ADVISORY COMMITTEE FOR AEEO?(AUTICS. REPORTS AND MEMORANDA, No. 146. #^ REPORT ON GYROSCOPIC THEORY. BY Sir G. GKEENHILL. LONDON : I'lUNTElJ [TNDEIt THE ATITIIOEITY OF HIS MAJESTt'S STATIONERY OEEroj') By DARLING and SON, Limitkd, Bacon Strjoet, E. To be puroliasecl , either directly or througii auy Bockseller, froni W'VMAN AND SONS, LmiTKD, 29, Bkeams Bu]L])i\(_;s, Fetteu Lane, E.G., and 54, St'. Mahy Street, Carhiff; or II. M. STATIONERY OFFICE (Scottish Branch), 23, Foirrri Street, EDiMiauuii; or E. PONSONBY, Limited, 116, Graeton Street, Dublin; or from the Agencies in the ]5j'itish Colonies and Dependencies, the United States of America, tlie (Joiitinent (d' Europe and Abroad of 'V. FISHER IINWIN, London, W.C. 1914. f'rice Ten ShilJuu/s. BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF fletirs W, Sage 189X 6:.'3c>33iOg, '^t.y^Xs. arY778 ^°'"*" ""'^^''y Library '^ »iii? ]ili »S.1,„SY,r°.5pop|c theor olin,anx 3 1924 032 182 200 The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924032182200 ADVISOET COMMITTEE FOE AEEONAUTICS. REPORTS AND MEMORANDA, No. 146. K E P O R T ON GYROSCOPIC THEORY. BY Sir G. green hill. LONDON : PEINTED UNDEE THE AUTHOEITT OF HIS MAJESTY'S STATIONEET OFFICE By DAELING and SON, Limited, Bacon Street, E. To be purchased, either directly or through any Bookseller, from WYMAN AND SONS, Limited, 29, Breams Buildings, Fetter Lane, E.G., and 54, St. Mary Street, Cardiff; or H.M. STATIONEEY OFFICE (Scottish Branch), 23, Forth Street, Edinburgh; or E. PONSONBY, Limited, 116, Grafton Street, Dublin; or from the Agencies in the British Colonies and Dependencies, the United States of America, the Continent of Eul'ope and Abroad of T. FISHEE UNWIN, London, W.C. 1914. Price Ten Shillings. E.V. Gyroscopic Theory has been explained in an elementary manner in Perry's Spinning Tops, and the mathematical treatment extended in Crabtree's Spinning Top and Gyroscopic Motion ; but for the full analytical development the Theorie des Kreisels must be studied of Klein and Sommerfeld, where no mathematical difficulty is passed over or ignored. The dynamical theory begins with Euler, 1748, and is carried on by Segner, Lagrange, Poisson, Poinsot, Pinseux, Kelvin, Routh, Klein, and others to recent times. The present Report is intended to have the same scope as the Kreisel-l'heorie, and to be consulted for reference of the mathematical formulas required for a practical problem as it arises, where the numerical data can be assigned, and an answer is wanted without delay ; even where Exact Theory is not yet prepared, and an approximate treatment must be made to serve. These practical problems are important in the discussion of the Stability of the Flying Machine, and the movement of the accessories, such as the gyroscopic influence of the motor and air screw. In the diagrams on the plates at the end an attempt has been made to give the scale and shape of a figure, without entering into mechanical detail, so as to make them serve as model of a blackboard drawing, in an explanation of facts to be recognised when met with out of doors on a wider area. Mr. J. L. Nayler, of the Aeronautical Department in the National Physical Laboratory has given valuable help in revising the proof sheets and verifying the results. CONTENTS. Chapter 1.— Steady Gyroscopic Motion „ II.— Gyroscopic Applications III. — General Unsteady Motion op a Top or Gyroscope „ IV. — Geometrical Representation op the Motion op a Top v.— Algebraical Cases op Top Motion „ VI.— Numerical Illustrations and Diagrams „ VII. — The Spherical Pendulum „ VIII. — Motion repbrrbd to Moving Origin and Axes „ IX. — Dynamical Problems of Steady Motion and Small Oscillation Page- 1 28 42 81 116 158 17.S 201 ON ... 239 Plates op Diagrams. 28570 Missing Page REPORT ON GYEOSCOPIC THEOET. This Report is intended to serve as a collection in one place of the various methods of the theoretical explanation of the motion of a spinning body, such as shown in a top, and other familiar examples, large and small, of gyroscopic action. The treatment will be carried out with analytical completeness, as far as is possible in the present state of mathematical development. The subject is likely in .the future to have important utility in its bearing on the steering and stability of a flying machine ; and the report will serve for reference, whenever theory is invoked in support of a practical application. CHAPTER I. Steady Gyeoscopic Motion. The simple toy spinning top, in its curious behaviour of standing up on a table ixgainst gravity, has directed attention to the distinction between dynamical and statical stability, in their opposite manifestation. " To those who study the progress of exact science, the common spinning top is a symbol of the labours and the perplexities of men who have threaded successfully the mazes of the planetary motions. " The mathematicians of the last century (XVIII), searching through nature for problems worthy of their analysis, found, in this toy of their youth, ample occupation for their highest mathematical powers." (Maxwell, Scientific Papers, I., p. 248, " On a Dynamical Top," fig. 1.) Analytical theory has been guided thereby in the consideration of the allied problems on a large scale, such as the motion of a gyroscope, hoop, bicycle, wheeled carriage, and ship, culminating in the Precession of the Equinox. In statical equilibrium the centre of gravity of a system seeks the lowest position it can assume ; but in the dynamical equilibrium of the sleeping top the centre of gravity rises as high as it is able. A top spinning steadily upright in dynamical equilibrium is said to sleep ; but man or an animal sleeps lying down in statical equilibrium. For ease of progression in movement a man assumes the upright attitude of a biped, or rides upright on the back of a horse or on a bicycle ; and any burden he prefers to carry as high as possible, on the shoulder or head. Mounted still higher on stilts his progress is not more difficult, with the confidence of experience. A confusion of the distinction between dynamical and statical stability has led to serious mistakes and the misapplication of theory ; such as, ballasting a ship too low and making it bottom-heavy and uneasy among waves, or spreading the railway gauge so as to lower the boiler or carriage body between the wheels ; also in the design of a soldier's knapsack low down on his back, suitable only for a halt, uncomfortable and fatiguing on the march. Theory based on imperfect assumptions has found itself in such cases in conflict with the opposite practical experience ; as also for instance in the old stage coach, loaded with passengers and luggage all high up on the roof as seen in the old coaching print, and running freer in consequence with a slower oscillation on the springs ; and to-day in the modern design of the locomotive engine, with the boiler as high as possible to pass under a bridge. It is the limitation of the loading gauge of the bridge and tunnel which has hampered railway developement, and not the restriction of the gauge, to 5 feet outside measurement, of the original Roman gauge ; and 7 -ft. gauge is all pulled up. Practical experience has not borne out the early opinion that a low centre of gravity is conducive to steady running of a locomotive or railway carriage ; or to quote from the report in the Record of the 1862 Exhibition by Rankine and MacConnell : — (28570— Aer,). Wt. 37036—592. 500. 1/14. D & S. A " Experience has demonstrated the error of the opinion held formerly, that it was essential to safety, especially at high speed, to keep the centre of gravity of a locomotive at a very low level ; an opinion by which the designer of a locomotive was very much fettered when planning an engine for high speed on the narrow gauge, in which great toiler power had to be combined with large wheels. " The locomotive engineer of the present time (1862) being freed from this restraint, does not hesitate to place the weight of the engine as high as may be required tor the convenient arrangement of its parts. " Of this the engines exhibited in division A of the classified list afford several examples, of which the most striking is 1280, a narrow gauge express engine with driving wheels 8 feet 2 inches diameter, being the largest pair of driving wheels exhibited, and 1269, the Wolverton engine, coming next with wheels 7 feet 8 inches high." The modern motibus, with passengers on the top, is another instance of easy running with high centre of gravity. For experimental illustration on the table the Maxwell top (fig. 1) will serve, spun by the finger and thumb in its agate cup. On a larger scale the bicycle wheel of fig. 2 is recommended ; spin enough can be given by the hand, and the delay is avoided of winding up the string of a top, with the risk of entanglement and failure, and the use of string in these experiments is undesirable. But if unavoidable, better use a short length of fiddle string, so short that the string is free at half the stretch of the arms. The bicycle wheel is large enough to be visible on the lecture table, of the usual 28 inch diameter ; but a larger wheel, say 52 inch diameter, is more effective still ; and they will bear hammering with a stick with little flinching. The axle of the wheel is prolonged in a stalk or shaft, and the end can rest in a small cup 0, so as to realise the motion of a top spinning about a fixed point ; and if the axle is carried through the wheel on ball bearings, the rotation is affected by the friction only to a slight extent. The gyroscope motion in fig. 2 comes to a stop when the rim of the wheel touches the table ; and to realise the motion when the axle is inclined at a greater angle with the upward vertical, the stalk of the bicycle wheel is pivoted in fig. 3 in a lug at 0, screwed to the axle of a bicycle hub, fastened vertically in an iron bracket bolted to the under side of a heavy beam or sleeper, stout enough to absorb the vibration quietly ; the beam can rest across on two supports, like step ladders. The wheel now hangs down vertically at rest and it can 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 rest. Take up the wheel of fig, 2 by the stalk and brandish it, a muscular sensation is obtained thereby of the gyroscopic influence of the wheel as it is spun. And generally, it is advisable to bring in as much simple experimental illustration as possible on a scale large and small, as a relief and assistance to the interminable analysis, and the strain of the mental picture ; the motion being in three dimensions it is not always clear and easy to represent it in a single diagram. Friction is apt to mask the features of an experiment on a small scale, so in these experiments of illustration, the scale should be made as large as possible when anv quantitative measurement is attempted. To quote Maxwell again {Scientific Papers, II., p. 242, Introductory Lecture ,»i Experimental Physics). " Let me sa^ a few words on these two classes of experiments : Experiments of Illustration, and Experiments of Research. The aim of an experiment of illustration is to throw light upon some scientific idea, so that the student may be enabled to grasp it. " The circumstances of the experiment are so arranged that the phenomenon which we wish to observe or to exhibit is brought into prominence", instead of beino- obscured and entangled among other phenomena, as it is when it occurs in the ordinary course of nature. To exhibit illustrative experiments, to encourage others to make them and to cultivate in every way the ideas on which they throw light, forms an important part of our duty. The simpler the materials of an illustrative experiment, and the more familiar they are to the student, the more thoroughly is he likely to acquire the idea which it is meant to illustrate. The educational value of such experiments is often iiiverselv proportional to the complexity of the apparatus. The student who uses home-made apparatus, which is always going wrong, often learns more than one who has the use of carefully adjusted instruments, to which he is apt to trust, and which he dares not tal- ^ to pieces." The distinction here between the two classes of Experiment is equivalent to that oiven by Aristotle, as Kara (jtvaiv, and napa fvaiv. ^ 8 The axle OG in fig. 3 has the two degrees o£ freedom of the theodolite or astronomical Altazimuth telescope ; the stalk OG can be placed at any altitude angle 9 with the vertical OC, and thevertical plane GOG can be set at any desired angle 4' ^^ azimuth. There is still the freedom of the rotation of the wheel on the axle in its ball bearings, allowing a relative angular displacement (j> ; and 0, ip, (f,, denote the three angles first introduced into the dynamical treatment by Euler in his Introductio in analysin in/initorum, 1748. (P. Stiickel, Encyc. Math. lY., 6, 28., Klein-Sommerfeld-ISroether, Kreisel Theorie, p. 938.) 1. To give the exact theory, with as few symbols as possible, of the Steady Motion of a Spinning Top or Gyroscope, moving at a constant angle with the vertical, a knowledge is presupposed of velocity and momentum, linear and angular, and their, vector representation ; and use must be made of the Dynamical Lemma — " The vector A'elocity of the momentum, linear or angular, is equal to the 'S'ector of the impressed force or couple." The vector representation of a couple as well as of a force is thus required also, the cou23le being due here to the preponderance, leverage, or moment of gravity : the notion too and determination of moment of inertia, or second moment. It is unfortunate that the word moment should be used in Dynamics with such a variety of meaning. The reader not familiar already with these dynamical quantities should study them in the Appendix, or else, for a simple explanation, in Maxwell's Matter and Motion. The representation by a vector of an angular velocity or momentum (abbreviated in the sequel to A.Y. or A.M.), as well as, a couple, will require a direction to be associated with the rotation ; this is effected by imagining a screw to be cut on the vector, right or left handed, and tlie vector is drawn in the direction of advance along the screw caused by the rotation ; and we shall select the right handed screw. Thus the rotation of a clock hand seen in front would be represented by a vector drawn towards the clock face. In crossing the plane of the face the rotation would appear reversed, as if seen by reflection in a mirror, but the representative vector would still be drawn in the same direction, now away from the back of the face. 2. In theoretical investigations of Dynamics the C.G.S. (centimeter-gram-second) system of units is now usually implied. Thus the wheel and stalk in Figs. 1, 2, 3 is supposed to scale M grams (g) in the balance so that M is the quantity of matter {Qu'antitas Materice) in grams; the preponderance about 0, Mh in gram -centimeters (g-cm) can be measured experimentally in fig. 3 by hooking up the stalk horizontally by a spring balance, and noting the distance of the hook from as well as the scale reading, the product of which will be Mh g-cm ; and thence h is inferred from the measurements, the distance of G the C.G. (centre of gravity) from 0. The M.I. (moment of inertia) about the axle OG is denoted by C, and measured in g-cm^, and about an axis through perpendicular to OG by A (g-cm^) ; and A is determined experimentally by letting the wheel hang vertically at rest in fig. 3, and giving it a small plane oscillation or conical revolution. Noting the length I (cm), which synchronizes, of the equivalent simple pendulun; (S.E.P.) made with a plummet at the end of a plumb line thread ; noting also T the time in seconds of a single beat of the pendulum, or 2T the time of a complete revolution in the conical movement ; then in accordance with elementary dynamical theory (1) , = 4 A = mi; and if n denotes the angular velocity, in radians/second, in the conical revolution, (2) n = |5 =' yi, An^ = g^ = gMh, g denoting, in cm/sec^, the acceleration of the gravity field, about 981 cm/sl The quantity gMh or An^ is what is denoted by P in Klein's Kreisel-theorie. To measure (7, detach the wheel and hang it by the inside of its rim from a supporting- knife edge, and set it swinging in its plane as a pendulum, as in fig. 4 ; measure the length, /cm, of the simple equivalent pendulum which synchronizes with the oscillation of the wheel, Then if a denotes the radius of the inner rim circle, (■^) /' = ^Z±i'l«^^ g.-^a C=Ma (/' - a\ Ma Ma^^^' _ ^ ^\ and the radius of gyration of the wheel about its axle is ^/ {a. I' - a). These quantities, M, h, A, C, I, give the physical constants of the apparatus in fig. 1, 2, 3 ; together with n T in the gravity field g. 28570 -*■ 2 3. But next spin the wheel about the axle with angular velocity B (rad/sec) ; this will give it kinetic energy ^ CR^ (ergs), and A.M. about the axle given by Cn (erg-sees), represented by the vector OC in fig. 5, 6. In the absence of gravity or preponderance j making G coincide with 0, the vector OC would not move ; but suppose OC is describing a cone round the vertical Ob at a constartt inclination 6 to the vertical, as the axle of a top in Steady Motion ; the vertical plane COC will then revolve round 0(^ with precession u, a constant angular velocity. Referring to fig. 2 and 5, where 6 is the angle which OG makes with the upward vertical, the precession n about OC is resolved into a component angular velocity fi cos about OC, which merely affects the axle and does not influence the rotation R of the wheel ; but the other component is ^ sin 6 about OA' perpendicular to OC in the plane <^0C, and this will give the wheel an additional A.M. about OA' of magnitude A ft sin 6 ; -and if this is represented by the vector CK, then OK is the vector of resultant A.M. Then in fig. 5, (1) CK = OC sin e - CK cos 6», and the velocity of K is (2) u. CK = ^ {CR sin 6 - A^l sin d cos 6), «,nd this is the magnitude of the couple in the plane COC required to maintain the motion. If the axle is free, and the couple is supplied by the preponderance of gravity, of moment (3) g3Ih sin = An^ sin (dyne-cm, or ergs), the vector velocity of K is to be equated to this couple ; so that (4) An^ sin = fx. CK = CR fi sin 6 - Afi^ sin 9 cos 6, and dividing out sin 6, and discarding this factor, (5) An^ = CR /ii — A fi^ cos 6, the condition for the free steady motion, with precession u, of the top in fig. 2, 5, at a ■constant inclination 6 to the upward vertical. This condition (5) can be written ... . CR n' CR' ( CR n\' CR' so that a top cannot spin steadily in the upright vertical position unless CR > 2An, and then it is called a strong top (Klein) ; a weak top has CR < 2An, and in steady motion it ■ ^ CR^ •cannot rise to within an angle with the vertical whose cosine is ^ 4A'n" Writing (o) in the form the value of ^t will not be real unless (8) 2^>^^^°^^)' ^>v/(^'-«^); and at the critical value -^ = v^ (COS e) = " , ,. = M^' The top cannot sleep upright unless (10) CR > 2An, iCm>2^g M,, > ^^ > 2 1^ Mh ; 6 2g C ' or the energy of rotation of the wheel must be greater than ~ times the energy acquired in turning round from the upward to the downward vertical position. The relation (6) can also be written (11) {y - af = d' - X, with cos « = x, "- = y, -^' = a M 2An ' a parabola graph. Starting from .*; = - 1 on the parabola, the axle pointing vertically downward (12) *' = - 1, ; 1). in a weak top, a < 1 , and it x reaches a-, cos » = . .^ 2 1 ^-nd then y = a, n = ~fTn- Thus to make a pin stand on its point, spin it ; but push the pin first through a wad or disc, so as to give it extra moment of inertia ; a simple experiment to carry out with appliance ready to hand. The inertia of the pin may be neglected in comparison with the wad, so that if a denotes the radius of the wad, of thickness b, (14) C = ^Ma\ A = M (h^ + ia' + ib'), <1^) 4AM = 4g2¥hM (> + ^a' + ^ > 1' ^^^ "P"g^* ^*^^^^^*-^ '' and if v is the relative velocity of finger and thumb in spinning the pin, of diameter d, and V the rim speed, «' .<^/ h' 4 b\ r^ ,, h^ 4b\ (16) v = dR, 2^ > 2 -^(1 + 4 -, + 3^), p>.S(l + 4-,+ 3^-,). Thus when h/a and b/a are small, fourfold A requires double v and V. Incidentally we can observe the difference of spinning the pin on its head, or in a smooth dish ; also of hurrying or retarding the precession by a rotation of the card or plate on which the pin is spun, as in Kelvin's experiment with a gyroscope on trunnions , in a frame of wood held in the hands. 4. Take the wheel in fig. 2, free to rotate on ball bearings about the axle and grasping the stalk horizontally at the end P, let the wheel be spun about the axle with an A.M. CR, represented by the vector GC. When the stalk is swung to the left, A.M. is communicated about an axis drawn upward, and the axle swerves upward, unless held down by a couple ; and this couple is the vector velocity of the A.M., of magnitude CR/n, if the stalk is swung round with precessional angular velocity fi. Swung to the right, the axle would swerve downward, unless held up by a couple. When the angular velocity E is expressed by N the revolutions per second, R = 2TrN and if the stalk is swung round horizontally so as to make a complete turn in T seconds u = 2Tr/T ; so that the couple, expressed in the gravitation unit, is (1) C^=C^. Thus if C for a Gnome engine and propeller is estimated at 240 Ib-ft^ ; and the revolutions are 1,200 per minute, N = 20 ; and if it is making a turn in 20 seconds, r = 20 ; and the couple works out to about 300 ft-lb. With a left-handed propeller the couple is reversed ; so that turning to ther^J^ji. ) a couple reacts on the flying machine tending to make it f t- ] requiring correction on the horizontal rudder by putting the helm / , „) ; thus the horizontal rudder would be used to steer the machine to the right or left. With a right-handed propeller this steering could be reversed. Returning to the right-handed rotation of the wheel, swing the stalk ( j*^^^^ j) : then A.M. is communicated about an axis drawn to the ( .^n, j ; and the stalk would «werve to the y^^^ ) ; unless held by a restraining couple, with axis / "P^^^ \ If 01 cuts KN in J, (8) JN = OM - KJ = OM - (l - ^)0C' =^, OC - CM, JN A CR A fi.co?,Q _ R - aco&H ~^ ^ ~C' ~A^i ~ An ~ n Since the axial angular velocity of the wheel is R, and of the axle is ^ cos in the same direction, then i? - ^ cos is their relative angular velocity, and this is the rate ot variation of Euler's d). so that Draw SS' through- parallel to MN, cutting KM, KN in S, S' ; then (10) KS = 2KM, KS' = 2KN, KS . KS' = k' = 4a^• and the tangent of the hyperbola at K is parallel to SS', or MN. If the wheel is clamped on the stalk, as if the axle had seized, and the system moves bodily round the vertical OC, i? = m cos B and I comes to F ; and condition (5) §3 becomea (11) An^ = CRfi - Afx^ cos 9 = (C - A) ix^ cos 6, (12) ^ =(^'-l)%inecose, i;|^._(^-l)^sin0. 7. In the investigation above the inertia of the stalk has been ignored, but when taken into account, the condition (5) §3 becomes modified to (1) An' + yliV = {CR + Cifi cos 0) n - {A + A^) ^r cos = CR/ii - (^ + ^1 - C'l) |t^cos 6, Aitii^ = (/Ml hi, when Ml, hi, Ai, Ci refer to the stalk. Substitute for the wheel on a stalk in fig. 3 the front wheel of a bicycle held in its fork, with the pin at through the steering head, as in the Anschiitz (ryro-Compass. I. With the pin parallel to the axle of the wheel, the state of steady motion round the vertical at a constant inclination is unaffected by the rotation of the wheel, but a couple is required, supplied by the pin, of magnitude K/n about the horizontal axis in the plane COC, K denoting the A.M. of rotation of the wheel. II. With the pin in the plane of the wheel at right angles to the fork, tiiere is no reaction across the pin, but if Ci denotes the M.I. of the wheel and fork about the axis OC through the centre of the wheel, and Ai about the axis through perpendicular to the plane of the wheels, the horizontal component of the A.M. of the system is (2) ^1 fLi sin 6 cos 9 — Ci /ti cos 6 sin 9 + K cos 9 ; and yit times this A.M. is its vector velocity, to be equated to the gravity couple ; so that (3) (^1 - Ci) ^^ sin 9 cos 9 + K ^ cos 9 = g Mh sin 9 III. If the pin at is fixed at an angle (j, with the plane of the wheel, then (4) (^1 - Ci) fi^ sin cos + AT^i cos 9 cos (j> = gWt sin 9, and the reaction couple across the pin is /{/.t sin ^. The fork may be held in the sleeve of the steering head, so as to allow a variation of f, with the head pinned through ; but the motion would not now be steady with the angle 9 constant. 8. In fig. 3, 6, the greater part of the motion of the wheel lies below the level of ; and a change is made to a measurement of 9 from the downward vertical, or nadir. This is in accordance with Darboux's practice (Despeyrous, Mt'canique, II., Note XIX.), but in the Klein -Sommerfeld Kreisel-theorie, the measurement of 9 is' from the zenith or upward vertical ; a change from one method to the other is made by a change in the sign of cos 9, leaving sin 9, n, and CR unaltered ; and in fio-. 3 6, (1) CK = OC sin 9 + CK cos 9, OC = CR, CK = .4 ,,"sin 0, ' (2) An' sin 9 = fi. CK = ,i {CR sin 9 + A^i sin 9 cos 9), (3) An' = CRjx + .i^i cos 9 ; the precession « being still represented when positive by an upward vector 00. And to the geometrical scale, ]^ = M KN = /"-CK _ An' sin 6 ^ « , ^ ' a n^ a Anfi sin 9 An/n sin 9 ^' (5) KM. KN = a^ = ^F, as before in (3) (4) § 5. n n • In the Maxwell top of fig. 1, as employed for a simple experiment, the C.G. is usually a httle below the support 0, and the " stalk OP points upward at rest ; then is taken as the inclination of OGr to the downward vertical, or of OP with the zenith upward ; and it is the movement of the point P which shows to the eye the path of the axle. So too with Newton's Precession Top in fig. 7. 9. When the wheel is spun rapidly, so that OC in the figure is large compared with C R, the point Ki is close to F, and fii is large, giving a violent motion. But K is close to C, so that we may take (1) . CK = OC sin 6, ^ = ^ sin 6, ^ ^ a An An^ (2) An^ sin 9 = CRu sin 9, fx = yr^, making fx small and independent of 9. This relation is true accurately in (5) § 3 when the axle of the top in Steady Motion is horizontal and cos = 0; and the approximation is useful in popular elementary explanation of gyroscopic motion, as given in Perry's Spinning Tops. The approximation amounts to assuming that the axis OK of resultant A.M. is undistinguishable from the axle OC, so that the velocity of C may be equated to the impressed couple. This is the case with the rotation of the earth, where the variation of latitude is insensible ; and the approximation was employed by Poinsot {Connaissance des TempSy 1858), in his investigation of Precession and Nutation. Precession of the Equinox. 10. The earth behaves in precession like a large spinning top, and the motion can be illustrated in fig. 1,. 