‘ba was opened every Monday and Thursday; in the time of Ibn Jubair it was opened with considerable ceremony every Monday and Friday, and daily in the month Rajab. But, though prayer within the building is favoured by the example of the Prophet, it is not compulsory on the Moslem, and even in the time of Ibn Baṭūṭa the opportunities of entrance were reduced to Friday and the birthday of the Prophet.20See De Vogué,Syrie centrale: inscr. sem.; Lady Anne BluntPilgrimage of Nejd, ii., and W. R. Smith, in theAthenaeum, March 20, 1880.21Ibn Jubair speaks of fourteen steps, Ali Bey of four, Burckhardt of three. The surrounding ground no doubt has risen so that the old name “hill of Safā” is now inapplicable.22The latter perhaps was no part of the ancient omra; see Snouck-Hurgronje,Het Mekkaansche Feest(1880) p. 115 sqq.23The 27th was also a great day, but this day was in commemoration of the rebuilding of the Ka‘ba by Ibn Jubair.24The sacrifice is not indispensable except for those who can afford it and are combining the hajj with the omra.25On the similar pelting of the supposed graves of Abū Lahab and his wife (Ibn Jubair, p. 110) and of Abū Righāl at Mughammas, see Nöldeke’s translation of Tabarī, 208.

2Mecca, says one of its citizens, in Wāqidī (Kremer’s ed., p. 196, orMuh. in Med.p. 100), is a settlement formed for trade with Syria in summer and Abyssinia in winter, and cannot continue to exist if the trade is interrupted.

3The details are variously related. See Bīrūnī, p. 328 (E. T., p. 324); Asma‘i in Yāqūt, iii. 705, iv. 416, 421; Azraqī, p. 129 seq.; Bakrī, p. 661. Jebel Kabkab is a great mountain occupying the angle between W. Namān and the plain of Arafa. The peak is due north of Sheddād, the hamlet which Burckhardt (i. 115) calls Shedad. According to Azraqī, p. 80, the last shrine visited was that of the three trees of Uzzā in W. Nakhla.

4So we are told by Bīrūnī, p. 62 (E. T., 73).

5Wāqidī, ed. Kremer, pp. 20, 21;Muh. in Med.p. 39.

6The older fairs were not entirely deserted till the troubles of the last days of the Omayyads (Azraqī, p. 131).

7This is the cross-road traversed by Burckhardt (i. 109), and described by him as cut through the rocks with much labour.

8Iṣṭakhrī gives the length of the city proper from north to south as 2 m., and the greatest breadth from the Jiyād quarter east of the great mosque across the valley and up the western slopes as two-thirds of the length.

9For details as to the ancient quarters of Mecca, where the several families or septs lived apart, see Azraqī, 455 pp. seq., and compare Ya‘qūbī, ed. Juynboll, p. 100. The minor sacred places are described at length by Azraqī and Ibn Jubair. They are either connected with genuine memories of the Prophet and his times, or have spurious legends to conceal the fact that they were originally holy stones, wells, or the like, of heathen sanctity.

10Balādhurī, in his chapter on the floods of Mecca (pp. 53 seq.), says that ‘Omar built two dams.

11The aqueduct is the successor of an older one associated with the names of Zobaida, wife of Harūn al-Rashīd, and other benefactors. But the old aqueduct was frequently out of repair, and seems to have played but a secondary part in the medieval water supply. Even the new aqueduct gave no adequate supply in Burckhardt’s time.

12In Ibn Jubair’s time large supplies were brought from the Yemen mountains.

13The corruption of manners in Mecca is no new thing. See the letter of the caliph Mahdi on the subject; Wüstenfeld,Chron. Mek., iv. 168.

14The exact measurements (which, however, vary according to different authorities) are stated to be: sides 37 ft. 2 in. and 38 ft. 4 in.; ends 31 ft. 7 in. and 29 ft.; height 35 ft.

