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scope in fig. 6 and a gyrostat in fig. 7. The gyroscope, as represented in figs. 2 and 3 by Max-well's dynamical top, is provided with screws by which the centre of gravity can be brought into coincidence with the point of support.

It can Fig. 3. Fig. 4. then be used to illustrate See also:

POINSOT, LOUIS (1777–1859)

Poinsot's theory of the motion of a body under no force, the gyroscope being made kinetically unsymmetrical by a setting of the screws. The discussion of this See also:

MOVEMENT

movement is required for See also:

Jacobi's theorems on the allied motion of a top and of a body under no force (Poinsot, Theorie nouvelle de la rotation See also:

des See also:

CORPS (pronounced as in French, from which it is taken, being a late spelling of tors, from Lat. corpus, a body; cf. " corpse ")

corps, See also:

Paris, 1857; Jacobi, Werke, ii. See also:

Note B, p. 476). To imitate the movement of the top the centre of gravity is displaced from the point of support so as to give a preponderance. When the motion takes See also:

PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

place in the neighbourhood of the down-See also:

ward See also:

VERTICAL (from Lat. vertex, highest point)

vertical, the bicycle wheel can be made to serve again II uyr„y~ um:u~uiou iinfll;IR~llitlil! Fig. 2.

mounted as in fig. .8 by a stalk in the prolongation of the See also:

AXLE (in Mid. Eng. axel-tre, from O. Norweg. oxull-tre, cognate with the O. Eng. exe or eaxe, and connected with Sansk. dksha, Gr. a wv, and Lat. axis)

axle, suspended from a universal See also:

JOINT (through Fr. from Lat. junctum, jungere, to join)

joint at 0; it can then be spun by See also:

hand and projected in any manner. The first See also:

PRACTICAL

practical application of the gyroscopic principle was invented and carried out (1744) by Serson, with a spinning top with a polished upper plane See also:

SURFACE

surface for giving an artificial See also:

HORIZON (Gr. 6pfTwv, dividing)

horizon at See also:

sea, undisturbed by the motion of the See also:

SHIP

ship, when the real horizon was obscured. The instrument has been perfected by See also:

ADMIRAL

Admiral Georges Ernest Fleuriais (fig. 9), and is interesting theoretically as showing the See also:

cor- rection required practically for the rotation of the earth. See also:

Gilbert's barogyroscope is devised for the same purpose of showing the earth's rotation; a de-R, scription of it, and of the latest See also:

FORM (Lat. forma)

form employed by Foppl, is given in the Ency. d. math. Wiss., 1904, with See also:

BIBLIOGRAPHICAL

bibliographical references in the See also:

ARTICLE (from Lat. articulus, a joint)

article "See also:

MECHANICS

Mechanics of See also:

PHYSICAL

Physical Appar- atus." The See also:

ROTA, COURT OF

rota- tion of the fly-wheel is maintained here by an electric motor, as devised by G.M. See also:

Hopkins, and described in the Scientific See also:

AMERICAN

American, 1878. 'T o demonstrate the rotation of the earth by the constancy in direction of the See also:

AXIS (Lat. for " axle ")

axis of a gyroscope is a See also:

SUGGESTION

suggestion that has often been made; by E. Sang in 1836, and others. The experiment was first carried out with success by See also:

FOUCAULT, JEAN BERNARD LEON (1819—1868)

Foucault in 1851, by a See also:

SIMPLE

simple pendulum swung in the See also:

DOME (Lat domus, house; Ital. duomo, cathedral)

dome of the See also:

PANTHEON (Lat. pantheum or pantheon; Gr. srav& ov, all-holy, from re, s, all, and Oeor god)

Pantheon, Paris, and it has been repeated frequently (Mentoires sur le pendule, 1888). 9 A gyroscopic fly-wheel will pre- serve its See also:

ORIGINAL

original direction in space only when See also:

LEFT

left absolutely free in all directions, as required in the experiments above.

If employed in steering, as of a See also:

TORPEDO

torpedo, the gyroscope must See also:

ACT (Lat. actus, actum)

act through the intermediary of a See also:

LIGHT

light relay; but if See also:

DIRECT

direct-acting, the reaction will cause pre-cession of the axis, and the original direction is lost. The gyrostatic principle, in which one degree of freedom is suppressed in the axis, is useful for imparting steadiness andstability in a moving body; it is employed by Schlick to mitigate the See also:

ROLLING

rolling of a ship and to maintain the upright position of Brennan's monorail See also:

CAR (Late Lat. carra)

car. Lastly, as an application of gyroscopic theory, a stretched See also:

CHAIN (through the O. Fr. citable, chcene, &c., from Lat. catena)

chain of fly-wheels in rotation was employed by Kelvin as a See also:

MECHANICAL

mechanical See also:

MODEL (0. Fr. modelle, mod. modele; It. modello, pattern, mould; from Lat. modus, measure, standard)

model of the rotary polarization of light in an electromagnetic See also:

field; the apparatus may be constructed of bicycle wheels connected by See also:

SHORT, FRANCIS JOB (1857– )

short links, and suspended vertically. Theory of the Symmetrical Top. 1. The physical constants of a given symmetrical top, expressed in C.G.S. See also:

units, which are employed in the subsequent formulae, are denoted by M, h, C and A. M is the See also:

WEIGHT

weight in grammes (g) as given by the number of gramme weights which equilibrate the top when weighed in a See also:

BALANCE (derived through the Fr. from the Late Lat. bilantia, an apparatus for weighing, from bi, two, and lanx, a dish or scale)

balance; h is the distance OG in centimetres (cm.) between G the centre of gravity and 0 the point of support, and Mh may be called the preponderance in g.-cm.; Mh and M can be measured by a See also:

SPRING (from " to spring," " to leap or jump up," " burst out," O. Eng., springau, a common Teut. word, cf. Ger. springer, possibly allied to Gr. QnrFpxecOac, to move rapidly)

spring balance holding up in a horizontal position the axis OC in fig. 8 suspended at O. Then gMh (dyne-cm. or ergs) is the moment of gravity about 0 when the axis OG is horizontal, gMh See also:

sin 0 being the moment when the axis OG makes an See also:

ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle 8 with the vertical, and g=981 (cm./s2) on the See also:

AVERAGE

average; C is the moment of inertia of the top about OG, and A about any axis through 0 at right angles to OG, both measured in g-cm.2. To measure A experimentally, See also:

SWING, DAVID (1830-1894)

swing the top freely about 0 in small plane oscillation, and determine the length, 1 cm., of the See also:

EQUIVALENT

equivalent simple pendulum; then (I) l=A/Mh, A=Mhl. Next make the top, or this simple pendulum, perform small conical revolutions, nearly coincident with the downward vertical position of equilibrium, and measure n, the mean angular velocity of the conical pendulum in radians / second; and T its See also:

period in seconds; then (2) 4ar2/T2=n2=g/l=gMh/A; and f=n/21r is the number of revolutions per second, called the frequency,T =See also:

EAR (common Teut.; O.E. are, Ger. Ohr, Du. oor, akin to Lat. auris, Gr. ovs)

ear/n is the period of a revolution, in seconds. 2.

In the popular explanation of the steady movement of the top at a See also:

CONSTANT, BENJAMIN (1845-1902)

constant inclination to the vertical, depending on the See also:

composition of angular velocity, such as given in See also:

Perry's Steady Spinning Tops, or Worthington's Dynamics of Rotation, motion of it is asserted that the moment of gravity is always the top. generating an angular velocity about an axis OB perpendicular to the vertical plane COC' through the axis of the top OC'; and this angular velocity, compounded with the resultant angular velocity about an axis OI, nearly coincident with OC', causes the axes 01 and OC' to keep taking up a new position by moving at right angles to the plane COC', at a constant precessional angular velocity, say a rad./sec., See also:

ROUND (O. Fr. rond, Lat. rotundus, the Fr. is the source also of Du. rond; Ger., Swed., Dan. and Nor. rond)

round the vertical OC (fig. 4). If, however, the axis OC' is prevented from taking up this precessional velocity, the top at once falls down; thence all the ingenious attempts—for instance, in the swinging See also:

CABIN

cabin of the See also:

Bessemer ship—to utilise the gyroscope as a mechanical directive agency have always resulted in failure (Engineer, See also:

OCTOBER

October 1874), unless restricted to actuate a light relay, which guides the mechanism, as in steering a torpedo. An experimental verification can be carried out with the gyro-scope in fig. I; so See also:

long as the vertical spindle is free to rotate in its socket, the rapidly rotating wheel will resist the impulse of tapping on the See also:

GIMBAL

gimbal by moving to one See also:

side; but when the pinch See also:

SCREW (O.E. scrue, from O. Fr. escroue, mod. ecrou; ultimate origin uncertain; the word, or a similar one, appears in Teutonic languages, cf. Ger. Schraube, Dan. skrue, but Skeat, following Diaz, finds the origin in Lat. scrobs, a ditch, hole, particularl

screw prevents the rotation of the vertical spindle in the massive See also:

PEDESTAL (Fr. piedestal, Ital. piedestallo, foot of a stall)

pedestal, this resistance to the tapping at once disappears, provided the See also:

FRICTION (from Lat. fricare, to rub)

friction of the table prevents the movement of the pedestal; and if the wheel has any preponderance, it falls down. See also:

FAMILIAR (through the Fr. familier, from Lat. familiaris, of or belonging to the familia, family)

Familiar instances of the same principles are observable in the movement of a hoop, or in the steering of a bicycle; it is essential that the handle of the bicycle should be free to rotate to secure the stability of the movement. The bicycle wheel, employed as a spinning top, in fig. 4, can also be held by the stalk, and will thus, when rotated rapidly, convey a distinct See also:

MUSCULAR

muscular impression of resistance to See also:

CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. «a urmu', to bend)

change of direction, if brandished. 3 A demonstration, depending on the elementary principles of dynamics, of the exact conditions required for the Elementary axis OC' of a spinning top to spin steadily at a constant de onstrainclination 8 to the vertical OC, is given here before See also:

pro- tioa of the ceeding to the more complicated question of the See also:

general See also:

CONDITION (Lat. condicio, from condicere, to agree upon, arrange; not connected with conditio, from condere, conditum, to put together)

condition motion, when 8, the inclination of the axis, is varying ofsteady by See also:

NUTATION (from Lat. nutare, to nod)

nutation. motion. It is a fundamental principle in dynamics that if OH is a vector representing to See also:

scale the angular momentum of a See also:

SYSTEM

system, and if Oh is the vector representing the axis of the impressed couple or See also:

TORQUE, or TORO (Lat. torquis, torques, a twisted collar, torquere, to twist)

torque, then OH will vary so that the velocity of H is represented to scale by the impressed couple Oh, and if the top is moving freely about 0, Oh is at right angles to the vertical plane COC', and (I) Oh=gMh sin o. FIG In the See also:

case of the steady motion of the top, the vector OH lies in the vertical plane COC', in OK suppose (fig. 4), and has a component OC --=G about the vertical and a component OC' = G', sup-pose, about the axis OC; and G'=CR, if R denotes the angular velocity of the top with which it is spun about OC'.