7 by the Maxwell or Newton top on the table, by twisting the stalk OP with the left hand against the clock and sun, the preponderance being made small so that the precession fi is slow, and with the clock and sun. Astronomical observation has revealed that the polar axis of the earth describes a cone round the pole of the ecliptic, of mean angular semi-diameter 9 about 23°^, in a period of nearly 26,000 years, giving an annual precession of 50", or 1° in 72 years ;. l°-4 = 1° 24' in 100 years ; 30° in 2,160 years, through one sign of the Zodiac. With these numbers (1) : - = 26,000 X 3661, taking 365^ solar days in the year, the estimate of Evdoxus, B.C. 400, and sidereal days- one more ; and the mean couple producing precession fi, a retrograde regression round the ecliptic, is round the axis Oif, where "v is the symbol of the First Point of Aries, where the ecliptic crosses the equator at the Vernal Equinox ; and the magnitude of the couple is (2) CR, sin« = CB?^^= g^' sJoSfLl =Tirf?i53= "-X*'* >< i^*'. about one 12-millionth part of ^CR^, the kinetic energy of rotation of the earth, taking sin 23^=1 = 0-4. The calculation of this couple is resumed later, in Chapter VI ; the effect of the couple in precession is to cause if to move to meet the sun, and so shorten the year by about 20 minutes, the fraction of the year which the annual precession 50" is of the whole circumference. If a, /3 denote the angular deviation of the axis of rotation 01, and of A,M. OK, from OC the polar axis, the earth moves in this uniform precession as if OC was the axis of a cone of semi-vertical angle a rolling inside a cone of semi-vertical angle h- a, of which the axis is through the pole of the ecliptic; so that M sin 23°-^ (3) i? sin a = ^ sin 9, sin a = -^ sin 9 = gg qoq x 366^- = 10'^ x 4*2; 28570 B 10 and a is a very small angle, which expressed in seconds, amounts to (4) „" = 206,265 X 4 • 2 X 10-^< 0"-01 ; and tan /3 = ^ tan a, so that a and /3 are quite insensible. . r ^u ^u f r..A^,-,^ But I describes a circle round the pole on the surface of the earth, of radius (5) l^ sin a = 1^=27 (cm), the quadrant of a meridian being 10« cm ; thus the approximation of Pomsot is justihed Milton's Paradise Lost, X., 668, gives the legendary description of the origm ot tne <)bliquity of the ecliptic, before which it was supposed there was eternal toprmg— " Some say he bid his angels turn askance The poles of earth, twice ten degrees and more, From the sun's axle ; they with labour pushed Oblique the centric globe — " This can be imitated on the table with the Maxwell or Newton top ; twirling the axle with finger and thumb of the left hand (against the clock and sun, widderstiins), so as to spin the top upright and asleep, and then allowing the stalk to press gently against a finger, so as to delay the precession, when the top falls and obliquity is produced. But the Precession is in the opposite direction, moving with the clock and sun, deasil, a.s described in Waverley, I., Chapter 24. A reverse action will bring the axle upright again, as promised bv Milton for the Earth at the Millenium. Herodotus describes (II. § 142) the account given by^the Egyptian priests ot the change of the astronomical seasons in a record of about 12,000 years, which is supposed to mean the half period of precession ; and Homer's allusions to_ the stars are definite enough for us to calculate back from the precession a lower limit of the date of his observations. . Hipparchus, B.C. 150, was the first astronomer to describe and explain Precession ; he seems to have been guided by the observation of Canopus, which in his time had about 37°J S.P.D., but according to Evdoxus, 250 years earlier, the star was then on the invisible circle of Athens, latitude 38° N. ; this implies in the absence of refraction that the S.P.D. of Canopus had increased half a degree in 200 years. Looking in Fig. 8 at the concave side of the celestial sphere round the N. pole P, as a star chart of the northern half of the sky bounded by the ecliptic, with pole at n, the component angular velocity of the pole P towards a star S, whose right ascension _^PS is denoted by ^, is given by fi sin 6 cos M, and this is the rate of increase of N. declination,, or S. polar distance. In 250 years, with sin = 0-4, and at the rate 50" sin 9 cos M = 20" cos M a year, the amount of change of S.P.D. would be (6) 250 X 20 X cos iR = 5000" cos M. Between 150-400 b.c. the right ascension of Canopus would be about 83°"5 or five hours and a half; giving a precession in S.P.D. (7) 5,000 X 0-1,132 ^ 566", not quite 10', instead of half a degree ; but refraction would account for this discrepancy in a star on the horizon. At the present time Rigel and a Hydrae, lying on or near the solstitial colure '^"P^, have a precession in declination of the full amount of about 20" in a year, quite large enough to be detected on a photographic chart at a year's interval ; and amounting in 30 years to 1 0'. Serson-Fleueiais Gyroscopic Horizon. 11. A spinning top with a polished upper plane surface, when asleep will provide an artificial horizon at sea, when the real horizon is obscured. The first instrument of the kind was constructed by Serson ; it is described in the Gentleman's Magazine, vol. XXIV, 1754 ; also by Segner in his Specimen theorice turbinum (Halaa, 1755). Serson the inventor was sent to sea by the Admiralty to test his instrument, but he was lost in the wreck of the Victory, 1744. A copy of the Serson top, from the royal collection, is now in the museum of King's College, London. Troughton's nautical top (1819) is intended for the same purpose. The instrument is in favour with the French navigators, as improved by Admiral Fleuriais (fig. 9) ; but the horizon given by the top is inclined to the true horizon at a small angle E, given in latitude A by (1) smE=^^^^h^ gMh ju denoting now the angular velocity of the earth, n = 2i: ^ 86,400. 11 The reason is seen from elementary gyroscopic theory ; for CRfi cos X is the vector velocity of A.M. due to the earth's rotation, which must be produced by a preponderating couple gMh sin E. If Ml denotes the precession of the top in the steady motion about its vertical, and 7\ = 27r//ii the period of its complete revolution, then when spun rapidly, as in (2) § 9, (2) CR Ml sin E = gMh sin E, (3) sin^=AeosX=gg^cosX; and Ti is observed by the rise and fall of the lines engraved on the lens L^ in fig. 9, seen through the opposite lens Lg. Expressed in minutes, or sea-miles of latitude, rA\ „ 360 X 60 . „ T, . (4 ) E = -r sm E = 5^ cos X, in which Sw may be replaced by 25 in practice ; and when the top is spun widdershins, against the apparent diurnal motion of the sun, and so with the earth's motion, this angle E is positive towards the south, so that the true latitude is E minutes or sea-miles to the north of the gyroscopic latitude. Thus if 2\ = 120 seconds, E = 4'.8 at the equator. The gyroscopic horizon can thus detect the rotation of the earth, better than the Foucault pendulum. Consult also A. Foppl, Kreiselversuch zur Messung der Umdrehungsgeschwindigkeit der Erde. Miinchen J 904, The top in the gyroscopic horizon is spun by a jet of air from a pump, acting on the vanes of a turbine wheel : when sufficient spin is attained, the pump is used again to exhaust the air, and so the top spins a considerable time in the vacuum, while aGSxed to a sextant. 12. We shall find it convenient in the sequel to investigate the motion of G the centre of gravity, and then the motion round G ; the final result is of course the same. The components of the reaction at in steady motion are Z = gM vertically upward, and X = n^M. GA horizontal ; and t-heir moment round G is (fig. 10) (1) gM. AG + ,1^. GA, AO = gMh sin + ^^ Mh^ sin cos 0, denoting OG by h, as before, A suffix must be employed to distinguish Aq^ the M.I. about an axis through G per- pendicular to the axle, from Aq, the former A, M.I. about a parallel axis through ; and (2) ^o = -^G + Mh' ; but C remains unaltered. The reaction at has the direction Op parallel to GB, where AB is drawn vertically upward of length X = g/fi^^ the height of the equivalent conical pendulum of the preces- sional motion fi (fig, 10) ; and the moment of this reaction about G is the vertical component at the arm Gp, or (3) gM.Qp = gM (GA + Ap) = gMh sin + gM. AO ^ A = gMh sin + tx^M¥ sin Q cos 0, as before in (1). Drawing Gc' = OC in fig. 10, to represent CR, and c'k to represent Aq u sin 9, then the velocity of k relative to G, or the vector velocity of G^, ignoring the homogeneity-, is (4) lu. ck = n { CR sin d — Aq /a sin cos 6) = vector of the moment of the forces round G = gMh sin 9 + /^.^Mh^ sin 9 cos 9 ; (5) CRfi sin 6» - (Ag -t- Mh^) ^^ sin cos = ghM sin 9, equivalent to (4) (5) § 3 above, after rejecting the factor sin 9. m n ft Ic The homogeneity factor, — or ^-j-j may usually be ignored in the treatment, until it is required, and the suffix may be omitted from Aq in the sequel, when the origin is transferred to G, Then in fig. 10, ((Cs 2^ - U + it^^'V _ ^ + M¥ ^^ :Gn Jm ~ Z ' 28570 B 2 12 .^. OL _ A + JfA^ _ ON '<^^>* . GL " ^ ~ Gn ' so that L?^]Sr is a straight line. From the dynamical relations /9) qM. Go = fi. ck, gM. Go = /u. kn, gM. GA :- ^. CK , ^M. GO = ^. KN, Gj9 GfO_ _ ^ _ Gm (10) GA = GO ~ KN - GM" so that pm is parallel to AM'. Draw Gr perpendicular to G^ cutting in r the horizontal through V at a vertical depth Gr = X' ; and draw rl perpendicular to GL, cutting AG in (? ; GZ c'k _ fi. ck _ m' a sin _ -A' sin _ A ('^) W ~ Tc ~ gM. (}p ~ gM. Gp ~ MX. Gp MX. Go' (12) GL Go = Gp. Gq = ^= OG. GL ; so that g is determined otherwise by drawing the circle round OjoL, cutting pG in ^ ; and G[ _ GO _ GA (1^) GL ~ Go ~ Gp' so that At is parallel to pL. 13. Take the problem of the wheel in rotation, suspended by a thread ED, fastened at E to the axle, and to a fixed point D, the axle of the wheel, swinging round in a vertical plane DGC with constant precession^ about the vertical OAD (fig. 11), in the same time as a simple conical pendulum of height (1) X = AB = ^. Experiment with a spinning bicycle wheel, and a thread attached to the stalk. If E was taken at G, the axle would preserve a constant direction in space, and the body would swing round like a simple conical pendulum GB of height AB, A being the centre of the circle described by G ; and the axle GC points in a fixed direction. But with E elsewhere between and G on the axle the thread ED is parallel to GB and the tension of the thread has a moment about G which must be sufficient to make the precession /j. of the axle the same as the angular velocity of the plane DGC about the vertical, if GC is to remain in this vertical plane. With E below 0, the thread ED would have to act as a strut ; with E above G the precession ^ is reversed. ' The vertical component of the tension of the thread being gM, the moment of the tension round G is gM. Gp, if ED cuts the horizontal line AG in p ; adjusting this to the vector velocity of G^, (2) gM. Gp = ^u ck, gM. Go = ,.. kn, if the vertical through p cuts the axle in o. Draw Gr perpendicular to G^ to cut the horizontal line through I' at a vertical deuth Gr = X m r ; and draw rl perpendicular to Gc' to cut the horizontal line AG in q • (o\ d^ _ kn _ Gl ^^ kc ~ km - Gl' and from (2), with km representing A/n, (4) ao = ^^ = ^^d^ ^ =^ A'' GZ' ^ " 9^^^ gM- Gl M~g Gl- (5) Go. Gl = Gp.Gq =. ^= OG. GL (Q-) G| ^ GO _ GA ^ ^ GL Go ~ G^' so that M is parallel to Lp : and {p,q), (o,l), (0,L) are pairs of convertible centi-es of suspension and oscillation. ^liicB oi 13 Thus to obtain a representative figure, with the axis GI. at a given angle 0, and with a given 0, L, L'; and a given position ED of the supporting thread j draw AI parallel to pL ; draw Iqr at right angles to OL cutting in r the horizontal through I' ; then Grk is perpendicular to Gr, and the geometry is as before. Taking moments as at first about 0, the moment of the weight vertically through G and of the reaction along ED is gM. GE sin 9, which is to be equated to fi. CK. Draw OR perpendicular to OK, to meet the horizontal line through L' in R, and draw RLi perpendicular to the axle, then (fig. 11) ^ . CK OL' (^) KC ~ OLi' M- CK _ ^.. KC OU _ {A + Mh') m'^ sin 6 OV _ Mj^, (») G±. - ^^ g.j^ g - ^j^ g.^ ^. QLj - gM sin B ■ OLi OLi (9) GE. OU = ~ + h' = OG. OL. Spun as a top in the small cup at 0, the points E,o coalesce with 0, and Li,l with L, making relation (9) the same as before in § 6, and fig. 5, 10. Representative figures of the dynamical diagram are shown in fig. 11, 12, 13, 14,. 15, with the thread in tension, and E above ; other figures can be drawn where E is below 0, and the thread must be replaced by a stiff rod, to act as a strut and transmit a pressure. A smooth spherical cup or boss, centre D and radius DE, may replace the thread or strut, so that the top spins steadily on the surface, with the point at E. When the flywheel is enclosed in a light case, as a gyrostat, the thread may be attached to a point on the case, and the conditions are as before, with E' the point where the line of the thread cuts the diametral plane of the case, as in fig. 13. 14. If the wheel is not spun, but still free to turn on the axle, CR = 0, and c', C coincides with G, ; G^, OK is at right angles to the axle, so that r, R lies on the axle, coinciding with Z, Zj. Thus if 6 and AB = Gr is given, r or I and p,o determinate, and EpD is parallel to GB in the plane of the thread. Figures are given in fig. 16, 17, 18. Although the wheel is not spinning now, and comes to rest whenever the axle is held, it will be found that the wheel rubs round on the axle, and that the third Eulerian angle (j> is not zero ; and the wheel does not return to its original po-^ition when the axle has made a revolution of the cone. The angle through which it has turned on the axle is found from (1) ^ = R + fi cos e = fi cos 9, with R = 0, to be given by (2) 2ir cos 9 = 2Tr — conical angle described by the axle = length of intersection of the reciprocal cone with unit sphere. This can be shown experimentally with a penholder, held between the fingers, and moved round in a cone by the tip of a finger applied at the end. We can make CR = by taking C = 0, so that the wheel may be replaced by the stalk, of which the inertia has been ignored so far; then fig. 16, 17, 18, will be representations of the motion of a rod, swung round at the end of the thread DE. 15. If 01, the instantaneous axis of rotation of the stalk, is drawn to cut the thread DE in P, and PJ is drawn perpendicular to the axle, then J may be taken as the centre of a circular base or rim fixed on the axle, which may be supposed to roll on a table at P ; and so a figure may be drawn of a rolling body, corresponding to the dynamical diagram ; but P must lie below the axle for the motion to be realised practically. When the wheel and axle make one solid body, then 01 is its instantaneous axis of rotation ; but in the general case the wheel may have an independent rotation on the axle, as in a motor car or flying machine. Fig. 19, 20, 21, 22, 23, 24, shows a body of revolution GP, and the geometry of its motion, like a disc, coin, plate, dish, chalk or china egg, wine glass, rolling steadily on a horizontal table, or on a cone or other surface of revolution, having a vertical axis OB passing through A the centre of the circle described by G the centre of gravity of the body. Experiment with a coin spun on the tabl-e, or rolled round the inside of a conical lamp shade, or a cup ; and compare it with a bicycle on a banked concial track, or inside a sphere. 14 Drawing G^' vertically downward of length ^//x^, the height of the equivalent conical pendulum for precession /n, the reaction at P the point of contact is along PED parallel to Al' ; and the motion is the same as before, where the body was swung round_ by the thread ED, and OP is the instantaneous axis of the axle, or of the body when it forms one solid, and the wheel is not spinning independently. Make pQ. Gq = -g, where PE cuts AG in p, so that p and q in AG are convertible centres of suspension and oscillation when the body is swung as a plane pendulum about p and q, and pq is the equivalent pendulum length. Then ql perpendicular to the axle and the horizontal through l' will meet in a point r, and G^ perpendicular to Gr will be the axis of resultant A.M. about G. The component vector Go' representing the axial component of the A.M., then c'k will represent J/^ sin 9; and with c'i = C/n sin 9 = -j c'k, Gi will be the axis of resultant angular velocity about G ; and then OP, the instantaneous axis of rotation, through P, and where the axle cuts the vertical through A, should be parallel to Gi. In making a representative figure to satisfy these conditions, it is advisable to fix the position of P lasl;, and then to fill in the outline of the rolling bodv, as in fig. 20 to 24. Begin with an arbitrary G and axle OG, vertical OA, and Gr = g/fi^ ; and assume an arbitrary p in AG, drawing the line DpE parallel to A^ to represent the line of reaction at P, on which P must be taken. A Making Gp. Gq = — , and drawing q/ perpendicular to the axle cutting the horizontal line through /' in r, then G^ is at right angles to Gr. C , . Make c'i = — c'k ; the line through parallel to Gi will be the instantaneous axis, and will cut ED in P the point of contact. Draw PJ, PH perpendicular to the axle OG and vertical OA ; P will describe a circle round the centre J fixed in the body, and round the centre H, fixed on the table or in the axis o£ the fixed surface of revolution. The shape o£ the rolling body can then be filled in to taste, as in fig. 24. A figure can always be made to serve if an internal flywheel on the axle is at our disposal, to trim the dynamical figure with an appropriate amount of A.M., imparted by spinning the wheel, as in a gyrostat. When the axle carries a flywheel, rotating with angular momentum A*, we take (1) OC = CR + K, and making (2) Kl= CR + K-AE, Jl=AR, then 01 is the instantaneous axis of rotation of the case ; and 01 must pass throuo-h P he point of contact, and the case can roll on the horizontal plane HP (or other surface of revolution about the vertical axis OH) : and the motion can be imitated by rolling the cone lOG fixea m the body on the cone lOA fixed in space about a vertical axis With K o£ a flywheel at disposal, any arbitrary figure can be made to suit the dynamical conditions. But when the body has no internal flywheel, and rolls ns one solid, it_ IS not so easy to draw a figure which will be possible dynamically as well ns geometrically as we have to make it. •' . 16. In trundling a hoop, or in the wheel race of military sports, with the axle nearlv horizontal, the circle is large described in steady motion ; so that GI' is too lar^e t^ appear m a figure to scale, the precession m being small *= For guiding the gun wheel in the race by the hand on the hub, the pressure must be applied m a plane at right angles to the direction in which it is desired to move the .fxle! The man running alongside with the wheel on his rio-ht side a (''^ownwardN . ^^ , n ° ' \ upward ) '^^ '^^ the hub will steer the wheel to the f !^ , ) \ right/* If the wheel i. turning to the (^jg*) on a circle, and leaning over to the r'}^]^^) I forward \ ^ ^'^it /' ^ P"'^ Uackwardj ''^ ^^^" ^"b will bring the wheel more upright. 15 When the wheel or hoop has fallen over, nearly flat on the ground, and it is desired to raise it again, a push on the rim downwards at the rising part is required ; this can be practised with plate or dish spinning on a table, or with a hoop on the ground. Steering an upright wheel to the left, by a downward push on the left hub, is equivalent to a preponderance as if the centre of gravity was at Gr in fig. 25, out of the plane PJ of the wheel, J its centre. Then with PE parallel to AZ', and o, p, r, E in coincidence, also q and I, and A ; and with (1) • EG.G^=^, Ir is vertical, and G^ perpendicular to Gr ; (2) tan, = :^, tan/3 = f = ^^-^, tan a = f tan /3 = ^j, where b denotes the radius of the wheel. With GJ = A, GE = «, EJ = A - ic, OG = OJ - A, (3) tany = ^-^, x = h-^{OJ-h), A and so that Mb where, V denoting the velocity of the wheel, (6) OJ = ^, X = 4, 11, 1 ,- rr • 2A \/ (gb) 1,9 1 1 /^T 2A and the least value oi ( is — p '^-^, when b fi^ = g, and then UJ = —^ • With the C.G. on the axle at G, out of the plane PJ of the wheel, or with a vertical force applied to the axle, the wheel can run straight, with the axle GJ inclined, and G vertically above P. Then if it is required to bring the plane P J of the wheel into the vertical, an increase of the vertical downward force on the hub will not raise the plane, but cause the wheel to take up a precession about an upward vertical axis. To make the wheel resume a vertical plane in fig. 22, with the vector of A.M. changed into the position in this fig. 25, a change of A.M. is required in the vertical plane cGc', and in a downward direction, so that a couple must act, represented by an axis downward, and this can be supplied by a forward push at G, or a backward push on the other side of J. So if the man is ( ,=„ j handed, and running on the ( • u^j hand side of the wheel, and the wheel is falling ( , J him, he must f " ,, j the hub horizontally to bring the wheel up to the vertical plane. This action will be found in accordance with the Kelvin rule of § 5, /HurryN ,, . j xv i /rises towardsN ,, ,. , [t\ -I "^ 1 the precession and the axle ( f ii r 1 the vertical position. The wheel, like a bicycle, should be steered towards the side it is tending to fall, either by a forward push on the higher end of the hub, or a pull backward on the lower end. Experiment with a boy's hoop, or better with a bicycle wheel detached, with the axle projecting both ends, and running freely inside the ball bearings ; the axle can be pressed in any direction, to show the result and effect. 16 The Top on the Top of a Top^ 17. This is an experiment suggested by Maxwell {Life, p. 232) ; it can be carried out easily when the point E o£ the small gyroscope wheel of fig. 3 is placed in a small cup fastened at P the upper end of the stalk of large wheel of fig. 2, provided the stalk OiP is not too long ; the point Oi of the large wheel resting in a fixed cup (fig. "iQ)], thus forming a gyrostatic chain of two links when both wheels are set spinning. Distinguishing the lower wheel by a suffix 1, and putting OiE = a, it is required to determine the condition of steady motion, with the two axles in one vertical plane, moving steadily round the vertical with precession ^. Denoting by X and Z the horizontal and vertical component of the reaction at E, then for. the motion of G in the upper wheel, (1) X= ^i^M (a sin 0i + A sin 0), Z = gM; and taking moments round G, (2) u . ck = fi (CE smd — Ap sin d cos 0) = Xh cos 6 + Zh sin = /x^M (a sin di + h sin 0) cos 6 + gMh sin 6. For the motion of the lower wheel, taking moments round Oi, (3) u. CiKi = yu [CyRi sin 9^ - (J., + M^hi^) fi sin 9i cos flj = fjMifii sin Of + Xa cos 9^ + Za sin 9^ = gM^hi sin 9-^ + jx^M (a sin 9^ + h sin 9) a cos 9^ + gMa sin 9i ; and (2), (3) are two equations connecting ^t, 0, 0^, 5, R^, and the physical constants. With Ml, ill, Ci zero we arrive at the previous case in §13. When the axles are near the vertical position, put cos = 1, cos 0i = 1, but retain sin 9 and sin 0i, then (4) [A + Mh^) 11? - CEfx + gUK\ sin + ^^ Mah sin 9^ = 0, (5) [{A + M, h\ + Ma')fx' - CiR, ^ + gMA] sin 9^ + fx'Mah sin = ; and eliminating sin 9 / sin 9i, (6) [{A+ Mh') ^2 - CR^^ + gSfh] [{A, + M, h\ + Ma?) ^^ - C,R^ a + qMJiA ~fxmVh' = 0, a biquadratic equation for fi, but Hnear for CR or CiRi. With i? = 0, Sj = 0, and g reversed, a double conical pendulum is obtained giving the fundamental plane oscillation of a double pendulum. When required to act like a bell and the clapper, the extra condition 9 = 9, must be introduced into the equations. For a geometrical representation in the preceding manner, make in fig. 26, (7) i¥i. Gi A, + M.Op, = M. Oi p,, M. GjA^ = M. j?,«„, then '^ ^ (8) «. ck = g.M. Gp, f,. CiKi = gM. Op„ 9i^^9Pl ck Grp ' where 0,p, = lhp,,0,p, =p,p and thence K,G\, perpendicular to O^C ; this settles the right amount of CiRi to trnn the motion. Gyroscopic effect in a Wheeled Cahriage, Paddle or Screw Steamer. .1, /?; J'°°!''''/ ^^''''^' *^' T^*°'^^ ^•^- °^ ^^^ wheels of a carriage is drawn across to the left hand to represent CR, C denoting the total M.I. of thf wheels^ndTthir angu ar velocity ; i? = «/;,, when movmg with velocity^, h denoting the radius of the wheels, taken as equal, as on a railway carriage (fio-. 27). Study the motor-car, motibus, and electric tramcar ' (1) as, = el^ (absolute, C. G. S. ergs), but '2 f/s^. For D = %°,N = 150, 21 = 30, T = 10, ~ = 3^, this fraction works out to about one in 400. Similarly for the rolling of a paddle steamer ; this can be imitated experimentally by the oscillation of a gyroscope flywheel, mounted across a body with a rounded base, resting on the table, as in fig. 28. Thus a screw steamer against waves ahead, and a paddle steamer in a beam sea, should show a tendency of swinging to one side and the other of the course, called yawing ; and to a modified extent when crossing the waves on a slant. So also on a motor car over a hump or dip in the road, as on a switchback railway. But the resultant gyroscopic effect is zero of the oars of a boat, or of the twin screws of a steamer or airship. Worthington's Dynamics of Rotation may be consulted for other practical and applications. 20. So far the gyroscopic effect of the wheels has been concentrated into a single :fly wheel of the same A.M., set across the carriage, or longitudinally for the motor. But consider more closely the case of a pair of wheels on the same axle, which can rotate independently by the insertion of a differential gear, or as in the ordinary cart on the road, required for turning a sharp corner. On a curve of radius a and a gauge 21, the angular velocity of the ^?'^^^*'^ wheel is (-,\ cy\ a ± I a ±1 ;at a velocity v ; and the A.M. about the axle is /a\ ^ ^ a + I , a — I ^au) ^ v l «' ^ 2 ^ —J CO = ^ -^ = ^ ^, and the vector velocity is 6' ^, as before ; and M is shared between M^ and Mo on the rails, In a railway carriage a pair of wheels is keyed to an axle, making the grinding velocity on the rail vl/a ; and then the extra resistance against a friction coefficient of •one in n. would be equivalent to an incline of one in n 2 ^h + t\ At 30 miles an hour round a curve of 10 chains radius, this wwks out to an ennival^nf anchne of one in 1,200, taking n = 5 ; but 4.0% steeper on a l-^^ngT ^ -^^^^^^^^^^K^ and the components of momentum are still Mu, Mv, having a risul anT.k.i^o- OO Iw V' momentum is unaltered, and no force is required to hlld the bodv ^ ' '"^ *^** *^^ ^- 1 5 ^{^^^ ^^ '• ^'^^^o^^'ied by a fluid medium, and M' is the on.i.fit^ .f ^• displaced, the medium is set in motion ; and when the bodvll "1'''^''* ^^ ?\ medium to the direction of the component u and v the comnoneSf '^'^^^^t^^al with respect can be expressed by aM'u\ (3M\ where I Tare Ttain ' "'•°^'"*^^"^ ^^ ^^^ ^«dium the shape of the body, and 6 > „ if the bodv ?« 1 "^icaUactors depending on broadsid'e movement .,^iike an;i?shipsidtgthrouoh1he^ "^' ^ " the'fact«r%or The resultant momentum F of the hndv nnri fK^ j- ' OF, will not be in the direction 00', and will chan' to O'F'' '^^'^'^'^'^i by the vector ■of momentum will be the impulse couple (fig. 29) ^ ^^ at u ; so that the change (1) -^'i = OF. 00' sin FO 0' ■ and putting .lOO' = 6, ^.QF = ^ foO'' ^ a F cos ♦ = (.1/ + ,..Mn„, J, ,i„ ^ _ (^„ ^ ^,,,^^, . 19 (S) Ni = OF. 00' sin (i, - 9) = OF sin f. 00' cos - OF cos f. 00' sin 6 = M' {(3 ~ a) uvt, and iVi, the impulse couple, is the accumulation in the time t of the finite incessant couple N, vector product of velocity and momentum, (4) N='^'=M'i(i- a) uv = {m^^W- MTW') ^' '^ ^ ^"^ ^- This N is the couple which acts on the medium when the body is held and moved parallel to itself ; the medium reacts on the body with an equal and opposite couple, tending to place the body broadside to the stream. Thus a body like a plate, settlmg m water, will choose the position of greatest resistance. In gravitation measure the value of N is (4) must be divided by g, and the magnitude of the couple is given in ft-lb by (5) N = M' (/3 - a) ^. The determination of a and /3 must be carried out on hydrodynamical theory, and so far has been eflFected for a few simple surfaces only, including the quadric surface. Thus for instance if fig. 29 is the representation of the cross section of an elliptic cylinder, we find by hydrodynamical theory. (7) N = M' ^ ~^ "w = Trp {c? - b'^) m = M"uv, where p is the density of the medium, and M" the quantity of it displaced by a circular cylinder, on a diameter equal to the distance of the foci of the ellipse. If the shape of the body, an airship or submarine boat, is an elongated surface of revolution, and if the body is moving in a direction nearly axial, so that 6 and ^ is small, the effect of the couple N is to remove the point of application of the resultant force of resistance from the centre to another centre of effort E. In a submarine boat the effect of the couple N is felt as a diminution of the longitudinal metacentric height. The body moving freely at an average uniform depth, the vector OF will have a fixed horizontal direction ; and the couple N will cause a loss h of longitudinal metacentric height, given by 1 1 M + aM' ~ M + (3M' and with no buoyancy, M' = M ; and with small, u = V, F = MV, The same would apply to an elongated airship, and account for the difficulty of keeping a straight course. 22. An elongated shot in the air spinning with angular velocity p about OA, and component A.M. Cj p, and precessing with angular velocity ^ about OF, will have component angular velocity /n sin f about OB, and A.M. component C^, /" sin (j>. Here Ci is Mki^, the M.I. of the shot about OA, as the component p does not set the medium in motion ; but C2 is Mk2^, the M.I. of the shot about OB, together with a term e M'k'^, due to the stirring up of the medium by the component ^u sin (^ about OB ; this term however is insensible for a shot in the air, where M'/M is small. The condition for a steady precession about OF is then (1) {Ci p sin .'cosi>~ C,p,. + {jf^^^p - MTJW') (^+ «^^-^')^ "' «««^ = 0- The least admissible value oi p for stability is that which makes the 'roots equal- of this quadratic in fi ; (3) " = J 7T P »ec