15The Ka‘ba of Mahomet’s time was the successor of an older building, said to have been destroyed by fire. It was constructed in the still usual rude style of Arabic masonry, with string courses of timber between the stones (like Solomon’s Temple). The roof rested on six pillars; the door was raised above the ground and approached by a stair (probably on account of the floods which often swept the valley); and worshippers left their shoes under the stair before entering. During the first siege of Mecca (A.D.683), the building was burned down, the Ibn Zubair reconstructed it on an enlarged scale and in better style of solid ashlar-work. After his death his most glaring innovations (the introduction of two doors on a level with the ground, and the extension of the building lengthwise to include the Ḥijr) were corrected by Ḥajjāj, under orders from the caliph, but the building retained its more solid structure. The roof now rested on three pillars, and the height was raised one-half. The Ka‘ba was again entirely rebuilt after the flood ofA.D.1626, but since Ḥajjāj there seem to have been no structural changes.

16Hobal was set up within the Temple over the pit that contained the sacred treasures. His chief function was connected with the sacred lot to which the Meccans were accustomed to betake themselves in all matters of difficulty.

17See Ibn Hishām i. 54, Azraḳī p. 80 (‘Uzzā in Baṭn Marr); Yāḳūt iii. 705 (Otheydā); Bar Hebraeus on Psalm xii. 9. Stones worshipped by circling round them bore the namedawārorduwār(Krehl,Rel. d. Araber, p. 69). The later Arabs not unnaturally viewed such cultus as imitated from that of Mecca (Yāqūt iv. 622, cf. Dozy,Israeliten te Mekka, p. 125, who draws very perverse inferences).

18The oldkiswais removed on the 25th day of the month before the pilgrimage, and fragments of it are bought by the pilgrims as charms. Till the 10th day of the pilgrimage month the Ka‘ba is bare.

19Before Islām the Ka‘ba was opened every Monday and Thursday; in the time of Ibn Jubair it was opened with considerable ceremony every Monday and Friday, and daily in the month Rajab. But, though prayer within the building is favoured by the example of the Prophet, it is not compulsory on the Moslem, and even in the time of Ibn Baṭūṭa the opportunities of entrance were reduced to Friday and the birthday of the Prophet.

20See De Vogué,Syrie centrale: inscr. sem.; Lady Anne BluntPilgrimage of Nejd, ii., and W. R. Smith, in theAthenaeum, March 20, 1880.

21Ibn Jubair speaks of fourteen steps, Ali Bey of four, Burckhardt of three. The surrounding ground no doubt has risen so that the old name “hill of Safā” is now inapplicable.

22The latter perhaps was no part of the ancient omra; see Snouck-Hurgronje,Het Mekkaansche Feest(1880) p. 115 sqq.

23The 27th was also a great day, but this day was in commemoration of the rebuilding of the Ka‘ba by Ibn Jubair.

24The sacrifice is not indispensable except for those who can afford it and are combining the hajj with the omra.

25On the similar pelting of the supposed graves of Abū Lahab and his wife (Ibn Jubair, p. 110) and of Abū Righāl at Mughammas, see Nöldeke’s translation of Tabarī, 208.

MECHANICS.The subject of mechanics may be divided into two parts: (1) theoretical or abstract mechanics, and (2) applied mechanics.

1.Theoretical Mechanics

Historically theoretical mechanics began with the study of practical contrivances such as the lever, and the namemechanics(Gr.τὰ μηχανικά), which might more properly be restricted to the theory of mechanisms, and which was indeed used in this narrower sense by Newton, has clung to it, although the subject has long attained a far wider scope. In recent times it has been proposed to adopt the termdynamics(from Gr.δύναμιςforce,) as including the whole science of the action of force on bodies, whether at rest or in motion. The subject is usually expounded under the two divisions ofstaticsandkinetics, the former dealing with the conditions of rest or equilibrium and the latter with the phenomena of motion as affected by force. To this latter division the old name ofdynamics(in a restricted sense) is still often applied. The mere geometrical description and analysis of various types of motion, apart from the consideration of the forces concerned, belongs tokinematics. This is sometimes discussed as a separate theory, but for our present purposes it is more convenient to introduce kinematical motions as they are required. We follow also the traditional practice of dealing first with statics and then with kinetics. This is, in the main, the historical order of development, and for purposes of exposition it has many advantages. The laws of equilibrium are, it is true, necessarily included as a particular case under those of motion; but there is no real inconvenience in formulating as the basis of statics a few provisional postulates which are afterwards seen to be comprehended in a more general scheme.