If µ denotes the constant precessional angular velocity of the vertical plane COC', the components of angular velocity and momentum about OA are µ sin B and Aµ sin 0, OA being perpendicular to OC' in the plane COC'; so that the vector OK has the components (2) and the horizontal component (3) CK=OC' sing—C'K See also:

cos 8 =G' sing—Aµ sinBcos0. The velocity of K being equal to the impressed couple Oh, (4) gi‘/Ihsin0=µ.CK=sing, (G'µ—Aµ2cos0), and dropping the See also:

FACTOR (from Lat. facere, to make or do)

factor sin B, (5) Aµ2 cosU—G'µ+gMh =o, or Aµ2cos0—CRµ+An2=0, the condition for steady motion. Solving this as a quadratic in A, the roots µr, µ2 are given by 22 G 4An (6) µr, µ2 =2Asec 0 rI ~ (I — G•2 cos B) ; and the minimum value of G'=CR for real values of µ is given by z (7) 4 ,2n2 = cos B,AR = 2 J (cos B) ; for a smaller value of R the top cannot spin steadily at the inclination B to the upward vertical. Interpreted geometrically in fig. 4 (8) µ= gMh sing/CK=An2/KN,andA=C'K/A sing=KM/A, (9) KM. KN =A2n2, so that K lies on a See also:

HYPERBOLA

hyperbola with OC, OC' as asymptotes. 4. Suppose the top or gyroscope, instead of moving freely about the point 0, is held in a ring or See also:

FRAME

frame which is compelled to rotate about the vertical axis OC with constant angular velocity µ; then if N denotes the couple of reaction. of the frame keeping the top from falling, acting in the plane COC', See also:

EQUATION (from Lat. aequatio, aequare, to equalize)

equation (4) § 3 becomes modified into (1) gMh sin 0—N =µ.CK=sin 0 (G'µ—Aµ2 cos 0), (2) N =sin B (Aµ2 cos B—G'µ+gMh) =A'sin8cos0(µ=µr) (µ—µ2); and hence, as µ increases through µ2 and Al, the sign of N can be determined, See also:

POSITIVE (or PORTABLE) ORGAN

positive or negative, according as the tendency of the axis is to fall or rise. When G' = CR is large, µ2 is large, and (3) irr^=gMh/G' =An2/CR, the same for all inclinations, and this is the precession observed in the spinning top and centrifugal See also:

machine of fig. Ic. This is true accurately when the axis OC' is horizontal, and then it agrees with the result of the popular explanation of § 2. If the axis of the top OC' is pointing upward, the precession is in the same .direction as the rotation, and an increase of from al makes N negative, and the top rises; conversely a decrease of the procession µ causes the axis to fall (Perry, Spinning Tops, p.

48). If the axis points downward, as in the centrifugal machine with upper support, the precession is in the opposite direction to the rotation, and to make the axis approach the vertical position the precession must be reduced. This is effected automatically in the See also:

WESTON, THOMAS (1737-1776)

Weston centrifugal machine (fig. so) used for the separation of See also:

WATER

water and See also:

MOLASSES

molasses, by the friction of the indiarubber cushions above the support; or else the spindle is produced downwards below the See also:

DRUM (early forms drome or dromme, a word common to many Teut. languages, cf. Dan. tromme, Ger. Trommel: the word is ultimately the same as " trumpet," and is probably onomatopoeic in origin; it appears late in Eng. about the middle of the 16th century)

drum a short distance, and turns in a hole in a weight Centel- fugal resting on the bottom of the case, which weight is dragged machine. round until the spindle is upright; this second arrangement is more effective when a liquid is treated in the drum, and See also:

WAVE

wave See also:

ACTION

action is set up (The Centrifugal Machine, C. A. Matthey). Similar considerations apply to the stability of the whirling bowl in a cream-separating machine. We can write equation (I) (4) N = (An2 sin B —µ.CK = (A2n2 —KM.KN) sin a/A, so that N is negative or positive, and the axis tends to rise or fall according as .K moves to the inside or outside of the hyperbola of free motion. Thus a tap on the axis tending to See also:

HURRY (or URRV), SIR JOHN (d. 165o)

hurry the precession isequivalent to an impulse couple giving an increase to C'K, and will make K move to the interior of the hyperbola and cause the axis to rise; the steering of a bicycle may be explained in this way; but Kr will move to the exterior of the hyperbola, and so the axis will fall in this second more violent motion. Friction on the point of the top may be supposed to act like a tap in the direction opposite to the precession; and so the axis of a top spun violently rises at first and up to the vertical position, but falls away again as the motion See also:

DIES, CHRISTOPH ALBERT (1755-1822)

dies out. Friction considered as acting in retarding the rotation may be compared to an impulse couple tending to reduce OC', and so make K and Kr both move to the exterior of the hyperbola, and the axis falls in both cases. The axis may rise or fall according to the direction of the frictional couple, depending on the shape of the point; an See also:

ANALYTICAL

analytical treatment of the varying motion is very intractable; a memoir by E. G.

Gallop may be consulted in the Trans. Camb. Phil. See also:

Soc., 1903. The earth behaves in precession like a large spinning top, of which the axis describes a circle round the See also:

pole of the See also:

ECLIPTIC

ecliptic of mean angular See also:

RADIUS

radius 0, about 231°, in a period of 26,000 years, so that R/µ=26000X365; and the mean couple producing precession is (5) CRµ sing=CR2 sin 231°/26000X365, one 12 millionth See also:

PART

part of 1CR2, the rotation See also:

ENERGY (from the Gr. ivipyela; iv, in, ipyov, work)

energy of the earth. 5. If the preponderance is absent, by making the C.G coincide ' with 0, and if Aµ is insensible compared with G', (I) N= —Gµ sin B, the See also:

formula which suffices to explain most gyroscopic action. Thus a See also:

CARRIAGE

carriage See also:

RUNNING

running round a See also:

CURVE (Lat. curvus, bent)

curve experiences, in consequence of the rotation of the wheels, an increase of pressure Z on the See also:

OUTER

outer track, and a diminution Z on the inner, giving a couple, if a is the See also:

GAUGE, or GAGE (Med. Lat. gauja, jaugia, Fr. jauge, perhaps connected with Fr. jale, a bowl, galon, gallon)

gauge, Gyro- (2) Za = G'µ, tending to help the centrifugal force to upset the See also:

TRAIN (M. Eng. trayn or trayne, derived through Fr. from Late Lat. trahinare, to drag, draw, Lat. trahere, cf. trail, trace, ultimately from the same source)

train; and if c is the radius of the curve, b of the wheels, C their moment of inertia, and v the velocity of the train, (3) is =See also:

vie, G' = Cv/h, (4) Z = Cv2/See also:

abc (dynes), so that Z is the fraction C/Mab of the centrifugal force Mv2/c, or the fraction C/Mh of its transference of weight, with h the height of the centre of gravity of the carriage above the road. A Brennan carriage on a monorail would lean over to the inside of the curve at an angle a, given by (6) tan a=G'µ/gMh=G'v/gMhc. The gyroscopic action of a See also:

DYNAMO (a shortened form of " dynamo-electric machine," from Gr. Sbvaµis, power)

dynamo, See also:

TURBINE (Lat. turbo, a whirlwind, a whirling motion ,or object, a top)

turbine, and other rotating machinery on a steamer, See also:

PADDLE

paddle or screw, due to its rolling and pitching, can be evaluated in a similar elementary manner (Worthington, Dynamics of Rotation), and Schlick's gyroscopic apparatus is intended to mitigate the oscillation. 6. If the axis OC in fig.

4 is inclined at an angle a to the vertical, the equation (2) § 4 becomes (I) N=sing (Ap2cosO—G'µ)+gMh sin (a -0). Suppose, for instance, that OC is parallel to the earth's axis, and that the frame is fixed in the See also:

meridian; then a is the co-See also:

LATITUDE (Lat. latitudo, latus, broad)

latitude, andµ is the angular velocity of the earth, the square of which may be neglected; so that, putting N = o, a—0 = E, (2) gMh sin E—G'µ sin (a—E) =o, (3) tan E _- G'µ sin a G'µ . gMh+G'µ cos a gMhsm a. This is the theory of Gilbert's barogyroscope, described in Appell's Mecanique rationnelle, ii. 387: it consists essentially of a rapidly rotated fly-wheel, mounted on See also:

KNIFE (0. E. cuff, a word appearing in different forms in many Teutonic languages, cf. Du. knijf, Ger. Kneif, a shoe-maker's knife, Swed. knif; the ultimate origin is unknown; Skeat finds the origin in the root of " nip," formerly " knip "; Fr. canif is a

knife-edges by an axis The See also:

hero- See also:

west; to its axis of rotation and pointing See also:

east and west; spun with considerable angular momentum G'. gyroscope. and provided with a slight preponderance Mh, it should tilt to an angle E with the vertical, and thus demonstrate experimentally the rotation of the earth. In Foucault's gyroscope (Comptes rendus, 1852; Perry, p. I05) the preponderance is made zero, and the axis points to Foucault's the pole, .when free to move in the meridian. Generally, if constrained to move in any other plane, gyroscope. the axis seeks the position nearest to the polar axis, like a dipping See also:

NEEDLE (O. Eng. ncedl; the word appears in various forms in Teutonic languages, Ger. Nadel, Dutch naal, the root being ne-, to sew, cf. Ger. nahen, and probably Lat. nere, to spin, Gr. vi)ats, spinning)

needle with respect to the magnetic pole. (A gyrostatic working model of the magnetic See also:

COMPASS (Fr. corn pas, ultimately from Lat. cum, with, and passus, step)

compass, by See also:

Sir W. Thomson. See also:

BRITISH

British Association See also:

REPORT (O.Fr. report or raport, modern rapport, from O.Fr. reporter, mod. rap porter, Lat. reportare, to bring back, in poetical use only, of bringing back an account, news, &c.)

Report, See also:

MONTREAL

Montreal, 1884. A.