= -^^ = 4^(/3-«) ^1^ i + ^if neglecting the square of M' /M ; and then Energy of rotation Energy of translation " u Angular momentum q, , \ fo \^'~\ Linear momentum L MA If the shot is moving as if just fired from a gun of calibre d, in which the pitch of the rifling is n calibres, and S is the angle of the groove with the axis of the gun, The chief theoretical difficulty here is the determination of a, j3, and j3 — a ; the «hape of the body must be assimilated to an elongated spheroid, x = 2a/2h diameters long, and then hydrodynamical theory shows that /S^ _ A _ ^ p, B \-A " ~ 25 ~ 1 - ^ where, with a = ha'., \ = h^z, 1 - B~ I + A' (9) <10) Jo (A + a'y{X +b')- j, (z + ^xdz {2 + a?f{z + 1) 1 X' B -h laVdX \(z + / - 1' ixdz (X + a=)*(x + by - ](z + x'y{z + ly {x- - ly "- ' s? - 1' <11) .4 + 25 = 1, x'A + 25 = - ^/'_^^ sh-i ^ (..^' - 1), ■changing to (12) ^U + 25 = ^ ^^ ^ ^^ sin-i V (1 - ^2) for an oblate spheriod, x < 1, )3 < „. The following table has been calculated by Mr. A. G. Hadcock from the formulas lABLE. 0-0 0-5 1-0 2-0 2-5 3-0 3-5 4-0 /3 — a — 00 -2-215 O-OOO 0-494 606 682 737 778 4-5 5 6 7 8 9 10 CO /3 — a 0-810 835 872 897 015 929 939 1-000 For the sphere <13) or = 1, .4 = 5 = i, „ = /3 = 1 Thus a spherical bubble starts up in water with acceleration double a For the cylinder ■ (14) a; = 00, .4 = 0, 5=1. " = 0, |3 = 1 ; and for a very elongated body we can takp 9, f^.,^ *v,' *. n value X, and 'replace a and /3 by and 1 ~ " ' ^''^^' corresponding to the 21 For the disc (15) ^ = 0, ^ = 1, B = 0, l3 = 0, a = oo, (3 - a = - oo. For small values of S, expressed in degrees, the formula in (7) (16) n tan S = ir, may be replaced by nS = 180, which will serve for all practical purpose ; thus if n = 30, 8 = 6°, but otherwise tan « = - 0. 1047 = tan 5" .59'. i n The Whitehead torpedo is of the nature of a submarine boat, as well as the Howell- Obry described in Webster's Dynamics, in which the stability of the course must be secured automatically, by means of a revolving flywheel which is the source of the propulsive energy. The axle of the flywheel is placed across the axis of figure, but the theory of stability is essentially the same as before. 23. The influence of the couple N on an elongated body moving axially in removing the point of application of the resistance R of the air from the centre of figure to a centre of effort E some distance in front of is shown in the Report of the Gottingen Aeronautical Laboratory, published in Engineering, August 11, 1911, with the graphical result in fig. 30. As in § 21 (8), (9), and if R is taken as a frictional drag on the surface S, assumed to vary as the square of the velocity F, and expressed by the formula where the velocity CI is determined by experiment. Consider an elongated body of diameter d, n diameters long, in which we may take the surface as equivalent to the enveloping cylinder, S = ir (Pn ; and in air of specific volume C, M' = ^ t (Pn -=- C ; so that ,.. OE 1 ,^ a + a U' With F.P.S. units, we may take C = 12.5 ftyib, and g = 32 f/s^, making .^C = 400 ; and (^) ~d- = 6^^ -"^rrj (20) • In fig. 29, for a body about seven calibres long, we may take fi — a = 0*9 from the Table on p. 20, and replace o by 0, and j3 by 1 in ^, making it ^ ; so that OE r...r..r/U' (2) ^ = ^{tj) <6) ^ = 0-075 (I^J d V20> Thus if the figure shows OE = l'2d, this implies that U is about 250 f/s. This is too small a value to agree with experiment, which makes U about 360 f/s ; but the discrepancy must be explained as due to neglecting head resistance, and taking the whole resistance as due to fi-ictional drag. Thus if we take U = 360, we should find OE = 24c?, double the value shown in the figure. The curious evolutions of a leaf or card falling in the air or of a plate sinking in water are explained by the couple due to the reaction of the medium, also the action of the Boomerang (G. T, Walker, Math. Encyclopcedia). Experiment with a card, flying horizontally through the air with spin ; the card is seen to turn over gradually as the direction of motion becomes inclined to its plane, and finally it gyrates like a top. Balance the card on a finger, and project by an excentric tap with a pencil or conjurer's instrument ; it is dangerous to flick it with the thumb nail, as inflammation is -easily set up. 22 Appendix on Genekal Dynamical TnEORy. 24. The Lemma employed in the dynamical treatment was stated in § 1 — " The vector velocity of the momentum, linear or angular, is equal to the vector of the impressed force or couple." To this may be added " The vector of hnear or angular momentum determines the impulse or impulse couple required to start the motion from rest ; or if reversed to stop it dead again." Thus in Hydrodynamics the velocity function gives the impulse required to stop the motion of the liquid. These principles require to be translated into the equations of a dynamical problem, expressed either in the abridged vectorical system in accordance with the modern pro- cedure of Maxwell, Clifford, Gribbs, Burali-Forti and Marcolongo, Foppl, Jahnke, Grans, Silberstein, and others ; or as formerly, with the ordinary scaffolding of Cartesian and Eulerian co-ordinates. (Consult Maxwell Scientific Papers II., p. 257, on the Classification of Physical- Quantities.) In Maiter and Motion, these fundamental principles of Dynamics are proved by Maxwell first for a single particle, and then for a system of a pair of particles ; thence the extension is made to three or more particles, either discrete or solidified into a body, or an aggregation of continuous bodies, rigid, elastic, or fluid, as in the Solar System. A particle at P, ^ grams, moving with velocity q cm/sec, is said to have momentum f.iq, g-cm/sec, or dyne-sees, in the C.G.S. system. If no force acts on it, the particle will retain this momentum unchanged, in accordance with Newton's First Law of Motion. Newton's Second Law asserts that, if a force acts on the particle the chano-e of motion (momentum) is proportional to the force and in the same direction. Thus if a force F dynes acts for t seconds, it will generate momentum Ft dyne-sees, represented by a vector in its direction ; and this vector must be added by the vector law to the original vector of momentum to obtain the resultant vector of momentum. The vector velocity of the momentum of the particle is given then bv the force vector ±1. <=> J ■ ^J *^«^.Tbird Law of Motion, the action and reaction is equal and opposite between a pair of particles; so that if the vector momentum of two particles is taken this is unchanged by their mutual action, and the vector momentum velocitv is equal to the vector of the impressed force, acting on either of the particles. " The extension is then made to a system of three or more particles, or to an ao-o-rea-a- tion of particles, solidified in continuous bodies ; so that, as stated already in § 1 ""^ the iZtlTZ:^^''''' '' '^' "^""^*"" '' ^ ^^'^''^' ^y^*^- - g-- ^T the'vector of 26. The angular momentum (A.M.) of the particle ,. at P about a noint is through perpendicular to the plane OPQ ^" .he oo";ir,f„rr;;elf ?Qtr^^^^^^^^^ *^ r''-^' °' " »^ between pg and 01. peipenaiculai to 01, multiplied by the perpendicular OFQ.^^KT^r^h^^,^^^^^^^^ ^? ..presented by twice the area A.M. about 01 is given by twicf thefrea 1^.7 ' ^ perpendicular to 01, the Then if is the angle between OH nnrl HT +i,^ t A.M. about 01 is cos 6^ time, the H about' OH TtbS X T' ^^i^^ " '""''^'^^ '^'^ vector law ; that is, if OH represents the resiUtent 1 M I A^' '""'l^f ''T^''^ ^>' *'^^' perpendicular to the plane OPO and OT IhTn . d^*" V^^'^^<^^^ about the axis ingle with OH, ^ ^' ^'' ^^ *^' component A.M. about an axis 01 at au (^) 01 = OH cos H. 23 Draw three rectangular axes Oai, Oy, Oz through 0, and project the area OPQ on to the three co-ordinate planes into OXX', OY Y', OZZ' ; then by a fundamental theorem of Solid Geometry, the projection Ipq of OPQ on any other plane at an angle 0, is the sum of the projections of OXX', OYY', OZZ' on this plane ; and this proves the theorem of the composition of A.M.. For if Aj, 7i2) ^s denote the components of A.M. of unit particle about the axes ■Ox, Oy, Oz, represented by the area OXX', OYY', OZZ' ; and if G denotes the resultant A.M., and L, M, N the direction cosines of OH, (2) h^ = LG, h = MG, h = NG, and if the line 01 has direction cosines I, m, n, (3) G cos 6 = Ihi + mhz + nh^, from the rule for the composition of vectors ; this is equivalent to "the theorem of Solid Geometry (4) cos d = IL + m M + nN. 27. When the motion takes place in one plane, say Oxy, with P at («, y), and with •components u, v of the velocity q, then in fig. 80, the A.M. of unit particle at P about Oz .(1) h = PQ, OD = twice the triangle OPQ = parallelogram PRR'Q = rectangle LNN'L' = rect. PN' - rect. PL' = rect. PW - rect. PM' = XV — yu. Generally in space, if P at (^, y, z) has velocity components u, v, w, then in ^addition (2) hi = yw — zv, h^ = zu — xw ; and (3) becomes (3) G cos Q = I {yw — zv) + m {zu — xw) + n {xv — yu). This can be proved independently by the formulas of Solid Geometry ; for the •component A.M. of unit particle at P about an axis 01 is U. PI, where U is the component velocity perpendicular to the plane OPI, and PI is the perpendicular on the axis 01. Then if I' , m , n , are the direction cosines of the normal to the plane OPI, (4) I'l ¥ m'm + n'n = 0, (5) I'x + m'y + n'z = lO ; thence (%-) ^ _ "^' _ ^' _ ^ ^ ' zm — yn xn — zl yl — zm PI ' :and (7) U = I'u + m'v + n'lc, so that, as expressed in (3), ■(8) U. PI = (I'u + m'o + n'w) PI = {zm — yn) u + {xn — zl) v + {yl — xm) iv = I {yw — zv) + m {zu — xiv) + n {xv — yu) = Ihi + mhi + nhy By addition, for an aggregation of particles such as ^i, (9) hi = 2^ {yw — zv), h^ — 2/a {zu — xw), Ag = 2^ {xv — yu) 28. For the single particle of § 25, according to the treatment of Matter and Motion LXX, the moment of the impulse about an axis appears as the increase of A.M. about the axis. If a pair of particles is taken, the force between them arising out of the mutual action is, by the Third Law of Motion, equal and opposite on such particle, and so does not •change the A.M. ; so that the change of A.M. of the pair due to an external impulse on either particle is equal to the moment of the impulse ; and so on generally for any number of particles, and their aggregate. Expressed by the Vector Law, the vector of additional A.M. is the vector of the moment of the applied impulse couple ; and the vector velocity of the A.M. is equal to the vector of the applied couple. 9A Suppose the aggregation of particles to be solidified into a single rigid body^ rotating about a fixed axis Qz with angular velocity R ; then for the particle w at r in fig. 30, q = OP. -R, perpendicular to OP, (1) u = - yR, V = X R, w = ; (2) XV - yu = («^ + y^) R, and for the whole solid body (3) h, = ^fi(w' + f) R = CR, C denoting 1 fi {x^ + y^), the M.I. about Oz. An angular velocity w about any other axis through 0, which we may take to be 01, having direction cosines /, m, n, will give P the velocity w. PI perpendicular to the plane OPI ; the vector product of the vector OP and 01 the vector of angular velocity will give the velocity of P, having components (4) u = wt. PI = w (zm — yn), V = wm'.PJ = 0) (xn — zl), w = wn . PI = b) (yl — xm) ; as in (8) §27 ; so that, if we put (5) wl = P, com — Q, ton = R, (6) u = zQ — yR, V = xR — zP, w = yP — xQ, the same as if the body had separate components P, Q, R of angular velocity (A.V.) about 0£, Oy, Oz ; hence the composition and resolution is proved of angular velocity. With the three components P, Q, R of A.V., and for a single unit particle at P, (7) hi = yw — zv = {y^ + z^)P — xyQ — xzR, Aa = zu — xw = — yxP + {x^ + z^)Q — yzR, A3 = XV — yu = — zxP — zyil + (a;^ + y^^)R. and for the aggregate solid rigid body (8) hi= + AP - FQ- ER, ho = - FP + BQ - BR, h = - EP- DQ + CR, where A, B, C denote S/. {y^ + z'^), 2|t (^^ + x^), ^fi (x^ + y'^), the M.I. of the body about Ox, Ov, Oz ; and D, E, F denote I,fxyz, ^fizx, 'Efixy, called ijroducts of inertia 29. Twice the kinetic energy (K.E.) of unit particle at P is, in ergs, (1) u' + „2 + ,„2 _ (^2 + ^2)p2 + (^2 + ^2)Q3 + (^^,. + ^^^2)2?2 _ ^y^QR _ 2zxRP - 2xyPQ ; and twice the K.E. of the whole body is therefore (2) 2T=AP' + BQ' + CR' - 2DQR - 2ERP - 2FPQ. But twice the K.E. is also (3) 2At. PL' 0,=' = M;V, where m^ is the MI of the body about 01, the axis of rotation : and if / m n are the direction cosines of 01, with P, Q, R = o. {I, m, n), ' ' (4) _ Mk' = AP + Bm' + Cn' - wLn - 2Enl - 2Flm, so that if the quadric surface is drawn whose equation is (5) K^' = Ax' + Bf + Cc' - 2Dyz - 2Ezx - 2Fxy, and if the hne 01 cuts the surface in the point I {x, y, z), where x, y, z = 01 (I m n) (6) M1^ = K-^^^ and k varies inversely as 01. Since M and ifF is always positive with u Dositivp for r*.n1 n.of*-^.. ^1 c be an ellipsoid, called the momental ellipsoid (M.E) ^^ ''''''^ Then in (8) §28 «^ c?Q ^ dR' ^^'^ 2r = Ph, + (ih, + Rh,, the scalar product of the vectors of A.V. and A.M 25 30. In the Hamiltonian method, T is expressed as a quadratic function of Ai, Agj ^3> in the form (1) 2r = Ahy^ + B'hi + C'h\ - Wh, h, - 2E'h, h, - 2F'h, h, ; and then by an algebraical procedure with the equation (8) §28, (2) A', B\ C\ D\ E\ F = {If - BC\ W - CA, F' - AB, AD + EF, BE + FD, CF + DE) -H {AD^ + BW+ CF - ABC + 2 DEF) dT (3) P = Jh,= ^'^1 - ^'^2 - E'h,, Q=~ = - F% + B'h^ - JJ'h,, i? = ^ = - EK - D'h + C'h ; /A\ a m T dT dT dT 31. If the origin is itself in movement with velocity components U, V, IV, (1) 2T=^a[{U + zQ- yRf + {V + x R -_z Pf ^ {W + yP- x Qf] = M{U^ + ^^ + W^) + 2M [(y W - zV)P + {z C - xW)Q+ {x V - y U) R] + AF + BQ' + CR' - 2 DQR - 2 ERF - 2 FFQ, where x,y, z denote the co-ordinates of the centre of gravity (C.G.) defined by (2) Su X = Mx, 'Euy = My, S^<2 = Mz. When is placed at the C.G., this expression for the K.E. is simplified to (3) 2T = M{TP + V^ + W) + AF + BQ' + CR' - 2 DQR - 2 ERP - 2FPQ, so that the K.E, is composed of two parts, (i) the K.E. of the whole body condensed into a single particle M a.t the C.G., (ii) the K.E. relative to the C.G. ; and these two parts may be treated independently. This is true generally for any system of bodies — rigid, elastic, or fluid ; the piinciple is proved in Matter and Motion, Ixxix., Ixxx., for a system oftwo particles ; and thence by extension to any number of particles, forming a material system. 32. Suppose the body is revolving about the axis Oz, held fixed, with angular velocity R ; the centrifugal force (C.F.) of the particle ^ at P (x, y, z) has the components (1) -M■R^ f.yR^ 0, having moments about the axes (2) - fizxR'-, fjizyR\ 0, and the moments for the whole body are (3) - ER\ DR', 0. These are the component couples with which the rotating body acts upon the axis Oz. The couples vanish when D = 0, E = 0, and the axis Oz is then called a, principal axis (Segner) ; a principal axis is a permanent axis of rotation when released and free, as the C. F. has no tendency to make it alter. 33. The axes of the M.E. are called the principal axes of the body at ; and changing to these as co-ordinate axes, the P.I.'s, D, E, F, are made to vanish, and (5) § 29 becomes (1) ■ Ax' + By' + Cz' = KS% the equation of the M.E. referred to principal axes. 28570 ii L^6 If g is taken as the length of the perpendicular from on the tangent plane of the M.E. at I, and K to correspond so that K$^ is kept constant, the direction cosines of this perpendicular are Ax, By, Cz <2) M ' with ,(3) A'a? + Bhf + CV = m\ Then we can write (4) 2r = ilfX,V = K ^2 u? = Kh\ where h is an angular velocity, the component about OH, making ,(5) h^J^=^ = ^ = ^- 8 01 .« y z ' and then if G denotes the resultant A.M., (6) 0' = hi' + hj + hi = A'F^ + B'Q + air- = {AV- + By + CV) ^= K'¥. V 34. When the motion of the body about a fixed point is started from rest by an impulse couple G, whose vector is OH, the angular velocity is about an axis 01 -conjugate to the plane of G with respect to the M.E. ; and to stop the motion again, the impulse couple must be applied in a plane conjugate to the axis 01 of resultant A.V. So also an applied couple generates Angular acceleration and velocity about an axis conjugate to its plane, with respect to the momental ellipsoid. When the body is rotating about an axis 01, the momentum of a particle fi at P is changed into the C.F. by multiplying by the A.V.&j, and turning the direction backwards through a right angle from the perpendicular to the plane POL The resultant C.F. couple has no component to correspond with il/Pw, the A.M. •about 01 ; but the resultant C.F. couple is w times the component A.M. perpendicular to 01, about a vector perpendicular to the plane lOH, where OH is the vector of resultant AM.G. ; so that the C.F. couple is w G^ sin lOH, in the plane lOH, the vector product of the A.V. and' the A.M.; and this is the couple felt by the axis 01 of rotation, when it is held fixed. The body feels the couple the other way, and the axis of the C. F. couple is a vector ■equal to the velocity of H by its rotation round 01 ; the rate of change of A.M. in space being equal to this C.F. couple. But if the axis of rotation is released, the vector OH. of A.M. becomes stationary, the former velocity of H due to rotation round 01 being neutralised bv an -equal and opposite velocity due to the couple of C.F. ; and 01 now moves, in space .and in the body, the velocity of I being parallel to the diameter conjugate to the plane lOH ; because the C.F. couple m the plane lOH will generate angular acceleration and velocity about an axis conjugate to the plane lOFI. 35. This is the foundation of Poinsot's representation of the motion of a rigid body .about a fixed pouit under no force ; realised if a body is tossed in the air in anv manner, when the C.G. describes a parabola, and the body moves about the C G as if ft was a fixed point 0. ■ . o u it ...ted by.o„i.g .he M.E, o„ a fixed plLJe a.,. St ^t's r'o.'mS the A.V. about 01 proportional to 01. ' ""^"^5 27 36. The C.F. couple, vector product of components P, Q, B, of o>, and AP, BQ, CR of 6r, has the components (1) {B - C) QR, {C - A) RP, (A - B) PQ ; so that the components of velocity through the body of the change of A.M. are given by (2) A^ = {B-C)QR+L, B^={C-A) RP + M, C^ = (A-B) PQ + N ; or (3) ~^ = Rh,- Qh,+ L, "^^ =Ph,-Rh + M, ^=Qh,-Ph + N; where L, M, N, denote the components of an applied couple ; and these are Euler's equations of motion. Under no force, L, M, N vanish, and we find the integrals of which are given in (4) and (6) § 33, with K and h constant. Also dw pdP dQ , pdR (B-C C-A A-B\-p^f, B - C. C - A.A- B p^j. , and from the three equations (6) I^ + Q^ + R' = u>\ AP' + BQ' + CR' = Kh', A'P' + B'Q' + CR' = K'h', we find by solution jK-B- C) Kh' + BCu.' _ -BC , (') ^'= A-B.A-C A-B.A-C^'^'' " " -*' (K - C - A) Kh' + CAu,' ^ - CA , ^ - B-C:B -A B - C.B - A ^'^*' "'"''' _ (A' - A - B) Kh' + ABu,' _ - AB , ^^ - C - A.C - B C - A.C - B ^'^'^ ~ '^'^ ,^, 2 ., , f- K + B + C - K + C + A - K + A + B\ ..,, [oj w„ , Wft-, w,-, = (^ -g^ , ^j^ ) jj^ J J^^fi ; so that (o) may be written d ' (9) -^ = ^ (4. w„' - w'. Wt' - w'. w,'— w'), an equation discussed in Chapter IV. 37. As K and S are arbitrary, the scale of the M.E. is not determinate, the shape is given, not the size ; and K is changed in the sequel to i>, as the letter is required no longer to denote a P.I. The inverse surface of the M.E. is called the ellipsoid of gyration (E.G.) ; and its equation is written in the homogeneous form ^^> A'^ B'^ C~ M' this ellipsoid has a definite size ; and such that the radius of gyration about any line through its centre is the length of the perpendicular on the tangent plane at right angles to the line. The artiele by Stackel, Elementary Dynamics, in the \lath. Encycloptedia will give the historical notices of the writers quoted ; Newton, Euler, Segner, Dalembert, Lagrange, Poinsot, Hamilton, Maxwell, Clifford. 2S570 ^ 2 28 CHAPTEK II. Gyroscopic Applications. 1. The instrument of fig. 32 is called a gyroscope, in which a flywheel is mounted in gimbal rings, which permit universal direction o£ its axle in the two degrees of freedom. The arrangement is open to view, as implied in the name gyroscope, and not ■concealed in a case, when it is called a gyrostat, as shown in fig. 33. A cheap, effective form is shown in fig. 34, 35, price one shilling, acting as a .gyroscope or gyrostat. The flywheel is spun by a string, but the treacherous use of string in a lecture is avoided in the larger gyroscope of fig. 36, the apparatus of fig. 3, or a duplicate, with the beam laid on its back on the lecture table ; a new stalk is used prolonged past the pin 0, so as to carry a counterweight to give any desired preponderance, or to counterbalance in equilibrium. Spin sufficient can then be given to the wheel by hand, as before ; and the apparatus is large enough to be visible to the audience, and a length of thread can be used attached to the tail at P to be pulled in any desired direction, and the subsequent :action studied of the axle. The axle remains at rest if counterbalanced ; but with a spin of the flywheel represented on the right-handed system by the vector OC of A.M., pulhng the thread horizontally to the (j.i^jj^) applies a couple to the body about an axis drawn (.^^ ), ;and the wheel and its axle (f^?f^). ^ ( 7pJ^f) P"" «^ *h^ tb^^^d '« equivalent to an (^l^^'^Xn) "^ *^« P^^P^^^' -derance at P, and apphes a couple about an axis drawn to the (^*^i^^), so the wheel ■and axle swing to the (^^^f^ ). When the pull of the thread ceases, the axle comes to rest again apparently, except tor a slight nnperceptible nutation. ^ As soon, however, as the axle touches a stop, the gyroscopic influence ceases, and the wheel moves unmfluenced by the rotation ; the same also, if the pinch screw at E i. tightened m fig. 31. So too a flying machine, with a right handed screw propeller of considerable -angular moment^um^ as in Chapter ], §4, can be steered in a horizontal plane to the ("St ) ^y (d%res8?ng) ^ ^""^^^^^^^ '"'^'^^e^ '-^^ ; but if the rudder is forward, as in a Wright machine, the action is reversed. The use of the vertical rudder will cause the axle of the machine to (f^^^) according -as the helm is put to ('*^^^^°^'"'^). If the axle rises, the way is checked, and stability is apt to be lost • so that this motion requires caution m the pilot, and it is better for him to use the horizontol ruddt Consult E. W. Bogaert, Veffet gyroscopique et ses applications (Dunod, 1912). 29 Any oscillating body, with one or two degrees of angular freedom of rotation, can be called a gyroscopic pendulum, when it carries a flywheel set in rotation, and is capable of independent motion. To study the motion experimentally, place a gyroscope, such as fig. 31, on a table ■at the top of a pendulum, oscillating about a fixed horizontal axis, as in fig. 44 ; and examine the motion of the axle of the flywheel, and its influence on the oscillation. Examine also the wrenching cfl'ect on the axis or knife edges of the pendulum when the gimbal rings are clamped. An analogous eff'ect is felt on a motor car, running over a high backed bridge, or down and up a dip in the road, tending to cause side slip with fore and aft A.M. ; the magnitude of the couple can be calculated by the rules of Chapter I. A dynamo in rapid rotation, bolted down to the deck, will tend to shear the bolts as the vessel ( -^ i^ ), according as the axle of the dynamo is placed (^ d ft) ^"^ *^^ ship. If the bolts are removed and the dynamo can revolve about a central pivot, it behaves like the flywheel of a gyroscopic pendulum with freedom in ammuth, and the ■controlling action on the oscillation is increased by the additional freedom as with the Schlick sea-gyroscope (Gray, Dynamics pp. 520, 529). So also in the Howell and Obry torpedo, and to a lesser extent in the Whitehead torpedo, where the chief improvement has been made in the thermodynamic efiiciency of the compressed air by internal superheating ; here a flywheel in gimbals acts on a light relay, which works the rudder to preserve the mean course in a series of zigzags. The behaviour of the gyroscopic pendulum is useful in explaining the action of the monorail carriage, fig. 3-7, on a railway gauge reduced to zero, holding itself upright -against gravity by the domination* of an interior flywheel. 