The whole subject rests ultimately on the Newtonian laws of motion and on some natural extensions of them. As these laws are discussed under a separate heading (Motion, Laws of), it is here only necessary to indicate the standpoint from which the present article is written. It is a purely empirical one. Guided by experience, we are able to frame rules which enable us to say with more or less accuracy what will be the consequences, or what were the antecedents, of a given state of things. These rules are sometimes dignified by the name of “laws of nature,” but they have relation to our present state of knowledge and to the degree of skill with which we have succeeded in giving more or less compact expression to it. They are therefore liable to be modified from time to time, or to be superseded by more convenient or more comprehensive modes of statement. Again, we do not aim at anything so hopeless, or indeed so useless, as acompletedescription of any phenomenon. Some features are naturally more important or more interesting to us than others; by their relative simplicity and evident constancy they have the first hold on our attention, whilst those which are apparently accidental and vary from one occasion to another arc ignored, or postponed for later examination. It follows that for the purposes of such description as is possible some process of abstraction is inevitable if our statements are to be simple and definite. Thus in studying the flight of a stone through the air we replace the body in imagination by a mathematical point endowed with a mass-coefficient. The size and shape, the complicated spinning motion which it is seen to execute, the internal strains and vibrations which doubtless take place, are all sacrificed in the mental picture in order that attention may be concentrated on those features of the phenomenon which are in the first place most interesting to us. At a later stage in our subject the conception of the ideal rigid body is introduced; this enables us to fill in some details which were previously wanting, but others are still omitted. Again, the conception of a force as concentrated in a mathematical line is as unreal as that of a mass concentrated in a point, but it is a convenient fiction for our purpose, owing to the simplicity which it lends to our statements.

The laws which are to be imposed on these ideal representations are in the first instance largely at our choice. Any scheme of abstract dynamics constructed in this way, provided it be self-consistent, is mathematically legitimate; but from the physical point of view we require that it should help us to picture the sequence of phenomena as they actually occur. Its success or failure in this respect can only be judged a posteriori by comparison of the results to which it leads with the facts. It is to be noticed, moreover, that all available tests apply only to the scheme as a whole; owing to the complexity of phenomena we cannot submit any one of its postulates to verification apart from the rest.

It is from this point of view that the question of relativity of motion, which is often felt to be a stumbling-block on the very threshold of the subject, is to be judged. By “motion” we mean of necessity motion relative to some frame of reference which is conventionally spoken of as “fixed.” In the earlier stages of our subject this may be any rigid, or apparently rigid, structure fixed relatively to the earth. If we meet with phenomena which do not fit easily into this view, we have the alternatives either to modify our assumed laws of motion, or to call to our aid adventitious forces, or to examine whether the discrepancy can be reconciled by the simpler expedient of a new basis of reference. It is hardly necessary to say that the latter procedure has hitherto been found to be adequate. As a first step we adopt a system of rectangular axes whose origin is fixed in the earth, but whose directions are fixed by relation to the stars; in the planetary theory the origin is transferred to the sun, and afterwards to the mass-centre of the solar system; and so on. At each step there is a gain in accuracy and comprehensiveness; and the conviction is cherished thatsomesystem of rectangular axes exists with respect to which the Newtonian scheme holds with all imaginable accuracy.

A similar account might be given of the conception of time as a measurable quantity, but the remarks which it is necessary to make under this head will find a place later.