S. Chessin, St See also:

Louis See also:

Academy of See also:

SCIENCE (Lat. scientia, from scire; to learn, know)

Science, See also:

JANUARY

January 1902.) A spinning top with a polished upper plane surface will provide an artificial horizon at sea, when the real horizon is obscured. The first instrument of this See also:

KIND (O. E. ge-cynde, from the same root as is seen in " kin," supra)

kind was constructed by Serson, and is described in the See also:

GENTLEMAN (from Lat. gentilis, " belonging to a race or gens," and " man "; Fr. gentilhomme, Span. gentil hombre, Ital. gentil huomo, in its original and strict signification, a term denoting a man of good family, the Lat. generosus (its invariable transl

Gentleman's See also:

MAGAZINE

Magazine, Gyroscopic vol. See also:

XXIV

xxiv., 1754; also by Segner in his Specimen theoriae horizon. turbinum (Halae, 1755). The inventor was sent to sea by the See also:

ADMIRALTY, HIGH COURT OF

Admiralty to test his instrument, but he was lost in the See also:

WRECK

wreck of the " Victory," 1744. A copy of the Serson top, from the royal collection, is now in the Museum of See also:

King's See also:

COLLEGE (Collegium)

College, See also:

LONDON

London. See also:

TROUGHTON, EDWARD (1753-1835)

Troughton's Nautical Top (1819) is intended for the same purpose. The instrument is in favour with See also:

French navigators, perfected by OC' = G', and C'K = Aµ sin 0, See also:

Con-strained motion of the gyro-scope. scopic action of railway wheels. Admiral Fleuriais (fig. 9) ; but it must be noticed that the horizon given by the top is inclined to the true horizon at the angle E given by equation (3) above; and if µl is the precessional angular velocity as given by (3) § 4, and T =22r/µ, its period in seconds, (4) tan E =µ cos See also:

lat =T cos lat or E =T cos lat 8640o 82r if E is expressed in minutes, taking µ=22r/864o0; thus making the true latitude E nautical See also:

MILES, NELSON APPLETON (1839— )

miles to the See also:

south of that given by the top (Revue maritime, 1890; See also:

Corn pies rendus, 1896). This can be seen by elementary See also:

CONSIDERATION (from Lat. considerare, to look at closely, examine, generally taken to be from con-, and the base seen in sidus, sideris, a star, the word being supposed to be originally an astrological or astronomical term)

consideration of the theory above, for the velocity of the vector OC of the top due to the rotation of the earth is (5) IA .0C' cos lat =--gMh sin E =al .0C' sin E, sinE=µcoslat, ETc8Flat in which 8r can be replaced by 25, in practice; so that the Fleuriais gyroscopic horizon is an See also:

ILLUSTRATION

illustration of the See also:

INFLUENCE (Late Lat. influentia, from influere, to flow in)

influence of the rotation of the earth and of the need for its See also:

ALLOWANCE (from "allow,' derived through 0. Fr. alouer from the two Lat. origins adlaudare, to praise, and allocare, to assign a place; so that the English word combined the general idea of "assigning with approval")

allowance. 7.

In the ordinary treatment of the general theory of the gyro- scope, the motion is referred to two sets of rectangular axes; the one Ox, Oy, Oz fixed in space, with Oz vertically up- ward; and the other OX, OY, OZ fixed in the rotating wheel with OZ in the, axis of figure OC. x The relative position of the two sets of axes is given by means of See also:

EULER, LEONHARD (1707-1783)

Euler's unsymmetrical angles 0, 4', f', such that the successive turning of the axes Ox, Oy, Oz through the angles (i.) ¢ about Oz, (ii.) 0 about OE, (iii.) 49 about OZ, brings them into coincidence with OX, OY, OZ, as shown in fig. 11, representing the See also:

CONCAVE

concave side of a spherical surface. The component angular velocities about OD, OE, OZ are (1) 4' sin 8, 9, 4,+4 cos 0; so that, denoting the components about OX, OY, OZ by P, Q, R, (2) P= 9 cos 41-4. sin 9 sin ¢, Q= -0 sin4++Gsin0cos4, R= ++4'cos0. Consider, for instance, the motion of a fly-wheel of preponderance Mh, and equatoreal moment of inertia A, of which the axis OC is held in a light ring ZCX at a constant angle y with OZ, while OZ is held by another ring zZ, which constrains it to move round the vertical Oz at a constant inclination 0 with constant angular velocity µ, so that (3) 8=o, >t =a; (4) P =µ sin 0 sin ¢, Q =µ sin 9 cos ¢, R +µ cos B. With CXF a quadrant, the components of angular velocity and momentum about OF, OY, are (5) P cos 7-R sin y, Q, and A(P cos 7-R sin 7), AQ, so that, denoting the components of angular momentum of the fly-wheel about OC, OX, OY, OZ by K or G', h2, h3, (6) h1= A(P cos y-R sin -y) cos y+K sin 7, (7) h2= AQ, (8) ha = -A(P cos -y-R sin -y) sin 7+K cos 7; and the dynamical equation (9) dl -h1Q+h2P=N, with K constant, and with preponderance downward (to) N =gMh cos zY sin y=gMh sin 7 sin 0 cos 41, reduces to The first integral of (i I) gives 2 (17) -2A d~) sin 7+ IAA' sin y sine 0 sin' 4, -Aµ2 cos y sin 8 cos 8 sin 4+(Kµ+gMh) sin B sin ¢-H =o, and putting tan ('IZr+14,) =z, this reduces to (r8) dt=n'slZ where Z is a quadratic in z2, so that z is a Jacobian elliptic See also:

FUNCTION

function of t, and we have (19) tan ('-,1Zr+14') =C(tn, dn, nc, or cn)nt, according as the ring ZC performs See also:

COMPLETE

complete revolutions, or oscillates about a sidelong position of equilibrium, or oscillates about the See also:

STABLE

stable position of equilibrium c/9--= =Ir. Suppose Oz is parallel to the earth's axis, and µ is the diurnal rotation, the square of which may be neglected, then if Gilbert's barogyroscope of § 6 has the knife-edges turned in See also:

AZIMUTH (from the Arabic)

azimuth to make an angle 13 with E. and W., so that OZ lies in the horizon at an angle E.$.N., we must put 7=22r, cos..e=sin a sin 93; and putting 4, = iir-6+E, where S denotes the angle between Zz and the vertical plane ZT through the See also:

ZENITH (from the Arabic)

zenith i', (20) sin 9 cos S =cos a, sin 0 sin b =sin a cos ,B; so that equations (9) and (To) for relative equilibrium reduce to (21) gMh sin E=K~=Kµ sin B cos 4 Kµ sin 8 sin (5-E), and will change (3) § 6 into Kµ sin a cos 0 (22) tan E=gMh+Kµ cos a' a multiplication of (3) § 6 by cos d (Gilbert, Corn pies rendus, 1882). Changing the sign of K or h and E and denoting the revolutions/second of the gyroscope wheel by F, then in the preceding notation, T denoting the period of vibration as a simple pendulum, Kµ sin a cos 1 = F sin a cos /3 (23) - tan E=gMh-Kµ cos a 8640o A/TZC-F cos a' so that the gyroscope would See also:

REVERSE

reverse if it were possible to make F cos a> 8640o A/T2C (Fo.ppl, Miinch. See also:

Bet., 1904). A gyroscopic pendulum is made by the addition to it of a fly-wheel, balanced and mounted, as in Gilbert's barogyroscope, in a ring movable about an axis fixed in the pendulum, in the vertical plane of motion. As the pendulum falls away to an angle 0 with the upward vertical, and the axis of the fly-wheel makes an angle 4 with the vertical plane of motion, the three components of angular momentum are (24) =K cos ¢, h2 =A0+K sin 49, h3 =Aq, where h3 is the component about the axis of the ring and K of the fly-wheel about its axis; and if L, M', N denote the components of the couple of reaction of the ring, L may be ignored, while N is zero, with P=o, Q=9,-K=o, so that (25) M' =h2 =A6+K+ cos 4, (26) o=h3-h18=A+-K9 cos ¢. For the motion of the pendulum, including the fly-wheel, (27) MK29=gMH sin B-M' =gMH sin 0-Ae-K¢ cos ¢.

If B and ¢ remain small, (28) A¢= K9, A‘'K(e-a), (29) (MK2+A)e+(K2/A) (8-a) -gMH0 =0; so that the upright position will be stable if K'>gMHA, or the rotation energy of the wheel greater than ZA/C times the energy acquired by the pendulum in falling between the vertical and horizontal position; and the vibration will synchronize with a simple pendulum of length (30) (M K'+A)/[ (K2/gA) - MH]. This gyroscopic pendulum may be supposed to represent a ship among waves, or a carriage on a monorail, and so affords an explanation of the gyroscopic action essential in the apparatus of Schlick and Brennan. 8. Careful See also:

SCRUTINY (Fr. scrutin, Late Lat. scrulinium, from scrutari, to search or examine thoroughly)

scrutiny shows that the steady motion of a top is not steady absolutely; it reveals a small nutation General superposed, so that a complete investigation requires motion of a return to the equations of unsteady motion, and for the the top. small oscillation to consider them in a penultimate form. In the general motion of the top the vector OH of resultant angular momentum is no longer compelled to See also:

lie in the vertical plane COC' (fig. 4), but since the axis Oh of the gravity couple is always horizontal, H will describe a curve in a fixed horizontal plane through C. The vector OC' of angular momentum about the axis will be constant in length, but vary in direction; and OK will be the component angular momentum in the vertical plane COC', if the planes through C and C' perpendicular to the lines OC and OC' intersect in the See also:

LINE

line KH; and if KH is the component angular momentum perpendicular to the plane COC', the resultant angular momentum OH has the three components OC', C'K, KH, represented in Euler's angles by (I) KH=See also:

ADO (d. 874)

AdO/dt, C'K=A sin ed'/dt, OC'=G'. See also:

DRAWING

Drawing KM vertical and KN parallel to OC', then (2) KM=Ad¢/dt, KN=CR-A cos Bd¢/dt=(C-A)R+Ady/dt so that in the spherical top, with C =A, KN =Ad4,/dt. Buler's coordinate angles. (II) A sin 7+Aµ2 sin y sine 0 sin ¢ cos ¢ +Aµ2 cos y sin 8 cos 8 cos 0-(Kµ+gMh) sin 8 cos 4i =o. The position of relative equilibrium is given by (12) cos 49 =o, and sin =Kµ+gMh-Ap2cosycos0 Aµ2 sin 7 sin 8 For small values of µ the equation becomes (13) Aat~ sin y—(Kµ+gMh) sin B cos 4=0, so that 4= 3 r gives the position of stable equilibrium, and the period of a small oscillation is zarI (A sin y/(Kµ+gMh) sin Oj. In the general case, denoting the periods of vibration about ¢=}r,-iv,and the sidelong position of equilibrium by 22r/(ni, n2, or n$), we shall find (14) (15) (16) ni' = sin B A sin y( gMh+Kµ-Aµ2 cos (7-0) {, sin =A-ii-T-71-gMh-Ka+Ays' cos (7+8)I, n$ =n1 n2/µ sin V.