3. To obtain a muscular sense of the forces called into play, hold in the hand a gyroscope flywheel in a ring, as in fig. 38, and twist the outer ring in its own plane yOz, backward and forward. In a conventional representation, the flywheel A in fig. 39 might be held at the points of the axle between a finger and thumb ; but mounted in a ring B in fig. 40, it -can be held in the hand as in fig. 38, and the flywheel can spin fi-eely with little friction between the conical points of the axle. A third outer ring C in fig. 41 joined up with B by short axles at y, y', at right angles to the axle of the flywheel, completes the Cardan, gimbal ring, suspension ; :and a stalk D, fixed to C, can hold the apparatus with three degrees of freedom, as in fig. 31 of the gyroscope, complete as manufactured for sale. Holding the ring B in the hand, in fig. 38, an actuating couple A -jr must be supplied in its plane, to give the angular velocity w about the axis Ox perpendicular to the ring, A denoting the M.I. of the flywheel and ring about Ox. If the wheel is spinning with angular velocity R and momentum CR, the vector velocity of the A.M. is CRw, and the ring B provides a constraining couple CRm on the wheel about the axis Oy, which is felt by the hand. But when the ring B is mounted inside the ring C by the short axles y, y' in the line yOy', this constraining couple is absent ; and the ring B will start out of the plane of C with angular acceleration-jj- about Oy, given in fig. 42, 43 by <1) A^= CR. = CRf. Thus when C is turned through a small angle 0, <2) A^^ = CRe; and if held again at this angle 6, the ring B with the wheel A will continue turning about Oy with angular velocity. <3) t = ^««. 80 But if 6 oscillates through a small extent, ^ and gMh, ^ > ^^ Mh ver 60°, that is, when the energy of rotation of the wheel is greater than A/C times the energy acquired by the pendulum in faUing through 60°, or h A/C times the energv acquired in falling from the vertical to the horizontal position ; and A/C is about | for" a wheel like a disc. Hence for the stability of a Brennan monorail carriage, calculate the enerov the carnage would acquire in falling over on its side from the vertical to the horizontal position ; then rather more than one-half of this energy must be imparted to the o-yro scope fly-wheel to ensure the stability of the position uprio-ht. It will be noticed that the axis 0?/ is placed vertical in the stilt gyroscope • and generally the result is mdependent of the angle at which Oy is held, so long fis it is at 'rio-ht angles to the pendulum axis. 1 he axle is shown horizontal in the Bmman monomil carriage, and vertical m the fechlick sea-gyroscope. al •carriage ; this appears to be similar to the brass Blondin, mentioned in Maxwell's " Life,^'' p. 332, in connexion with the experiment of the Top on the top of a Top of Chapter I. § 17. If a.gimbal ring comes against a stop, the gyroscopic influence disappears at once ; hence the wheel must be left quite free ; it must not be required to actuate any mechanism, as in the Bessemer saloon, or in steering a torpedo, except through a light relay. (^Practical Engineer, July 8, 1904. Recent improvements in steering mechanism of a Torpedo, Engineer Gazette Supplement.) In the Schlick sea- gyroscope the oscillation is damped by a hydraulic buffer, the effect of which is investigated later. The arrangement on the Franconia is given in Engineering p. 805, Dec. 15, 1911. 5. A fly-wheel will have the three degrees of freedom of the gimbal rings of the gyroscope if floated in a vessel of water or mercury, or if supported on a rounded base on •a table ; and so the gyroscopic effect of the machinery of a paddle or screw steamer, or motor car can be investigated experimentally. Thus for example, if the Brennan stilt gyroscope has a cylindrical base of radius a and rests on a flat table, we must replace h in (4) § 4 by A — a ; or if it rests on another ■cylindrical surface rounded to a radius 6, then h must be diminished by ^, so that is the condition of upright stability ; as in Gray's model, with small wheels running on a wire, when the sign of a must be changed. The action is improved by giving preponderance to the flywheel, and the theory is discussed on p. 74. Consider the problem of the gyroscopic influence required to steer a ship on its ■course ; also a railway train at full speed. The equations for small displacement are, writing D for the operation d/dt, ,(2) AD'^d + CRD^ = 0, ADd + CRip = 0, (3) (A + A) D^ - CRD9 = couple = (— _ ^') ^, {Chapter I. §21) ; and then (4) (A + A) D^ ^{^^R^^^ + E^^ = ^vhere A^ denotes the M.I. of the ship about a vertical axis, and A of the flywheel and gimbal frame about a diameter. Therefore for stabilitj- Assimilate the ship to an elliptic cylinder surrounded by liquid, progressing over a ■smooth horizontal plane with axial velocity V ; L \' ) A \o a) -.„ ,„, a Wb + W a W + W 7 1 CR' A , Wa + Wb ^W'U'^ C^ Wb + Wa' ■and in a ship, with W = W, energy of flywheel A ^ A.") energy of ship "" C ' in which we may take the ship long enough for e = I ; also -^ = i, in the flywheel. 32 The Theory of Oscillation, Fkee, Damped, and Forced. 6. Representing the operation ^ of a differentiation with respect to the time t by the single letter D (Perry's 0), then the operation of Z) on a simple periodic term, such as E °?® at causes it to change into oE ^?^ {at + Jtt) ; that is, it multiplies by q and gives a lead of ^ir to the phase angle qt. A second application of D has the same effect, so that (1) D'EZ^'"tEZ(V^^)--'tEZj>, and so is equivalent to a mere multiplication by — q^. 1 r*os • • Conversely the inverse operation, expressed by -=^ applied to E ^.^^ qt, divides by q, and gives a lag ^t ; so that /o\ 1 ET cos , E cos , . 1 \ (2) -^E . qt = — . (qt — iTT). ^ '' J) ava^ q sm ^ '' ■ The sine or cosine is used according as we start with t = at the middle of the roll or oscillation, or at the end. More generally, the operation on E cos ^^ ol P + QD, where P and Q are alge- braical constants, is equivalent to a multiplication by s/ (P^ + C^2 + 2/Z> + n^) = 0, with a solution. (7) 6) = V "■'* cos [ V (n^ - /') t + £]. In the case where the viscosity co-efficient / is so large that / > n, the solution of (6) changes to (8) % = Ae^^' + Be -"'^ (9) ' ni,n3=/± V (/^-n^), representing a state of motion called Dead Beat, where the body creeps back to the position of equilibrium, and does not shoot through and oscillate. Illustrate with a hydrometer, in water or vaseline, to show the diiference between a damped vibration, oscillating through the position of equilibrium, with amplitude dying out gradually, and a dead beat motion, creeping up gradually to the position of rest. The damping term is given the form 2/Z> d for analytical convenience, not so much because it represents very accurately the physical facts. In the analytical discussion of a damped vibration it is convenient to put (10) / = n cos y, v/ {n^ — f^) = n sin y, and then the naperian log decrement is 2?? cot y ; and in the solution of the diifcrential equation of damped vibration (11) ^ '^ ^^ ^°^ ^T "^ ^'^^ ^' ^-^' + 2n Z> cos y H- n^) 6 = 0, (12) 6 = V^'^'^cos {nt sin y -h e) and, dropping t, do (13) lTt~ " '^V""*™'^ cos {nt sin y — y) d^% = nXe-««<=osr cos {nt sin y — 2y), J- df reducing to undamped vibration when y Represented graphically by the curve in fig. 48, (14) . y = ae-»-oosr cos {nx sin y), the curve is sinusoidal, with half waves of equal length — = , enclosed between the ^ ° n sm y' exponential curves, (15) y = ± Qfe-«-^oos7j which (14) touches at the mid point of a half wave ; and the log decrement is the ratio of the wave length to the constant subtangent in the exponential curve (15). At a maximum or minimum v, the x is — ? — in advance of a node, and then ^ n sm y ' (16) y ■= ±_ ag-»!a!oos7 gin ^ _ + (3,g-na;co3r + logsinr ; and at an inflexion, the x is t^ — in advance of a node, and then ' n sm y ' {\1\ II = + /yg- «a: cosy + log sin 2y Thus an advance of the curve (14) a distance — ^ '^, —^ ^ will bring all n cos y n cos y dv djV the points where #- = 0, j^= on to the limiting exponential curves in (15). Another graphical representation of the damped vibration can be given by the equiangular spiral, in which the radius vector reA'olves with constant angular velocity n cos y, and the motion of the point on the spiral is projected upon oblique co-ordinate axes inclined at an angle y. 28570 E u 8. The jfree damped vibration in (1) (5) (7) §7 will die out at compound discount ; lout when the vibration is maintained by a periodic forcing term E cos qt, the forced vibration B is given by the differential equation (1) B^e + 2ncosy i>0 + n^O = E cos qt, with a solution (2) ^ = SqCOs {qt - e), having a lag e„ and amplitude 9^ ; and these are determined from the substitution of (2) in (1), makmg (3) - qX cos {qt - e) - %iq% cos y sin {qt - e) + n\ cos {qt - s) - :E cos qt = ; and equating to zero the coefficients of sin qt and cos qt, (4) (n^ — q^) sin e — 2nq cos y cos e = '0, U (5) (n^ - q^) cos e + 2nq cos y sm e - — =0, «o that 2uq cosy (6) tauE = — # /, (7) ^ = ^ [(n^ - ^2)2 ^. 4^2^2 j,og2 ^] = ^ (n* + 2nV cos 2y + ^*). Or immediately, by Perry's symbolical method, E (8) 6 = -f^ ^r^n '<> cos qt, in which i)^ may be replaced by -q^ ; and then in accordance with (4) §6, with P = 2/, Q = n^ - ?^ <9) ^ = — rr^ Iv2 77^-21 COS W - *^'^"' -^^X 9. Schlick's Sea Gyroscope, described in the Transactions of the Institution of Naval Architects (I.^.A.), March, 19U4, for wave control and mitigating the rolling of a ship, is designed to act in the same way as the flywheel of the gyroscopic pendulum •described in (2) §4 ; and with no damping of the vibration, the motion would be given by the same equation, the ship rolling in the small oscillation of a simple pendulum, but a little quicker. The axle of the flywheel is mounted vertical in Schlick's apparatus, in fig. 49, 50, held in a gimbal ring moveable about an axle fixed across the ship ; and to guard against a violent motion or lurch, the damping is made by a piston moving in a cylindrical hydraulic buffer, filled with oil or water, which percolates through an adjust^able hole in the piston from one side to the other. The rolling, too, of the ship is damped by the friction of the stirring up of the water ; and supposing it is found experimentally that the damped oscillation of the ship, rolling freely in still water through a small angle B can be given by (1) B = %e~'* aospt, where w/p is the time in seconds of a roll from side to side, and e""2,r//^ jg ^j^^g decrement factor of the angle Oq in a return in time 2 Tr/p to the same side of the vertical ; then B satisfies the differential equation. (2) Z)^e + 2/2)6 + (/2 + /)fl- 0, the equivalent of the dynamical equation. (3) {AD^ <), + FD . . • - qMhFq „ (4) 0, sxn .. = 00 sm . = ^^ Af + gMhf + F'Y ^ 7. „ AY - Aq'qMh + F'q' ^ (o) 0,cos. = E+ 0ocosa = ^.Af + gMhy + F'f^ (6) tan. = ^^^^ AY - Aq^gMh + F^q (7) Q{- = E^ + HEO.cos, + e,^ ,2 (AY + r-) y^ - (_ Aq' + g3Ihy + FY' Relatively to the ( ^ ^^^^ ) the ship rolls \° .\ ) the wave, according as the •' Vwave suriace/ ^ \ with / ' ° lag (-) is ( P ,. I, or as Aq^ is ( , ) than [qMh, or qMh — -^ ). ° \jj/ Vnegative/' ^ Vgreater/ \if i j a J This can be shown experimentally by a pendulum with a heavy bob, of period 27r/q to represent the long waves of a ground swell, carrying a pin on which a light cardboard midship model section is hung at the metacentre, with a moving weight to raise or lower the centre of gravity. The model should be hung on a revolving crank pin to imitate motion in deep sea waves. The apparatus shows how a ship can be made easier among waves with a small metacentric height, and a free period longer than the wave period. (Consult Maxwell on Grovernors, Scientific Papers, II., p. 105.) ScHLiCK Sea Gyeoscope. II. With a suffix to denote the corresponding quantities for the Schlick gyroscope,. the motion of the axle, when there is no spin and the ship is still, is given by (1) AiD^f + F^Dc^ + gMA ^ = 0, where Fi is the damping factor of the hydraulic buifer ; and the solution of (1) is of the same form as the free damped vibration in (1) §7. The reaction of the damping couple causes the ship to pitch slightly at the same time. But now spin Schlick's flywheel with A.M. K ; then the motion of the ship and of the flywheel will call up a couple KD d acting on the flywheel to increase ^, and KDf on the ship, to decrease 6 ; so that the equations of motion are (2) (AD^ + FD + gMh) 8 = - KD - The dynamical theory can be enunciated in a series of Propositions : — • I. The component angular momentum, CR, of the top about the axle, denoted by G' and represented by the vector OC, is constant in magnitude, because of the uniaxial symmetry ; but OC varies in direction. II. The component angular momentum about the vertical is constant, because the axis of the impressed couple of gravity is horizontal ; denote this vertical component by G, and represent it by the vector OC in fig. 5, repeated in fig. 51. III. Then OK is the vector of component angular momentum in the vertical plane COC, if the planes through C and C, perpendicular to OC and OC, intersect in the horizontal line KH. IV- If KH is the vector of component angular momentum (A.M.) perpendicular to the plane COC, then OH is the vector of resultant A.M. ; and OH can be resolved into the three components ' (1) KH = .4 ~, CK = A sine ^, OC = G' = CR. ' But we must choose carefully between the two ways of measuring 0, either from the upward or downward vertical. Klein, following Euler, selects the upward vertical ; so that V. Projecting the components in (1) on the vertical OC, as in figs. 5 and 51, (2) CK sin + OC cos = OC, (•^) .4sin^0^ + Ci2cos0= (?, (4) > 63, Z2 = cos 62, z^ = cos 6^ ; and with ^i < — l,,we can put Zi = — ch 61 by symmetry ; and then substituting ^^ = 1, — 1, and — E in (29), (42) ( ^' "^ ^ y = ^(l- z,.l-Z2.1 - Z2) = 4 sin2l03 sin^ifl^ ch^Jfli, (43) i ^' ~ ^ )' = § (1 + ^.v 1 + ^2- - 1 - %) = 4cos2i03 cos^JOs sh^S^, (44) {--^~)' = k{- E-z,. -E-Z2.-E-Z,); and from (32), (37), (40), /J.ti^ _ f^™ 2 ^3 sin |- 02 ch J 6*] dz _ rcos J ^g cosing sh ^ 0, dz ^^ "^ ~ J ^ sin^ 10 x/(2Z) J cos^e V(2Z) = L + K(^-^^-^-^^----^--') ^^ -^ sin- --l^)- ■/^ A^ + 2 V{2Z) ■ ^/ (1 -^l^ + 2)* 46 XL The vector of H being p exp tsy/, and its vector velocity ^ p exp m, this velocity is to be equated to the vector of the impressed couple of gravity ; in other words, the hodograph of H is described by the vector axis of the gravity couple. The axis of the gravity couple is of magnitude gMh sm 9, in a direction perpen- dicular to the vertical plane COC, so that the vector axis of the couple is (46) gMh sin Be ii' + i^')^ = iAn? sin Qe^ ; and so the axis describes a curve like the projection on a horizontal plane of a point L, C, or P, fixed in the axle OC of the top, but turned forward through a right angle, ihus d (47) iAn- sin Qe"^ "^ Jt^P ^^P ^^'^ ' thence the motion of the axle can be inferred when the curve of H is known, by means of a differentiation with respect to ^. i. i • This holds equally well for the general unsymmetrical top, but here the analytical difficulty is not yet overcome of the determination of the curve of H, except in a few restricted cases. Drawn to a geometrical scale we take OC = S, OC = S', and choose a length k such that («' w = *ir = ¥' « + 2? = ^ = « + 4:^ and then (49) Jm sin Be^^ = ~ i exp Tui and the projection of P is obtained by taking OP = \k. XII. This curve of H will be identified in the sequel as a Poinsot herpolhode,' of Chapter I., §35, the curve described by the point of contact when a quadric surface with centre at is rolled on the fixed horizontal plane through C. Poinsot shows that the motion of a body supported at its centre of gravity can be realised by rolling the momental ellipsoid on a fixed invariable plane. The Maxwell top can be used to realise this motion experimentally by moving the screws so as to put the body out of kinetic symmetry, and to bring the centre of gravity to the point of support ; also the Prandtl apparatus, described in the Zeitschriftf. Math. u. Physik^ 1912. Denoting CK by p, and making the equation (18) homogeneous, (50) cos e = 2 g - iJ = 2 ^ - i^, and (51) p sin = Scos + S' ; so that, eliminating 0, (52) /[l-(2e;-^)] = (2S^ + 8' - E%^\ the (p, jo) equation of the curve of H. Relatively to the origin 0, this may be written 0H2 „n2i / 0H2 2 (53) (OK^ - «^) L 1 - ( 2-T - ^) ] = (2 S^ + S' - i^S ) From (51) (54>v ^ = _ ^ + ^' cos ^ ^ _k_ h + h' cose _ k d4 So also, putting C'K = p\ /KriN / S + 8' COS0 ^^'^ P = -^irTT-- (56) -f = _ if^iljl ^ _ 1 Acos£+A; kdi. ^ ' ^^ «in'^ » ^^^^J~=- 2^dT' in the Spherical Top, so called when C = A nnd OW ic +u„ • c ■ ^ A.V., as well as of a!m. ' ^""^ ^^ '' *^« =^^'^ of instantaneous 47 Since r.„. . ,^ CC'2 OC^ + 2 OC.OC cos 6 + OC'2 [07) sm- e = Q^ = — ^^^2 , solving the equation as a quadratic in cos 6, ,-., ^ „0H2 ^ OC.OC ± ^/ (0K2 - OCl 0K2 - QC'^) (58) cos0=2-^-ir= -^ — ^^^2 '- so that the stationary values of OK are OC and OC ; and then (o9) cos 6 = -rwT- or OC "' OC" When OK = OC, CK = 0, and OK is the perpendicular oti the tangent at a point of inflexion on the curve of H ; and then the curve of H round C is looped, and (60) 2^'-^^g&- But with OC -= OC, OK = OC, and CK = 0, and the tangent at H passes through C and CH is stationary. The curvature is useful in drawing a curve of H ; denoting the radius of curvature by r, dp' ,„-.. _ dp _ x_d^ _ |_F_sin^ ^^ > '' ~ ^d^ ~2^~g f 8'cos0' de (62) *" TT ~ ^^^ ^^^ ^' *'^^ velocity of H, , . r^ ^F sin^ e p _ . 8 + g' cos e 8 cos fl + 8' ^^ p (8 + 8' cos 0) (8 cos + 8')' r kBinH ' k sinH ' In the plane KCH, representing the angle KCH by x', ^dj ^d9 ,(,A\ ^ ' KH dt dt de (64) tan )( = "-^ — — CK G + (r cos e ^ sin ^ s™ ^ <^^' sin 0- dt so that the axle OC is moving at an angle Jtt — ^ with the vertical plane COC ; and, ^^ as in (40), Q A + h'z ^^'^ ^^' ^ = .^^e^iln.zy "° ^ = sinev(g+.) > 2 ^_±Z5 + i ^ (2Z) (66) sin e%'' = %--^^ ^ '-. Differentiating with respect to t, (67) ^ = 2 ^' .\l' - ^±^ = 'it-^4 - 2(^ - l) A', ^ ^ dt sm'' ^ ^ 2 c?^ dt \C / ' where, as in (36), with (^o', ct') the polar co-ordinates of H in the plane KCH', //JON djs' Ti + h'z 7/ h — Hh' ^^^) -di = fi^TT = ^ ■" fl-^- In accordance then with (40) and (67), r A - ^A' dz (69) (70) .J s/(2^) _, o M 1 \ ^'^, H + z V {2Z) ^ ''"" ^h + h'z^ "" ^ ^ \C n r-h'- Eh n dz , V(2Z) E + z V (2Z) + ^^"^ h' + A^ + ^^' n thus exhibiting the addition and subtraction of the two Elliptic Integrals in (32), of the Third Kind ; and showing the symmetry between t// and ^ in the spherical top. 48 The Elliptic Integeal and Function. 2. The First Elliptic Integral (I. E. I.) in §1 (31) is normalised to the standard form of Legendre by writing it (I) mi = f' /^'j[J'^'^' ^ ™ = '^v^CK^s - -i)] ; and then, putting (2) Zi- z = {z^- z^) sin^ <^, z - z^= {z^ - z^) cos^ ^, z - z^ = {z-^ - Zt) A%, Zo Zo /9 '^9 "~ ^X (3) AV = 1 - k' sin= i>, k' = ^^-^, -c - = ^J-3^, equation (1) becomes (4) mt = j'^^^ = F{,t>), or F{i>, k), in Legendre's notation, where k is called the modulus, and <{, the amplitude ; and i^^ is calculated and given in Table IX of his Fonctions elUptiques, t. II, by double entry, for every degree in the modular angle sin'^K, and every degree of the amplitude (p. Thence, by Abel's inversion, and in Jacobi's notation, with Gudermann's abbreviation, (5) ^ = am mt, sin ^ = sn mt, cos ^ = en mt, A

, a comparison with the Spherical Trigonometry formula (25) cos c = cos (a — h) cos^ ^C + cos (a + b) sin^ JC, suggests another mechanical illustration of the nutation of the axle, by taking a wheel, put out of balance to an appropriate amount, revolving freely under gravity about a smooth axle fixed at an angle J (02 + ^3) with the vertical, the line OP to a point P on the wheel making an angle ^ {O2 — 63) with the axle. The angle 6 made by OP with the vertical will vary in the appropriate manner, with the angle ^ (tt — C) = am mt, as in revolving pendulum motion ; but the motion in azimuth ^ will be different, and not comparable. When a pendulum axis is nearly vertical, the oscillation is very slow, as of a door on hinges not quite plumb ; it acts here as the horizontal pendulum, so called, used for the measurement of a slight variation of gravity, such as due to lunar disturbance. Writing u for mt, and (26) 1 + cos = 2 cos^ 102 sn^u + 2 cos^ ^63 cn^w, 1 — cos = 2 sin^ |02 sn^M + 2 sin^ ^03 cn^w, the integration of (13) (14) §1 can be written - _ /A^ 1 \ ;,' h' — h p du (27) 2 (^ - ^ ) = .^^ - 1 j A ^ + -^^ J ^ ^^g2 xe^ gn^ ^ + COS'' 103 cn'u' (28) U + i^) =(i - 1) k'i + ^^^ r -^^T-a 2 '^'^ • 2 1 fl — 2- ; ^ ^ 2 vv rj \Q ) 2m U sm^ J02 snVM + sm^ ^ 03 cn^M ' /'9q^ h - h' ^ sh |0i cos |-02 cos ^03 h + h' ^ ch ^O^ sin ^02 sin ^3 ^ ^ ^^2^ V (sh^ ^0^ + cos^ 103)' 2m n/ (ch^ I0i - sin^ pj)' Thus in the Spherical Pendulum, with h' = 0, /OAN xuifl * ifl * 1/3 ua 1 + cos 02 cos 03 , „ _ sin 02 sin 03 (30) th i0i = tan M^ tan *03, ch di = ^, sh. 0i =- 5—7— — -J > ^ ^ 21 2 - ^ rf' i cos 0, + cos 03 cos 02 + cos 03^ (31) *=fj^ sin 09 sin 0, dz (1 - z^) y/lzi - JH. z - 22- {22 + 23) z + '^ + ^2^3] 28570 50 3. If s' denotes the length of the arc described by a point fixod in the axle of the top at a distance a from 0, 1 / ds'\^ _ H + z jL _ T' / ^ + ^ (2) ~dAlL^) Z~' a -J.V' dz^ so that a new Elliptic Integral makes an appearance. But this integral for s is non-elliptic, \iH + 2 is a factor of Z, say -H" = - Zy or - z^^ never ^^g. With fl" = - 2^1, (3) — = -, — T = = am mt : ^ ^ a J, V {z^ - z .z - Z2) ^ and with H = — Z2, when the axle comes to a cusp, dz ^ . . / Zi - z a J = 2 sin V: = /f sn mt, cosg-^ = dn mt, z cos V z — ^1 ■^3 — ^1 sini— = /f sn m^, cosj- a (4) (5) so that s is a times the angle made with the vertical by the plane pendulum in §2 of length L ; and the length of the arc from cusp to cusp is 4 a times the modular angle, sin ~^ k. Fig, 53 is selected as typical where C is below 0, and S must be taken negative ; the axle comes to a cusp, and the motion is complete with six cusps,.and a modular angle -^^ ir. Then the total length is twice the circumference of the circle through the cusps. To reduce the integral in (2) to a standard form employed in the sequel, substitute a new variable s, not the s of §6 below, by putting z + H Z s z — z^ z~^'W z., + B S — S3 S" (11) k' = Sj — S3 Z2 . — Zi - II Zi . Zo + W Si — S2 H •Sl - «3 ^2 - ^1 ■ 23 + -0' If V denotes the velocity of H in the herpolhode curve given in (62) Si bv V =^ ^nk sin 6>, the rectification of the herpolhode is given by v / s , j- lk\^mednt= ^k j y^f^dz, (12) but this is hyper-elliptic ; except for the rosette curves and an intermediate orbit, §14 where 1 ± ^ is a factor of Z, when the integral degenerates into the lI.E.I. ' ensures the Steady Motion— Slightly Tremulous. 4. In the Steady Motion of Chapter I., and the small oscillation which en' dynamical stability, z, an4 z, close m upon z, so that we may write (28) (29) §1 d^z <2) ^ == - 2'^ (^3 - ^1) (^3 - 2) + 3 n^ (^3 - .)^ = _ 4 „,2(.^ - 2). ... , making the number Of beats per second, up and down, in small nutation of the axis ' (3) ~ Ij^^JL /flJILii . 27r TT V 2 ' thus, as in § 2, keeping time with the single beat of a pendulum of Un^th (4) Z = H^3 .)• 51 Here the modulus k is zero, K = Jtt, and the Elliptic Function degeij L- 2 - 4 OM.ON' ' ' = 4 OM . ON ~ KS . KS" and the number of beats per second is m _ MN n "^ _ / ^ ^^^ TT - 2 v/ (OM. ON) TT ' n "V X' For a spherical pendulum, OC = 0, ^1 /OM „ ^, ONn , , „ „ ^, 4-3 sin^ (9) Z = nt)N + ^ ^°' ^ + "OTJ = i (^'^ + 3 cos e) = 4eos0 ' • The wheel making Rfiv revolutions per second, the ratio . beats/sec _ MN n _ MN An C__ _C MN ^^^' revs/sec ~ V (OM. ON) 5 ~ v^ (OM. ON) CR A ~ A' OG ' With precessional velocity yu, and Af-i represented by ON, the apsidal angle js I TT Afi n ^ ON 2 ^/ (OM. ON) , ON ^^^^ ^ ~ " m~ An m-2'^~ V (OM. ON) MN 2 '^ ^ MN and the height A. of the equivalent conical pendulum is given by U-; Z - V ? " ON ~ KC ~ OL ' A - ui. , if OR in fig. 6 is drawn at right angles to OK to meet LE at right angles to OC in R, and RL' is drawn horizontal to meet the OC in L'. , , ^^ : , In the limiting case where K in fig. 5 is the mid point of C'F beats/sec C G'¥ _ C sin 0' ' -' ^^^' ' revs/sec ~ 2^ ' OU ~^- 1 + cos^ 0" ■ ' '^ The, Apsidal Anj3ile. ■ 5. The apsidal angle of tt in (37) §1 is denoted by fl, and it is obtained by integ- rating in (37) between the limits 2-3 and 2^, and the complete III. E. I. is required. This can be evaluated when it is Identlified with Legendre's {m') p. 138, t.I,' Fonctions elliptiq'ues, by putting 1 (1) ' z''= Z.2 sin^ <^ + ki cos^^, ' ' as before in (6) §2 ; and further, in Legendre's ijiotation, ,) , (2) E + z = E + Zi - (zi - ki) sin^^ U(^ + z^) ,(1 + n mi^), ,n\ Z-, — Zo -, , E +' Zo , K — E — z. IT, r,-\ E + z^' ■ ■• E + 23' E + z,. - E n (4) „-=(i^„j(l+i.)=^ ^1 3. ^3 — -1 • and putting z = — E in Z oi (29) §1, ■ (5) C-^-^)' = * (^'+ ^b) (E + z,){-E - z,), ' z, > ^, >' - ^ > z,:^ Then, integrating in (37) §1 through, the apsidal angle of the herpolhode, ,^. „ ., rp h' ^ Eh f'" i 1 dz _ r s/ (|. E +-4 E + Z2. -E - Zi) dz '■'■■■) - J {E + z,) {l+n 8in=^ *) ; V (^^);:: , d ,. = 1 •■ :■ ■' u -x^ = n/ a n, (ni c) (1- .. . of Legendre, in his notation. i ': : ■'''■'■ '*' 28570 G 2 f ■: (■ 52 Employing Jacobi's notation of k, k', K, K' , H, H\ for Legendre's c, b, F^ (c), F^ (b), E^ (c), E^ (b), and keeping his 0, f distinct from Euler's angles, we can write Legendre's (ra') (m) Van'(n,c) = ^^+{K-H)Fie,K') + K[>/ a - E (0, >c')], when Legendre's 6 is given by (7) - 1 +'b^ sin^e = n, which becomes in our notation (8) An0,K')=-n=|^ /Q^ T /2 • 2 fl -, '^' /a~ E{9,k') ==E(w,k') - H'. Employing the Legendrian relation (15) H'K + HK' - KK = i,r, the equation (m') can be written (m') VaU'(n, e) = H'K + HK' - KK + {K - H) [K - F (a,, ^')1 + K[E {^, k') - H] = KE (c., k') - (K- H)F (o,, k'). This is simplified further in the Jacobian notation by putting (1*^) "^ = am fK\ = am (1 - /•) K', so that (17) fK' = F{a,,K') = fV/" , = r^ ^{^s-z,)dz (18) (i-/)^' = i^(0,.')= re^L,^^ r yifajz^^Oi^. Jo A(0, ^) J_^ s/(- 4Z) • ^^^^ J-^~7T4:F)— = J_^ + 1 =(l-/)/r/+/v: and (20) ^(o,,.') =/:£?' + zn/A- zn being Jacobi's Zeta Function ; and then (m)^aU^ in, c) = K(fH + zn/A') - (Z - H)fK = ^./ + 7^,^/7^, m which -^ ' (21) ^=-dnMl-/)A', 1 + n = /^ sn^ (1 _ ;-)7^^^ 1 + -^i^ = ,. ,„.y^.. Also with '^ (22) m^ = 2eK = P -^^ifL.-_iiIi^ J^ S/C4Z) ' (23) Jmr= if = (''.^n^is:Jiydz J.. s/UZ) ' v/(4Z) {4Z) we have finally in (6) ^'^^ " = M^+4T/+/^zn/7f, and >P, the apsidal angle of ^, will be the same as n, or sometimes U - T. 53 We replace / by f when E replaces E^ and use 11' to denote the apsidal angle of w' in (68) §1. Then if A - EK is negative,/' < 1, (i^o) n' = p'r - i,r/" - Kznf'K' ; and cos x' passes through zero as z decreases from z^ to z^, so that x increases by ir ; and the apsidal angle 4> of <^ in (69) §1 is (26) O = P'T +^ - i,r/' - KzvifK + (^ - l) h!T. But we must take /' > 1 when h. — Eli is positive ; and then (27) n' = lAT- 4,r(2 -/') - Zzn(2 - /)Z' but here cos x' does not vanish between the limits, and ^ is the same as before in (26). The apsidal angle is important in giving variety to the pattern of the gyroscopic ■curve. The complexity of a curve, drawn by the gyroscopic pendulum of §2, Chapter II., is increased still more by the variation in the apsidal angle, due to the damping effect of slight friction. Elliptic Function Notation. 