The following synopsis shows the scheme on which the treatment is based:—Part 1.—Statics.1.Statics of a particle.2.Statics of a system of particles.3.Plane kinematics of a rigid body.4.Plane statics.5.Graphical statics.6.Theory of frames.7.Three-dimensional kinematics of a rigid body.8.Three-dimensional statics.9.Work.10.Statics of inextensible chains.11.Theory of mass-systems.Part 2.—Kinetics.12.Rectilinear motion.13.General motion of a particle.14.Central forces. Hodograph.15.Kinetics of a system of discrete particles.16.Kinetics of a rigid body. Fundamental principles.17.Two-dimensional problems.18.Equations of motion in three dimensions.19.Free motion of a solid.20.Motion of a solid of revolution.21.Moving axes of reference.22.Equations of motion in generalized co-ordinates.23.Stability of equilibrium. Theory of vibrations.

The following synopsis shows the scheme on which the treatment is based:—

Part I.—Statics

§ 1.Statics of a Particle.—By aparticleis meant a body whose position can for the purpose in hand be sufficiently specified by a mathematical point. It need not be “infinitely small,” or even small compared with ordinary standards; thus in astronomy such vast bodies as the sun, the earth, and the other planets can for many purposes be treated merely as points endowed with mass.

Aforceis conceived as an effort having a certain direction and a certain magnitude. It is therefore adequately represented, for mathematical purposes, by a straight line AB drawn in the direction in question, of length proportional (on any convenient scale) to the magnitude of the force. In other words, a force is mathematically of the nature of a “vector” (seeVector Analysis,Quaternions). In most questions of pure statics we are concerned only with theratiosof the various forces which enter into the problem, so that it is indifferent whatunitof force is adopted. For many purposes a gravitational system of measurement is most natural; thus we speak of a force of so many pounds or so many kilogrammes. The “absolute” system of measurement will be referred to below inPart II., Kinetics. It is to be remembered that all “force” is of the nature of a push or a pull, and that according to the accepted terminology of modern mechanics such phrases as “force of inertia,” “accelerating force,” “moving force,” once classical, are proscribed. This rigorous limitation of the meaning of the word is of comparatively recent origin, and it is perhaps to be regretted that some more technical term has not been devised, but the convention must now be regarded as established.

The fundamental postulate of this part of our subject is that the two forces acting on a particle may be compounded by the “parallelogram rule.” Thus, if the two forces P,Q be represented by the lines OA, OB, they can be replaced by a single force R represented by the diagonal OC of the parallelogram determined by OA, OB. This is of course a physical assumption whose propriety is justified solely by experience. We shall see later that it is implied in Newton’s statement of his Second Law of motion. In modern language, forces are compounded by “vector-addition”; thus, if we draw in succession vectorsHK>,KL>to represent P, Q, the force R is represented by the vectorHL>which is the “geometric sum” ofHK>,KL>.

By successive applications of the above rule any number of forces acting on a particle may be replaced by a single force which is the vector-sum of the given forces: this single force is called theresultant. Thus ifAB>,BC>,CD>...,HK>be vectors representing the given forces, the resultant will be given byAK>. It will be understood that the figure ABCD ... K need not be confined to one plane.

If, in particular, the point K coincides with A, so that the resultant vanishes, the given system of forces is said to be inequilibrium—i.e.the particle could remain permanently at rest under its action. This is the proposition known as thepolygon of forces. In the particular case of three forces it reduces to thetriangle of forces, viz. “If three forces acting on a particle are represented as to magnitude and direction by the sides of a triangle taken in order, they are in equilibrium.”

A sort of converse proposition is frequently useful, viz. if three forces acting on a particle be in equilibrium, and any triangle be constructed whose sides are respectively parallel to the forces, the magnitudes of the forces will be to one another as the corresponding sides of the triangle. This follows from the fact that all such triangles are necessarily similar.