The velocity of H is in the direction KH perpendicular to the plane COC', and equal to gMh sin 8 or See also:

Ant sin 0, so that if a point in the axis OC' at a distance An2 from 0 is projected on the horizontal plane through C in the point P on CK, the curve described by P, turned forwards through a right angle, will be the See also:

HODOGRAPH (Gr. &56s, a way, and ypagav, to write)

hodograph of H ; this is expressed by (3) where See also:

PEA (Pisum)

peA'i is the vector CH; and so the curve described by P and the motion of the axis of the top is derived from the curve described by H by a differentiation. Resolving the velocity of H in the direction CH, (4) d.CH/dt=An2 sine sin KCH=An2sin8 KH/CH, (5) d. ICH2/dt=A2n2sin See also:

ODE (Gr. uiSri, from aei&ew, to sing)

Ode/dt. and integrating (6) 1CH2 = A2n2(E-cos 0), (7) 1OH' = A2n2(F-cos 0), (8) 1C'H2 = A2n2(D-cos 0), where D, E, F are constants, connected by (9) F = E +G2/2A2n2 = D +G'2/2A2n2. Then (I o) KH2 = OH2-0K2, (II) OK2sin2 B=CC'2=G2-2GG' cos 8+G'2, (12) A2sin26(de/dt)2=2A'n'(F-cos6)sin' 0-G2+2GG'cos 0-G'2; and putting cos 0 -z, (13) ( ) 2=2n2(F-z) (1-z') - (G2-2GG'z+G'2)/See also:

A2(z)

A2 =2n2(E-z) (1-z2) - (G'- Gz)2/A2 =2n2(D-z) (1-z2) - (G - G'z)2/A2. =2n2 Z suppose. Denoting the roots of Z o by z2, z3, we shall have them arranged in the See also:

order (14) z1>I>z2>z>z3>—I. (15) (dz/dt)2=2n2(zi-z) (z2-z) (z-z3). (16) nt= f:2dzld(2Z), an elliptic integral of the first kind, which with (17) m=n&,z1-z3 sz2-za V 2 ' zi — z2' can be expressed, when normalized by the factor '((Z1-z3)/2, by the inverse elliptic function in the form (18) mt = (z Al (zi-z3)dz J zed [4 (z1 z) (zz-z) (z–za)] sn i V z2-z3-cn i ~z2-Z3-dn i -z3 (19) Z— 23=(Z3—z3)sn2mt, z1-z=(z2-za)cn'mt, zi – z = (zi - za)dn2mt (20) z = z2sn2ml+zacn2mt. Interpreted dynamically, the axis of the top keeps See also:

time with the beats of a simple pendulum of length (21) L =l/i suspended from a point at a height 1(zi+z3)l above 0, in such a manner that a point on the pedulum at a distance (22) 3 (zi-z3)l = 12/L from the point of suspension moves so as to be always at the same level as the centre of oscillation of the top. The polar co-ordinates of H are denoted by p, le in the horizontal plane through C;. and, resolving the velocity of H perpendicular to CH, (23) (24) (25) an elliptic integral, of the third kind, with pole at z=E; and then (26) -,'=KCH=tan-iKH/CH iA sin Bdb/dt _ i d (2Z) =tan- G'-Gcos9tan- (G'-Gz)See also:

JAn' which determines Otherwise, from the See also:

GEOMETRY

geometry of fig. 4, (27) C'K sin B =OC-OC' cos 0, (28) A sin2 04/dt =-G-G' cos 0, G - G'zdt G- G'dt G +G'dt (29) '=) I–Z2 A=1) 1–z A+5 ('G +G' the sum of two elliptic integrals of the third kind, with pole at z = =1 ; and the relation in (25) (26) shows the addition of these two integrals into a single integral, with pole at z=E.

The motion of a See also:

SPHERE (Gr. vclsa?pa, a ball or globe)

sphere, rolling and spinning in the interior of a spherical bowl, or on the top of a sphere, is found to be of the same See also:

CHARACTER (Gr. xapareri7p, from xap&crew, to scratch)

character as the motion of the axis of a spinning top about a fixed point. An1sin = iAn2sin 8e4' = dt (pe i) pdi /dt =An2 sin 8 cos KCH. p2dvr/dt = An' sin 8. CK =An2(G' -G cos 8) =if G'-Gz dt ( (G'-Gz)/2An dz E-z A ) z3 E-z (2Z)' while from (33-39) (42) (41) 773' i=zk2(za-z)=1)2--Pa', See also:

Pie < o <P3' <P2 <P32, F = zi +z2+z3, X-2µ+v=k2z, A-v=k2, A-µ I+z li-v I-Z A-v- 2 ' A-v=—2 with z=cos 6, 0 denoting the angle between the generating lines through H ; and with OC =5, OC' =&', the length k has been chosen so that in the preceding equations (4o) 6/k = G/2An, S'/k =G'/2An; and 6, k may replace G, G', 2An; then 2Z 1 d9 24KH2 I-z2 n2\di) k' ' The curve described by H can be identified as a Poinsot herpolhode, that is, the curve traced out by rolling a See also:

QUADRIC

quadric surface with centre fixed at 0 on the horizontal plane through C ; and Darboux has shown also that a deformable hyperboloid made of the generating lines, with 0 and H at opposite ends of a See also:

DIAMETER (from the Cr. &a, through, µErpov, measure)

diameter and one generator fixed in OC, can be moved so as to describe the curve H ; the tangent plane of the hyperboloid at H being normal to the curve of H ; and then the other generator through 0 will coincide in the movement with OC', the axis of the top; thus the Poinsot herpolhode curve H is also the trace made by rolling a line of curvature on an See also:

ELLIPSOID

ellipsoid confocal to the hyperboloid of one See also:

SHEET

sheet, on the plane through C. See also:

Kirchhoff's Kinetic Analogue asserts also that the curve of H is the See also:

PROJECTION

projection of a tortuous elastica, and that the spherical curve of C' is a hodograph of the elastica described with constant velocity. See also:

Writing the equation of the See also:

FOCAL

focal See also:

ELLIPSE (adapted from Gr. EXkei 'tc, a deficiency, iXXeirely, to fall behind)

ellipse of the Darboux hyperboloid through H, enlarged to See also:

double scale so that 0 is the centre, (30) x2/a2+y2/132+z2jo = 1, with a2+a, $2+X, denoting the squares of the semiaxes of a con-focal ellipsoid, and A changed into µ and v for a confocal hyperboloid of one sheet and of two sheets. (31) X>o>,u>-,82>v>-a', then in the deformation of the hyperboloid, X and v remain constant at H; and utilizing the theorems of solid geometry on confocal quadrics, the magnitudes may be chosen so that (32) a'+A+Q2+k+v=OII2= 1k2(F-z) =- p' +00, a2 +!x = z k2 (zl – z) = P2 _ (33) (34) 132+µ=sk2(z2–z)=P2—P22, (35) (36) (37) (38) (39) 2Z _4(a2+u)(/32+01L – z2 k2(µ– X) (p.- v)' which verifies that KH is the perpendicular from 0 on the tangent plane of the hyperboloid at H, and so proves Darboux's theorem. Planes through 0 perpendicular to the generating lines cut off a constant length HQ=6, HQ'=6', so the line of curvature described by H in the deformation of the hyperboloid, the intersection of the fixed confocal ellipsoid A and hyperboloid of two sheets v, rolls on a horizontal plane through C and at the same time on a plane through C' perpendicular to OC'. Produce the generating line HQ to meet the See also:

PRINCIPAL

principal planes of the confocal system in V, T, P; these will also be fixed points on the generator; and putting (43) (HV, HT, HP,)/HQ=D/(A, B, C,), then (44) See also:

AXE (0. Eng. aex; a word common, in different forms, in the Teutonic languages, and akin to the Greek 41v17; the New English Dictionary prefers the spelling " ax ")

Axe+By'+Cz2 = D6' is a quadric surface with the squares of the semiaxes given by HV. HQ, HT. HQ, HP. HQ, and with HQ the normal line at H, and so touching the horizontal plane through C; and the direction cosines of the normal being (45) x/HV, y/HT, z/HP, (46) A2x2+82y2+C2z2 = D262, the line of curvature, called the polhode curve by Poinsot, being the intersection of the quadric surface (44) with the ellipsoid (46).

There is a second surface associated with (44), which rolls on the plane through C', corresponding to the other generating line HQ' through H, so that the same line of curvature rolls on two planes at a constant distance from 0, 6 and 6'; and the motion of the top is made up of the See also:

COMBINATION (Lat. combinare, to combine)

combination. This completes the statement of Jacobi's theorem (Werke, ii. 48o) that the motion of a top can be resolved into two movements of a body under no force. Conversely, starting with Poinsot's polhode and herpolhode given in (44) (46), the normal plane is See also:

DRAWN

drawn at H, cutting the principal axes of the rolling quadric in X, Y, Z; and then (47) a2+u=x.OX, lg2+tl=y.OY, =z.OZ, this determines the deformable hyperboloid of which one generator through H is a normal to the plane through C; and the other generator is inclined at an angle 0, the inclination of the axis of the top, while the normal plane or the parallel plane through 0 revolves with angular velocity d,p/dt. The curvature is useful in drawing a curve of H; the diameter of curvature D is given by _d¢2_Ik2sin38 ID_ ik2 (48) D-dp 6-3'cos0' p =KM. KN' The curvature is zero and H passes through a point of See also:

INFLEXION (from Lat. inflectere, to bend)

inflexion when C' comes into the horizontal plane through C; 1G will then be stationary and the curve described by C' will be looped. In a state of steady motion, z oscillates between two limits z2 and z3 which are See also:

close together; so putting z2 =z3 the coefficient of z in Z is (49) 2Zlz3+zza= _I+A GGn 2= _I +(OMcose+ON) (OM+ONcose), OM .ON OW +0N2 OM2+0N2 (5°) 22153OM . ON cos e, z1= 20M . ON ' OM2-20M .ON cos B+ON2 MN2 2 (z,.-z3) _ OM . ON _ OM . ON' With z2 = z3, K =o, K =1r ; and the number of beats per second of the axis is m=n zl-z3 __ MN is ,r ,r 2 ./ (OM .0N) 22r' beating time with a pendulum of length (53) __ l _40M .ON L 2(zl-z3) MN2 1. The wheel making R/2,r revolutions per second, (54) beats/second MN n C MN revolutions/second -,I (OM .ON) = R A' OCC" from (8) (9) § 3; and the apsidal angle is Z,r Ap, n ON 2s/ (OM.ON).1 _ON (55) µm=An m 2'r al(OM.ON) MN , 2"~MN"' and the height of the equivalent conical pendulum X is given by 6 X_ g =n2_OMKC OL (5) i _lµ2 µ2 ON KC OC if OR drawn at right angles to OK cuts KC' in R, and RL is drawn horizontal to cut the vertical CO in L; thus if OC' represents 1 to scale, then OL will represent X.