6. In the discussion of a III. E. I., such as in (32), (37) §1, it is advisable to begin Tvith the algebraical form, and, only after the sequence of the quantities has been settled, to select a substitution appropriate to the notation of Legendre, Jacobi or Weierstrass. Thus, in a figure to geometrical scale, we take _2 (1) i(£,,)=|: = ^, = ^, where a; or s is a new variable, and M is a homogeneity factor (not the M of Chapter I). The suffix a = ] , 2, 3, referring to a root oi Z = 0, (2) *(^ + ^") =" P =^2 = -— g^, and the sequence runs (3) \> z^> z> z^> - E, - F, - E, - 1> z^, (4) p/ > f? > pi > > /,l^ (5) x-i > X > x-i > > *i, (6) So, < S < $2, < o- < Sj, c''' dz cp^kdfT' n'^^Mdx r" Mds ^^^ ""^ ^ J.VlJz) = j,VB = Lvx= J„V^' <8) R = 4. Pi' - p\p' - pi. p' - pi, (pi negative) (9) X = 4. x^ — X. X — X2. X — «i, (xi negative) (10) S = 4. «! — S. S2 — S. S — S3, {S2 > S > S3). Uij 2^ - p - j^6 - ,]/e- As in equation (4), (7) §2, and (22) §5, where e = ;=,, 2eK 02) mt = (%t = Ff ■/' ""w'^r ''""v/ref =-vf^ =-'"■' y.-r^- = f^isd^^ = - -y-'^-< = --y 4^ = V S V S2 — S3 n/ .Sj — S3 "" ^ s^ — s.' .54 Then from equation (17) §5, (13) fK' = F(a,y) = f f,A(w,K') fzo + E '^^ = sn- /zA^iL = en- 7^-?-:^^ = dn- A-L±^ — 4Z) V 5^2 — ^1 \/ Z^ — 2^ >/ Z^ — Zi •° v/(p/-p,^)V j; / - Pi '-' ^ ^n-1 / P/ ^ dn-1 / ..» v/(-i?) -«" v/ p/-p,2 Vp/-Pr v/ ■° ^(^--^^i^ = sn- /_JI^ = en- /-^^ = dn- /-A_ I°3 2 2 P3 - Pi = '"-'v/-^ = en ^ V s. Sx — §2 = dn' 1 / "^ - ■S 3 V s, — Sd' ^ r v(gi -g3)ds' J. n/(--5) ■ ■ This collection of formulas will be found useful for reference, to save repetition in the sequel. 7. Employing the same selection of variables for the III. E. I. in w, which in Legendre's form n (n, c, ^) is normalised by the factor ■J a, and putting (1) (2) 2 An " ' n • /• '" M ' G^^EG _h' - Eh _ S' - E^ _ L - EL then from (37) §1, (3) 2An n M r^a I' Tir h' - Eh n ch + * rf^ ^ f dz E + z VIJZ) ^ n'i ^ (2Z) 'P- g' - g g dp-" ^ cMp' J p = r'" L' -EL dx r Ldx ^j!e_ ^X } ^X J . ■'■ rj_ - EL ds — LL ds r Lds Since ]\P (4) (5) (6) h' - Eh n = k {E + z,. E -^ z.,. - E ^ z,) k ) k' EL y--4-i 2 3 M J * -Mf = - 4 ■^ ~ h- — ^) and ^ (/> + i/-) by 111 and Ha, fh' + h (1) n, = I " _^f_, n - J^ J 1 - z V (2Z)" Denoting by r^, r^ and Ti, Tg the value of s and i? for ^ = — 1, + 1, in (1) §6, (2) ^(1+^)=^', J(l-^) = '^' Z with the sequence ,^s Zy< - 1 T, > Sg > S > S3 > ^2 )' Here ITi is of the same form as Tl — ^AT'in (6) (24) §5, so that we can put (6) ni = J,r/i + Zzn/iZ', , where, with z^ < — 1 < z^, and E in (17) §5 and (13) §6 replaced by 1, = sn -' / = en ' / = dn ^ / \/ Z2 — Zi V ^2 — ^1 V 23 — fi <8) (1 - /,) K'=F («„ .') -y y^^iztzy -I 7 ^3 - ^1- 1 ^ ^2 = cn-1 / ^3 - ^2- - 1 - -^i -jp-i /^lTLj!? V ^^2 — •2'l- 1 + ^3 V ^2 — ^'i- 1 + 2-, V 1 + ^g" = sn 9. But rig is to be identified with Legendre's (k') p. 134, Fonctions eUiptiques f., with '(1) I — z = 1 — Z3 + (Zi — Z2) sm^(j> = (1 — ^'s) (1 + nsin^^), and introducing Legendre's 9 (not Euler's) ,(2) cotH =^ n ^ '^--^, 1 ^-1+^ = 1:112 ^ ^ 1 — Z3 sin'' ft 1—^3 T I '^'- l-^i_ AU^') A(fl,>c') . n ^3 — 2'] cos^ ' " sin cos 9 ' Also = &m (1 — f2) K', where .(3) (i-f,)K'=Fi,,,) =j; /S"(-jf' = sn V i. — -2^2 V 1 — ^2 V 2'3 — ^'i- 1 — ^2 (^3 - Zi) a /fL.^.^en- /Lr^^dn- / ^^. VI— 2^] Vi— ^1 Vl— ^1 <4) Aa = J^ {w.i, k) = J = sn ' /'\ wta '^ /i'^ sin « cos 6/ (o) E(e, k) = ^ + H - E (w2, /c ). We find then Legendre's equation, in his notation (k') V' « ni (n, c) = i TT + r^ A (b, 0) - ^ (ft,0)] F\c) + [i'-'Hc) - ^(c)] F{b,B) l_COS 1/ becomes changed in our notation to r^TT, T^nr TTTT, rsin 0A0 k'^ sin ft cos „, „ . ,v 1 r, (6) Hs = HK + B'K - KK' + \j^^^ - ^q H' + E (o.^, k') J K + iK-B)[K' -F{^,,k')] '^^0^^ + ^^ ("^' ""> -iK-H)F (.2, .') 56 = K fcq^^^'y. + ,n^3/r) + [H'K -{K-H) r] f, \ snj2K ' ' = Kz%fzK' + iff/a, where cufoK dn/gA „ ^,, (7) -• ,,f,K' = -^-^^Jr—^znf,k, in accordance with Glaisher's notation cnw dnw d , „ tt/ ,n (8) zm = ~^^ + zn ^. = ^ log H(^.), H(m) being Jacobi's Eta Function, and 9 {u) his Theta Function, with (9) zn M = ^ log Q{u). Then for the \p apsidal angle (10) ^F = n^ - Hi = Jtt (/a -/i) + A' (zs/iZ' - zn/i/r) from (24) §5 ; and for the (j> apsidal angle, in a spherical top, (11) • ,T > S2 > s > s, ; and in the reduction to Legendre's form ,-x .0 S — S-, So — S a2^ *1 ~" ^ (1) sm^ d, = ^ , cos^ ./. = -? , A-.^ = -^ , ^ ' ^ S2- s^' ^ S2- Ss Si - S3 (2) sin^a. = !LZL^, cos^, = ^^^, A^^, -c') = ^-^^^ , A' {6, k) = '±^^^ , ^ Si — S2 Si — S2 Si — 53 ff — S3 /ON 2 S2 - S3 ,2 gi - Sa j^ _ r's^{si-j3)ds jr, _ f' n/ ( si - S3) ds ^A> /JT' ff'^ '^ r w(gi-s3)c^s n 7;-' p^ft J^ r ^^^^ -S3)<^S (4) /A = F (<,., ic ) = J ^ ^ ( _ ^) ) (1 -/) A = i' (0,K ) - J^ V(-*S) ' i^ (0, k) + F (w, k') = K', (5) E (a,, k') = |"a (o,, .') rfo, = J^ ^ "^^ J^^^j ^(I^)' ^ (J-, -') = 5^', zn/Z' = ^ (,.,/) -^,Fiw,K') ; then Legendre's Table IX can be used to calculate the numerical value of (A). This formula (A), the equivalent of Legendre's (m') can be proved by means of the Lemma, (B) ^s^ i^-v(-^) f i^^-=^ = .-s, ^ '' as a — S ^ ' da a — S ' which is verified immediately by the differentiation. "~ 7^' **"" \/( - 2) Integrating with respect to the elliptic differential elements, — r, and . _ „c , and between the limits Sg and Sg of s, and JJ,3 s/( -2) s/^^'^rfs (t' > S2> S, s' > S3 > T2 ; ^ ^ J S3 S — rg n/ /S J tr' — Si. tr' — 5'' s/ *§ _ f' 2 v^( - 2') ds' / (t' - Sg. g' - S3 ry (si - S3) ds' J.,, ff' — s' s/ S' V 5i — 0-'. Si — S3 J n/>S' = J,r/2 +'Kznf,K' + K ^ cn/adn/g sn/2 since, as in (2), = Jt/2 +Kzsf.J{:', S3 > Tg > - CO ; ^^3 ~ ^1 Si — S3 Si — o-' (12) sn¥,Z' = ^^ ^ 1 - •S'l 5l - T2 ^1 - S2 12. Write in (6) (24) §5, from (6) §8, (6) §9, . 1 + ^ 1 + ^ (2) Ha = K(i. 1 - ^a- 1 - ^2- 1 - ^1) J X — Z , cnyg^' = g- — S2 Si - Sa' dn%K' = g — S3 Si - «s (1) n, = r V (i 1 + ^.. 1 + ^2- - 1 - ^1) dz X . ^ r. fj.. and introduce the substitution (3) - 2., = ^2 - ^1- ^2 - ^ 3 ^ - ^2 s/(2Z) + = V (^3 + ^j _ ^r _ 2') = - - (^2 dz' z — z. (4) z then with (5) 1 2 — ^2 n/ i^ s/Z' ^2 ^3 - I + CO + CO - Zl - 1 ^2 Vl + ^. -1-2 v^ 1 + ^.. _ I (6) ^2 = sin-' v/ ll!"i I t = co«-' y 1 - ^2- ^3 + — ^ — 2r we shall find (7) (8) ^i — z — z 1 - z' 1 f ^ >/(2Z) + V (i- 1 + fs. 1+^2.-1 - z,) dz' -l-z' - .^^liJLZ^fa^Lziii- 1 - ^i) n/(2Z')' dz 7X2Z) v/(2Z')* 59 Integrating between the limits z<, < z < ^3, and — w < z < z^^ (9) i, . n. . n/, n/ . ]'[ ^ ■/»■ i ^ ^i ^^. - ' - ^ _i|^, (10) J. = n,-n;. n, ^ |-^ ^ (i- ^ --J T -'•'-) ;^; and going back to the variable s, the two remaining forms of the complete III. E. I., of the circular form, are given in (D) n/ = r^LllIll iL = 1, _ Hi = 4.(1 -/O - K^nf,K', •> Hi S — Tj V o Si > ri > s„ m' fJC = ^-LJ^Il, cn^ A = iL^ii, dn^ f, = ^^-^^^^ ; Sl - 52 •' Si - S2 Si - S3 (E) n^' = r ^^~^^^ ^ = u,-l^ = K z,f,K' - 1. (1 -/,), J Si S — T2 V O S3 > rs > - 00, 8n2 /2/^' = '-^^^^ cn^ /2 = ^i-=-I?, dn^ /2 = ^^^^^. Si — r2 * Si — Tg Si — Tg These theorems are useful in the sequel in assigning limits of the apsidal angle ; and also in proving Legendre's relation (15) §5 ; because by (4) we can write (A) §11 afi-esh as ^^^ J., and similarly (20) (1 3F sin e^) ^ = [ ^/ ( - X, X,) + L ^ x^f (21) (^X3PBhe,)' = x,'- {D^ A)x,-2LQ + B = X2X3 - 2 L ^ (-XiX2Xs) - Ux^ = [ V (^'2 ^3) - X s/ ( - x^ f. Thus M"^ can be written in the three forms (22) M'={^ 2x, -D- Af + 4 [v/(- x,x.^ + L^x,Y = {- 2x2 + D ^ Ay + 4{s/ {- x^x,) + L^x^f =^(-2x, + D + Ay-4[^ (X2X,) - L,{-}x ,)f =-i\L+ s/(-^Or+ (v^^3+ ^^2y][\L- s/(-^OP+ (s/%- s/a:,yi relations to be interpreted geometrically in the sequel. We make the apsidal angle n, or ^ = ^^f when we take (23) Z, = - P = _ ^ (^3 _ ^^) ^^j.'x'^ and then (2^) J-1^' sin 03 = V ( ^'3. a;3 - X,) zn (1 - /)A" (2^) J-^^' sin 02 = ^ ( X2. X, - x^) zs (1 -f)K' 61 A Rosette Curve. 14. This is the name given by Klein to a curve described when the axle of the top passes periodically through the vertical, downward or upward ; and then 2:3 = 1, or ^2 = — 1 ; and when Zi= — 1, an intermediate curve is described ; and in these curves (1) E = H, Z'2 = L\ (2) L'HP = {D + AL+ 2Qy = DM' = {U + ALf + SQD - ABL\ - QD + BU + AQL + Q2 = 0, <=» u^^L) u^-i) (y^-^)=«. when resolved into the factors, equations equivalent to sin 63, sin 621 or sh 0i = 0. In the lower rosette (4) sin 03 = 0, L = -L= J^^, ,^ + V'=.0,in(14)§l; /2 = l,/ = /' =1-/1, (5) z, = 1, M' = 2x,- D- A = x,- x,-x^ + ^' =^ '--^ J-3 .63 (6) M^- (1 - cos 0) = 2{x^ - x), M sin J0 = n/ (^3 - x) (7) sin je = / '"'' ^^ ~ ^ = dn/K' sn mt, sin U. = dn ^'K' ; <8) iM2(l + ch 0,) = 2 (a;, - ^'1), J/ch 101 = s/{x,- X,), ch 101 = J^^fzr^^ = ^^ = dn(l-/K ' «^ "^ (1 - ^) ^'' ^^ ^^ ^^3) §6- ™- 7 ^3 - ^i _ V (a;., - x^ _ / a;3 A 8 , r- v/ —2 M n/^7=^2= ''^*^^' ;^ = I = - «h i0i cos 4^ <9) Z + P = ^ (^3 - ^1) zn/r - / ^^ V a-3 = s/(^3 - ^1) (zn /iT' - " 'd//"'^ ) = - n/ (*3 - *i) ™ (1 - f)K (10) ^-jLZ^,^^^^2A^^=-2Z^zn(]-y)Z'; and the apsidal angle (11) n = i:r/-Zzn(1 -./)Z'; also <12) ^-^^ = /(/,) =/(!-/), 1=^^. In the upper rosette (13) sin 02 = 0, L' = L = - / Z-^}^^ ^ - ^A = 0, in (L3) §1, /l=l, /=-/=l-/2, (14) ^2= -1, i/^ = - 2*2 + i^' + ^ = - ^2 + ^'1 + *3 - ^i^ = ^' - -^2 • ^- - ^1 ^ P=-c», ^ = 0, r=co, ^ = - n^ so that the motion is very sluggish, and has eternity for a period. In the intermediate curve (27) .,= -1, L'^L~y':t^^, *-*-«. /.= '. f*f''' Xo X^ _ «3 - Xi. X2 - Xi (28) M^ = - 2xi + L' + A = - xi + X2 + os^ - -^ ZTJ^ ' (29) M' (1 + cos 6) = 2{x - x,), M cos x q = ^ {x - x{), (80) -H=y,^^^:^;;^ffv.=--^'^'^"'»'' (31) cos 1 03 = sn/r, 03 = TT - 2 am//r, (32) cosMs =/^-' sn//r, sin i 02 = dn//v', as in the lower rosette (38) m n i/ V X2 - Xy i + P = V (^3 - ^i) (='^n/^^" + "^^) = ^ (-^'^ - ^^^ ^^■^^^'' (34) (35) p^ = ^^n^=2/4zs/A-', (36) n = ^ = i7r/+ Azs//r (37) >P - p^ = I (/s) # _ S^ = „ cn/A^dn/A" (^^) ^ ~ *" sin^ |e 1 - sn^/A' dn^ mt In the separating case of the upper rosette and intermediate curve, (39). zi = Z2=-- 1, K = 1, K = 00, and the axle rises to the upward vertical asymptotically in an infinite time, where it will be in unstable equilibrium and fall again. Here (40) cos ^9 = cos ^03 sech {nt cos ^©3), 1 1^ , • -1 /cos 0o — cos ^ ' ^ V 1 - cos ' , ^x , ,, _, sin i03 , .1 th (inicosiflo) ^42^ -dj — ht = cos '^ — — 4-7? = tan ^ — ^ , ^ ^ ^' ^*^^ ^ smje tan ^03 _ ^ .1 1 — cos 03 ch {nt cos ^03) _ 1 • -1 sin ©3 sh (?ii cos ^©3) ~ ^ ch (nt cos ^03) — cos 03 ' ^ ch {nt cos -^03) — cos ©3' a relation connecting true anomaly 2()/- — ht) with hyperbolic excentric anomaly nt cos ^03, in a hyperbolic orbit of excentricity sec 03 ; (43) sin 40 sin {4' - ht) = v/ (cos^ J03 - cos^ ^0) = cos ^©3 th {nt cos ^63), (44) ^^ 2l " " ^"' ^^'■ I ■' t m A CusPED Figure. fr:.-. , 15. The precession o£ -ip is zero and changes sign where (fig. 53) dt ~ ^' sec tf - ^ , ^ 2^{~ QL' + BD + AQL + Q') d^ . G' r L' + AL + 2Q (1) ^^ = 0' sec0=-^=-^=- ^jjp (2) tan0= ^^-j^, _ 2-j[L',/ai3 + V - W1X2. L^Xs + s/ - X1X3. - L^ (— Xi) + %/ 3-2*3] and at the corresponding point of inflexion on the herpolhode of H ^^^ , A;2 - 2 l,^ - ZV ~ i¥^ (X^ + AL +. 2Q) ~ J/^- (4) -i^ =2 (^B + ^^ + 2Q) M' = 2iifS sm-e = JZsine, (5) p = S' sin 0, with C and K in coincidence. If the axle comes to a cusp on the upper circle, 6 = 62, (6) W ~ ^ when z = Z2, h + h'z.^ = 0, (7) Z={\- <) {H +z)-2 ^1, (z - Z2f, so that H = - Z2, h - Hh' = 0, /' = 1 ; (8) zj' = h't + (-p — 1) 2h't, or A7 in the spherical top ; (9) p'2 = 4 P (ZT + 5) = i P (^3 - ^2) cn^ mt = p^^ cn^ mt, p'2 = 0"; A' _ . (10) X = sm ' ^ — ^ = cos ' : — . ^ J sm sin sin sin \ = ■y/ (^3. — z, z — z^). 1 + ^3^1 „ ;i'2 Z'2 ^^^^ ^2 = J^IT^' ^3 + ^1 = - 2 ^ = - 2 ^2, sm 02 = V (^3 - •Si- ^2 - *i) 2 ^ (^3 - ^2)^ ,2 ^ sin'^ 02 "^ 1 - 2^2% + z^' "" 1 - 2z2Zs + zf Thus if ^3 = 2^2, 2'2 = /c. _ .„ Wich cusps (12) Z2 '' ^ ^ -^''' G' K L' U + AL + 2Q i^n) m, = 2.., -u-A= -i^(i^-+^r-8Qz^ + 4^i: ^ ^ U + AL + 2Q reducing to (14) (Z +J^^) [(Z V .^2 + V - *i ^3)^ - (^3 - ^-2) (*2 - ^1)] = 0, of which the second factor must be taken, the first factor giving an upper rosette ; then (15) • J M^ sin 02 = n/ (a^a — x^. X2 — Xi) <16) ^M'= ^ {x,- x.,. X2 - X, ) ^ ^^^- ''^ - ^'1) - ^/ ( - ^1- ^3 - ^l) X2 ^3 — -^i V a;3-«2 ^ .r2 - a-'i 64 Then with cusps, and/ = 1 in (13) § 6, (20) cos03 = sn/, sin 03 = en/, d^ = ^ tt - 3.mf K' ; (21) cos 02 = /c'2 sn / - K dn/, sin 02 = /ck' sn f + k' dn f ; (22) eh e, = ^ = dTCr:^ ) = ""^ (' - ^ ^' '^ '^ = '^ "^' ,9„sm / ^B - ^1 / dn/+,csn/ /).'_ /nls^^i ^ /dn/-Ksn/ Thus if/ = 0, 03 = 1 :r, eos 02 = - k, 02- i tt is the modular angle ; and in (24) § 5, SO that the mean precession is the same as if the axle was moving steadily round m a horizontal position. ^t half time to a eusp (24) z — dn^ Zi = ^/ mt = (^3 - 2l. Z2 - ^1) (25) tan x'-yz Z-^ — Zy • Z^ — Zi ^A sn / + ■ sn / + dn/ K dn/" 1 V7 sn / + '^dn/ sn f ~ "dn/ =y^ + en (1 - en(l -f) X' = J TT - ■ 1 am (1 --/) ^'. From eusp to eusp the top will have revolved through an angle meaning (27) f^ = ^ ^^ tan 1 am (1 - /) A' revolutions. We can have also (28) H = - z, (not - z,\ h -Hh' = 0, /' = 0, a.' = A'/, in the spherical top, (29) p" =lk'{z-z,)=^ P {z, - z,) dn'mt = p'^ dn^ ^ .0 (30) sin eX'* = ^ [z^ - z, z - z^) + i^^ (2. z - z.) n ' ^2 + ^3 ' ^2 + -, - - 2 ^ = - 2 ^„ (31) -I=T^fv^ .2 + ^'. = -2^^= -2 (32) cos 03 = en /; 03 = am //v', cos 02 = - dn/ ch 0^ = ' -''«^/ + ^"^ t ^ /Qox ™ _ / ^n/ + en/ A' / dn/ - en /• 65 The Notation or Weierstrass. 16. This notation is useful in the reduction of the Elliptic Integral to one general standard form, before proceeding to discuss special cases, such as (m') and {h') of Legendre, and our Xly and Ilg ; and it is used freely in the Kreisel-iheorie. Schwartz's Formeln und Lehrsdtze sliould be consulted for a complete explanation ; but so far as required for our purpose, three functions are introduced, of a variable m, (1) pw, tu^ all, the p, zeta, and sigma function of Weierstrass, defined by (2) s = s {u) = pw + c, (7 = s {v) = pi? + c, ^ = -3 (*i + ^ + ^3)5 s — Sa = pw — ea, a = 1, 2, 3 ; (3) u = IJ -^, ^l = p'. = - ^ ,S', p'. = - s/ 2, in accordance with our previous notation of § 6 ; and Weierstrass, for his theory, takes (5) S = p'^M = ifu - (/2 PM - gs, (6) 61 + 62 + 63 = 0, 02= - 4(^363 + 6361 + 6^62) = 2{e^ + ei+ ei), .^3 = 4 6162%- These new functions are called of the First Stage, as not requiring the resolution of S in their definition ; but ■J {S - So) = s/ {\>U — Ba), occurring in the preceding applications, is expressed by a one-'valued elliptic function^ and of the Second Stage so called, because the factors of S are required. A period rectangle is drawn, OAj Ag A3 (fig. 55), in which OAi and OA3 represent wj, and W3, the real and imaginary half period of s or pz<, defined by r"-'' rfs K {'"'' (is K'i fr,s r-" as K C-'' .'s,.*, VO V (Si — 5o Js,.-(. 's^.ia-yS ^/ (Si — 53)' Js„-00 V »S n/ (Si — 53) the function s or pw assumes all real values for any representative point on the boundary of the rectangle, such as P or Q, but it is complex at a point in the interior. In our previous applications we have to take P, the point for u, on the side A2A3,, and Q for v on AjAg ; and representing them by the broken vectors OA3P, OAjQ, (8) w = 0J3 + e(i>i, V = (Di + fw3, where e and / denote real fractions ; / a constant, and e the unrestricted indeperident variable proportional to the time, so that P traverses the line Ag A3 to an unlimited, extent. In the (k') or Ilg form, the representative point for v would be Q3 on OAj, v = f2 w^. The Addition Theorem of Weierstrass (Schwarz, p. 13, §11) is (9) ^--2^-- -=ipv-^u, and by a logarithmic differentiation with respect to v, with the definitions in (4); (10) tin . V) - tin - V) ,- 2lv = ^^. Then from the integral I in (I) §13, writing s{v) for a, to avoid confusion, (11) f ■ = M;^JzA) _ Pi = _|PJ^ _ Pi ^ ' an, s(v) — s Tpv — pw Then integrating (11) = ^t{u + v) ~ ll{u -v) -Iv - P. du ^ a{u — V) (12) (J+P„).= |t-^ = Jl„g-^__j,-2., 28570 66 and from (9) p^ r, — S a (u + v) rr (u — V) (15) •'^i«"-"-^^^«-""' (16) ■ l=.-P-t-^n., «fexp(.-S«>- = ^^^«-% Changing to the Jacobian Eta Function (Schwarz, §39, p. 52) ,, 1 HmV(si-S2) 1 Uu^/ (Si - S3) (17) e-^-.-'.u = 4^(,:_,^.^73i;) eZ - s/(s, -63) H(Z) ' where 17 = Z^w, and c r(M + ?;) • -2^" _ H(m + z;) V (si - S3) (1^) ^(m - vf " H(m - v) n/ (si - 53) Differentiating (17) logarithmically (20) lu --U= sj{&^- S3) ZS M n/ (Si - S3), so that, with (21) u = 0,^ + /wg, ?; V (si - S3) = if + fKi, (22) rw - % = V (si - 53) zs (Z - //rO = - ^V (si - S3) ( X + ^'i/-^')* Then if we take, as in (2) §13, (23) P = n/ (si - S3) zn//r, dli , ^ , H {eK + r / + Z + /gp Itt^ A^^) ^1^ = 2 ^ Jog H (eZ + Ki - K- fK'i) + 0.1 5 and integrating from m = wj to ;< = e^i -•- wj, (^5) (7 a^r/e) ^ - 2 iog jj ^^^ ^ ^^, . _ ^ _ ^^, .y The logarithmic term returns to its original value as e increases from to 1, so that I increases by ^irf. For {k') and Tig we must take (26) V = V2 =/20>3, uv/(si - S3) ^^ f^K'h 1 / (27) Iv — - V = V (si - S3) z^fji'i = — i-y {Si — S3) f zs/a^' + ^^j, ^nd then take (28) P = n/ (si - S3) zs/2/r. But the complex iargument in these expressions of the sigma and theta and eta function makes them useless for numerical purposes, and so the Weierstrass notation breaks down in an application unless we make the Third Elliptic Integral pseudo-elliptic, as in a later chapter, by taking / a simple rational fraction ; and so explore the general analytical field. 17. We arrive immediately at the Jacobian form of the Third Elliptic Integral by a double integration of the formula (1) dn^ (u + v) - dn^ (u-v) = " 4/ c^ sn ^ en 7; dn ^ sn u en t^ dn u . (i — k'' sn" w sn" uy for integrating with respect to m, ,^ en V dn v (2) P am (w + w) - i; am (w, - v) - 2 P am ?; = 2 5^'^-^'^— '' - sn^^ sn V 1 - k2 sn2 X) %\^ »,' (3) zn {u + w) - zn {u - w) - 2 zn w = 'l^l^^ ",^^^^ f ^" '" sn^l* . I — K^ su" 1: HIV a and integrating agam (4) i l°g e fe^ + u zn . = f^," "^^ ^^^ ^ ^^^''' "■ ''" . k^ + 'V Jo 1 — K- sn' r sn^ ?< and this integral is Jacobi's Tl (m, v). Also en V dn V (5) sn V du , , Q{u — v) 1 2 — 9 5— = i loff ^7 — ; — V + u zsv. 1 — K" sn' V sn" t« ^ s> q^^^ ^ ^^ Thus, in a spherical top, (13)§1, ^''h' - h (6) ^{f - ^) = n dz ■J {I + z^. \ + Z2. — \ — Zi) dz 1 + z v/(4Z)' 1 + ^ V (2Z) and expressed in the the Jacobian notation, with u = mt = 2eK, v = K + fiK'i, (7) (8) 1(1 + z) = cos^ J0 = cos^ 103 cn^ mt + cos^ |02 sn^ mt = co%^^di{l — k^ sn^ v sn^ m), . COS'* Jflj' 1 + ^•g cos i0o V 1 + ^, ^ -^^ COS ^03 /• cn u dn w (9) (10) (11) (12) (13) (14) sn du "'J'l -1 sn V •A^ ■ / - 1 - ^1 en w = ^ / r , V 1+^3' ■ ^'j. 1 + 2'2> K^ sn^ ?.' sn^ u + ^3. ^^3 - ^'l ^ v^ (a -_^i)^ = J^(<^ - '/')2V 1 + 2 V4Z COS COS 1 + Z3 ,2 ifl _ „o^2 ifl e^o e(^ - ^) e(^ + u) e(0 -+)i = Q(^ — li) ^2u zs u e(T; + u) ' ifleJC*-'/')^ = COS M3 QO Q(^ - ^) e"-^-^, Kz^v = K zs (Z + f^K'i) = i^fii + iK zuf^K', ^cosi 0e4(?'-'P> =icos4-0. and this is Klein's function y, when a change is made to measuring from the zenith, and J gives a lower rosette curve, with ^ + r/. = 0. Again, in a spherical top, (14) §1 C' h' + h (15) where (16) (17) J + ^) = n dz •^{\ —z-i. \ — Z2. 1 — Zi) dz 1-Zs/{2Z)~\~ 1-z v/(4Z) ^ (1 — z) = sin^ J = sn^ J 03 cn^ mt + sin^ ^ ^3 sn^ mt = sin^ J ^3 (1 — K^ sn^ u sn^ u) K^ sn^ ?) = 1 — sin^ 1 e. 2 "2 sin' i 03 = 1 - dn^ w 1-^2 1 -^3' en' V = (18) en ?; dn w sn w 1 — K^sn^ V sn^ 1 rfw = 1 - Z2 1-2-3 1 -^i 1-^3 ^2 1 -03' sn^y = Zo — Zj - enV2^" Zi. 1 — 2^2 w = fsK'i ; Z-i. Zo, — Zy) -J (^3 — 2i) dz 1 - ^ 1 - ^-o V(4Z) J + 'A) ', (19) (20) (21) (22) sin^ g- = sin^ J ^3 g (if> + 0) i =. B'O e(w - m) e(t' + v) e' w e-* u 0{v - u) „2UZBV (v + u) K ZS V — K zsf^K'i — hTrf^^ + i K zsf^K', . ,. ,. ,s- . ■,. eO OiAK'i - mt) ,, , „ j,j.,.mti sin J0e4C0+*> = sin iflj e(/2ZV) 9m^ ^^P ^^^"-^^ "^ zs/gZ) -^, and this is Klein's a, with a change to the upward vei^tical, and a gives the motion of an upper rosette with ^ — 4>. 2857C 68 18. In the notation of Weierstrass, with u = wg + ewj, Wj = w^ + fioi-^, and from (11) §16, /i\ w ,\ • r'lvTt ds rir)'vidu , , (t(u + Vy) (1) K^ -^)^= J^^^^, V^- = j{^± p, = i log ^^^ - utv, C^j o(l + .) - cos ie - -^^2 - ,1^2 - i/Vw^ r .Qx _ to(M + Vi) e""^'"' Then, differentiating logarithmically, or, as an integral again, P» h' — h I .n n _ ^^2^ ra dz <5) log , = i [pj^-lp:^ d^. = 1 r ^^^1^:1:^ ^ = -- , ^^^^^. ~ J p?; - pw -J S3 Ti - 5 s/5 j^ 1 + ^^ n/(2Z) Differentiating (4) again in which «o that (8)^ ^|r = 2p. + pz,, Lame's differential equation, of the first order. Similarly, with u = Wg = yj ^jg, <9) log a = ^\-^:_±_^+ v(2Z) , — ^7-(2Z)' i(^ + ^)^ = J / ,-./ -7:s'^ J. 1 - - 'V-*^*^) — T-^i = 2 pw + pu2. Interpreted dynamically, with 0^) ^l^M = n^ + ., M? = a; + v^ the Lame differential equation of the first order, (12) I d^w ^ is the vector equation of a central orbit in which (13) ^o = ^^^A,-„,r ^2 + o _ pv^jpu and (^ (« w) ^^^^ ~r~ = (2 pw + p?;) (*■, y) = (3 py - 2 iliV) r (cos 0, sin 6) so that the acceleration to the centre is <1^) P = 2 .1/y - 3 pp.. Thus, for instance, the H curve {p,^ - ht) is a central orbit, with, (39) § 1 n (16) P _ IL „ / 2 2 2 ■>% The general Lame equation of the n th order is ^ '-^ ^•5^ = »(» + 1)P« + A, in which z. is the product of simple elements, such as in (13), and ^''^ ^'-f-^'-^^, A = (2.-1).V. equatln^fXtok'^rdt!'^^"^^^ '' ''^ '''''''''' P^^^^'^™ -^^ -^-^- the L:une 69 The Lame Function. 1 9. The function \^) w = d> (u. v) = —^ U a U a V is called the Lamd element or function, and is such that (2) log ^ (..,.) = 1 [ll^il^du = 4 f ^-l-:!^ % a standard form of the III E. I., as in (5) (9) § 18 ; and (3) (j, {u, v) (j> (u, — v) = pv — pw. It provides also the solution of the Lame differential equation (8) § 18, of the first order, in the form (4) w = A (m, v) + B (^ (u, — v) ; for, differentiating logarithmically either term, (5) - ^ = I (u ■¥ v) — lu — tv = h ^ 2J^ = V [p (m + u) + pw + pwl w du ^ ^ pv - pu "-^ ^ / r r J <^) wd^- \w 5^) = - P(" + "^ + P"' w ^^= 2P^ -^ P^- In the Jacobian notation, following Hermite, we can take a typical Lame element as (7) . = ^(.,.) = ^^|(V^.-«^-', " QuQv satisfying the relation (8) <^(m, v) (j>{u, - w) = 1 — K^ sn^ M sn^ c, and the Lame differential equation in the form (9) ~~= 2k' sn^ u + h. w du' For differentiating (7) logarithmically, and representing differentiation with respect to u by an accent, w' , . cn u dn ?; (10) — = zn (m + w) — zn M — zs w = — /c^ sn M sn u sn (m + v) w su V (11) !^ - C^)' = dn^ (u + v) - dn'u = k' an' u - k' sn^ (u + v) (li) !f^' = ^2 g^2 ^ _ ^2(2 _ ^2 gn2 ^ gjj2 „) gn2 (^ + t)) + 2^^ SU M CQ U du W SU {u + ?') cn^ V dn^ u , + V, ; sn'' V , s cn^ V dn^ v = K^ sn^ M + /c^(sn M en r dn u — sn w en M dn m) sn (m + u) + - — — g- ^ oil V , . cn^ V dn- w = k' sn- M + /c2(l - K^ sn^ u sn^ w) sn (m - v) sn (m + w) + — ^ = K^ sn^ z< + K:^(sn^ u — sn^ v) + sn' ?j cn^ V dn^ z? cn^ w 9 dn^ w sn" V c o 51 , 7 7 cu- ■(? 2 dn 1/ 1 sn'' u sn V Lame's equation, suitable for Klein's a, /3, y, S, with (13) v = K+fyK'i, ovf^K'i. A change of v into w ± iT'i will give the other element employed by Hermite, (14) w = ^ K.) = ""^Hf^y^ e-^-^ i> (u,v) ^ K - .) = 1 - J^, (15) — = zs (m + v) - znu — znv w ^ en (m + ?;) dn {u + «) _ ^2 g^ « sn « sn (m + r), sn («/ + v) (16) ^ = r^V - Sii!L±4 _ dn^z. = 2.^sn^^ + dn^ . w \w j sn^ (m + v) after reduction. 70 If 10^ denotes the second solution of the Lame differential equation in (6) 1 dhv, 1 dhv (Pivi cPw and integrating dwi dw ^ , , -^ w — Wi-r- = 6, a constant : du du d fw{\ ^C_ ^ fdu du \w I w Si Wi (18) (19) Some special values of h lead to an exceptional form of w and w^ ; thus pw = e« in (6) gives a solution (20) w = n/ (pt< - ea), and then w^ = s/ (pw - «„) \t (u + w«) + eau], introducing an E.I. of the second kind. The Hermite element in (7) has a special value with v at a corner of the. period rectangle in fig. 55, shown in the table : V I- sn V h W Wi K K + K'i K'i K 1 - 1 - 1 -k' dn u en u sn u 1 dn u en u sn u — JT — zn (K — u) — ^u ZB (K-u) - f^ -k"^U — ZSM +( 1 — ^j u So also for (21) to = _c s^ 1 1 1 rf' d' d' c' s' Co-ordinate Axes, Fixed and Moving. 20. Eeturning to the measurement of 6 from the zenith, and Klein's parameter functions a, )3, y, 8, (1) a = COS J exp ^ { + ^) i-, 8 = cos J exp ^ {—

„ Z. The spherical surface in fig. 56 is regarded as a star chart, looked at from the centre of the sphere ; not as a sea chart or map on a terrestrial globe. The relative displacement of the frames xyz and XYZ can be replaced by a single rotation through an angle w about an axis OQ through 0, making angles «, b c with X, y, z, or X, Y, Z, such that on the unit sphere, (2) a = Qx = QX, b = qy = QY, (^ = Q^ = QZ ; and Q is the point of concurrence of the three great circles bisecting xX, yY, zZ at rio-ht angles ; and the quaternion versor which turns xyz into XYZ is written *' (3) where (4) J. = sin^ ocosa, (5) Q = Ai + Bj + Ck+D, B = sin ^ u, cos b, C= sin ^ w cose, Z> = cosi(o A-' + B' + C' + D' = l in which case the quaternion is called versor. 71 Let ZX, zx in fig. 56 intersect in E ; then EZ^ = J tt - ^, E^Z = J ;r - i/-. With > i^, EQ is the exterior bisector of the angle ZE^, because the perpendiculars from Q on EZ and Ez are equal ; and Q lies on x^ QR, bisecting Zz at right angles in R, so that QZz = QzZ. Also QZE = Q2:E, because the spherical triangles QZY', Qzy' are congruent by a rotation round Q, QY and % cutting ZX, zx at right angles in Y' and y'. Thus QZz is the half sum, and QZE is the half difference of EZ^, E^rZ ; or (6) QZ^ = 1 TT - 1 (^ + ^), QZE = H^ - 'i^), QZ^i = ii + ^)- Thus when <^ - ^ = 0, Q is at E in ZX and zx ; and when ^ + i/- = 0, Q is at x^ in X Y and xy. Then in the right-angled spherical triangle QZR, ZR = 10, QZR = i^r -1 (0 + i^), QZ = c, RQZ = i.,, (7) Z>=cos^w = cos RQZ = cos ZR sin QZR = cos J0cos ^ (^ + )/.), (8) C = sin |w cos c = sin RQZ cos QZ = cos ZR cos QZR = cos J9 sin ^ (<)> + iP) ; and in the quadrantal triangles QZX, QZY. (9) A = sin Joj cos a = sin ZQR cos QX = sin ZQR sin QZ cos QZX = sin QR cos QZE = sin ^d cos J (<^ — "A) (10) B = sin 1(0 cos h = sin ZQR cos QY = sin ZQR cos QZ cos QZY = sin QR sin QZE = sin 10 sin i U — \P) Thus (11) - " + ^ JD = 2 ' a = £> + Ci, n =- 1-7 + /3 2i ' B y-(3 2 2i ' (12) a = D + Ci, ^ = D - a, I3='-B + Ai, y = B + Ai, so that A, B, like j8, y, are Lam^ functions of the first order, with v = K + f^K'i ; and C, -D, are like a, S, with t? = /j K'i ; and expressed as integrals, in our notation, .h' + h dz r" (13) log(a,g) = n + v/(2Z) iog(r,/3) = 1 + .A' - h n 1 - ^ v/(2Z) s/(2Z)- (14) (15) 1 ^ 1 d^ B df = n^ ^yu + V^ 2 M' IJlJ = L ^ = 2 2 pM + Y>vj y df (3 df "^ AP ■ d^a df i3 — 1^ ^j2 d?Q 2 ^ {Kreisel-Theorie, p. 238 ; Schonflies, Jahresb. DJI.V., p 457. Nov.-Dec. 1909.) 21. The change of cartesian co-ordinates from the one frame of reference to the other is shown in the matrix : — X Y Z X COB Xo; cos Yx cos Z« y cos X.y cos Y(/ cos Zi/ 3 cos X^ cos Ya' cos Z2 and the cosine of the nine angles between x, ?/, z, and X, F, ^, can be given in terms of a, b, c, w, or A, B, C, D, four functions, but three only independent, since (1) cos h + cos'b + cos 'c = 1, A' +^B' + C + D' = 1. The expressions can be traced back to Euler, in his symmetrical parameters (2) X, /u, V = (A, B, C) J D = (cos a, cos b, cos c) tan J w. They were given later, by Rodrigues in Liouville, V., 1841, and Cayley, Fhil. May. 1845, 1879 ; and the proof can be given by spherical trigonometry, as before. 72 In the triangle ClZR in fig. 06 (3) sin 4 Z^ = sin ZR = sin J w sin c (4) cos Z^ = 1 - 2 sin^ J ^ sin^ c = 1 - 2 sin^ i o, (cos^ a + cos^ b) = 1 - 2A' - 2B' = -A' - B'+ C' + D'; = a8+ jSy = COS 6 and similarly (5) ■ cosX.^= A^-B'-(y + D\ = i (a^ - /3^ - y^ + S^) = cos