As a simple example of the geometrical method of treating statical problems we may consider the equilibrium of a particle on a “rough” inclined plane. The usual empirical law of sliding friction is that the mutual action between two plane surfaces in contact, or between a particle and a curve or surface, cannot make with the normal an angle exceeding a certain limit λ called theangle of friction. If the conditions of equilibrium require an obliquity greater than this, sliding will take place. The precise value of λ will vary with the nature and condition of the surfaces in contact. In the case of a body simply resting on an inclined plane, the reaction must of course be vertical, for equilibrium, and the slope α of the plane must therefore not exceed λ. For this reason λ is also known as theangle of repose. If α > λ, a force P must be applied in order to maintain equilibrium; let θ be the inclination of P to the plane, as shown in the left-hand diagram. The relations between this force P, the gravity W of the body, and the reaction S of the plane are then determined by a triangle of forces HKL. Since the inclination of S to the normal cannot exceed λ on either side, the value of P must lie between two limits which are represented by L1H, L2H, in the right-hand diagram. Denoting these limits by P1, P2, we haveP1/W = L1H/HK = sin (α − λ)/cos (θ + λ),P2/W = L2H/HK = sin (α + λ)/cos (θ − λ).It appears, moreover, that if θ be varied P will be least when L1H is at right angles to KL1, in which case P1= W sin (α − λ), corresponding to θ = −λ.

P1/W = L1H/HK = sin (α − λ)/cos (θ + λ),P2/W = L2H/HK = sin (α + λ)/cos (θ − λ).

It appears, moreover, that if θ be varied P will be least when L1H is at right angles to KL1, in which case P1= W sin (α − λ), corresponding to θ = −λ.

Just as two or more forces can be combined into a single resultant, so a single force may beresolvedintocomponentsacting in assigned directions. Thus a force can be uniquely resolved into two components acting in two assigned directions in the same plane with it by an inversion of the parallelogram construction of fig. 1. If, as is usually most convenient, the two assigned directions are at right angles, the two components of a force P will be P cos θ, P sin θ, where θ is the inclination of P to the direction of the former component. This leads to formulae for the analytical reduction of a system of coplanar forces acting on a particle. Adopting rectangular axes Ox, Oy, in the plane of the forces, and distinguishing the various forces of the system by suffixes, we can replace the system by two forces X, Y, in the direction of co-ordinate axes; viz.—

X = P1cos θ1+ P2cos θ2+ ... = Σ (P cos θ),Y = P1sin θ1+ P2sin θ2+ ... = Σ (P sin θ).

(1)

These two forces X, Y, may be combined into a single resultant R making an angle φ with Ox, provided

X = R cos φ, Y = R sin φ,

(2)

whence

R2= X2+ Y2, tan φ = Y/X.

(3)

For equilibrium we must have R = 0, which requires X = 0, Y = 0; in words, the sum of the components of the system must be zero for each of two perpendicular directions in the plane.

A similar procedure applies to a three-dimensional system. Thus if, O being the origin,OH>represent any force P of the system, the planes drawn through H parallel to the co-ordinate planes will enclose with the latter a parallelepiped, and it is evident thatOH>is the geometric sum ofOA>,AN>,NH>, orOA>,OB>,OC>, in the figure. Hence P is equivalent to three forces Pl, Pm, Pn acting along Ox, Oy, Oz, respectively, where l, m, n, are the “direction-ratios” ofOH>. The whole system can be reduced in this way to three forces

X = Σ (Pl), Y = Σ (Pm), Z = Σ (Pn),

(4)

acting along the co-ordinate axes. These can again be combined into a single resultant R acting in the direction (λ, μ, ν), provided

X = Rλ, Y = Rμ, Z = Rν.

(5)

If the axes are rectangular, the direction-ratios become direction-cosines, so that λ2+ μ2+ ν2= 1, whence

R2= X2+ Y2+ Z2.

(6)

The conditions of equilibrium are X = 0, Y = 0, Z = 0.