9. The gyroscope, motion in fig. 4 comes to a stop when the rim of the wheel touches the ground; and to realize the motion when the axis is inclined at a greater angle with the upward vertical, the stalk is pivoted in fig. 8 in a See also:

lug screwed to the axle of a bicycle hub, fastened vertically in a See also:

BRACKET

bracket bolted to a See also:

beam. The wheel can now be spun by hand, and projected in any manner so as to produce a desired gyroscopic motion, undulating, looped, or with cusps if the stalk of the wheel is dropped from See also:

REST (O. Eng. rest, reste, bed, cognate with other Teutonic forms, e.g. Ger. Rast, Riiste, rest, and probably Gothic Rasta, league, i.e. resting or stopping place)

rest. As the principal part of the motion takes place now in the neighbourhood of the lowest position, it is convenient to measure the angle 8 from the downward vertical, and to change the sign of z and G. Equation (18) § 8 must be changed to mt=nt1 '''3z1= ('3s/ (z,-zl)dz Vy2 1, V(4Z) Z = (z-F) (I-z2)-(G2-2GG'z+G'2)/2A2n2 = (z-D) (I-z2)-(G-G'z)2/2A2n2 = (z-E) (I-z2)-(G'-Gz)2/2A2n2 = (z3-z) (z-z2) (z-z,), (3) 1> za>z>z2>-I,D,E>zl, (4) zl+z2+z3 = F = D-G'2/2A2n2 = E-G2/2A2n2, and expressed by the inverse elliptic function (5) Mt = /ELL cn-1 Z22 = 1 z-zt sn I=dn tl J3_Z2 Z3-Z2 z3-zi (6) z =z2sn2mt+z3cn2mt, K2 = (Za z2)/(za-Zi)• Equation (25) and (29) § 8 is changed to W =1 G'-Gz dt l G'-GE dt Gt (7) 2) z-E A-2 J z-E A 2A' (8) = f( G'z-G-dt _ 1 ('G'+G dt - 1 (G'-G dt () J I-z2 A- 2 I-z A 1+z while 4, and w change places in (26). The Jacobian elliptic parameter of the third elliptic integral in (7) can be given by v, where vI E (s3_z )dZ,, 1 +P2 K+(I-f)Ki', ^4z) Z2 where f is a real fraction, (10) (1_f)K'= z2V(z2-zl)dz E (-4Z) ' (11) fK.= I E(za-zi)dz ziJ (4Z) = sn-1 IE—z` = cn-1 Iz2—E =dn_I `Iz3-E Z2-Zj Z2-Z1 za-z1, with respect to the comodulus K'. Then, with z=E, and (12) 2ZE=-1(G' -GE)/An]2, if 1I denotes the apsidal angle of a, and T the time of a single See also:

BEAT (a word common in various forms to the Teutonic languages; it is connected with the similar Romanic words derived from the Late Lat. battere)

beat of'the axle, up or down, II+GT (z3dz 2A j z2 z-E 3,1(2Z)' = if +Kzn f K', in accordance with the theory of the complete elliptic integral of the third kind. Interpreted geometrically on the deformable hyperboloia, flattened in the plane of the focal ellipse, if OQ is the perpendicular from the centre on the tangent HP, AOQ=amfK', and the See also:

ECCENTRIC (from Gr. F,c, out of, and Kivrpov, centre)

eccentric angle of P, measured from the See also:

minor axis, is am(1f)K', the eccentricity of the focal ellipse being the comodulus K'. A point L is taken in QP such that (14) QL/OA=znfK', (15) QV, QT, QP=OA(zs, zc, zd)fK'; and with (16) (17) (18) By choosing for f a simple rational fraction, such as z, I, an algebraical case of motion can be constructed (See also:

ANNALS (Annales, from annus, a year)

Annals See also:

MATHEMATICS (Gr. /saBnµar1Kil, Sc. vOcvn or E7rlQTt'µn; from p iN.La, "learning" or "science ")

Mathematics, 1904). Thus with G'-GE=o, we have E=z, or z2, never z3; f=o or and P is at A or B on the focal ellipse; and then (19) p=G/2A, (20) I(i+pt=tan-12 (27) p (z-E) (21) sin 0 exp (P+pt)i=See also:

ill [(-z2-z3)(z-z/(,)]+J[(z3-z) (z-z2)], 1 +Z2Z3 Z2-'Zg _ G _ C _ G' ZI 22+Z3 ' 2 2An n 2Anz1' or (22) sin 6 exp(''+pt)i =[(-zi-z3) (z-z2)]+ [(z3-z) (z-zl)], +zlz Nis-z3= G _p= G' 22 zl+z3' \ 2 2An n 2Anz2' Thus z2=o in (22) makes G'=o; so that if the stalk is held out horizontally and projected with angular velocity 2p about the vertical axis OC without giving any spin to the wheel, the resulting motion of the stalk is like that of a spherical pendulum, and given by ,h2 (23) sin 0 exp ('+Pt)i =i./ (V-' cos8)+ ,~ I (sin28-2 [cos B) =isinas/(sec acos8)+ l[(seca+coso)(See also:

COSA

cosa-cos8)], if the axis falls in the lowest position to an angle a with the down-ward vertical.

With za = o in (21) and v2 =- cos i3, and changing to the upward vertical measureinent, the motion is given by (24) sin O e'4 =en's/ _cO9(3[A/(1-cos cos 0) +is/ (cos 13 cos8-cos28)], and the axis rises from the horizontal position to a See also:

SERIES (a Latin word from serere, to join)

series of cusps; and the mean precessional motion is the same as in steady motion with the same rotation and the axis horizontal. The See also:

SPECIAL

special case off = z may be stated here; it is found that (25) aexp(to-pt)i= - I(1+x)2(K-x) +i I(1-x)2K+x), (26) p22=a'(K-x2), (27) 4X2sin0exp('G-pt)i=(L-+K-x)\I(-x)2(K+x) +i(L- 1 +K+x) \((1 +x2)(K -x) ' L= z(1-K)+Xp/n, so that p =o and the motion is made algebraical by taking L = 1(1-K). The stereoscopic See also:

DIAGRAM

diagram of fig. 12 drawn by T. I. See also:

DEWAR, SIR JAMES (1842- )

Dewar shows these curves for K= ., , and a (cusps). to. So far the motion of the axis OC' of the top has alone been considered; for the See also:

SPECIFICATION (from Med. Lat. specificatio, specificare, to enumerate or mention in detail)

specification of any point of the body, Euler's third angle 4, must be introduced, representing the angular displacement of the wheel with respect to the stalk. This is given by (1) d~+cos0--=R , d(4'+4,) _ rr C G'+G (2) dt - \1 A) R+ A(1+cos8)' d(4'-0= (1_C,) R+ G'-G dt \ A A(i-cos8)' It will simplify the formulas by cancelling a See also:

SECULAR (Lat. saecularis, of or belonging to an age or generation, saeculum)

secular See also:

TERM

term if we make C =A, and the top is then called a spherical top; OH becomes the axis of instantaneous angular velocity, as well as of resultant angular momentum. When this secular term is restored in the general case, the axis 0I of angular velocity is obtained by producing Q'H to I, making HI A-C HI _A-C (3) Q'H_ C ' Q'I A ' (51) (52) (I) (2) (9) (13) mT=K, m/n= (z3-z,)/2=OA/k, GT_ G k =QHK 2AA 2An OA OA ' II=f+Q OAQHK=2f+~AK. of I; (28) and then the four vector components OC', C'K, KH, HI give a resultant vector 01, representing the angular velocity w, such that (4) OI/Q'I =w/k. The point I is then fixed on the generating line Q'H of the de-formable hyperboloid, and the other generator through I will cut the fixed generator OC of the opposite system in a fixed point 0', such that IO' is of constant length, and may be joined up by a See also:

LINK

link, which constrains I to move on a sphere.

In the spherical top then, (5) 1(0+0 = I+z A, 1(4,—4,) = J G —z 2A depending on the two elliptic integrals of the third kind, with pole at z= = I; and measuring 8 from the downward vertical, their elliptic parameters are: (6) v1 — _ (Z3-(4Z) zl)dz=flK,• I l a, v2=I' =1 21(z3—z))dzK+(1—See also:

F2(X2+Y2)

f2)K'y l(4Z) f1K'= flZ3—z1)dz I 1 J (—4Z) sn 1~zi .1 =cn-1 \I1—zl=dn 1 ~I—z1, ,(9) (I—f2)K'= z /'(z(3--z14Z))dz esn 1 —zl =.- Z2 2 I —zl z2—zl I1 +23 z3-zl•a= cos le exp ( t3=i sin ZB exp 2(—0+4')i, y=i sin ZB exp 2( 0—02, (15) S= cos zB exp Next See also:

KLEIN, JULIUS LEOPOLD (1810–1876)

Klein takes two functions or co-ordinates X and A, defined by x=x+yi__ r+z r—z x—yi' and A the same function of X, Y, Z, so that A See also:

PLAY

play the part of stereographic representations of the same point (x, y, z) or (X, Y, Z) on a sphere of radius r, with respect to poles in which the sphere is intersected by Oz and OZ. These new functions are shown to be connected by the bilinear relation (17) >,=aA+13 yA+saS—fly=I, in accordance with the annexed See also:

SCHEME (Lat. schema, Gr. oxfjya, figure, form, from the root axe, seen in exeiv, to have, hold, to be of such shape, form, &c.)

scheme of transformation of coordinates- 2 2a/3 y2 S2 2yS os2 248afyS)Ry=, , - aS—/3y=I, aS-i-(3y=cos B, aS =~/c(—= while, for the motion of a point on the axis, putting A =o, or oo , (20) X = B/S = i tan 48e91, or X = a/y = —i cot 2BeY2, and (21) a/3 = 2i sin See also:

Bel'2, ay =1i sin See also:

Bed'i, giving orthogonal projections on the planes GKH, CHK; and (22) See also:

aaa d Q=nesu*isin, ineB; ZB the vectorial equation in the plane GKH of the herpolhode of H for a spherical top. When fl and 12 in (9) are rational fractions, these multiplicative elliptic functions can be replaced by algebraical functions, qualified by factors which are exponential functions of the time t; a series of quasi-algebraical cases of motion can thus be constructed, which become purely algebraical- when the exponential factors are can-celled by .a suitable arrangement of the constants. Thus, for example, with f =o, f' =I, 11=1, f2 = z, as in (24) § 9, where P and P' are at A and B on the focal ellipse, we have for the spherical top (23) (I+cos 0) exp (sec f —cos B) Al (cos (3 — cos 0) +i(-sl sec fl I cos /3) I/ cos 8, (24) (I —cos B) exp (4,—1p—g't)i Al (sec/3—cos 0) I (cos )3—cos ©)+i('l sec 13 —/ cos)3)Al cos 8, (25) g,.g'=n,/ (2sec13)tn/(2cos13); and thence a, /3, 7, S can be inferred. The physical constants of a given symmetrical top have been denoted in § 1 by M, h, A, C, and 1, n, T ; to specify a given state of general motion we have G, G' or CR, D, E, or F, which may be called the dynamical constants; or e, v, w, v1, v2, or f , f', f1, f2, the analytical constants; or the geometrical constants, such as a, /3, S, k of a given articulated hyperboloid. There is thus a triply See also:

INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

infinite series of a state of motion ; the choice of a typical state can be made geometrically on the hyperboloid, flattened in the plane of the See also:

LOCAL

local ellipse, of which K is the ratio of the semiaxes a and /3, and am(' —f) K' is the eccentric angle from the minor axis of the point of contact P of the generator HQ, so that two analytical constants are settled thereby; and the point H may be taken arbitrarily on the tangent line PQ, and HQ' is then the other tangent of the focal ellipse; in which case B3 and 92 are the angles between the tangents HQ, HQ', and between the focal distances HS, HS', and k2 will be HS.HS', while HQ, HQ' are S, S'. (7) (8) =dn ' Then if . v' = K+(I —f')K'i is the parameter corresponding to z=D, we find (to) f=f2f'=f2+fi, (II) v=v1+v2, v'=vl—v2. The most symmetrical treatment of the motion of any point fixed in the top will be found in Klein and See also:

SOMMERFELD

Sommerfeld, Theorie des Kreisels, to which the reader is referred for details; four new functions, a, f, y, S, are introduced, defined in terms of Euler's angles, B, 4,, ¢, by (I2) (13) (14) (16) Z EI a2 ay SS a6 +#7 where (I$) = x+yi, n=— x+yi, =—z, E=X+Yi, H= —X+Yi, Z= —Z; and thus the motion in space of any point fixed in the body defined by A is determined completely by means of a, 13, y, S ; and in the case of the symmetrical top these functions are elliptic transcendants, to which Klein has given the name of multiplicative elliptic functions; and (19) As H is moved along the tangent line HQ, a series of states of motion can be determined, and drawn with accuracy. II. Equation (5) § 3 with slight modification will serve with the same notation for the steady rolling motion at a constant inclination e to the vertical of a body of revolution, such as a disk, hoop, wheel, cask, See also:

WINE (Lat. vinum, Gr. oivos)

wine-See also:

glass, See also:

PLATE

plate, dish, bowl, spinning top, gyrostat, or bicycle, on a horizontal plane, or a surface of revolution, as a coin in a conical See also:

LAMP (from Gr. Xag ras, a torch, Xaµaav, to shine)

lamp-shade. The point 0 is now the intersection of the axis GC' with the vertical through the centre B of the horizontal circle described by the centre of gravity, and through the centre M of the horizontal circle described by P, the point of contact (fig. 13).

Collected into a particle at G, the L R body swings round the vertical OB as a conical pendulum, of height AB or GL equal to g/µa=a, and GA would be the direction of the See also:

THREAD (0. Eng. praed, literally, that which is twisted, prawan, to twist, to throw, cf. " throwster," a silk-winder, Ger. drehen, to twist, turn, Du. draad, Ger. Draht, thread, wire)

thread, of tension gM(GA/GL) dynes. The reaction with the plane at P will be an equal parallel force; and its moment round G will provide the couple which causes the velocity of the vector of angular momen- tum appropriate to the steady moment will be gM.Gm dyne-cm. or ergs, if the reaction at P cuts GB in m. Draw GR perpendicular to GK to meet the horizontal AL in R, and draw RQC'K perpendicular to the axis Gz, and KC perpendicular to LG. The velocity of the vector GK of angular momentum is times the horizontal component, and (1) horizontal component /Aµ sin a=KC/KC', so that (2) gM.Gm =Aµ2 sin a(KC/KC'), A KC' (3) _R-C' sin aGm = GQ.Gm. N~ µa The instantaneous axis of rotation of the case of a gyrostat would be OP; drawing GI parallel to OP, and KK' parallel to OG, making tan K'GC' = (A/C) tan IGC't; then if GK represents the resultant angular momentum, K'K will represent the part of it due to the rotation of the fly-wheel. Thus in the figure for the body rolling as a solid, with the fly-wheel clamped, the points m and Q move to the other side of G. The gyrostat may be supposed swung round the vertical at the end of a thread PA fastened at A' where Pm produced cuts the vertical AB, and again at the point where it crosses the axis GO. The discussion of the small oscillation super-posed on the state of steady motion requisite for stability is given in the next See also:

PARAGRAPH

paragraph. General 12. In the theoretical discussion of the general motion Genera rl of a gyrostat rolling on a horizontal plane the safe and a otion of shortest See also:

plan apparently is to write down the most general IY'stat o equations of motion, and afterwards to introduce any romng on a plane. special condition. Drawing through G the centre of gravity any three rectangular axes Gx, Gy, Gz, the notation employed is u, v, w, the components of linear velocity of G; p, q, r, the components of angular velocity about the axes, ht, ha, ha, the components of angular momentum; 01, Os, 0'a, the components of angular velocity of the co-See also:

ORDINATE

ordinate axes; x, y, z, the co-ordinates of the point of contact with the horizontal plane; X, Y, Z, the components of the reaction of the plane; a, (3, -y, the direction cosines of the downward vertical. The geometrical equations, expressing that the point of contact is at rest on the plane, are (I) u—ry+qz=o, (2) v—pz+rx=o, ' (3) w—qx+py=0.

The dynamical equations are (4) du/dt -03v +Oaw = ga+X/M (5) dv/dt—0tw+Osu=gt4+Y/M, (6 See also:

DW(i+at)X587

dw/dt—0su+0iv =gY+Z/M, and (7) dht/dt —Oahs+0sha = yZ —zY, (8) dhs/dt—01ha+0aht =zX —xZ, (9) dhs/dt -0aht+Ochs =xY —yX. In the special case of the gyrostat where the surface is of revolution round Gz, and the body is kinetically symmetrical about Gz, we take Gy horizontal and Gzx through the point of contact so that y=o; and denoting the angle between Gz and the downward vertical by 0 (fig. 13) (Io) a=sin 0, 0=o, -y=cosO. The components of angular momentum are (II) hl=Ap, ha=A , ha=Cr+K, where A, C denote the moment of inertia about Gx, Gz, and K is the angular momentum of a fly-wheel fixed in the interior with its axis parallel to Gz; K is taken as constant during the motion. The axis Gz being fixed in the body, (12) 01=p, =q =—dO/dt, Oa=p cot O. With y=o, (1), (2), (3) reduce to (13) u=—qz, v=pz—rx, w=qx; and, denoting the radius of curvature of the meridian curve of the rolling surface by p, =pcos0d=—gpcos0, dt=—psin0 =gpsin0; du dq dt See also:

dtz —qap sin 0, dt dtx+pqp sin 0+qrp sin 0, dt d x—qap cos O. The dynamical equations (4) ... (9) can now be reduced to (18) dtz—paz cot O+q2(x —p sin 0) +prx cot 0 —g sin 0, (19) M d z dtx—pq(x+z cot 0—p sin 0)•fgrp cos 0, (20) M d x+qa(z—p cos 0) +p2z—prx—g cos 0, (m+ xa) at-x4+pgx(x+z cot0—psin0)—grxpcos0=0, (M+x2) —xz-j—px(x+z cot 0—p sin 0)+rxpcos 0=0. Eliminating Y between (19) and (21) (25) ((+z2) —xzdc—Mpq cot 0+qM —pgz(x+z cot 0 — p sin 0) +qrzp cos 0 = o, (B) —x4+(M+za)(2+ 1pcot0—M +pz(x+z cot 0 —p sin 0) —rzp cos O =o. In the special case of a .gyrostat rolling on the See also:

sharp edge of a circle passing through G, z=o, p=o, (A) and (B) reduce to (26) p (IbC7x2 + Ide )dr = (M1x2 I)dhd0a +C ' dp ha d.psin0=hasin0, (27) dB+p cot 0 =A, do A A (17) —zY =Adt —Apq cot 0+qha, —zX —xZ = Aj +Apacot 0 — pha, xY= at -See also:

cat= —Cgig. Eliminating Y between (19) and (23), (24) (A) (C) (M +x2+z) —pM + (+z)pa cot 0+paxz daq A, +q2p(x cos 0—z sin 0) — prx(x+z cot 0) —g(x cos 0 —z sin 0) =o, and this combined with (A) and (B) will See also:

lead to an equation the integral of which is the equation of energy. 13.