+ cos cos ^ cos »//. Again, (7) cos Za; + cos Xz = sin Z2^(cos E^Z + cos EZ^) = 2 sin Zz cos QZR cos QZE = 4 sin RZ cos RZ cos QZR cos QZE = 4 sin^ J, since (10) cos RZ sin QZR = cos ZQR cos ^w, cos QY cos 6 sin QZE = - cos QZY = - ^ sin QZ sin c Thence cos Zx = 2AC + 2BD = i( - a|3 + 78) = sin 9 sin ^, cos Xc = 2AC - 2BD = J(- ay + ^S) = sin d sin ^. (11) (12) _ _ Similarly (13) cos X?/ = 2AB + 2CD = U{ - a^ + jS^ - y' + 8^) = cos <^ sin >/, + cos fl sin ^ cos ^, (14) cosY'a,' = 2AB - 2Ci>= \i (a^ + /S^ - y' - S^) = - sin ^ cos »^ - co&0 cos ^ sin i^, (15) cos Y^ = 2BC+ 2 AD = J?: (- ay - (38) = sin cos .^, (16) cos Z^ = 2BC- 2AD = J? (a^ + y8) = - sin 6 cos ^. Thence the semi-symmetrical relations (17) cos Xx + i cos Xy = a^ - /3^ = (cos ^ + / cos 6 sin <^) e^\ cos Xc = - ay + (38 = sin e sin ; (18) cos Yx + i cos Yy = i (a^ + j3^) = ( - sin + / cos cos ^) e"*^, cos r^r = i ( - ay - (38) = sin e cos (^ ; (19) cos 7.x + I cos Zy = - 2a)3 = - / sin Q e"^, cos Zz = a8 + /3y = cos ; (20) cos Xx + i cos Y« = - |3^ + 8^ = (cos ;/- - / cos 6 sin i/-) e— **, cos Z.t; = — a/3 + y8 = sin 6 sin i// ; cos Xy + / cos Yy = l (|3^ + 8^) = (sin i^ + i cos 6 cos j/-) e— #, cos Z?/ = i (o|3 + y8) = — sin cos ^ ; (21) (22) cos Xz + z cos Yz = 2j3S = « sin f^—fi" cos Zj = a8 + /3y = COS G. Thence, or directly by spherical trigonometry on fig. 56, the matrix for Euler's unsymmetrical angles 0, i/-, (j>, X Y Z X cos (^ COS i// — COS e sin i^ siu i/- — sin ^ cos 4* — cos 6 cos ^ sin ^ sin sin t^ y cos a sin ;// + cos 9 sin ^ cos v// — sin (^ sin ^ + cos 6 cos ^ cos ^ —sin e cos \^ z sin 9 sin i^ sin fi cos cos B In the order of displacement i//, 6, »//, o£ §20, (i) X + iy,z is changed into x^ + iy^ = (a- + iy ) e-'/'*, z ; (ii) ?/i + /z, x^ „ y, + /Z = (Vi + ?^ ) e-oi, A'l ; (in) Xi + iy^, Z „ X + /F = (a;, + iy.,^ e-^'-,Z. Thus, for example, (23) X = Xi cos + ?/2 sin <^ = x^ cos ^ + (y^ cos + j sin 9) sin = (a; cos i/i + y sin xp) cos <^ + ( — .x' sin 4' + y cos ;^) cos 6/ sin (j> + z sin sin ^. and so on, for the other relations above. 22. To realise these angular displacements mechanically, and to make them visible, take two gimbal (Cardan) rings zx, Zx, concentric at 0, as frames of reference, and let the outer ring zx be moveable about a fixed (vertical) axle through Oz, and the inner ring Zx about two short axles in the line Ox, pinning the two rings together ; and mount the flywheel body on the axle through OZ, fixed in the inAer ring, as in fig. 57 of Grilbert's barograph. With preponderance, the motion of the flywheel under gravity w^ould be similar to the gyroscopic motion shown in fig. 3, neglecting the inertia of the stalk and gimbal rings. Starting with the two rings in the same plane, and making the movement about an axle one at a time, (i) turn the ring zx into the position zxi, by a rotation \p about Oz, carrying with it the ring Zx and the body ; (ii) turn the inner ring into the position Z^'i by a rotation 6 about Ox, carrying the body with it ; (iii) turn the body about OZ through an angle f so that Zx comes into the position ZX, and XYZ is taken as the new frame of reference. The components of angular velocity of the inner ring are then 6 about Oa'i, \p about Oz, equivalent to components, (1) 6, sin 6 Tp, cos Q-ip, about Xi, r/i, Z ; or equivalent to components (2) cos (^ 61 + sin (p sin 9 i^, — sin ^ + cos ^ sin p, cos Q ;/-, about X, Y, Z ; and the frame of reference XYZ has the additional angular velocity <^ about Z. Then if 0„ 627 %i denotes the components of angular velocity of the frame of reference XYZ, (to be distinguished from the former use in § 6 and elsewhere) (3) 61 = co&(j> f) -r sm(i> sin9 -p, ,6.^ = — sin^ 6 + cos<^ sinfl ^, 63 = <^ + cos0 i/-. If P, Q, R denote the components of angular velocity of the body, movable about the axle OZ, P = 0^, Q = 62 ; also i? = ^3 if the frame of reference XYZ is fixed to the body; but as they can rotate independently, it is useful sometimes to keep R and '63 distinct. Experiment with these gimbal rings in fig. 57, suspended by a thread attached at z. The above is Klein's order of rotation ; but Routh takes the inner ring as pinned to the outer zx about two short axles in the line Oy perpendicular to the plane zx, carried by another gimbal ring zy or xy fixed at right angles to zx ; and makes the second rotation about Oy, the position of Oy after the displacement \P about 0^^, as in fig. 58. The final position will be different according to the order the displacement is made, of 6, 1//, (j,. In showing the angular displacements with the wheel of fig. 3, the order will be d, -p, (j>, as in fig. 59 ; and. here (i) keeping the vertical spindle fixed, and moving the .stalk through an angle & about the pin at 0, xyz is changed into x^yz^ by a rotation 9 about y ; (ii) holding the wheel and the axle together, x^yz^ is changed into argyaZ by a rotation ^ about z ; ^ . (iii) releasing the wheel from the axle, ajg^/aZ is changed into XYZ by a rotation <(> about Z ; and then (i) changes a; + tz, _v into Xi + izi ^ (a; + iz) e-^\.y; (ii) changes all three ; (iii) changes X2 + 23/2, ^ into X + iY = {x^ + iy^) e-^\ Z. K 28570 74 If the wheel is not held to the axle in the (ii) movement, but is free to revolve, ^ + i// cos e = 0, if is kept constant, while ?// varies ; thus in a complete revolution of Tp with the axle at an angle 0, the wheel will have turned round backward on the axle through an angle 2t cos 9, the length of intersection of the reciprocal cone with the unit sphere ; as before on p. 13, §14, Chapter I ; and in this reckoning the restriction can be removed of making the axle return to its original position so as to close the conical angle. Here R = ; otherwise + ip cos = Bt, R being constant ; meaning that if the axle is arrested in any position, the wheel is left revolving with angular velocity R. Kelvin's trunnion rings, or knife edge gimbals (Natural Philosophy §345 ; Gray Dynamics, p. 506), can be realised in fig. 3 and 36 by dividing the stalk OG- in a joint at 0, at right angles to the stalk and the pin at ; and then clamping the vertical spindle at 0. Taking the pin at in the direction of?/, the order of displacement from the vertical is on fig. 60 through (i) i&n ungle about y, changing xyz into Xiyzi ■; (]]) » ,, ^ „ ^i, „ '^iyzi „ XjJ/oZ ; (in) _ „ „ /3 y A+ S' as demonstrated in Klein-Sommerfield Kreisel Theorie, p. 26 ; and thus the motion in space with respect to the fixed frame xyz of any point fixed in the frame XYZ defined by the parameter A, is determined completely by X in terms of a, j3, y, g. Thus for a point on the axle we take A = 0, or x • 'and then (3) ^ " S " ' *^" 2^ ^''''' or X = - = - /cot |0 e ^, in the stereographic projection on the horizontal plane xf/. But in the orthogonal projection on the horizontal plane of (47) (49) §1, (4) aj3 = ^i sin Oe^', y S = ^i sin 0e-+'. In the associated motion, where and X are interchanged ^^) _ ^ = aani0r-*S «y = Ji sin fle^', /38 =1/ sin «e-*'. Also, with measured from the zenith, <®) "^ ^J'^'^'i^^ ^7 = - sin^ he, ,,S - 3y = 1, aS + /3y = COS 0, V ( - ~a,3yS) = J sin ; 75 and in (49) § 1, and (8) (10) §18, (^) dt \k "^P ^0 = *'" '^^ ^'^' = '^"^ = nVdF-^ W P (10) ^^exp... =^-(a^-^^),and^exp^. = -(3^-S-^). ■ In the Weierstrass notation, / , 1 \ P / I \ • T (m + v) „ a('U + V2) ^ <^ (W' + Vi) „v„ (11) rexp(CT - ht)i = -W ^e-<'- a = -W -e-vzv,, )3 = V „ ,. e-«^''^, (12) ??f/;; = il/f/w, iliw = 1/ (w3 + ewj), A^/ = m^w — utv^ — utv^i (13) V = V^ + 1-2, V2 =f2<^i, Vi = wi -/lO^s ; so that the equation in (9) is the dynamical interpretation of Halphen's relation in his Fonctions elliptiques^ I. p. 230, /-. .N C? fff (// + Vi + V2) .. ., ~\ a (u + V2) ., a iit + v-i) ., - ^ ' C?M |_ ^2) i> (", ?'l)- Many demonstrations have been proposed for this theorem in (2), for which reference may be made to Cayley, Phil Mag., 1845, vol. 26. Lacour, Nouvelles Annates, Dec, 1897, Campbell, Mess, of Math. W. Burnside, Acta. Math. 25, 1902. A. R. Forsyth, Theory of Functions, p. 621. A. Schonflies, Jahresbericht, Kov., Dec, 1909, p. 457. : i Wellstein, Jahresbericht, June, 1910, p. 169. G. T. Bennett, Proc. L. M. S. vol. 9, 1910. ■ Poincare, Lectures. .■ ! Rotation about the given diameter OQ, through an assigned angle w, of the frame xt/z into the position XYZ, gives a linear relation between the coordinates of the same point with respect to the two frames, as shown in §21 ; and stereographic projection is a conformal operation, in preserving an angle unchanged. ■ Hence a linear relation must be expected between X and A, as in (2) ; and it remains to be proved that the a, (3, y, S there can have the values assigned in (1) §20. This is shown in Forsyth's Theory of Functions, p. 621, by considering the points ou OQ, unaltered by the rotation, and so the fixed jjoints of the linear substitution. Otherwise, considering points on the polar axis Oz or OZ in fig. 56, in the transform- ation (2), and first at Z on OZ, (15) A' =0, r = 0, Z = r, A = 00 ; (16) X = r sin 9 sin 4-, V = — ''" sin cos -ip, z = r cos 6, ,,_, , X + v! r sin (sim/-— «cos i//) — / sin Qe'^' . , ,„ ,, a (17) X = -^--^ = ^:j — ^ J- — ^ = -j —= - / cot J0e'l'» = -, ^ ' r — z r (1 — cos d) 1 — cos 7 At the antipodes Z' on OZ, (18) X= 0, r = 0, Z = - r, A = ; (19) X = — r sin B sin ipi V — '"' sii^^ cos 1^, z = — r cos %, (20) ^ ^ sin e ( - sin ^ + ^^ cos ^) ^ ■ ^^^ ^^^^, ^ ^ ^ 1 4 cos I At z on the axis 0^, (21) X ^0, 2/ = 0, z = r, X = -^ ; (22) X = r sin sin ^, Y = r sin cos «^, Z = r cos ; (23) A = ^ + V = ^ sin g (Bin ^ + ^" COB .^) ^ • ,^t i0,_,, ^ _ ^ ^ ' r — Z , r (1 — cos 0) - 7 and at the antipodes z on Oz, (24) « = 0, 2/ = 0, „- = - r, X = ; (25) A' = T-r sinfl sin (^, • F = — r sin eos , ' Z= — rcos^; (26) A =.Bine(-sin^-.cos^) ^ _ .^^^ ig^_^, ^ _ ^ ^ ^ 1 ■+ cos ^ a Thence the values are obtained of a, |3, y, S, as in (1) §20. 28570 '^ ^ 76 Kirchoff's Kinetic Analogue of the Twisted Elastica. 24, The projection on a horizontal plane of the path of a point, C, L, or P, fixed on the axle of a top, has been identified in XI, §1 with the hodograph of the curve described by the vector OH of resnltant A.M., a curve in a horizontal plane through C. The spherical curve described by a point, L or P, will then be the hodograph of a twisted curve on a vertical cylinder, whose cross section is the curve of H, when this twisted curve is described with constant velocity ; and the curve can be identified wdth a tortuous Elastica, made by a uniform round wdre, bent and twisted in a certain manner by a wrench applied at the ends ; this is the statement of Kirchoff's Kinetic Analogue for our case, where the axle of the top moves so as to keep parallel with the tangent of the elastica, as the point of contact moves with constant velocity. Take the axis Oz vertically downward as the axis of the applied wrench, consisting of a thrust Z along 0^", and a couple iV in a plane perpendicular to Oz ; denote the torsional couple about the tangent of the wire by iV', and the flexural rigidity by B. The component couples of resilience about the axes 0.K, Oy, Oz are taken to be B times the curvature of the projection on the co-ordinate planes ; that is (1) B{y'z"—y"z', z'x"—z"x\ x'y"-x"y'), an accent denoting diiferentiation with respect to the arc s. The equations of of equilibrium are then (Binet and Wantzel, Comptes rendus, 1844) (2) B(y'z" - y"z') = N'x' + Zy, (3) B{z'x" - z"x') = JN'y' - Zx, (4) B{x'y" - x"y) = Kz + .V. Differentiating each equation with respect to s, multiplying by x\ y\ z\ and adding gives (5) = — — , so that N' is constant, the analogue of (r' Hi/ Br ' ?; = 2eK + K'i, v = K + fK',\ v' = j{ +f'K'i^ vi = A' + f\K'l, v., = t\K'i, for cos 0= + 1 ; f2 -/i =/, /2 + /i = 2 -/, ;; = 1 - ^(/ + /), /, = 1 - i (/' - f)^ (10) ^ {-J. exp ct/) = i in sin f''/'' = ?i « f3, (11) (12) o- (» + t'l + <''j) >■ ./ V '"-11.' ij ^ — iili-\—1lZr. = ^l!L±ij),-».-,.,^_OL±_^A),-«o, = (w, n) «?> (», i^a), 90 e («/ + v)>. "977 ec J ^•'' algebraical if /,r is congruent to a period. 81 CHAPTER IV. Geometrical Representation of the Motion of a Top. An important theorem— in this subject of gyroscopic motion is due to Darboux {Comptes rendus 1885, reprinted in Despeyrous, Mecanique IT., Note XVIII., XIX.) which asserts that the top in the motion ot its axle can be accompanied by a deformable hyperboloid, built up of its generating lines, as in the model designed by Wiener of Darmstadt, and published by Teubner, shown in fig. 62, to a scale of one to eight. In the ideal realisation the lines of the hyperboloid would have to penetrate each other, and the figure turn inside out. An actual model of rods of finite thickness will begin to jam near the extreme position of focal ellipse and hyperbola, like other linkages, as in fig. Q2, A and B ; and it would require to be taken to pieces to avoid the position. The Deformable Hyperboloid of Generating Lines. 1. The hyperboloid passes through the shape o£ a series of confocals in such a manner that OC, OC is one pair of generating lines through 0, ,while the opposite pair is made by the parallel lines HQ, HQ' through H. The hyperboloid changes so that H moves in a direction KH perpendicular to the tangent plane at H, or to the parallel plane COC, and if moved with, velocity J nk sin d, as in Chapter III, §1 (62), it will give the plane COC the appropriate precessional velocity in azimuth, with OC held vertical. But the principal axes of the hyperboloid move independently of the axle of the top. The synthetic method must be adopted again ; and it is convenient in the explanation to take the hyperboloid enlarged to double scale, so that becomes the centre, and OC a fixed vertical diametral line, called the invariable line, and then OC is called the conjugate line. Writing the equation of the focal ellipse (fig. 62A, 64) and a confocal surface, with a, /3 instead of the usual a, 6, which are required elsewhere in Darboux's notation, then X + a^, X + j3^, X can denote the squares of the semi-axes of a confocal ellipsoid, with X changed into fi and v for a confocal hyperboloid of one sheet, and of two sheets ; arranged in the sequence (2) 00 > A > > M > - /3'^ > V > - a^ Then in the deformation of the hyperboloid of one sheet, fi ranges from to — /3l while X and v remain constant for the point H on the surface. Utilising the theorems of Solid Geometry on confocal quadrics, the magnitudes may te chosen so that <8) X + a^ + ^ + j3^ + V = OH^ = iP (cos e + F) = p" + S^ measuring ^ from the downward vertical. Replacing cos d by ^^ as before in Chapter III, but keeping it distinct in mind from the z co-ordinate in (1) we are able to write, (4) ,^ + a^ = W{^--^i)=p'-p'= «' dn^ mt= ^- ^-j - ^ x\ (5) u + /32 = JF {z - z,) = p- - pi = a' K^ cn^ mt = ^~ ^/>f ~^ f. (6) .. = JF {z - z,) ^p'- pi = - a' k' sn^ mt = ^ ~ \p^ ~ ^ ^^ (7) pi HQ - A, B, O where A, B, C, D are new constant quantities ; these must not be confused with the A, C of Chapter I, or those of Chapter III ; this overlapping of notation appears unavoidable in our subject, where we wish to follow the system of former writers. A new quadric surface is given by (2) Ax'+ Bf-v Cz^=D^, as if it was the momental ellipsoid of a body with principal M.I. A, B, C ; or from (1) lit 1/^ Z^ (•^) H V + HT + HP = ^Q' with the squares of the semi-axes given by HV.HQ, HT.HQ, HP.HQ, and with HQ the normal line at H, and so the surface touching the horizontal invariable plane throuo-h C The wire line of curvature lies on this surface (3), and the principal planes "cut off the constant length HV, HT, HP on the normal HQ ; the direction cosines of the normal are (^) HV' HT' HP' «^ m ' so that squaring and adding, .2 „2 V (5) gy2 + HP + HP2 = 1' or (6) A^x" + BY + Cz" = i)2g2 83 a second quadric surface, an ellipsoid ; and so we arrive again at the Poinsot polhode as their curve of intersection, the wire line of curvature, intersection of the confocal surfaces defined by X and v. Conversely, from (2) (4) (5) (6) §1, we can put (7) lAx' = (5 - (7) (/x + a'), IBf ={C -A) (u + /3^), ICz'^ = {A - B) m, where (8) {B-C)a^ + {C -A)^^ = Wl\ (9) A{B- C) a^ + B{C - A)^^ ^ Wl\ and then r^f^^ ^' v' ^' (B-C C-A A - B\ I , 2 9 9 (11) ~2 2 + -5-^ a + T 5 = 1 ^ ^ p' - Pi P^ - P2 P' - p/ on taking, as in (4), (5), (6) § 1, ns^ 7 B-C.C-A.A-B , Z). i^ - i). C - ^ Z).Z)-A^-C , (3^ (13) a= 7^7, 8, (3 = TWn ^' " = -■' ABC "' ^ ~ ABC .2 > and then 9 ») 9 9 9 9 (i"^) nr~^ + ^ 1 2 + Y = 1' ^ 2 + ' -i + - = I, ^^ A + a A + jO A ' v+a V + p V ' provided X, v are the roots of the quadratic in p, B-C C-A A-B (i-'^) -A_ + -^ + -^ = • p + a^ p + P^ p ' so that the polhode curve is the line of curvature, intersection of the confocals in (14). Thus if -D = C, a^ = (3^, and the line of curvature is a plane curve, central section of an oblate spheroid, Poinsot's separating polhode. 3. Drawing the normal plane to the Poinsot polhode cutting the principal axes in X, y, z, (1) ju -I- a' = X. OX, /.I + (3^ = y. OY, /u — z. OZ, .-- ^^ _ _^ BC f ][' _ CA ^ + «- ^ p2- p^ ~ A - B. A - O ^ J, ^^- f? - p^- B - C.B z" z^ AB „2 _ „ 2 - c - A.C - B' p P - Pi as in (4) (5) (6) § 1, and so the deformable hyperboloid is determined, of which one generator through H is vertical, normal to the horizontal plane through C, Poinsot's invariable plane ; the other generator is parallel to OC, and the plane COC will revolve round OC with angular velocity -~ when H moves with velocity ^nk sin 0. The second pair of surfaces associated with the 'conjugate line OC may be written (3) A'x^ + B'f + Cz"^ = D' g'2, (4) ^'V + BY + C''^ = D'^ S'', intersecting in the same polhode curve or line of curvature ; and the motion is such that (3) rolls on a plane through C perpendicular to OC, corresponding to the other generating line HQ' through H. Thus one polhode curve, or line of curvature in wire, will roll on two planes, at a constant distance from 0, S and 8', and the motion of the Top is made up out of the combination ; this completes the statement of Jacobi's theorem ( Werhe II., p. 480) that the motion of a Top can be resolved into two Poinsot movements of a body under no force. 28570 ^ 2 84 4. In the spherical top the vector OH of angular momentum may also be used to represent the angular velocity. But in the general top the vector 01 of angular velocity will be drawn to a point I fixed in HQ', such that (1) Ql=i ^ ' Q'H a where A, C have the original meaning of M.I. as in Chapter I. This point I can be joined to a fixed point G' on OC, and that IQ', IGr' are the generators through I of the hyperboloid ; and then IGr' is a fixed length, so that I describes a sphere about the centre G', and the motion of the Top is represented by rolling the curve of I on this sphere, with angular velocity proportional to 01. 5. Making a fresh start with Poinsot's motion in (2) and (6) §2, representing the motion of (2) rolling on a fixed invariable tangent plane, the horizontal plane through C, and differentiating with respect to the arc s. (3) ds ^ ds ds '' ds ^ ds ds HK dx ""Ts dy y^s - C - A- B dz ds B - C~ A ~ A - B C B - C.C - A.A - B. ABC because /ox dx dy dz „,^ the projection of OH on the tangent line HK. Combining (2) and (6) §2 with (4) «2 + f + ^2 = OH^ = p' + S', we find by solution (5) (l-^)(l-^).^.,..(l_|)(l_^),3 = ,._,. (6) P.' = - (^ - §) (i - ^>^ = - QW^ = - QT. QP, on fig. 64, where the hyperboloid is shown flattened in the plane of the focal ellipse as m fig. 62A ; and so also ' (') (i - s) (1 - l)y'= ^=- of: <■•!= - (i - §) (1 - 1) f-~ QP. Qv = 0Q.^ («> (' - Z) (l - §')»'= '-'-'>- rf= - (l - 2) (l - t) «■- Q- QT = 00=; C9X ^ ^ J^C HK ^ Jllv^r /^\-_ BC HK^ ^ ' ds A-B.A-Cx p^-f,;^' [dsj -'A- B. A-C J^^'^ • ' ■ • Substituting in (") <$h (Mh (S)'- 1. (IJ) -i SC 1 CA 1 HK=^ A~ B. A -C- ^~=r^^ + J5^^Tr^6"-Tr AB - , ,v - ''-'^' (12) -iR = p^- p;i p^ - pi ,. _ ,3., ^ ^ 4. ^. _ 3. . _ ._ , _ BC ' - r r 1 ^^^^ ^^A~Z B.A-C ^(^'- P^') ('>'- P^') + ... + ... To make the identification with (17) §1, where (1^) HK=J^j_J, HK^=i/.^^_, 2n dt ^ gin^ 0' P - S2 2 hp So with 6 measured from the downward vertical, as iii Darboux's treatment, we put (1.5) p'- pa'= JF(cos - z„), p'= P^(cos e + E), and then so that, in addition, (17) K = - ^k' sin^-e = Ik' cos^ 6 - i/^ ^{p^- ^EPy - It, requiring (19) i(^-l)S=f?.5.^-l^^-^-^^-^-^-^' 8* \A B r I ABC (-<■») ^¥-^[bc^ ca^ab) abcKa^b^ c~ ') Then, from (11), / , A - D. B - D. C - Z>.,V (21) CK^ ^ ,^ - HK^ =. ^- ^_ L ' /99N / 2 rf-^N' 1 _« 1 CK^ (^^A-D.B-D.C-D ..\' S' ' r2^^ ^"^ _ g , A-D.B-D.C-D 8^ _ QH QV.QT.QP. ^^ '^ dp' ^R ABC p^ V R s/R p' ^ R ' and changing the sign when a start is made from H3, (24) _,,^r^^Qv:QT.QP_^^ J p p" -J R the equivalent of (3), §7, Chapter III. On fig. 65, (25) CK sin = g cos + 8' = 2 g «^ + g' - ^g, and from (21) above, /^^N riT' ■ /, 0/2 A- D.B - D.C - D ^^\l (26) CK sm e = 2 (p^ + ^^^^^ ^ 8^ j-^ implying (27) g-^g-2 ^^^^^^^ -, ^^*>' g"~ \5^ ^ a4 ^i^ ABC)/c'' If v denotes the velocity of H in the Top motion, v = ^nk sin B ; while from (5) (7) (8) /- A ABC so that, from (9) dx _ BC M^ = ^g in^. HK sin 6 (^^) ^ ^ A-B. A-C "^" " .4 - 5. 4 - C w . BC 1./R _n B - C ^, ^ . ^9Q^ 17? ^ ^ - G. C -A. A- B \' ,.,2,2 (29) 4K- J7T7T )^y^ = k A- B. A- C X k A with two similar equations. y^ : 6. In Darboux's notation, p, q, r denote components of angular velocity about the axes and the resultant angular velocity w about OH is taken in a constant ratio to OH, such that r w h n p _ q _ _ = _ = _ (1) ^ ~ ^ ~ J ~ OH g ^'^ This converts (30) §5 into dp B - C (2) i=-^ 'i'' 86 exactly as in Euler's equations for the motion of a body under no force, of which A, B, C are the principal moments of inertia ; and (2), (6) §2, become (3) Ap' + Bq' + Cr' = Dh\ A^ + B^ + 6'V^ = D'h\ so that in the notation of Routh's Rigid Dynamics, T = Dh\ G = Dh, and h represents, the constant component of angular velocity about OC, the axis of resultant A.M. Putting further (4) Aa = Bb = Cc = Dh, the relations take the form given by Darboux and a, b, c represent constant components of angular velocity, such that, from (1), (4) and (1) §2, ,^. a, b, c, h HV, HT, HP, HQ (9) (10) With the Darboux notation, and in addition with his (11) ' a + b + c = P, be + ca + ab = Q, abc = R, the preceding expressions become more simple algebraically ; thus (20) §5 becomes so that Darboux's Q is our n'^ ; and then from (18) §5, (13) j;,'4(2f-Q.-2)=^f*-Q-a^', F.E+^^ = ^J1^, n'' \ h h^ / Q 9j2 Q ' and from (28) §5 (14) | = i'.(g-f^f^, h'=9!L^, ^^_j^,^ 2ia-h)ib-h) (c-h) k n VA' A'' /n^' Q Q • We can write (15) Q' = [Q-2h{F- h)Y -i(P-h){a-h){b- h) (c - h) = ^Q2 - 4 (P - h) {a -h)(b- h) (c - h) for the purpose of identification with iW in (15) §13 Chapter III., p. 60 ■ and in the notation there, where the Q must be distinguished by an accent n k p, q, r, h _ X, y. z. S n k p q r Ax, Hy, a' 6' c' m ('^'r. = (»-C)(^-I)0-|)0-;-). g=(i-l^)(.-f)(i-| I - _ h equivalent to (19) h R = hi K _ ^. «' _ G, G' M n ~l 2A,r- From (1), (4), and (6), (7), (8), § 5, (20) 'V = (l-f§:)A^ = A^-(A-6.A-e) = (6 + c);.-6o, -r= (c + a)h-ca, u.,' = {n + b)h-ab; 87 5 2 ^2 and, as in (2) § 3 (21) P = «' .g\ =, . = b? — lai^ a — h. a — c' w^ — wg^ b — c. b — a' w^ — w,^ c — a. c — b Putting, for a moment, as on page 27, (22) Q = 4. 0,^ - a>^ V - '^'. '^s' - -^^ = (^)^-R, /q„N c?a> .dp^^dq, dr b — c. c — a. a — b i , n, when the elliptic integral for mt is normalised. 7. The equation of the plane OCH is (1) (i^-(7)^+ {G-A)y-+ {A-B)'y=Q, while, from (9) §5, along the polhode curve of H on fig. 66, A dx B dy C dz ^^> W^C'^db^'C'^^^yTs^A'^^^^Ts' which proves that OE parallel to KH; the tangent at H, is conjugate to the plane OCH (fig. 65). This is seen geometrically by resolving the angular velocity w about OH into a component h about 00, and another component about OF parallel to CH. Considering the motion of H along the polhode in fig. 66 on the quadric (2) §2, the first component about OC the perpendicular on the tangent plane at H does not affect H ; but the second component makes the plane OEF, fixed in the quadric and conjugate to OH, to turn about OF ; so that if H moves to a consecutive position H' on the polhode, the plane OHH' is conjugate to OF, and so HH' is parallel to OE. Poinsot gives a proof depending on fundamental dynamical principles in his Theorie nouvelle de la rotation des corps, 1851 ; discussed already in §35, Chapter I., the treatment is resumed here. In the first place the components p, q, r of the angular velocity about OH of a body whose momental ellipsoid, referred to the principal axes, is (3) Ax'- + By" + Cz"- = m^, give rise to components Ap, Bq, Cr of A.M., so that the vector OC of resultant A.M. Dh is perpendicular to tlie plane conjugate to OH, with respect to (3). The impulse couple Dh about the axis OC will start the body with angular velocity w about OH ; or reversed will stop the body dead. — The momentum of a particle of the body is changed into its centrifugal force (CF.) from OH by multiplying by w, and turning the direction backward through a right angle. The CF. has nothing to correspond then to the component A.M. about OH, but the resultant CF. couple will be the component A.M. perpendicular to the axis OH of rotation multiplied by w, and turned backwards through a right angle in a direction against the rotation ; the CF. couple is therefore Dhu)sm COH,in the plane COH, the vector product of the A.M. and angular velocity, and this is the couple experienced by the axis OH if it is fixed. The body feels the couple the other way, and the axis of this couple is given by the velocity of C by its rotation round OH, and if OH is fixed the rate of change of A.M. of the body is equal to the CF. When the axis of rotation OH is released, the axis OC of A.M. becomes stationary, the former velocity of C being neutralised by the velocity due to the couple of CF., and OH moves in space ; and the CF. couple will cause H to move in the direction conjugate to its plane COH, In the Vector- Quaternion method, the product of the vector pi + qj + rk of angular velocity, and of Api + Bgj + Crk of A.M. is a quaternion. The scalar part is Ap^ + Bq^ + Cr^, a constant, twice the kinetic energy, which we have written DJr. The vector product is (4) {B - C)qri + (C - A)rpj + (.4 -B)pqk,^ giving the components of the CF. couple ; and so leading to Euler's equations of motion of the vector of A.M. through the body, (5) A^^(B-C)qr, , , 88 8. The geometrical property of the herpolhode curve of H can be obtained from the relations of the three conjugate semi-diameters OH, OF, OE, combined with OC = S, the constant length of the perpendicular from the centre on the tangent plane at H. Then for the surface (3) §7 (fig. 65, 66), (1) OH^ + OE^ + OF = {^ + -§ + §) ^', and with KCH = x, cos x = mj) as in Chap. III., §1, X, / n^ D^ D"^ \ (2) OF. OF^ cos^x + OK^. OE^ + OW. OC^ = (^ + ^r^+ -j^f^ (3) OCl OEl 0F2 cos^A = -^^. Then (4) OF + OF = (j + ^ + ^ - ly - p' (5) OKI OF . OC^. 0F= (^ + ^ + 3^ - j^y and by solution (6) CK'. OE- = {,' + - "''- "^/bc' '^"^ »y (7) OKI OP -[(£ + §+ ^ - l)?- - ,'] (CK' + S')-(;g+^ + ^-3^)s* and writing (3) (8) CK^ OF. 0F2 = -^ ^p\ CK^, we obtain, as before in (26) §5, (9) CK^ = ^ ^ ^, which makes J "- ABC ^ ^ nn pp. ABc' P ABC^P ^ ^ ^^ ~GK\ OW- 2 'A- D. B - D. C - D~' ^ ^ — ABC ^' (12) CK. OF = JgP sin e. The measure of curvature at H of the surface (3) §7, product of the principal radii of curvatuie, i?i and R2, is ^^^ ^' S2 HV. HT. HP ^^^ ABC^^ HQ ' and if HN is the normal chord through H, (14) i,4.1.. 