§ 2.Statics of a System of Particles.—We assume that the mutual forces between the pairs of particles, whatever their nature, are subject to the “Law of Action and Reaction” (Newton’s Third Law);i.e.the force exerted by a particle A on a particle B, and the force exerted by B on A, are equal and opposite in the line AB. The problem of determining the possible configurations of equilibrium of a system of particles subject to extraneous forces which are known functions of the positions of the particles, and to internal forces which are known functions of the distances of the pairs of particles between which they act, is in general determinate. For if n be the number of particles, the 3n conditions of equilibrium (three for each particle) are equal in number to the 3n Cartesian (or other) co-ordinates of the particles, which are to be found. If the system be subject to frictionless constraints,e.g.if some of the particles be constrained to lie on smooth surfaces, or if pairs of particles be connected by inextensible strings, then for each geometrical relation thus introduced we have an unknown reaction (e.g.the pressure of the smooth surface, or the tension of the string), so that the problem is still determinate.

The case of thefunicular polygonwill be of use to us later. A number of particles attached at various points of a string are acted on by given extraneous forces P1, P2, P3... respectively. The relation between the three forces acting on any particle, viz. the extraneous force and the tensions in the two adjacent portions of the string can be exhibited by means of a triangle of forces; and if the successive triangles be drawn to the same scale they can be fitted together so as to constitute a singleforce-diagram, as shown in fig. 6. This diagram consists of a polygon whose successive sides represent the given forces P1, P2, P3..., and of a series of lines connecting the vertices with a point O. These latter lines measure the tensions in the successive portions of string. As a special, but very important case, the forces P1, P2, P3... may be parallel,e.g.they may be the weights of the several particles. The polygon of forces is then made up of segments of a vertical line. We note that the tensions have now the same horizontal projection (represented by the dotted line in fig. 7). It is further of interest to note that if the weights be all equal, and at equal horizontal intervals, the vertices of the funicular will lie on a parabola whose axis is vertical. To prove this statement, let A, B, C, D ... be successive vertices, and let H, K ... be the middle points of AC, BD ...; then BH, CK ... will be vertical by the hypothesis, and since the geometric sum ofBA>,BC>is represented by 2BH>, the tension in BA: tension in BC: weight at Bas BA : BC : 2BH.The tensions in the successive portions of the string are therefore proportional to the respective lengths, and the lines BH, CK ... are all equal. Hence AD, BC are parallel and are bisected by the same vertical line; and a parabola with vertical axis can therefore be described through A, B, C, D. The same holds for the four points B, C, D, E and so on; but since a parabola is uniquely determined by the direction of its axis and by three points on the curve, the successive parabolas ABCD, BCDE, CDEF ... must be coincident.

as BA : BC : 2BH.

The tensions in the successive portions of the string are therefore proportional to the respective lengths, and the lines BH, CK ... are all equal. Hence AD, BC are parallel and are bisected by the same vertical line; and a parabola with vertical axis can therefore be described through A, B, C, D. The same holds for the four points B, C, D, E and so on; but since a parabola is uniquely determined by the direction of its axis and by three points on the curve, the successive parabolas ABCD, BCDE, CDEF ... must be coincident.

§ 3.Plane Kinematics of a Rigid Body.—The idealrigid bodyis one in which the distance between any two points is invariable. For the present we confine ourselves to the consideration of displacements in two dimensions, so that the body is adequately represented by a thin lamina or plate.

The position of a lamina movable in its own plane is determinate when we know the positions of any two points A, B of it. Since the four co-ordinates (Cartesian or other) of these two points are connected by the relation which expresses the invariability of the length AB, it is plain that virtually three independent elements are required and suffice to specify the position of the lamina. For instance, the lamina may in general be fixed by connecting any three points of it by rigid links to three fixed points in its plane. The three independent elements may be chosen in a variety of ways (e.g.they may be the lengthsof the three links in the above example). They may be called (in a generalized sense) theco-ordinatesof the lamina. The lamina when perfectly free to move in its own plane is said to havethree degrees of freedom.