The equations (A) (B) (C) are intractable in this general form; but the restricted case may be considered when the axis moves in steady motion at a constant inclination a to the vertical; and the stability is secured if a small nutation of the axis can be superposed. It is convenient to put p = Sl sin 0, so that 12 is the angular velocity of the plane Gzx about the vertical; (A) (B) (C) become a -82+d a coto=A(M +C).3, a See also:

DIFFERENTIAL

differential equation of a hypergeometric series, of the form of See also:

Legendre's zonal See also:

HARMONIC

harmonic of fractional order n, given by (29) n(n+i)=CMxa/A(Mxa+C). For a sharp point, x=o, p=o, and the previous equations are obtained of a spinning top. The elimination of X and Z between (18) (20) (22), expressed symbolically as (30) (22) —z(18)+x(20) =0, gives (28) (A*) (M +x2) -a - xz sin 8 -12.x(xsin0-2zcos9-psin' 0)+rxpcos0=o, (B*) -x4J+ (M+ Z2) sin e— oo ~7+2c2 (M+z2) cos e \ +ilz sin 0(x-p sin 0) - rzp cos 6=0, (C*) (M+x2+z2) a++q'p(x cos 0-z sin 0) - Nsin 9 +122 (1V +z2) sin 0 cos 0 +122xz sine 0 -12rx(x sin 0+z cos B)-g(x cos B-z sin 0) =0. The steady motion and nutation superposed may be expressed by (1) 0=a+L, sin 0=sin a+L cos a, cos a=cos a-L sin a, it=µ+N, r=R+Q, where L, N, Q are small terms, involving a factor en1~, to See also:

EXPRESS (through the French from the past participle of the Lat. exprimere, to press out, by transference used of representing objects in painting or sculpture, or of thoughts, &c. in words)

express the periodic nature of the nutation; and then if a, c denote the mean value of x, z, at the point of contact (2) x=a+Lp cos a, z=c-Lp sin a, (3) x sin 0+z cos 0=a sin a+c cos a+L(a cos a-c sin a), (4) x cos 0-z sin 0=a cos a-c sin a-L(a sin a+c cos a-p). Substituting these values in (C*) with dq/dt = -d2B/dt2=n2L, and ignoring products of the small terms, such as L2, LN, .. . (C**) (M+a2l-c2) Ln2-(p+N) (C M K+ ~Q) (sin a+L cos a) A -1-(µ2+2µN) (m+c2-ZLpc sin a) (sin a cos a+L cos a) +(µ2+2µN) [ac-Lp(a sin a-c sin a)] (sine a+L sin 2a) -(p+N) (R+Q) (a+Lp cos a) [a sin a+c cos a+L (a cos a-c sin a)] -g(a cos a-c sin a)+gL(a sin a+c cos a-p) =0, which is equivalent to (5) µC M Ksin a-1-µ2 (It' +0) A sin a cos a +µ2 ac sine a-,Ra(a sin a+e cos a)-g(a cos a-c sin a) =o, the condition of steady motion ; and (6) DL+EQ+FN =o, where (7) D= (M+a2+0) n2-, M K cos a- 2µ2pc sins a COS a A. +µ2 (.fq+c2) cos a—µ2p(a sin a-c cos a) sin2a +µSee also:

PAX (Lat. for " peace ")

Pax sin 2a—µRp cos a(a sin a+c cos a) -,uRa(a cos a-c sin a)+g(a sin a+c cos a-p), (8) E=-µm sin a-µa(a sin a+c cos a), F = -C VI+K sin a+2µ (M+C2) Sin a cos a +212ac sine a-Ra(a sin a+e cos a). With the same approximation (A*) and (B*) are equivalent to ( C 1+a2) Q -ac sin a L -µa(a sin a+2C cos a-p sine a) +See also:

Rap cos a=0, (13**) -acQ-}- ( 1+c2) sin aL C R+ K+2 ( -+c2) cos a +µc sin a(a—p sin a)—See also:

REP, REPP, or REPS

Rep cos a=O. The elimination of L, Q, N will lead to an equation for the determination of n2, and r''2 must be positive for the motion to be stable. If b is the radius of the horizontal circle described by G in steady motion round the centre B, (io) b=v/µ=(cP-aR)/µ=c sin a-aR/µ, and drawing GL vertically upward of length X=g/µ2, the height of the equivalent conical pendulum, the steady motion condition may be written (It) (CR+K)µ sin a-µ2 sin a cos a=-gM(a cos a-c sin a) +M (µ2c sin a-, Ra) (a sin a+c cos a) =gM[bX-1(a sin a+C cos a) -a cos a+c sin a] =gM. PT, LG produced cuts the plane in T.

Interpreted dynamically, the left-hand side of this equation represents the velocity of the vector of angular momentum about G, so that the right-hand side represents the moment of the applied force about G, in this case the reaction of the plane, which is parallel to GA, and equal to gM . GA/GL; and so the angle AGL must be less than the angle of friction, or slipping will take place. Spinning upright, with a=0, a=o, we find F=o, Q=o, and (12) -C Df K+2µ(1l+c2)-Rcp=o, A CR+K f A (13) (,,I+c2) n2=µ M µ2 (M+c2 / +µRpc—g(c—p), (14) (A +0) 2n2s (C M R+Rep) 2-g (M+e2) (c-p)• Thus for a top spinning upright on a rounded point, with K=o, the stability requires that (15) R>2k',/[g(c-p)]/(k2+cp), where k, k' are the radii of gyration about the axis Gz, and a perpendicular axis at a distance c from G; this reduces to the preceding case of § 3 (7) when p=o. Generally, with a=o, but ao, the condition (A) and (B) becomes (16) (M +a2) ?= 2µac-Rap, A -act =C M K+Rcp-2µ (g +a') , so that, eliminating Q/L, (17) 2[(M+c2)(M+a2)-a2c2] µ=(M+a2l )(C M K)+MRcp, the condition when a coin or platter is rolling nearly See also:

FLAT (a modification of O. Eng. flet, an obsolete word of Teutonic origin, meaning the ground beneath the feet)

flat on the table. Rolling along in a straight path, with a=ir, c =o, =o, E =o; and (18) N/L = (CR+K)/A, (19) D= (M +a2) n2+g(a-p), F =- CRR Z K -Ra2, N D (M -1-a2) n2+g(a-p) L=—P= (C 2) K' M+a R+M \M +a2) n2 = (CRA+K.) [ +at) R+M ] -g(a-p). with K =o, and rolling with velocity V = Ra, stability V2 a-p lA a-p 2g>2_-> 2C C+1' A Ma= Mae or the body must have acquired velocity greater than attained by rolling down a plane through a vertical height 2(a-p)A/C. On a sharp edge, with p=o, a thin See also:

UNIFORM

uniform disk or a thin ring requires (23) V2/2g> a/6 or a/8. The gyrostat can hold itself upright on the plane without advance when R=o, provided (24) K2/AM-g(a-p) is positive. For the stability of the monorail carriage of § 5 (6), ignoring the rotary inertia of the wheels by putting C=o, and replacing K by G' the theory above would require (25) A (aV+A) >gh. For further theory and experiments consult See also:

Routh, Advanced Rigid Dynamics, See also:

CHAP

chap. v., and Thomson and Tait, Natural Philosophy, § 345; also Bourlet, Traite des bicycles (analysed in Appell, Mecanique rationnelle, ii. 297, and Carvallo, See also:

Journal de l'ecole poly-technique, 190o) ; Whipple, Quarterly Journal of Mathematics, vol. See also:

xxx., for mathematical theories of the bicycle, and other bodies. 14.

Lord Kelvin has studied theoretically and experimentally the vibration of a chain of stretched gyrostats chain. (Prot. London Math. Soc., 1875; J. Perry, Spinning Tops, for a diagram). Suppose each gyrostat to be equivalent dynamically to a fly-wheel of axial length 2a, and that each connecting link is a light See also:

CORD (derived through the Fr. corde, from the Lat. chorda, Gr. xop(5rt, the string of a musical instrument)

cord or See also:

STEEL, FLORA ANNIE (1847– )

steel See also:

WIRE (A.S. wir, a wire; cf. Swed. vire, to twist, M.H.G. wiere, a gold ornament, Lat. viriae, armlets, ultimately from the root wi, to twist, bind)

wire of length 21, stretched to a tension T. Denote by x, y the components of the slight displacement from the central straight line of the centre of a fly-wheel; and let p, q, t denote the direction cosines of the axis of a fly-wheel, and r, s, 1 the direction cosines of a link, distinguishing the different bodies by a suffix. Then with the previous notation and to the order of approximation required, (1) (2) (9) (A**) (20) (21) Thus requires (22) Oi =-dq/dt, 02 =dp/dt, h1= A01, h2 =A92, h3=K, to be employed in the dynamical equations (3) dtl-Bahl+B2h2=L, .. in which 02h1 and 83h2 can be omitted. For the kth fly-wheel (4) -Aqk+Kpk=Ta(gk -Si) I T¢(qk-sk+i), (5) Apk+Kqk=-Ta(pk-rk)-Ta(pk-rk+i); and for the motion of See also:

TRANSLATION

translation (6) Mil, =T(rk+1-rk), M9s=T(Sk+l-Sk); while the geometrical relations are (7) xk+1-xk = a(pk+l +pk) +21rk+i, (8) yk+l-yk=a(gk+l+qk)+21Sk+.1. Putting (9) x+yi= w, p+qi=n- r+si= these three pairs of equations may be replaced by the three equations (Io) Au'Yk—Kr^Jki+2Tawk—Ta(Qk}1+Ok) =0, (II) Mk—T(°k+l—Qk) =0, (12) wk+1—wk—a(ak+l+ask)—21°k+1=o. For a vibration of circular polarization assume a See also:

SOLUTION (from Lat. solvere, to loosen, dissolve)

solution (13) wk, k, °k= (L, P, Q) exp (nt+kc)i, so that c/n is the time-lag between the vibration of one fly-wheel and the next; and the wave velocity is (14) U=--2 (a+1)n/c.