2 _ A + B + C , two simple exercises. ^ 9. Continuing with Poinsot's treatment, the direction ratios of OF, parallel to CH, through H {w, y, z) and C (^ x, ~y. ^ ^ ), are (1) _ U- D) X, (B - D) y, (C - D) z, and since (2) _ _ U- £>)V 4 {B - D)Y + {c - nyz' = z>2 ^2^ the direction cosines of OF are (3) ^- -P * J^- I> y C - D z thence the expression for OF in (11) §8. > 89 The direction ratios o£ the normal to the plane OCH are given by (1) § 7, (4) {B - C) yz, (C - A) zx, {A - B) xy ; and on reduction (5) {B - Cfyh^ + {C - AfzV + {A - Bya?f = D'By, so ihat the direction cosines are /g\ B — C yz C — A zx A — B xy ^ ' D %' D Ip' D Tp The direction cosines of OE or HK are given in (2) §5, (7^ ( ^ - G C - A A- B \ - ABC ^^ ^^ \ Ax ' By ' ~C^) B -C.C -A.A- B ' where from (11) (29) §5, .ox jrrr2 P /B-C.C~A.A- B\' X^z' (8) HK = -4^= ( ^^ ) p,L_, so that the direction cosines are B - C C - A A- B (9) -^— F, ^g— ^^> — ^-^^ . IF sin e and thence the expression for OE in (10) §8. The "angle between OE and the normal to the plane OCH is the angle HCK, denoted by ;)( in §1, X, Chapter III.; and so sm « sm ^ ^^ E + z Pp' >/ E (10) , / 3 ^ A-B.B-J.C-D^ g . „ \ ABL I sm » cos y^ = p — ■ ■ > and then from (10) §8, /n \ ^T- ASF sin (11) P cos X = CK = ^ Q^, . The C.F. couple, as determined in §7, is (12) Bh,^ sin COH = Dio^ cos COH sin COH = D,^^ ^^OT^ ^ ^^' ^T^ ' and acting in the plane OCH it generates A.V. about OE conjugate to the plane OCH ; and if this A.V". is a dt about OE in the time dt^ during which H moves along the polhode or herpolhode, jJarallel to OE, a distance ds^ , . adt ds (D ds n ds ^^^^ V = OH' '^ = OE dt ^ k df But a may be considered the A.V. about OE generated by the C.F. couple treated as an impulse couple ; and the A.M. due to the A.V. a is in the plane OCE, The component about OE being a times the M.I. about OE, which is D iryp} y the A.M. due to the A.V. a is /,.N n f ^ V ni^ 1 n , 0(1CH (14) JJa [?yp) sec X = C.r. couple = jU71'' — p — > ,OEl CHcosv , 2 ■ n ^« , / ■ ^ (15) It = n^ ^p = ^n^ sm 0, -r: — pik sm 0, as before in (6-^) §1, Chapter III. ; derivable also from squaring and adding the equations in (30) §5, and employing (9). Then (16) p'^ = v. CK = Jn/fc. CK sin = h (p^+ ^ ~ '^' ^~gf!' ^ ~ ^ S^) A'x'+ By+ C'z'- D'B' , {A - D)V+ {B - jD)Y+ (C - Dfz' -i- By - ^ ABC ~ '' ABC ' (17) OJ ^ = jj^,Sh = ^ h, 28570 M i)0 so that F describes a central orbit in the fixed plane through perpendicular to OC ; and in the spherical top d6 8cosfl + S' CK Sk aW-^-inSIr (18) i = ^ k.m9 = ^ F^i^ = ^^ OE^' OE^^-JnS^, ^ flo that E describes a central orbit in the plane through perpendicular to OC. 10. The algebra is not so heavy with Darboux's notation, where we write, in a homogeneous form. using r and j9 for a moment in a new meaning ; the equations become OE^ + OF^ _ (P - r)h <2) F - Q ' OF , 0F2 Qh-B <3) P-W +^^^ "0^' so that , , 0E2 Qh- R- ( P - r) h' (4) (F - ^) ^^ = ' Q ' OF^ (P-r)ph-Qh + R^ <5) (p-h)^-= ^ , :and substituting in (3) §8, or OW OP p - A _ Rh i^) k^ • k^ - r - h Q2' -we obtain <7) [(P - r) ph - Qh + R][Qh- R- {P - r) h'] = Rh {p - h) {r - h) the relation connecting this p and r in the polhode or herpolhode. Thence {Qh - RY - (Qh - R) (P - r)h^ - R (P - h) h^ <8) P - Ih [Q^ -[2{P-r)h- Qf] ^ V „ a - 2 (P - r) A <9) cos e = 2^,-JE ^^ "^ ^ <10) iQ' sin^ = - (P - A) P + (P - r) QA - (P - r)2 h' IQh-E-jP- r) h'Y [Qh-R-(P- r) h'f <11) p- h- ^j^^^2 _^2{P -r) h - Qf]- iAQ-sin^e [Qh- R - (P - r) h'Y - - (P -h) Rh + (P- r) Qh' - (P - ry ¥ Written as a quadratic in r, or rather P — r, <12) (P - r)2pA^ - (P - A== (Qp + Qh- 2R) + Rh{p- h) (P -h) + (Qh - Ry = 0, :and solving the quadratic, .(13) 2ph (P-r) = (p-h) Q + 2(Qh- R) ±Q ^ (p- h. p - h,) ^ ^ ' Q'h h ' /1-x fl -Qh + 2R±QV (p-h.p- h,) X>' ^, G = Dh, G' = D'h'. By symmetry a = ^" "" == «« = MM - 77 - " > a - 7 - Q' (18) ^- = 4-„ QQ' = aa' = /3/3' = yy = Q^ ''' " " with (19) Q' = n'^ = n^ = Q. We can write also (20) q2 = «= - 4i2 (a - A) = (2aA - af - ^a^ (b - h) (c - A), with a geometrical analogue (21) F = HS-l HS''^ = (OH^ + 0S')2 - 40KI OS^ = (OH^ + OS^)^ - 4HV=' QP. QT, and so on, (22) k' = (OH^ - 08^)2 + 40S2. KH^ = (AB" - OH^)^ t 40Q2. HP2 analogues of the equations in (22), §13, Chapter III. for MK 94 Constants of the Motion of a Top. 14. The physical material constants of a given Symmetrical Top have been denoted in Chapter I. by M, h, A, C,l = A/Mh ; and thence ti and T when swung as a plane pendulum, suspended at the point 0, in a gravity field /;. To specify a given state of general motion the additional constants, G, G' or CR^ and E, F or H, are required ; these may be called the dynamical constants. Or the motion can be specified by k, K, /, /', fi, h which may be called the analytical constants. Lastly the motion may be represented by the articulated hyperboloid, and by the constant lengths a, |3, S, S', h, called the geometrical constants of the motion. Suppose for instance the dynamical constants G^ G' are given, with An^ and their geometrical equivalents HQ = 8, HQ' = S', and HS. HS' = P ; and the angle 9, in addition, in the plane of the focal ellipse. Drawing the perpendiculars QO, Q'O to QH, Q'H will determine ; and the ellipse can be constructed geometrically with OH, OD as conjugate semi-diameters, by drawing HJ = HJ' = OD perpendicular to a bisector of the angle OHQ' ; the axes of the ellipse will bisect the angle JOJ' ; the confocal touching HQ, HQ' will be the focal ellipse, and the construction may be completed. If 02 had been given, the confoCal touching HQ, HQ' is the focal hyperbola of fig. 62B. A geometrical diagram differs from the algebraical form in that it should represent an actual case. The diagrams employed in the sequel have been chosen carefully with the intention of mapping out the general algebraical field ; and the shape and con- figuration is selected so as to admit of exact geometrical representation, and so serve to provide a number of actual states of gyroscopic motion, not drawn at random without any guidance of calculation. The choice of a typical state of motion can then be made geometrically on the hyperboloid, flattened for simplicity in the plane of the focal ellipse, as in fig. 6^A, where the modulus k is the ratio /3/a of the semi-axes a and /3. Then, in accordance with preceding notation and theory in Chapter III, the eccentric angle from the minor axis of the point of contact P of the generator HQ is am (1 — f)K and o) = &m fK' is the angle AOQ. Thus two analytical constants are settled thereby, K. and/; and the point H may be taken arbitrarily on the tangent line PQ, and HQ' i& the other tangent of the focal ellipse. Then 03 is the angle between the tangents HQ, BQ' ; and 63 is the external angle between the focal distances HS, HS' ; while HQ is 8, HQ' is g', and HS. HS' is F or OD^, where OD is conjugate to OH in the confocal through H. Expressed in the geometry of the ellipse, ,,. fl AB^ - OH^ . ^ „0Q. HP „0Q'. HP' (1) cos 03 = Q^ps , sm 03 = 2 ^p2 = 2 Qp^ ; ,^. « OS' - QH^ . ^ „ OS. OK „ OQ3. HT (2) cos 02 = ^^2 , sm02 = 2 Qp2 =2 ^^^, ■ (^\ .\. f> QS^ + QH' ua „ OS. OJ , QW. HV (3) ch0, = QJP2— , sh0, = 2-Q^^ = 2 ^ Qp, ; (^) exD - — rh 1 - HS + HS' HS - HS' (4) exp 01 - gg,, ch 5 0, - 2QJ) , sh J 01 = gQp ; (5) i^ = ch 01 - cos 02 - cos 03 = -^QH^ - AB^ ^ cos03 + i;=2gg, cos 03 + F = 2-gQ,\ Thus 01, 02 are dipolar co-ordinates of H in fig. 64, and HS, HS'= ke ±i^> OD = yfc - and for the co-ordinates of H, » ^ (6) °- = sh^i OK _ sin 02 OS ch 01 + cos 0.' OS ~ ch 01 + cos 0./ On fig. 64 of the focal ellipse (^ Ps^ = 0Q^ p,^ = 0Q2^ pi= = - qyx\ (8) P,' - Pi' = OCe + QW^ = OW^ = a\ (9) pi - pi = OQ^ - 0Q2= = OQ. PG = ^S ( 10) p\ - p,' = 0Q2^ + Q2S2 = OS^ =a'-^^ 95 Expressed by the x of §6, Chapter III, <11) <12) (13) (14) 0Q2 QV. QT X, OQ/ QP. QV X, F ~ yfc^ ~ i/2' QW^ ^ QT.QP ^ -X, /jgx QV, QT , QP ^ 1_ / /^2^3 / -X1X3 /-£i^\ .ne) QV, QT, QP ^ / x^x^ / — XiXs / - XiXj <17) PT. QV = 0B2, PT = g'^''^"'^ = uk' /-^i-^3-^i COS ft>Aft> V ^2'a?3 ' <18) PV. QT = OAl PV = a ?''^'^ =a /^2-'^3-^, sm wAw V — a;ia;3 /19) VT. PQ = OSS VT = a -^^"^ = aK'' /x^^x^-x^ sm w cos o) V — x-^x^ ' «,nd for the co-ordinates of P, /20) NP or OM = « """"^J^ = a ^ajL^irfi, MP = 8 ' ^ Aw «3 . «2 — a?! K Sin w 9 sin w = OK" <21) Ao) Zi(0 PT /c^ tan^ (o cos- <^ — ajj. x^— X2 QV ~ A^OJ ~ cos^ u> ~ X^X-i PV 1 siu^ ^ _ a;2. %— a;i QT ~ tan^ wA^w ~ sin"-* w — x-^x^ VT 1 a!2 — .i;! QV ~ cos^ w ~ X2 .gg. SW, S'W OQ + OQs ^x,T Vx^ '^^^> k - k - M ' QH _ Z HW, HW _ L±^ -X, ^'^^> k ~ M' k ~ M ' and the geometrical relation <24) (HW2+ WS^) (HW'2+ W'S'2) = HS^ HS'2= /i;S is equivalent to (25) [(Z + v/ -a;0'^+ (N/a?3 - Va;^)^] [(Z - V -a?0'+ (>/^3+ v/^2)''] = M\ reducing to the relation in (22) §13, Chapter III. If the tangent at P cuts the directrices of the ellipse in Z, Z' i^Q) SW2= PW. WZ, S'W'2= PW. W'Z', Tind hence the expression is inferred of ¥ in terms of lengths measured along the tangent HP. Other geometrical relations may be cited here, useful in the sequel ; S'Oa = OW = a, if QO2 = OQ, = P2 ; 0,W is parallel and equal to OS ; OT^ = TQ. TV, OV'' = VQ. VT, TQ. QV = OQ^ = V, VQ. QP = QOj^ = p^' ; VO2P a right ^ngle ; S'V = VS = VW, OS^ = OG. OT = PQ. VT, OS. OK = HV. QW, OQ. QP = OQ2. Q2S, QT.QP = QWS 0W2 = 0Q2 + QW^ = QV. QT + QT. QP = OA^, HQ^ - -QT. QP = HQ^ - QW2 = HO^ - OW^ = HO^ -OA^, QV. QT = 0Q\ HQ^ + QP. QV = HQ2 + QO2' = HOs^ and so on. 15. In the general expression for 0, when the hyperboloid has opened out from the focal ellipse, (1 ) sin^ 10 = ^^:^^, cos^ M = ^-1^, tan^ M = ^— -^. A — V A — V ft — V Put X = CO or V = - 00, cos ^0 = or 1, sin J0 = 1 or 0, - 180° or 0°. Flattened again into the focal ellipse of fig. 62A, <2) M = 0, = 03, tan ^63 = V^, X = F sin^ J03, v = - F cos^ JWj. 96 Flattened into the focal hyperbola of fig. 62 B, (3) A' = - /3^ = e,, tan ^0, = y A ±_^^. , X + (3^ = P sin^ ^e„ V + j3- = - k' cos^ p^. In the imaginary case, where (4) ^ = - as = ■/ ti„ th 101 = y ^, X + a^ = F Ch^ 101, V + a^ = P Sh2 J6I,. AtP, 00 (5) X = 0, M = 0, cos^SPS' = dn/A" = ^ ; and from (9), (10), (11), § 1 (6) snV, = 4^' = ,-^, c„./, = ^, d„V. = ^, (7) »■ (1 - /.) - j^. cnMl - A) - j^, dn' (1 - .« = ^^. /o\ 2 ^ — 1 — 2l V + a" — V — fi^ — V (8) sn^ /i = -— -- = ^— -^„ cn^ /^ = ^;r^^., dn^ ,A = ^, (9) sn'' (1 -A) = °;:^r_^^ , cn^ (1 - A) = /l^."', , dn^ (1 -/O = ^^- Then, patting (10) 4. X + a-. X + /3l X = L, 4. V + al V + |3^ V = .^', Thus in fig. 67, ^2 = ^^ fi^^' is the minor excentric angle of the points Ho and Hj, and 0*1= axnfiK' of H and Hj, the confocal ellipse through H cutting the tangent at A in H^j and H0H2, HHi being the confocal hyperbolas ; so that, if OQi, OQ2 is the perpendicular on the tangent to the ellipse at Hj, Hg, (12) AOQi = am (1 - /O K\ AOQ2 = am (1 - /,) K', (13) AOQ = a, = am /A", AOQ' = a>' = am /A", with the former relation of (12) §9, Chapter III., (14) / = /2-/i, /' = 2-/2-,A; and since /, / are constant along the two tangents, /, Kx dX dv ^ The roots of the ec| nation are X, v, so that (17) X, ,- = x^ + y' - a- - (3'' /f/r- + I/- - a- - )3-\2 -} '2 + /3'.r + o- //- - a^ /3^ Thus X + V = 0, if H lies on the orthoptic circle of the focal ellipse. From the relations in (13) §6, Chapter III., and from (b) (7) (8) §.), (>«) »^'^-^» = f'i. -vA-44' .c.a-h , „ A B-C D _a - b. o - h A C- B D A-C-B~D Then, writing u for mt, (23) I' = - ^C^__ P' - P.' _ BC p/ - p,^ ^^^, ^ 8^ 4-i?. 4-C- ~I' A-B.A-C W '^'' D.D - B A.A-B 1 dn^ M /94N /A-cf A. D - B , , JD B (24) (^^j = 2>.^-^ ^^ " = T-^ dn^ z. = snV dn^ u ; A~ B and similarly Then if I, m, n and /', m', n' denote the direction cosines of OC, 0C\ (26) I = L — = sn f (Jn u, I' = sn /' dn u q By J. , „ r Cz T J. I T /./ n = ~ = __. = dn / sn w, w = — dn / sn m (27) cos e = W + mm + nn' = sn/sn/' dn^ u + en fen f cn^ u — dn/dn/' sn^w as before in (10) §10, Chapter III. Then at the value u = 0, K, K + K'i, corresponding to cos Q — z^, Z2, z^, (28) sn^ tt = 0, - 03, cos Os = sn/sn/' + en /en/' = cos (w' - w), sin 63 = sin (w' - w) ; (29) cn^ u = 0, 6 = 0.,, cos 02 = k'2 sn/sn/' t dn/dn/', sin 02 = K' (sn/' dn/ ± / sn/dn/' ; (30) dn^M - 0, = 0l^■, '2 1 ' ch0i = - ^ cn/cn /' + -2 dn/dn/', sh 0i = |i (cn/dn/' - dn/cn/') In the plane of the focal ellipse, of fig. 64, (31) 03 = o,' - (o = am (f'K', k') - am (/'/l', «'), the angle between OQ, OQ' or between the tangents HQ, HQ'. In the plane of the focal hyperbola, of fig. 62 B, 03 = 0)2 — ^2, where ^2) ^^2' are the angles which OQ, OQ' make with the transverse axis ; and (32) cos (1)2 = Ao» =; en (/K'k, — j sin wg = «:' ^ii w = sn (/K'k, — ) (33) 02 = wg' — ft)2 = am iflvK, — j — am (/K'k, — ). In a similar manner it can be proved that (34) 0, ^• = am [(1 - /') K'i + K, k] - am [(1 - /) K'i + K, k]. From the relations in (2), (3), (4), with the notation of §10, Chapter III., (35) tan J03 = -^, cos pj = -~, sin 103 = ^ ; (36) tan ^62 = p^, cos |-02 = — — , sin J02 = ^ ; tan id, - ^, cos ^03 ~ A' '- *»= - A- COS |-02 k's2Ci ~ A ' th J0i = /c'siSa, ch40i 1 ~ A' A^ = 1 - k'\W. (37) th 101 = k's,S2, ch ^01 = ~, sh M, = " '''' (38) 28570 98 Thus, for example, with H on the auxiliary circle and OH^ = OA^, then (39) cos 02 + cos 9s =0, 02 + O3 = TT, d^ Co tan J02 tan ^63 = 1; ^'2^2 = K's2\di, d^ = j, Ci = ^, S2 d d 1 (40) tan jOs = I = sn (1 - /2), tan 102 = 7^ = 7^ = .' sn (1 - A)' sn (1 -/2) = '^' sn (1 -/i). From (6) and (8) , , F X - V ci ,„ A 1 ,„ „ (41) ;-^ = -^ = i + ^^^ = .7 = i?-'^'^ _ 1 + dn 2/2 _ 1 - dn 2f^ ~ 1 - en 2/2 1 + en 2/1 ' and with (42) 2/1 = 2-7'-/; 2/2 = 2-/+/,^ ^^ dn f dn f — en / en/' — F sn / sn /' (43) -, = 2 a (sn/dn/' + dn/sn/') 2 /..N ^' + S s n/ - sn/ g' -g ,2 , ,, ., ,, (44) = — ^-7 7^, = K 2 sn /] sn fo( sn / + sn /). ^ '' a sn/isn/2 ' a /I yjv / y/ The Apsidal Angle. 16. From (8) §1, (1) !L= / -1— = ^, nt= ^F «''c'' sin' w - yV cos^ u> - z'A'u> = ; and from (5) § 17, (4) w ^ a cos o^Ao, dn mt, y = a sin wA^ en mt, z = a/c'^ sin a. cos t. su m^ and the herpolhode is produced by rolling the line of curvature through Q on a plane through the centre 0, or by developmg the cone into a plane. Between Q and P, the rolling surface is a hyperboloid of two sheets, changino- to a hyperboloid of one sheet between P and T, and to an ellipsoid from T down^ o"l' faJ «5l:re H ijj'above q "'''' ""^°''''' ^"' "^'^ contrafocals of the first system' At V the rolling surface is an elliptic plate, 3 2 2 (^^ 0" + V^VT + V^VP = l' '^" = 0, 2/ = „-^cnm^, c = „ ^^ At P the plate is hyperbolic, sin a, ^" ""' ^~ = « ^- sn mt. ce) - + y^ + ^' _ 1 ^ cos w J ' ..2 „;,, , ^^^ PQ. PV ^ PQ. PT + ~ ^' "^ ^ " ~^^ di"^^^ y = ° ~^ci^mu c = ; and an elliptic plate again at T, _^_ ^ t ^ ^' , A. ^^) TQ TV + +TQnrP = l' ^ = a^,dnm^,y = 0, c = « -^^^i^ _^ sn mt. cos to 103 Thus for example, for/ = J, when P is Fagnano's point on the focal ellipse, (8) QT = a, QV = /3, qP = a-(3, and the surface (5) (6) (7) becomes (10) ^+i.+^ = ,_ft <") .4l3 ^ ^ - 3 = °- Fig. 68 has been drawn with (12) / = i /c - -I = 0-6 = sin 37°, ^= 1-115. 2""- AtY, (13) OX = * = 0-8' |l = 0-5 + 0-892 = 1-392 = ^. AtP, (14) ^ = 1 = 0-2, II ^ Q.g _ Q.223 = 0-237. AtT, {15) ^ = f = 0-8, " = 0-.5 - 0-892 = - 0-392. For the spherical pendulum, when H is on the pedal at Qi, we find ^ = 2 ^ 0-4, I^ = 0-5 - 0-446 = 0-054 = ^^^, - - - 0-973. OA ^ ' Jtt ^tt ' tt A little lower, where n = 0, ^ = - tt, H is at 0' where LO' = 4-48, with OA = 10. Associated States of Motion. 20. With H at L, the apsidal angle 11 or "^^ is ^irf; and when / is a rational fraction, the herpolhode of H is an algebraical curve. As H is moved up or down from L, the herpolhode may be supposed to expand or contract in a fan-like manner, while p keeps between the same limits p., and p2, but the HT apsidal angle is made to increase or diminish byy=-T-^! ^^^^ so it can be laid off graphically on a straight line. Seen stroboscopically, with appropriate rotation, all these H curves can be made to appear the same algebraical curve as when H is at L,'and the projection of a point on the axle is seen as an algebraical curve at the same time. With H at P, the herpolhode of H has cusps on the outer circle p — p^; and 6s = 0, a lower rosette. Below P, 63 is taken negative : and between P and T the curve of H has loops, so that w can be stationary, and retrograde, as when K and C are on opposite sides of OC ; and we take "^^ = 11 — tt. g' When 0) is stationary, S cos + S' = 0, cos = — -»-; c thus in the spherical pendulum with 8' = 0, cos 6 = 0, and H has loops if the spherical pendulum crosses the equator. The angle i/- is stationary when S' cos 6 + ^ = 0, cos 6 = — r? ; and the axle o g describes cusps if cos 62 = — »?■ Somewhere between P and I there is a position 0' of H where the apsidal angle n is zero, and 'i^ = — ir ; and the curve of H is a closed oval, described retrograde. Below 0' the apsidal angle 11 is negative ; and at T, cusps appear on the herpolhode at H2 on the inner circle p = p^. Somewhere also between P and T the pedal is crossed, and a spherical pendulum motion is obtained, with a looped herpolhode, and 63 > 90°. 104 H Si ~1! C3 thid, tan ^6*2 tan \e, I -1 -1 - 1 00 CO a' 1 + cog wA(0 Aw + COS 0) — Aw — cos w k' 8n= i/ I + Aw K^' sin w tan (^TT — ^(u) V - 1 -1+2 a:'^ sin w^ — cos 2w Aw K sin w tan (^TT — w) b A(i> k'^ sin w — Aw sin w — Aw + K k' cos w , 1 — sin w Aw K tan (J^TT - -Jwj Q ... ... ... 6 Ah> i.-'^ sin w — Aw sin w — Aw + K K COS W , 1 — sin w Aw — K tan (^TT — |w) L ... ... ... P 2 A^w - 1 1 — 1;' COS w Aw Aw K sin w a 1 — COS oiAw Aw — COB 01 — Aw cos w -K'sn^(l-i/) 1 + Aw k' sin w — tan \b> T - 1 cos 2w — k' cos w 00 — tan w Aw 6' A(i> — k' sin w — kAw — sin w — Aw + K k' cos w , 1 + sin w Aw — K — tau (^TT + ^w) r - 1 - 1 -1 — 00 — 00 H a /' A h 4 B ! G D I 00 2+/ -f 1 1 1 a' Aw + cos w sin w 2 */ -\f 1 + cos e. OOS 63 — ch fli cos Ss — ch ^1 COS 6, V cos wAw sin (i> 2-/ / . 00 cos* 0) A'w /> Aw — K sin M 1 \iX+f) Ki-/) cos flj COS e. 1 — ch ^1 cos (^3 — ch 61 cos Oj cos w Q ... . /^ Aw — />■ sin w 1 \(X+f) i(i-y) COS ^2 COS 1)3 1 — ch «! cos 6)3 — ch Oi COS 63 cos w L ... ... zn /' zs / zn/ ZS(l-/) zn f /-n (1-/) P (>•'* sin :"' sin= w — cos'-' 0) A(o 00 a Aw — cos w sin w i + i/ i-i/- 1 + cos O2 COS Oa — ch Oi cos 9, — ch Oi COS t„ T sin wAw cos w -/ 1+/ 1 sin- w 00 A'(» &' — Aw — K sin w -1 1 + H1+./") i + Hi-n cos 0,, cos &., 1 - oh ^1 cos 6,, cos w — ch 0, cos Os I' 00 -2+/ 2 2-/ 1 105 (e, = T - i SPS'j [ e, = TT - sps' j ^3 = 2'^ — (^ d, = \w- e, = 0, = f «s = I {d, = 7r- SPS' I f ^3 = - <. \ I 0^ = ^-^ SPS'j 6; = - 2oj 63 = — -i^r — ■ — sin wAd) / Ad) + K sin w cos w y — sin (1) cos w 1 — cos wAo) K- sm (1) / ^"^ + cos w _ - sin w cos ft) — sin ft^Aw cos fcl y - 1 Aw — K sin w y- w + cos (It cos (i)A(i( y Aw — k- sin u) y A6» — K sin - sin w cos w y._^ 1 . — Sin w cos (u yAiii + i: si 27~ sin w - 1 ft» Aw — cos la 2k^ sin" b) Abi + K sin sin wAw cos w — sin wAw — t: cos w 00 A/ D' B/ D' 9L D' ellipsoid elliptic plate hyperboloid of one sheet a cone, A : B : C = — i-''^ sin^ w : «'' cos^ w : A" i hyperboloid of two sheets hyperbolic plate hyperboloid of one sheet elliptic plate ellipsoid 1 00 1 1 . COS^ ft) sin' 0) 1 1 — cos-* ftJ CO 1 1 1 A' ft) 00 A'ft> 1 1 ellipsoid, circular cylinder. elliptic plate. circular cylinder, hyperboloid of one sheet, circular cylinder. hyperboloid of one sheet. hyperbolic plate. hyperbolic cylinder. elliptic plate. circular cylinder, ellipsoid. 106 Below T the curve of H is an undulating Sgure, described retrograde, and 62 > 180°, and ^ = n again. — - At b' on the tangent at B' of the focal ellipse, or on the asymptote of the focal hyperbola, 02 lias a maximum value, 180° + SJ'S' = 180° + VSB, and the axle describes a series of cusps, with 6^ = — Jtt — w, and tan Jflg = th Jflj tan ^62- Below b' the undulations are smoothed out of the herpolhode ; and finally at F, ^3 = - 180°, 62 = 180°, and the top motion is sluggish, in the neighbourhood of the upward vertical. Moving H upward from L, the apsidal angle IT increases from Jtt/ ; and if L is below b, the herpolhode has undulations which cease when b is reached on the tangent at B of focal ellipse, or on the asymptote of the focal hyperbola ; here 02 has a minimum value, and the axle has cusps on 6 = 62, while 63 = Jtt — o>, tan ^63 = th Jflj tan J02- The curve of H has an inflexion then between P and b, and also between T and b'. Above b, as well as below b', the undulations are smoothed out of the herpolhode, and the motion of the axle is featureless ; and finally with H at I, J" = 00, ^3 = 180°, 63 = 180°, and the axle motion is sluggish, on the verge of instability, in the upward vertical position ; the motion being the reverse of the motion corresponding to H at I'. The annexed table on pages 104, 105 gives the value of flj, 02) ^3 and other quantities for the guiding points, when H is placed at Q, P, T, V, b, b', a, a, . . . . , to serve as a check on a general result, and to show the associated state of motion for the same K andf. We notice that p2 and ,03 are nearly equal and the motion is featureless, unless k' is small and the focal ellipse nearly circular. Having selected then in an illustrative case the value of k\ and a definite tangent PQ, with AOQ = w= aiu/K', a series of states of associated top motion can be constructed geometrically, corresponding each to a definite position of H on PQ ; and the special positions are considered in the previous Table. When / is made a rational fraction the motion is pseudo-elliptic ; and purely algebraical when H is placed at L, as developed in Chapter V. The associated motion of the axle for another position will differ by a secular term in 1/- ; viewed stroboscopically this secular term can be made to "cancel, and the motion will appear algebraical again. Geometrical Eepresentation of the Steady Motion. 21. In the steady motion, with the axle at a constant inclination to the downward vertical, the diagram degenerates into HQ, HQ' passing through A, A', coincident with S, S', the focal ellipse shrinks up into the straight line SS', and the articulated hyper- boloid becomes a stiff framework, which revolves round OC parallel to RQ with constant precession ' (1) ^ = ^/il, and. 02 = = 9,, k = 0, K= i^ ■ (2) /c' - 1, /r = 00, W = 1; (3) fK' = J^ sec oj dw = ch-' sec to, sec a> = ch fK\ sin w = th//t'. Then if to is not a right angle, as in fig 72, (4) / = 0, zn/Z' == E'^ = sin 0. ; if Ls'^ilLiot"'' "^*' ' '• "'"^' ^'"^ ' ""'''' P^^y '- '^^ f— -^ ^o allow for the (5) ~=^, j.r=-^-(i«jis:)x-. It OS' 2 Qg -2T, f6) ^= /— M ^ HS m _ OS 00 ^ ^ n V HS" m OS' n " TlHSniSl' "''^^ = '^' ''^ = ^y-, 107 the geometrical interpretation of Ferrers's result, connecTtiing to/tt the number of small beats, up and down, and 27r//i, the period of a complete revolution of the axle round the vertical, their product being the number of beats in one circuit of the precession, (7) 2™==2g|; as given already in §4, Chapter III.; where m/n is the number of beats of the axle to one small beat of a pendulum of equivalent length I. The pedal degenerates into two circles, on diameter OS, OS' ; so that if H is placed at Q2 01^" Q3 on the circle OQ'S', the representation is given of the motion of a spherical pendulum, in steady motion, with the thread parallel to S'Qg or S'Qs- "With H at S, ^ = 0, ^1 = — 00, ^i = - 00 ; and ^3, 2^2 can assume any value with K = 0, so that no conclusion can be drawn as to the nature of the motion of the axle ; the velocity is very great, to be imitated by whirling a plummet very fast ; and employed with a milk separator, or sugar and drying centrifugal machine. Generally QH = 8, HQ' = S', with proper attention to sign on fig. 72, and if H is a little above S, the axle points down along OS, the velocity is great, g' G G' S S' (8) = Jtt — w, S = — QS, S' = OS, J = — sin (0 ; -^^ — = -V- are large. When H is a little below S, the axle points up along OS', (9) e = - ^TT - m, g = - QS, 8' = - OS, I = sin w, o so that G' changes from + 00 to — 00 as H passes down through S. And in each position (10) n = 0, , p = i CO* f^) ^ HS „ . M / HS (12) Y' = ()^ = 2 sm w, ^ = Trsmw ; - = / tjct = s/ tan w ; n«^ at- / QS' _ / g' + g'' 1 / fj'' ^' \ _ 1 / (^ ^V \ '^^^^ n - y HS. HS' ~ V ~~^~ = 2 y (^4 p + JgTgj - 2 y ^2„2 + C2Ji2), the beats of the axle to one beat of pendulum length I. But if/ is not zero, w must be a right angle, and SS' vertical ; because (14) fK' = 00 , sin w = th fK' = 1, am fK' = w = ^w ; and P, T, L He in the line SS', with P and T in coincidence with S, and (15) DF " ™ •^^' = ^'"^ ~ ■^^' = ^ -f- The downward vertical position of the axle is always stable, with 02= ^ = ^3= 0, and H somewhere between S and S', making ,,^. , HS , ,^ HS-HS' OH (16) eO,= gg,, th ie,= HS + HS' = " OS B^_ _0W OH^ g_ g;_,i. ^'■'^ F~ HS. HS' ~ 0S=^-0H2 - '''^ 2^1' k-~k~^^^"^' (18) i;=/^OS =^ "OS -OS =^-0S =.^ + *^2^^ ^ . LH , OL LH , OH 2' g' CR ^_^^ ~ ^ >/(r+ F) ~ -* - ^{C'R'+ AA'n') ^ ^ A 2n 1 a OA n OS n ^ ,^ n y/, C'R"^ (19) T^W "^ = 7r\/(HS. HS') =^''^^^'^^s/\} + I2W the number of beats per second of the axle, up and down ; that is ch J0; beats to one beat in small plane oscillation of pendulum length /. 28570 2 108 If the upward vertical position of the axle is stable, with 62 = 6 = 9^ = tt, then H must be taken somewhere beyond S, making HS , ,, OS (20) e^> = gg/, th J01 = - -Q^j, (^^^ F = HS. HS' = OH^ - OS^ - ""^ ^^'' k- k-'''' ^"'^ n TT + >P , LH , OL LH , OH , ^ ,, 1. (22) i-=^^ = /-OS=l-OS-OS=^- 0-8=1+ coth J0, jT -i- ^ Ci2 ^^ C RT OS • n , _ n // (7^7?^ (23) 17 = 1 - '^^^^ i^i = 1 + v((7^i2^-4X2n^ = ^ + Z 2^^' < ^' (2^) T = . ~V(HS. HS') = ;^^^ 5^1 = -y (4ZV -Ij the number of beats of the axle per second, sh ^^i beats for one beat of the pendulum length I, and RT/^tt is the number of revolutions in the time 7. A closer examination can be made when P and T are made to separate from S or A by taking / = 1, but (1 — /) K' finite and am (1 — f) K' = (jt, with K' = 00 , k' = 1, (25) (1 — f) K' = j\ sec (j> d (j) = ch"^ sec <^ ; and then L comes to 0, in fig. 72A, and OP = OS sin f, OT = OS sec. .^. The points b and b' can be marked off by taking Tb, Th' the G.M. of TO and TP. When H is placed below S, the figure is representative of a stable retrograde motion of the axle in the upward vertical position, 6 =Tr ; and H at b' gives a figure with cusps ; with H at T the axle passes through the upward vertical, 62 = tt. When H is below b', the motion at H3 and H3 is retrograde, and also at Pg and Pg ; When H passes above b\ the motion at H3, H2 and P3 is retrograde, but direct at P2 ; the cusps on the P curve have opened into loops which do not embrace the vertical, as in a pseudo- regular precession ; and the apsidal angle is as in (22). Between h' and T the H curve has inflexions, corresponding to a stationary value of ■^^ ; and at T- there is a cusp at Hg, and B.2 = ir. Above T, d2 a + /3-^ - T^ - T'>, ' ^ ^^ ~ T^ = - P'^ ; and thence , a - b. a - c. b - c. b - a f. - a. c - b ^^^^ a' P ' c^ ' are unchanged when the letters are accented as in (12). Since .^c^ OA^ PV. QT a-c.h-b v^yj 0D2 - 0D2 -^ q5 ' OB^ TP. VQ c-b.a-h (20) OD^ - OD^ - q5" > ^9n QS^ VT. PQ a-b.h-c y"^^) 0D2 - 0D2 = q5 , with similar relations for the other tangent H(^', we infer that (22) b - c.a - h, c- a.b - h, a - b. c - h, are unaltered when the letters are accented. ni From (23) cos 6*3 = 0^2 = Q 5 and similarly (24) cos 02 = ^^-^^ , - ch 01 = — ^^ — ; (25) F^ohe,- cos 02 - cos 03 = ^^^ - «^- ^ - 7 ^ ^^^^ ; as in (13) §6, unaltered for accented letters also^ h-b.h-c _ QT. QF _ QW^ h- c.h- a _ QP. QY - QQa' h- a.h-h ^ -QV.QT _ OQ^ r^',^ a^- h-h. h - c _ HVl QW^ OSl OK^ a'^ A' -^>'. h' -c ^ .