By a theorem due to M. Chasles any displacement whatever of the lamina in its own plane is equivalent to a rotation about some finite or infinitely distant point J. For suppose that in consequence of the displacement a point of the lamina is brought from A to B, whilst the point of the lamina which was originally at B is brought to C. Since AB, BC, are two different positions of the same line in the lamina they are equal, and it is evident that the rotation could have been effected by a rotation about J, the centre of the circle ABC, through an angle AJB. As a special case the three points A, B, C may be in a straight line; J is then at infinity and the displacement is equivalent to a puretranslation, since every point of the lamina is now displaced parallel to AB through a space equal to AB.

Next, consider any continuous motion of the lamina. The latter may be brought from any one of its positions to a neighbouring one by a rotation about the proper centre. The limiting position J of this centre, when the two positions are taken infinitely close to one another, is called theinstantaneous centre. If P, P′ be consecutive positions of the same point, and δθ the corresponding angle of rotation, then ultimately PP′ is at right angles to JP and equal to JP·δθ. The instantaneous centre will have a certain locus in space, and a certain locus in the lamina. These two loci are calledpole-curvesorcentrodes, and are sometimes distinguished as thespace-centrodeand thebody-centrode, respectively. In the continuous motion in question the latter curve rolls without slipping on the former (M. Chasles). Consider in fact any series of successive positions 1, 2, 3... of the lamina (fig. 11); and let J12, J23, J34... be the positions in space of the centres of the rotations by which the lamina can be brought from the first position to the second, from the second to the third, and so on. Further, in the position 1, let J12, J′23, J′34... be the points of the lamina which have become the successive centres of rotation. The given series of positions will be assumed in succession if we imagine the lamina to rotate first about J12until J′23comes into coincidence with J23, then about J23until J′34comes into coincidence with J34, and so on. This is equivalent to imagining the polygon J12J′23J′34..., supposed fixed in the lamina, to roll on the polygon J12J23J34..., which is supposed fixed in space. By imagining the successive positions to be taken infinitely close to one another we derive the theorem stated. The particular case where both centrodes are circles is specially important in mechanism.

The theory may be illustrated by the case of “three-bar motion.” Let ABCD be any quadrilateral formed of jointed links. If, AB being held fixed, the quadrilateral be slightly deformed, it is obvious that the instantaneous centre J will be at the intersection of the straight lines AD, BC, since the displacements of the points D, C are necessarily at right angles to AD, BC, respectively. Hence these displacements are proportional to JD, JC, and therefore to DD′ CC′, where C′D′ is any line drawn parallel to CD, meeting BC, AD in C′, D′, respectively. The determination of the centrodes in three-bar motion is in general complicated, but in one case, that of the “crossed parallelogram” (fig. 13), they assume simple forms. We then have AB = DC and AD = BC, and from the symmetries of the figure it is plain thatAJ + JB = CJ + JD = AD.Hence the locus of J relative to AB, and the locus relative to CD are equal ellipses of which A, B and C, D are respectively the foci. It may be noticed that the lamina in fig. 9 is not, strictly speaking, fixed, but admits of infinitesimal displacement, whenever the directions of the three links are concurrent (or parallel).

AJ + JB = CJ + JD = AD.

Hence the locus of J relative to AB, and the locus relative to CD are equal ellipses of which A, B and C, D are respectively the foci. It may be noticed that the lamina in fig. 9 is not, strictly speaking, fixed, but admits of infinitesimal displacement, whenever the directions of the three links are concurrent (or parallel).

The matter may of course be treated analytically, but we shall only require the formula for infinitely small displacements. If the origin of rectangular axes fixed in the lamina be shifted through a space whose projections on the original directions of the axes are λ, μ, and if the axes are simultaneously turned through an angle ε, the co-ordinates of a point of the lamina, relative to the original axes, are changed from x, y to λ + x cos ε − y sin ε, μ + x sin ε + y cos ε, or λ + x − yε, μ + xε + y, ultimately. Hence the component displacements are ultimately

δx = λ − yε, δy = μ + xε

(1)

If we equate these to zero we get the co-ordinates of the instantaneous centre.