Then (15) P(—An2+Kn+2Ta)—QTa(e`*+I) =0, (16) —LMn2 QT(eL1 I) =o, (17) L(e'1—1)—Pa(e"+I)—2Qle°ti =o, leading, on elimination of L, P, Q, to (18) cos c = (2 Ta+ Kn—An2) (1—Mn2l/T)—Mna2 2Ta+Kn—An2+Mna2 Mn' 2Ta(a+l)+Knl—Anil (19) 2 sin2zc= T 2Ta+Kn—An2+NIn2a2- With K=o, A=o, this reduces to See also:

LAGRANGE, JOSEPH LOUIS (1736-1813)

Lagrange's condition in the vibration of a See also:

STRING

string of beads. Putting (20) p=M/2(a+l), the See also:

mass per unit length of the chain, (21) K =K/2 (a+1), the gyrostatic angular momentum per unit length, (22) a = A /2 (a+l), the transverse moment of inertia per unit length, = (a+l)2n2p Ta+Knl—an2l (a+l)n 2 TTa+Kn(a+1)—an2(a+1)+pn2a2(a+l)' (25) I sin (a+l)n/U _T T+(Kn—an2) (1+l/a)+pn2a(a+1) p T+(Kn—an2)1/a In a continuous chain of such gyrostatic links, with a and 1 infinitesimal, (26) U2'-- T 4 1 + Kn-an2 p ( T +(Kn—an2)l/a for the vibration of helical nature like circular polarization. Changing the sign of n for circular polarization in the opposite direction (2/) - T Kn+an2 U'2 — I- P T—(Kn+an )l/a In this way a mechanical model is obtained of the action of a magnetized See also:

MEDIUM

medium on polarized light, K representing the equivalent of the magnetic field, while a may be ignored as insensible (J. Larmor, Proc. Lond. Math. Soc., 1890; See also:

AETHER, or ETHER (Gr. deli p, probably from caeca, I burn, though Plato in his Cratylus (410 B)

Aether and See also:

MATTER

Matter, Appendix E). We See also:

NOTICE

notice that U2 in (26) can be positive, and the gyrostatic chain stable, even when T is negative, and the chain is supporting a thrust, provided en is large enough, and the thrust does not exceed (28) (Kn—an' ) (1 +1/a) ; while U'2 in (27) will not be positive and the straight chain will be unstable unless the tension exceeds (29) (Kn+an2) (I +1/a). 15. Gyrostat suspended by a Thread.—In the discussion of the small vibration of a single gyrostat fly-wheel about the vertical position when suspended by a single thread of length 21=b, the suffix k can be omitted in the preceding equations of § 14, and we can write (1) A—K&i+Tack—Ta° = o, (2) See also:

MiD+T°=o, with T=gM, (3) w—aw—b° =o. Assuming a periodic solution of these equations (4) w, = (L, P, Q) exp nti, and eliminating L, P, Q, we obtain (5) (—An' +Kn+gMa) (g—n2b)—gMn2a2=o, and the frequency of a vibration in double beats per second is n/27r, where n is a See also:

root of this quartic equation. For upright spinning on a smooth horizontal plane, take b= oo and change the sign of a, then (6) An2—Kn+gMa =o, so that the stability .requires (7) K2>4gAMa.

Here A denotes the moment of inertia about a diametral axis through the centre of gravity; when the point of the fly-wheel is held in a small smooth See also:

CUP (in O.E. cuppe; generally taken to be from Late Lat. cuppa, a variant of Lat. cupa, a cask, cf. Gr. KuaeXXov)

cup, b=o,and the condition becomes (8) (A+;tIa2)n2—Kn+gMa = o, requiring for stability, as before in § 3, (9) K2> 4g(A+Ma2)Ma.. For upright spinning inside a spherical surface of radius b, the sign of a must be changed to obtain the condition at the lowest point, as in the gyroscopic horizon of Fleuriais. For a gyrostat spinning upright on the See also:

SUMMIT

summit of a sphere of radius b, the signs of a and b must be changed in (5), or else the sign of g, which amounts to the same thing. Denoting the components of horizontal displacement of the point of the fly-wheel by E, 77, then (10) br=E, bs=77, b, =i;++1i=a (suppose), (I1) w=atl+X. If the point is forced to take the motion (;, 77, i') by components of force X, Y, Z, the equations of motion become (12) —Aq+K))= Ya—Zaq, (13) Ap+K4= — Xa+Zap, (14) Mib=X +Yi, M(1"—g) =Z; so that (15) A$—Kti+gMaw+See also:

MAZE

Maze=Maz, or (16) (A+Ma2)w—Ktsi+gMa +MaX=Ma. Thus if the point of the gyrostat is made to take the periodic motion given by X=R exp nti, -=o, the forced vibration of the axis is given by vI= P exp nti, where (17) P{—(A+Ma2)n2+Kn+gMa}—RMn2a =o; and so the effect may be investigated on the Fleuriais gyroscopic horizon of the motion of the ship. Suppose the motion X is due to the suspension of the gyrostat from a point on the axis of a second gyrostat suspended from a fixed point. Distinguishing the second gyrostat by a suffix, then X = bn',, if b denotes the distance between the points of suspension of the two gyrostats; and the motion of the second gyrostat influenced by the reaction of the first, is given by (18) . (Ai+M1h12)C8i1-Klsli = —g(Mihi+Mb)ti—b(X+Yi) - —g(M1hi+Mb)i—Mb(a6+X); so that, in the small vibration, (19) b —(A1+Mih12)n2+See also:

KIN (0. E. cyn, a word represented in nearly all Teutonic languages, cf. Du. kunne, Dan. and Swed. kon, Goth kuni, tribe; the Teutonic base is kunya; the equivalent Aryan root gan- to beget, produce, is seen in Gr. 'yEvos, Lat. genus, cf. "kind")

Kin+g(Mrhi+Mb) =Mn2b(aP+R), (20) R{—(Al +MIh12+Mb2)n2+Kin+g(Mihl+Mb)}—PMn2ab2=o. Eliminating the ratio of P to R, we obtain (21) {—(.A+Ma2)n2+Kn+gMa} X {—(A1+M1h12+Mb2)n2+Kin+g(Mihl+Mb)}—M2n4a2b2=o, a quartic for n, giving the frequency n/27r of a fundamental vibration. Change the sign of g for the case of the gyrostats spinning upright, one on the top of the other, and so realize the gyrostat on the top of a gyrostat described by See also:

Maxwell. In the gyrostatic chain of § 14, the tension T may change to a limited pressure, and U2 may still be positive, and the motion stable; and so a motion is realized of a number of spinning tops, superposed in a See also:

COLUMN (Lat. columna)

column.

16. The Flexure Joint.—In Lord Kelvin's experiment the gyrostats are joined up by equal light rods and short lengths of elastic wire with rigid See also:

ATTACHMENT

attachment to the See also:

rod and case of a gyrostat, so as to keep the system still, and free from entanglement and twisting due to See also:

PIVOT (Fr. pivot; probably connected with Ital. pivolo, peg, pin, diminutive of piva, pipa, pipe)

pivot friction of the fly-wheels. When this gyrostatic chain is made to revolve with angular velocity n in relative equilibrium as a plane See also:

POLYGON (Gr. rroXus, many, and ywvia, an angle)

polygon passing through Oz the axis of rotation, each gyrostatic case moves as if its axis produced was attached to Oz by a flexure joint. The instantaneous axis of resultant angular velocity bisects the angle ir--0, if the axis of the case makes an angle 0 with Oz, and, the components of angular velocity being n about Oz, and —n about the axis, the resultant angular velocity is 2n cos 1.(ir--0) =2n sin 20; and the components of this angular velocity are (I) -2n sin 10 sin 20= —n(1—cos 0), along the axis, and (2) -211 sin 1-0 cos 10 = —n sin 0, perpendicular to the axis of the case. The flexure joint behaves like a pair of equal See also:

BEVEL (from an O. Fr. word, cf. mod. biveau, a joiner's instrument)

bevel wheels engaging. The component angular momentum in the direction Ox is therefore (3) L= —An sin 0 cos 0 — Cn(1—cos 0) sin 0+K sin 0, and Ln is therefore the couple acting on the gyrostat. If a denotes the angle which a connecting link makes with Oz, and T denotes the constant component of the tension of a link parallel to Oz, the couple acting is (4) Ta cos Ok(tan ak+i+tan ak)—2Ta sin 0k, which is to be equated to Ln, so that (5) — An2sin 0k cos 0k—Cn(1—cos 0k) sin Ok+Kn sin 0k —Ta cos Ok(tan ak+l+tan ak) +2Ta sin 0k=0. In addition (6) Mn2xk+T(tan akil—tan ak) =0, with the geometrical relation (7) xk+i—xk—a(sin Ok+l+ sin 0k)—2l sin ak+1=0. When the polygon is nearly coincident with Oz, these equations can be replaced by (23) c=(a+1)n/U, equation (19) can be written (24 sin (a+1)n/U}2 (8) (—An2+Kn+2Ta)Bk—Ta(ak+l+ak) =0, (9) Mn2xk+T(ak+l — ak) = o, (1o) xk}1—xk—a(Bk+1+9k) -2lak=o, and the rest of the solution proceeds as before in § 14, putting Of) xk, Bk, ak = (L, P, Q) exp cki. A See also:

HALF

half wave length of the curve of gyrostats is covered when ck=7, so that irk is the number of gyrostats in a half wave, which is therefore of wave length 2sr(a-j-l)/c. A plane polarized wave is given when exp cki is replaced by exp (nt+ck)i, and a wave circularly polarized when w, as, u of § 14 replace this x, 0, a. Gyroscopic Pendulum.—The elastic flexure joint is useful for supporting a rod, carrying a fly-wheel, like a gyroscopic pendulum.

Expressed by. Euler's angles, 0, 0, ¢, the. kinetic energy is (12) T=IA(92+sin2B>%i2)+iC'(1—cos9)Z'2+IC(¢+l. cos9)2, where A refers to rod and gyroscope about the transverse axis at the point of support, C' refers to rod about its axis of length, and C refers to the revolving fly-wheel. The elimination of 'G between the equation of conservation of angular momentum about the vertical, viz. (13) A sin291(r—C'(1 —cosO) cos9,G+C(¢+¢cos B) cos 9=G, a constant, and the equation of energy, viz. (14) T—gMh cos0=H, a constant, with 0 measured from the downward vertical, and (15) ('+iG cos 9=R, a constant, will lead to an equation for dB/ di, or dz/dt, in terms of cos 9 or z, the integral of which is of hyper-elliptic character, except when A=C'. In the suspension of fig. 8, the, motion given by is suppressed in the stalk, and for the fly-wheel ¢ gives the rubbing angular velocity of the wheel on the stalk; the equations are now (16) T=IA(92+sin2 BP) +IC' cos' 91y2+2CR2=H+gMlt cos 0, (17) A sin201G+C' cos' 0#+CR cos B=G, and the motion is again of hyperelliptic character, except when A = C', or C' =o. To realize a motion given completely by the elliptic function, the suspension of the stalk must be made by a smooth See also:

ball and socket, or else a See also:

HOOKE, ROBERT (1635–1703)

Hooke universal joint. Finally, there is the case of the general motion of a top with a spherical rounded point on a smooth plane, in which the centre of gravity may be supposed to rise and fall in a vertical line. Here (18) T= (A+Mh2sin29)92+ Asin29>G2+2CR2=H—gMh cos 0, with 0 measured from the upward vertical, and (19) ' A sin24+CR cos9=G, where A now refers to a transverse axis through the centre of gravity. The elimination of'Lleads to an equation for z, = cos 0, of the form ((dzl12 g Z _ g(zl —z) (zz—z) (za — z) (2O) ldtJ 2S I—z2+A/Mlt'—2h (z,—z) (z—zb) with the arrangement (21) z1, zt>/>z2>z>z3> —/>zc; so that the motion is hyperelliptic.

End of Article: GYROSCOPE AND GYROSTAT

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