§ 4.Plane Statics.—The statics of a rigid body rests on the following two assumptions:—

(i) A force may be supposed to be applied indifferently at any point in its line of action. In other words, a force is of the nature of a “bound” or “localized” vector; it is regarded as resident in a certain line, but has no special reference to any particular point of the line.

(ii) Two forces in intersecting lines may be replaced by a force which is their geometric sum, acting through the intersection. The theory of parallel forces is included as a limiting case. For if O, A, B be any three points, and m, n any scalar quantities, we have in vectors

m ·OA>+ n ·OB>= (m + n)OC>,

(1)

provided

m ·CA>+ n ·CB>= 0.

(2)

Hence if forces P, Q act in OA, OB, the resultant R will pass through C, provided

m = P/OA, n = Q/OB;

also

R = P·OC/OA + Q·OC/OB,

(3)

and

P · AC : Q·CB = OA : OB.

(4)

These formulae give a means of constructing the resultant by means of any transversal AB cutting the lines of action. If we now imagine the point O to recede to infinity, the forces P, Q and the resultant R are parallel, and we have

R = P + Q, P·AC = Q·CB.

(5)

When P, Q have opposite signs the point C divides AB externally on the side of the greater force. The investigation fails when P + Q = 0, since it leads to an infinitely small resultant acting in an infinitely distant line. A combination of two equal, parallel, but oppositely directed forces cannot in fact be replaced by anything simpler, and must therefore be recognized as an independent entity in statics. It was called by L. Poinsot, who first systematically investigated its properties, acouple.

We now restrict ourselves for the present to the systems of forces in one plane. By successive applications of (ii) anysuch coplanar system can in general be reduced to asingle resultantacting in a definite line. As exceptional cases the system may reduce to a couple, or it may be in equilibrium.

Themomentof a force about a point O is the product of the force into the perpendicular drawn to its line of action from O, this perpendicular being reckoned positive or negative according as O lies on one side or other of the line of action. If we mark off a segment AB along the line of action so as to represent the force completely, the moment is represented as to magnitude by twice the area of the triangle OAB, and the usual convention as to sign is that the area is to be reckoned positive or negative according as the letters O, A, B, occur in “counter-clockwise” or “clockwise” order.

The sum of the moments of two forces about any point O is equal to the moment of their resultant (P. Varignon, 1687). Let AB, AC (fig. 16) represent the two forces, AD their resultant; we have to prove that the sum of the triangles OAB, OAC is equal to the triangle OAD, regard being had to signs. Since the side OA is common, we have to prove that the sum of the perpendiculars from B and C on OA is equal to the perpendicular from D on OA, these perpendiculars being reckoned positive or negative according as they lie to the right or left of AO. Regarded as a statement concerning the orthogonal projections of the vectorsAB>andAC>(or BD), and of their sumAD>, on a line perpendicular to AO, this is obvious.

It is now evident that in the process of reduction of a coplanar system no change is made at any stage either in the sum of the projections of the forces on any line or in the sum of their moments about any point. It follows that the single resultant to which the system in general reduces is uniquely determinate,i.e.it acts in a definite line and has a definite magnitude and sense. Again it is necessary and sufficient for equilibrium that the sum of the projections of the forces on each of two perpendicular directions should vanish, and (moreover) that the sum of the moments about some one point should be zero. The fact that three independent conditions must hold for equilibrium is important. The conditions may of course be expressed in different (but equivalent) forms;e.g.the sum of the moments of the forces about each of the three points which are not collinear must be zero.

The particular case of three forces is of interest. If they are not all parallel they must be concurrent, and their vector-sum must be zero. Thus three forces acting perpendicular to the sides of a triangle at the middle points will be in equilibrium provided they are proportional to the respective sides, and act all inwards or all outwards. This result is easily extended to the case of a polygon of any number of sides; it has an important application in hydrostatics.

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