^ . C^7) -^ ^ 0^^ = Qj^j— = ^, - 5Sti Wi, r^o^ h\ h - c. h - a _ - HTl QO2' OP. OS^ h'\ h'-c. h'-a ^ . .,. (28) ^, ^^3 = oD^= S? =-ism-'tf2, (29) ^^— = ^, = OD^ =• Q^ = -i'^"^^'^ (30) Q^ = (a - 2ahy - 4a\ h - b. h - c = 1(3 - 2bh y - 4R h - c. h - a = ij - 2chf - 4c^ h - a.h -b = Q'-4E{P-h), as before in (12) §6, and with a geometrical interpretation, as in (24) §14. We can write, in the notation of (1) §6, (31) ^ = 75 = ~ — -. J, — -. -^ (cos % — Zy) = s 7 (cos — Zi\ ^ ' ^ P 2. A — B. A — C^ ^' 2. a — b. a — c^ ^' (32) *-T, = Tr\r5 = o J (cos — z^) ^ ^ a^ HV- 2. a — b. a — c ^ ^ % = h-^- 2.b-c.b: :-a''''''-'-^ (cos — 2^3) (33) HP^' " 2. c - a. c - b a a^ h' — b'. h' — c' a^ ~ a"'' ~ h — b. h — c ,„,, ,, , OA^ - OH^ QW^ - HQ^ QP. QT -m' _ {h-b.h-c) - h' V^*; ^(.^2 + ^3) = Qj)2 = Qjyi - OD'^ Q /o'\ 1/ ^ {h - c. h - a) - h? (3o) ^{z, + z,) = -' ^ '- , t'xa^ If N HQ' + OQ' _ OE _ HQ^ + QT. QV _ (h-a.h-b) -h' 1,6b) il^i + ^2; = - - oi)=~ ~ OD^ ~ OD' ~ Q ' and thence the previous relations in (19) (20) (21) by subtraction. Again be (h - a) _ - HT. HP. QV _ KH HP. OQ (37) aQ HV. OD^ OK" OD'^ .KH . „ _ KH HP'. OQ' _ b'c' (A'- a') -OK '''' ^'-~ UK- ~0W 7q 112 because HT. QV = OV. KH, 0^ = ^ ' ^^^'^^^^ ca (h - b) HP. HA^ TQ _ HP. OQ OK ^ c'a jh' - b') ^^^^ bQ = HT. OD^ " OD^ KH b'Q ab (h-c) _ HV. HT. PQ _ a'b' jh' - c') ^•^^^ ^Q HP. OD' c'Q Many more relations may be cited here, all derived as in (8) § 13 from the coincidence of the polhode cones and capable of a geometrical interpretation. Thus for example nn «'. h' - a Z ^'-h' - b' ^ j'- h' - c' ^ _a§y ^^^' h-a h-b h-c Q (A^-s 4L ^_ ^- = r.r A ^D 7^ = ¥ (Routh QJ.M.XKm) (4^) A{- A + B + C)~B{A-B + C^ C {A + B - C) ^^ D' 1 h^ K =1. = D-A + B+C-2DW, h B' CURVATUBE OF THE PoLHODE. 24. With u = mt, the equations in (:-50) §5 become ,. dx _ B - C yz dy _ C - A zx dz _ A - B xy _ ^ > d^l~ A a' du B a' du C a ' (2) y^ -.f = D- \- D ,^, ,f-x^ = ^^-^^y,. ^ ' ^du du BC a du du CA a dy dx D. C - D ^ x—^ — y — = r^= o- ; du ^du AB a ,^. d^z d^y o d'^x d^z d?y ^(Px ,3^,, Differentiating logarithmically the quotient of a pair of the equations in (1), and reducing dy d^z _ dz (Py __ dy dz , , , dy d^z _ dz ^y __ dy dz f\ dy 1 dz ^ ' du du du du^ du du \y du z du - C - A. A - B x^yz / C - A z_x _ A - B xy\ _ C - A. A BC a^ \ B ay C azi WC^ " -'^ ' - A. A - B. n - A. D ^x^ with a cyclical interchange for the others ; so that the direction ratios X, ^, v of the osculating plane o£ the polhode may be written /.N A A - B. A-C 3 B - C. B - A ^_ C - A, C - B ^ , a - b. a - c Jl 3 b - c. b - a ¥ ^ c - a, c - b h^ „ ^^ h-b. h-c a^*' h-c. h-a b^y^ h-a, h-b c*^' in Darboux's notation ; and the equation of the osculating plane is, as in Salmon's Solid Geometry, p. 333, ex. 1, 2, /gN " - (>■ « - g ^[^ + b - c.b - a y^ c - a.c - b ^ _ /8 \* ^ ^ h - b. h - c a" h - c. h - a b' h - a. h - b c' ~ \h) ' If we write 5, c, d for (sn, en, dn) u, and ('') *' = Pd, y = Qc, z = Rs, 113 (9) ■ ^ = - Pk' ((!' - s')d = - ^\c' - f) X ; (11) dy_ HO ~ AGqH + AG'qH ~ iHG. QQ + JHG'. QQ' ~ HG. QQ ~ HS. HS' (C. Taylor, Conies, ex. 322, 337). (8) HR. HF = HG. HG', HS. HS' = HF. HQ. At an inflexion on the herpolhode of H, QQ' = 0, HG' = oo ; and GG' cutting H J, HJ' in C, C, (9) HC = ^^ = gj cos^ifl, HC = -gj^ = jjj single. The normals at G, G', and GG' touch the parabola ; the circle round HGG' passes through F ; CFC is a right angle ; FH touches a confocal at its midpoint, on vertex tangent of the parabola ; the angles OHS, OSF are equal ; HF and JJ' are parallel ; OH. OF = OSl If HJ and HJ' cut the principal planes in /, m, n, and V, wi', n', HJ is normel to the confocal X and HJ' to v ; and (10) _ X + a2= HJ. HL = HQ. HV . . . Draw X^ perpendicular to QH, cutting QQ in ^ ; (11) QQ. Q^ = QH. Q^ = QX. Qx, and q is the orthocentre of QPT. Draw TM, PJST perpendicular to QX (12) dM. cZN = dx. dX, (13) QQ. Qq = i (QP2 + QP - FT) - ^ (qH^ + QX^ - HX^), (iA\ ^' - "■^' _ dx _YR V'H' ^ ' fi + a'~ QX~ dX~ V^ ~ V'^' ' QV is conjugate to the plane OQX, , . . and so on. The hyperboloid finishes by flattening again into the plane of its focal hyperbola (fig. 77) with 03 as the angle between HQ, HQ'. Focal ellipse and hyperbola can be represented in one diagram, in fig. 77, with HQ a vertical tangent common to the two focal conies ; and variety is given to the representation of a state of motion by an arbitrary choice of the position of H on HQ.jj] Corresponding points on the focal hyperbola and ellipse can^be denoted by the same letter, and distinguished when required by a sufiix, 3 and 2. 28570 p 2 116 CHAPTER V. Algebraical Cases of Top Motiox. 1. The dynamical constant!?, G, G' , and E, F, or H, being three in number, there is a triply infinite ( oo') assemblage of the states of motion of the top, so that the choice of an illustrative numerical example is at first sight an embarrassing one in its variety. But a representative diagram should be selected carefully, to illustrate some numerical case of an actual state of motion, to fix the ideas and to consult for elucidation on a figure ; and this can be obtained by choosing, in Chap. III., §5, an aliquot part of the elliptic period for the parameter of the III. E. I., so that/ is a rational fraction, such as ,,. 112 1,3 1,2,3,4 1,5 ^^ 2' 3' 3' T' 5 ' "6"' ■ ■ " The III. E. I. for ro- in §1, Chapter III. can then be replaced by the logarithm, of an algebraical function, in accordance with Abel's theory of the Pseudo-elliptic Integral : and in the H curve, p exp (tn- — pt)i can be expressed by an algebraical function of X, or z, or p^. Thence by a differentiation with respect to ?, the hodograph of H can be derived, and the associated motion of a point on the axle of a top, given by (2) ^ik sin Q exp {jp — pt)i = -j- p exp (ot — f() i in an algebraical form — "pour bakyer le champ analytique," and to make an oasis in the desert of analysis. The Theta function result of § 16, 17, Chapter III., although analytically complete, is of no practical value, as it would require the function to be tabulated to a complex argument, mt ■\- K -^ fK'i, or ^eK + fK'i. (3) But when/ is a rational aliquot n-th part, as in (1), and v = K + — K'L n eoe (m - r) ' ^ exp (tr - pYi is a rational algebraical function of sn m, en m, and dn u ; and the numerical value can be taken out of Legendre's Table IX. The construction of these algebraical cases was discussed in the FM. Trans., 1904, " Th.e Third Elliptic Integral and the Ellipsotomic Problem" which may be consulted for the theory of the results required in these applications. 2. Beginning with the simplest of such algebraical cases, take (1) Q - 0, EG - G' = 0, „ _ <57 7., S , g ^ ^ — ^-j = nt = - nt = - mt ; ^A k a and then /= or 1, and /? = - z, or - z,, but never - z, for real motion, because E + z must be positive. So also, with (2) G-ffG' = 0, ^' = h't, /' = Oorl, and then H — — ^j or — Z2. at P .^^'" ^' ^ ^' *' P''''''^°" ^' '''° ^'^"^ ' = ^2' ^"d the axle passes through a cusp Drawn up in parallel columns 117 E = Zi, pi = 0, and P is at A on the focal ellipse ; p^ = a, p^ = aK, (3) p = p3 dn mt = a dn -^ 7S7, Y = Sin ^ ^^-^ : -X — ^ Sin Q ,h ^{2.z = cos '■ ^^ : n sin I ^i) (7) sin 6 sin (./. — ht)= V {z^ — z.z - z^) sin 6 cos {\p — hi)' = - ^ (27 z — Zj) (9)y (11) = -y i- Z^ - Z2. Z - Zi). — z^ — Z2 h 8 S n ~ k ^ — kzi' I + Z3Z2 (13) k" = 2 Zi = ^2 ~ ^1 E = Z2, p2 = 0, V at B, on the focal ellipse ; 103 = /3, (4) p = 103 en m^ = /3 en ^ OT. (6) sin cos sin 6 _, A y/ (2. ^ - ^Tg) 23 -t 2^2 '2 l-zl sinOo Zr Z 4' (15) . = '2 K Z3 — «2 n ^B + ^1 " ^1-1 sh 0, 1 — ^-j' K sin 63' (16) ^= l,r + lAr=i,r + Z^. (18) cos = ^1 - v/ (^i - ^? . ^1 - zX). With H on the tangent at B, and /=1, (20)-HT = oo, HP = HV = HQ = g, B = 0, A = C = D, and the rolling surface is the circular cylinder (22) ■ x" + z''= l\ g dn mt. 2 = g /c sn mt, y (24) p = j3 en mt = (3 en -^ 7!T. c ^ OA (1) - = v/ ( — ^^2) = n/ 2*^5 n *_:=v(-K) = yi, (3) ^ = ^Z h' = m When H is at B, the axle oscillates in a plane like a pendulum, through an angle 202 = 4 sin"^ K, as before in § 2, p. 48 ; (26> g, 6!, (?' = 0, S- = i TT. 3. With ^^2 = 0, Z1Z3 = — 1 ; H is on the tangent at B where OH = OS, and g' = 0, A' = 0, so that the axle moves like a spherical pendulum, projected from the horizontal position with angular velocity 2h ; and '' g = ^fy - 2/3='), F = 2/3 ^/ y - (3') n l< he , / '2 2\ cos 03 = — , — = - = \/ {^K — K ) K ma (4) s. = j^ + ^ {r- - k2) k. 118 (5) sinflexp (jP-ht) i= V (1 -cos 63 cos 6) + ^ n/ (cos 02 — cos Q, cos Q) returning to the measuremeat of d from the upward vertical, (7) cos = cos 02 sn^ ^^j sin B cos (j/- — ht) = dn m^, sin B sin (j/- — ht) = k sn m^ en mf (9) tan {xf, - A;f) = K sn mf en m? dn mt 1 — dn 2mt = /Lz V 1 + dn 2mt' cos 2(1/- — A^) = dn 2m^, sin 2(%Ij — ht) = k sn 2m^. (6) sin exp ((// — ht) i = v^ (sec 03 — cos 03. cos 0) + iV (cos 03 — COS 0. COS + sec 03) measuring from the downward vertical ; (8) cos = cos 03 cn^ mt, sin cos ijP — ht) = sin ©3 en mt, . n • r , ij.\ sn mt dn m^ sin sin ()// — A#) = -, sn mt dn mt V (k'^ — K^) en TO^ — 1 / I - en 2m^ ~ V U' - /c^) V 1 + en 27w^ (10) tan (^ - AO = 1 The angle ^ associated with this spherical pendulum where /= 1, Z2= 0, may by an interchange of h and h' be taken to give xp for a top, which is spun with A.M. (11) CR = 2Ah = 2An- = 2An /'^-^ = 2An v/cot 2a, ^ K V 2kk ' if a denotes the modular angle ; the axle is held out horizontal and let fall, so that it starts from a cusp on the equator and falls to an inclination 03, where cos 03 = tan a ; rises again to a cusp on the equator, and so on. In the uniform precession fi with this spin R, and the axle out horizontal, (12) CRii^ = An', - ^^ rf^ _ CR cos dt An (13) -2 A"-'^' sin^ 0, cos A sin^ In this motion, /' = 1, and (H) sn/isr'= sin w = 4 = cos 03 = sech 0i, n^cos _ 9 u. sin^ cos 03 ' sin^ 0" dn/Z' = k\ (15) th2i0i = tan2403 = K + /C = tan (^TT -a). With / = 1, and H at Z on the auxiliary circle of the focal ellipse, where SZ is perpendicular to QZ, (16) (17) (18) (19) ^3 = ^2 — ^T, + 03 = TT, Z2 + Zi = 0, am/'Z' = to' = i,r tan 03 = - tan 02 = 2 1, K P = ZS. ZS' = A v' (4k'^ + k'), B = a^', g' = g sin o,' 4:A^n 2^2 (20) (21) S2_ k'' AA'n' B ~ kk"' /fc' GCE 4AV m' „2 V{'\k'' +■ k^ 4k 1 V (4,c'=^ + k'Y ^ = *7r + k'K. Thus if K = 0, 02 = 03 = 1 ^TT, m the axle then flies round m the horizontal plane of the equator with infinite velocity r?i%TthSjf\%L:^^^^^^^ ^^-^-' - -^^ ^ Pl— t whirled roun^d . _ • r'-\ ^Jl ? f />tersection of the tangents at A and B of the focal ellipse, and •^ 7 I'V ~An'{ ~ yl 2 '■ *^^second case being obtained by an interchange of d, first t'as? o7/ = 0, r= 1,' ''' '^^^ '""^ ^°"^^'^^^^ ^P^^^-^' -^ - combi^i^ wTth tht (1) P = «dnm^, p' = (i en mt, 119 = n/ (1 - sec 02 cos e) + i -y (sec 63 - cos 9. cos 6) = v/ (1 - cos 63 cos 0) + «■ n/ (cos 02 - cos 0. cos 0) (2) sin exp { + xp — ht — h't) i = n/ (sec 02 — cos 0. cos 02 — cos 0) + 2 ( n/ sec 02 + \/ cos 02) n/ cos (5) (1 — cos 0) exp { — -iP + ht — h't) i = v/ (sec 02 — cos 0. cos 02 — cos 0) + ^ ( n/ sec 02 — n/ cos 02) v/ cos and with h = kw,, h' = m, Klein's a and (3 are given by (6) a = [ J n/ (sec 02 - cos 0. cos 02 - cos 0) + 4 «■ ( v/ sec 02 + n/ cos 02) n/ cos 0]* exp ^ (1 + k) mti, (7) i3 = [- J n/ (sec 02 - cos 0. cos 02 - cos 0) + ^ ^■ ( n/ sec 02 - v/ cos Og) s/ cos 0]* exp ^ (1 — k) mti. Then, with measured from the upward vertical, (8) dip h — h' cos 'dt^ ^ S^"0 = 2^ 1 — sec 02 cos 1 -cos2 2A cn^ mt — 1 — K^ sn* m^ = A (1 + cn2m^) (10) ip = ht + ^ cos-i dn 2mt (12) ^ = /cK At half time (14) h't = mt=lK (16) 4> = J/ciT + 4 cos-V = ^kK+^ (J^ - 02) (9) d<^ ^ h! — h cos 57 " sin2 = 2A / 1 — cos 09 COS 1 - cos2 2A' dn^ m^ 1 —'f? sn*m^ = A' (1 + dn 2m0 (11) (p = h't + ^ am 2ni^ (13) <3) = K + i,r. (15) Ai = ^kK (17) = ^K + ^amK ,K + In the uniform precession ^ with the axle horizontal, and this value of OR and h', n ^ = _ = .A' = h. (18) Ci? A* = An\ With (c = 0, a small nutation is obtained on this steady motion. With K = 1, 02 = ; and the axle rises from the horizontal position into the vertical, but takes an infinite time. Bisection of a Period. 5, Next consider the bisection of a half period, by taking / = J ; then with K + ^K'i corresponding to mt when « = in the formula (12) Chapter III, §5, (1) (2) (3) (4) x = ^3 en" mt + X2 Sir mt, x^ Sua "^ iCt) = sn^ [K + IK'i) \ - 1 _ i _ / ■=^'3 - ^1 f N 1 1 1 Xi = s/ {X^ — X->. Xi, — Xi 3J3 i!B2 a — 2 + „ 2 tv ™5 """ ^2 tCa tt/o sn^m^ y U/o X - on2 /vv)/ ^ sn'' mt = K sn'' mi = y, suppose, and then (5) (6) X z = 1 - 3/^ X — X2 , X3 X2 Xi zl _ 1 _ Zl X' X — Xi x^ + 1. -y\ 120 Writing P, L, M for the former ^'j^^ , then from Chapter III, §13 preceding, p^i(^_ ^,), withA = V (1 + P^) = H-;^+^/'^). 1 M^ sin exp x^ = ^ (1 - 2/') ' < {\ 3P sin ey = (2P + L)2 + (4P - Z^ + 1) ^/^ - 2/^ ili^ cos d = AP'-L'+l- 2y' M^ = (4P^ - D +lf+ 4. (2P + i)^ M% = 4P2 - i^ + 1, if' sin 03 = 2 (2P + L) M^zo = AP' - L' + 1 - 2k = - L'- 1 + 4:PF„ y\ (8) (9) (10) (11) (12) (13) (14) (15) i¥2 sin 02 = 2 (^ + X) v/ (1 - /c), ili= sh 0i = 2 ( v/) - dx ^-P(iL)(i_,^)^%)j^ n/AJd VF 2/' , /I - 2Pv - «' 1 • -1 /I + 2Py = «°^ V 2.1 -y^ - i . = sin 'y^^ 1 + 2Py - y' _i 4 ") _ 2P2/ sin 2 (ct- — ^^) = y 2' P P3 1 = 7a7 — pt, P = ^ + ^ n pt=Pr:mt=(L + P)/(l—^) mt ={L + P) s/K. mt, ^ n m ^ \M . z^ — Zx' pT= {L + P) v/K. 2K, (18) (19) (20) (21) (22) ■ (23) ilf I exp (t^ - pt + \^)i= v/ (i . 1 - 2Py -f) + i s/ {I . I + 2Py - f) (24) ^M^ sin exp {^ — pt + ^7^)^■ = -^.W^ sin exp ■^^i exp (/ + ^tt)?' X + 2P -Ly^ + zyVF 7 (| . 1 - 2Pv - y^) + ,V (| . l + 2Pv - y ^) v'(l-2/^) ■ ^/(l-2/^) = (Z + 2P - 2/) s/ (i . 1 - SP?/ - f) + i{L + 2P + y) vih. I + 2Py -f) (25) {^M^ sin 0)^ exp 2 {^ - pt)i = [(Z + 2P)2 - y^] ^/ [(1 - 2/2)2 _ ^p,y2^ + 2iy \{L + 2P) + (Z + 2P)=' - (Z + P)?/^] The simplification of writing Z3 for Z + 2P can be made here and elsewhere, equivalent on the focal ellipse to measuring the distance of H from P instead of L, since (^^) Z ~ HQ- And we can write, for the curve of H, (27) M I exp (tjt - pt)i = [ V (* . 1 - 2Pv - y-) + / s/ (i . 1 + 2Py-f)]%ji^{- ^W) = i[ n/ (1 + 2Py - 2/') + s/ (1 - 2Py - 2/^)] + i/[ v/ (1 + 2Py + .v^) - ^ (1 - 2Pt/ - f)-\. 121 The Quadric TeanspokiMation. 6. At this stage the Landen Quadric Transformation comes in useful ; we put (1) mt = 2eK, e = ^ so that e increases by unity during one beat, and change to a new period and modulus, ,-, 2 \/ K / 1 — K (2) ^ = FT-.' ' = i^k' -'0 A {(o, c) where (4) am 2 eK = am eF + ^w - am {1 - e) F (5) sn 2 eK == (1 + c') sn eF m (1 - e) F (6) y= V . ^n 2eK = c sneF m {1 - e) F = y \-^^ll^ , (7) v/ (1 ± 2Py - 2/2) = en ei^ ± en (1 - e) F. These relations of the Quadric Transformation are shown on fig. 78, where (8) <^ = am 2eK, w = am eF, w' = am {1 — e) F (9). i> = ADR = DSC + DCS = ^ + ^tt - a.' CX CR. CS CF . (10) ^^'^ -^ = CD = CDTCF = CD ^'^ '^ '^^ '^ AD , FD 1 - /c CF , ^ , (^^^ '' = a:c' '=dc = i^^' gd = ^^'- (12) DR2 = D«2 + Bf = DC' cos^ w + DF^ sin^ w = DC^ (cos^ a, + c'2 sin^ to) = DC^ (1 - c^ sin^ to), DR = DC A (a,, c) (13) RS^ = CF^ - 4AQ2 = CF^ - 4AD2 sin^ = CF^ (1 - k^ sin' i>), RS = CF A (^, k). J. As RS turns about D with angular velocity -it, the velocity of R is given by either (U) CF|,orDRsecARD|.DBi§, so that 1 du) _ 1 d^ 1 (iw 1 d<^ 1 — du)' ^^"^^ DR rfT ~ RS 57' CD A(co, c) "^ CF A(^, /c) = CD A(a,', c)' and since ^ grows from to tt as w grows from to ^ tt, dw CD {^ di> 2K ^ ' Jq A(w, c) CpJoA((^, k) 1 + c ^ ■' ' ^^ = JoA;.' (1-^)^ = Jo A.' 2^^ = J„4' ^ as in Chap. III. § 6 ; and the other relations can be deduced when required. 7. Then witkthis quadric transformation (1) ilf£ exp (ot - pi) i = cneF + i en (1 - e) ^ (2) ]\P^ = en' eF + en' (1 - e) F =1 - c' sn' eF sn' (1 - e) iT- (3) p cos (t3- — ^0 = p3 en ei^, p sin (w — /)if) = p^ en (1 — e) F. Relatively to axes 05, Or;, revolving with angular velocity p, or viewed strobo- scopically, and replacing ^3 by a, (4) ^ = a en eF, n = a en {\ — e) F ; „' ^ c'' sn' eF _ c' - c'' en' gj^ a' (^) ^~ dn' ei'"' c''+c'cn'eF' ,'2 ^ ,2 5'~ c'^ + c^ '^ a" and the shape is shown in fig. 81. 28570 <6) (7) Since 122 a' , C'' ^- + r,2 c'2 = c v//cl = ^ = P C\\/k / ■J K the equation of the C4 in (7) can be written (9) (5+^)(^^+^0=^^- And from (17) §5, (10) ».(i.w = y4«w, ,,(z.w = y>-±i^', r _ V(l + 2Py - y') ^ {I - 2Py - y') 1-f (11) cos 21= ^ ^ ^ cn^eP-cn^(l-6)F^ = sn (1 - 2e) F, 1 - c' sn' eF sn' {I - e) F (12) / = 1^ _ 1 am (1 - 2 e) P. Thus at half time from P3 to P2, t = ^T, 2e = ^, ' 1 (14) V7 - \pT = / = Itt - i am iP = ;|Tr - * tan - 1 -^ = J tan - 1 s/ c' = I tan - ^ / : ~ '^ = 1 cos - 1 K = I; sin - 1 k'. 8. In a lower rosette curve, <1) 03 = 0, 2P + L = 0, M^=l, P=l, |=-L|=_i(i_^)= _i(l_,), (2) sin ^ = 2/ = V K sn mi = dn J iC' sn mif, (3) cos ieex^{xp-pt + ln)i= - n/ (J . 1 - 2Py - y^) ^ i ^ (i . 1 + 2Py - y') (4) cos J exp (tp — pt) ? = en (1 — e) P + 2 en eF = i fi exp ( — pti) . . , . _ en eF _ en eP dn eF _ / dn 2eP + en 2eP (5) tan (^ pt) - —^^—-^ 0' sn eF V dn 2eP - en 2eP / 1 + sn ( l-2e) P V 1-sn (l-2e) P <{G) cos 2 (;/- - ^0 = - sn (1 - 2e) F (7) 2 {4,~pf + lir) = am (1 - 2e) P (8) 4, - pi = ^ am (1 - 2e) P - i,r. In describing the apsidal angle, e grows from to J, (9) ^-- ^pT = - i^r, i|^r = - p vk. /r = - Iz:'^' /v = - A-zn 1 /r (10) * = - ^TT - z zn 1 /r, agreeing with (11) § 14, Chapter III. In the upper rosette (11) 92 = n, ^=-~, SP + Z^-v'K-, L + P=-P„ m - OA = ■" *<^^ + '^)' ^^' = 1 + K-, (12) cosi0=cniP'cnm<= y;^, sin iO ^ /^-^, 4 123 (13) sin 10 exp {xp - pt + ^tt)? = ^^ v (1 + /c) /, .X • 1 « / N . - sn(l - e)F + i sn eF (14) sm 10 exp (^ - ;)0« = ^ (i + ^) , , sneP /I - cn2eP (1^) *"° (^^ - '^) = sn (1 - e)F = yi+cn2,F (16) cos 2{pt - ip) = on 2eF, pt - i. = ^ am 2eF (17) ^pT = - Z zs JZ', ^ = - Itt - Zzs ^K', agreeing with (22) §14, Chapter III. The upper rosette for K = 0*77384, and the lower rosette for K = 0-42893 are shown in Dewar's stereoscopic diagram of fig. 79 ; these values of the modulus were chosen so as to make ^ = |7r and Itt, as nearly as possible, and so obtain a closed figure. In the intermediate motion (18) sh0i = O, L^s/ic, 2P + L = —, L + P=Pi, \//c (19) cos 10 = sn IZ' dn mt = ^Lz^", sin 10 = ^'^A-t-^' : (20) sin |-0 exp (x^ - pi)i = / , " [sn (1 - e)P + i sn eP] = a exp ( - p^") V 1 + K (21) ^-^^ = iam2eP, (22) ipr = Zzs ^K', ^ = Itt + Zzs j/r, agreeing with (36) § 14, Chapter III. The upper rosette and iatermediate curve both depend here on I {^K'i), which we see here can be expressed by (23) I (iz'o = J^4^(Lr/I _^ ~ ''''' V i 2. 1 + 2/2 y ^" " '''' V 2. 1 + ^2 — 5" (24) exp /^■ = «" (1 - ^) F +i sn .P V (1 + y^) (25) exp 2Ii = sn^ (1 - Q P - sn^ .P + 2z sn .P sn (1 -. e) F ^ 1 + c^ sn^ eP sn^ (1 - e) P = en 2eF + z sn 2eF {2' 2 ^^"^" 3A' ^ ~3' /3~3' This shows that there are cusps when (9) p^; = 1 - . = 1 1 p_ 1^ ""3' V3' (10) tan a. - tn J/r - ^ - v/ 3, ■J K oj = Itt, sn \K = n/ 6 — n/ 3, (11). c =1, tnli^= . y2, snii^' = v/3 - 1. Otherwise, there are cusps when HL is at the level of B on the focal ellipse, and then 1 3" (12) ux = *(i + '^) = ul ^^^ '^ = V^'' " = ^ For values of /c< -, P> — --, the cusp is blunted, and the curve has four waves. o V 3 3 14 The value k = -, P = — — , Pj = — — , was selected for fig. 81, given as the illustrative diagram in the Annals of Math. 1904, " Mathematical Theory of the Top " ; but it has the disadvantage of being too special, in ranking 63 = J"", and of bringing the point of half time into coincidence with the inflexion on the herpolhode. A better representative value is k = 4, which gives fig. 82 ; and other numerical illustrations are discussed in the next chapter • such as the case, more interesting arithmetically, of /c' = V 2 - 1, k = x/ (2 V 2 - 2), K/K = V 2 ; or k' = ( v/ 2 - 1)^, K = 2K'. 10. In the geometrical interpretation of / = i on the focal elliijse of fig. 83, P is the Pagnano point, and L is the mid -point of PQ ; (1) PV = QT = a, PT = QV = /3, OQ = v (a/3) ; and the collection of formulas following is required in the arithmetical discussion of the associated dynamics of a top, and interpreted on fig. 83 of the focal ellipse of the deformable hyperboloid. r2^ LP _ 1 LY ^ LT _ a + |3 1 + k A ^ ^ LQ ' LQ LQ a - /3 - f^^- = P ' (3) M = _l_ ^ = ^!L. 0Q_ 2./„f3_ 2Vk _ 1 ^' LQ 1 - k' LQ 1 - ;c' LQ a-^' ~ \-„~ F (4) tan «. = y ^ = j~. sin . = sn W = J ^- -|-- ^, cos . = y ^ - (5) pi = 0Q= = a/3, pi = OQ,^ = (i.„- 13, pi = _ ,^)W.= _ „. „ _ ^. P3 \ aJ ^ " K + K 125 (6) LW = s/ (a. a - |3) - 1 (a - |3), SW = OQ - OQ2 = n/ (ajS) - ^ ((3. a - (3) (7) 4LS2 = 4LW^ + 4SW2 = (a + /3) [5a - 3/3 - 4 x/ (a. « -/3)] 4LS'2 = (a + /3) [5a - 3/3 + 4 ^/ (a. a - /3) J ; In the purely algebraical case, where TI is placed at L, (8) . _ 16F = 16LSI LS'2 = (a + /3)2 (Ba^ - 14a/3 + 9/3^) (9) ' S = LQ = i(a - /3), ^ ' k' (a + (3y {da' - 14a + 9/3^) (1 + k)' (9 - 14/c + 9c^ (1 + k)s/(9 - 14/c + 9/c^)' . - 4(l-/c)v'/c 2P . /i - 2P (1 + /c) V (9 - 14/c + 9/c2) v/ (1 + P^ 1 + 9P2)' ' 1 + 3P^' ,^ .. _ 3 — 5/c ■ o _ 4x/ (/c . 1 — /c) sin 03 _ n/ (1 — k) ^^^ ^' " 7l9 - 14,c ^r9?)' ^'"^ ' ~ V (9 -14/c + 9/c2)' sirTei TT^' ,. ,,x — 5 4- 3/C -in 4 \/ (1 — /c) <^^'''' ^^~ ^ (9 _ 14^ + 9^2)' sn»i- _-^^__^__^ ,T.. p _ _ - 1 + 6/c - k' __ 1 - P2 (ib) i._-^i-^, ^^- (1 +,c) V(9-14. + 9/c^) - ^(1 +P2.1 + 9P^ E = F-2—-- -3 + 10,c-3/c2 1 - 3P^ P (J + /c) V (9 - 14^- - 9vc^) v/ (1 + PM + 9P^) H= F -2—= ~ ^'' + 68/c - 66/c2 + (^S/c^ - 27k^ 1 - lOP^ - 27P* k' (1 + /c) (9 - 14,c + 9/c^)3 V (1 + P^) (1 + 9P2)r (17) 24' = ^ + ^, 2Pi = E + z,= ^' 2 F " ' ^' ^K' '^ ' ^' (1 +,c)v(9-14 + 9/c2) V (1 + PI 1 + 9P2), 9pI = E + ^ - 8/c (1 - k) ^ 4P F ' (1 + ^) V (9 - 14,c + 9/c^) >/ (1 + P=. 1 +~9P^)' and from (23), (24), (28), § 5, (18) ^ = tan-. ^' y [i|i:J-: - i., ./ - ^« sn * F - yV J - -JPy - 2/2 _ 2P (1 + F')\/ ~ P' + {I + 3P) f - y" Thus y = P makes j// = ^tt, where the node lies, and <") ^^*- .:z\^}-r '-'-^' (20) cos = s/ J ^gp2 , sin 2z!y = j^3^2- On the focal ellipse, with H at L, ,^^. 0K2_l + /c OP _ 1 + /c OL^ _ (1 + /c)^ ^^•' '' 0A2 ~ 4 ' 0B2 4,c ' 0A2 4 ' /99^ OM^ 1 OW _ _jc__ GW^ ^ (1 - 3/c)^ ON'^ _ . (3 - /c) ^ ; l^-j OA^^TT^' OB^^'l+zc' OA'^ (1 + 0" OB^ (1 + 0' /90X OQ^ _ OQ'^ _ . (1 + 0^ . Ul_ ^ (I -0^ ^ ■> (JA^ " "' OA^ ~ 9 - 14/c + 9/c^ ' ()A=^ 4 ' (24) LQ'2 = L(l - '^')' OA^ 4 (9"- 14/c '+ 9/c2) LP' = LI - '0' LF' = (1 - kY (9 - 1 4/c + 9/c^) (JA? X" ' OA'^ ' 4 (1 + Kf 5 K " 1 126 A LQ 1-K ^' _LQ^_ 3(1^^^013^^ (23) :d "^ LV " r+"/c' X>' ~ LV ~ 9 - 14/c + 9k2 ' B LQ - 1 + /c 5' _ LQ' _ 3 (1 - Q (1 - 3. ) (26) D^ ~ T7T= 1 + K ' S' ~ LT' ~ 9 - 14k + 9^2 " ,«,, C LQ , C LQ- 3 (1 + .y 3 (1 + J«) . (27) :d " ~ LP " ~ ' 5' ~ LP' ~ 9 - 14k + 9k^ ~ 1 + 9P2 ' (^^) i) - 1 + K 4', 5', C _ 3 (1 - k) (3 - k), 3 (1 - k) (1 - 3k), 3 (1 + k)2 D' ~ 9 - 14k + Qk'' ' (29) \ + V = OU - a' - (3'^ = ^a' {- S + 2k - Sk') (30) X - V = LS. LS' = 1 a^ (1 + k) v/ (9 - 14k + 9k'^) ^' '' _ - 3 + 2k - 3 k^ ± (1 + k) V (9 - 14k + 9k2) (^^) §2 - 2 (I - kY - 1 - 3P ^ ± v/ (1 + i^. 1 + 9P^) ,^n^ -. fi2^ ^ 2 = 2^ -, \>0>n> - (i->v> - a\ and from §17, Chapter IV, (32) g2 = ^ _ ^xa dn^ mt, |r = q _ ^■, cn^ m^, ^ = k sn^ mi?. At the inflexion of H, where ^ is stationary, Z V (9 - 14k + 9k2) (33) i+z:'. = o, ^ = - r - % (1 + .) — ^' sin 4 s/ 2k 2 x/2. 3(1 + k) ~ 3v'(l + P') 12\5 2p2 16k 4 (34) p- - ^ + ^ - 3 (1 + ^) ^ (9 _ 14^ + 9^.2) - 3 V (1 + P'. I + 9 P')' .2 _ 2 „ 2 ,;2 — 1 (35) ^2 ^ g^2 _ g2 32k (1 -kY 8^P F ~ ^~F~ ~ (] + k) (9 - 14k + 9k^Y ~ (1 + P'y (1 + 9^)' ' m) ,0,2 ^. _ ;>^ 12 (1 - k)^ 12P^ (^^^ ""' ^ ~ /- 9 - 14k + 9k^ - 1 + 9P^ ' , (3 - k) ( - 1 + 3k) 1 - 3P^ t!^n^ X - 12 (1 - kY ~ 12F' ' , /I - P V 3\S . ^ 3n/3(P + P3) (37) ^ = i^r ± tan "^ ( i ^ p ^ 3 ) , sm 2 ^ = ^ \^ 9p2 ' . At the equator, where z = 0, (38) 2/^ = Hl + 3P^), sin 2 ^ = i^^-^^f^|ig±^) At an inflexion of P, according to §37, ,„„, r'2 or orzj A « V3 (1 + 9P^) - s/(ll - 26P^+ 27P*) (39) Z .^ + 3Z. + 2LH = 0, cos 6 = 2 v3 j (1 + k 1 + 9P^) We find also, by the formulas of the next chapter for r and P, the radius of curvature at H and P, r4o^ "^ = -^- -^^ - (1 + ^y ^*"^ P3 1 + P^ ' a sin y, - P2(_ 1 + p2) , ^ ■ P2 1 - 3k' a sin «2 1 + 6k - Hk^' and on the equator, where z = 0, R_Kju) __ 2P^(1 + P^) '^ '' a H 1 - ]0P2 - 27P*' For the deformable hyperboloid, to set out the rods on the plane ot: the focal ellipse take the tangent lines YLPT^, V'P'L^, and the three other symmetrical pairs, as in fig. 74 or 80, and jom them where they cross by the universal ioint invented by Professor Wiener. ' -^ 127 Trisect;on. 11. For the trisection of a period, begin with / = f, and build up a solution on the pseudo-elliptic integral, which we write (1) f = {K + |/v'o = r -p^ + Q d^ = I cos-i (^:^6PW^^^^ _ ^ SP V (.-^3 - X. X - X,) where (2) xy = - 3P(3P + 4), X, + X, = 12(P + 1), a;3 «2 = 12P (3P -h 4) = - 4,2?! = 2Q, a^g - «2 = 4 V (6P + 9), ^■3 - xy. x^. - xy = 3P (3P + 4)-l : ; (3) P= V («3 -^i) znfA". The expression is rationalised by putting (4) 6P+9 = b% «i = - 1 (P - 1) (^.2- 9), b >3, «3 = (* - 1) (b + 3), ^3 = (* + 1) (b - 3), «3 - a^a = 46, xs- Xy = l{b - If {b + 3), «2-^i=i(& + ly (^-3) ^ ^ (^ - 1)^ (b + 3)' (6 - 1)B (b + 3)' (6) cnK = ^^, dn|A" = ^-A_, sn ^A' = ^-^ (7) zn |A^ _ P _ J / ^>^ - 9 \ n/k' V (a's — ajj. .Tg — a?i) ^ U^ — 1/ Then in the herpolhode curve of H, n ifcT ' p3^ .-cs (8) ^ - p^ = r, ^ = ^ + ^ P" - * and for the axle of the top (9) ]\PcosB = 2x - U + ^P" - n (10) {^M^ sin 0)2 = - a;2 + (^3 _ gpa + 12) ^ + 12P (3P + 4) (P + 3P + 2) (11) ilf* = {D - 9P2 + 12)2 ^ 48P (3P + 4) (P + 3P + 2) (12) i^2^i = - P2 _ (3P + 2) (3P + 6) = - P2 _ 1 (^2 _ 5) (j2 + 3) (13) if2 sh 01 = V (^'^ - 1. ^2 - 9) (P - 2) (14) M^ sin 02 = V (^^ + 1. 6 - 3) [2P + (^ - 1) (^, + 3)] (15) if2 sill 03 = V (6 - 1. 5 + 3) [2P + (6 + 1) (^, _ 3)] , „. sin 03 _ 2P + {h + 1) {b - 3) / 6 - 1. 6 + 3 ^ ^ sin 02 2P + (^. - 1) (6 + 3) V ^. + 1. ^, _ 3" Thus in a lower rosette (17) P = - H^ + 1) (^ - 3), iP = b{b- 1)2, in the upper rosette (18) P = - H* - 1) (* + 3), M' = bib + 1)2, and in the iBtermediate curve (19) L = 2, J/=3P+ 4 = i(^'- !)• 128 12. The lAotion of the axle o£ the top is given by (1) ^J\P sin e exp (^ - pt) i = MHm 6 exp x«- exp It and, introducing the simplification of (25) §5 by writing L = U + 2, equivalent to- measuring the distance of H from V instead of Q, (2) I M^ sin B exp x« = A v^.i" + — ^ ^-^ ^jlc ~ ' so that, writing i^ for 8F in (1) § 11, (3) -'«! exp Mi = {x - 2F) V (x - X,) + iF V (x^ - x. x - x.^ = A, suppose ; and multiplying (2) and (3), and arranging in powers of Zj, we find, /.N n 1/2 • />.=! Of, .\- fr , 2 (x - x,^ + i -J (x, - x. x - x^. x - x^ f (4) [i I'P sin ^f exp 3 {^ - pt)i = L^ + —^ '-^ ^^ — J {{x - 2F) V (.r - X,) +iF^ {x,- X. X - «o)] = L^' A + ZL^B + S^C + i>, (5) A = {^x - 2F) V {x - xi) + i F ^J («3 - «2- ^ - ^2) B= s/ [_x - x^ { (-^ + 2) X -2F (F + 4) + i V {X3 - ^-cs- *■ - ■«2- x - Xi)\ C = {x - X,) \[x - 2 {F + A)] V (x - X,) + i (F + 4) ^/ {x, - X. X - X2)\ D = (,r - x,y {{F + 6) X - 2 (F + 4) {F + 8) + i ^ {x-^ - x. x - x.. x - .1^1) \ ; or arranged in powers of x — x-^ = ?/^, x^ = — F (F + 4), with F = -y* + 2(F^ + eF + 6) f - F (F + 4)^ (6) A = p' - F{F + 6)y + iF ^Y B = (F + 2) y' - F{F + 4)^ y + hf n/ Y C = f - (F +2) {F + 4) y' + i{F + 4) y^ V F Z) = (F + 6) / - (i^ + 4)3 7/ + %* n/ F The substitution m „2_ ^ _ ,, x^^^j^^^^^^jc, . F(i^ + 4) [4 {F + 4)- xl will convert J. into X>, and 5 into C ; and the norm of J., 5, C, Z> is (8) ,r\ .-c^ [4 {F + 4) - x] {x - x{), x [4 (i^ + 4) - «]2 (,t- - x^f, [4 (i^' + 4) - aO^ (.r - x,Y. Also 4i) = BC, B^ = .-1.2 Z), C^ = AD"^ ; showing that (4) can be written in the condensed symbolical form, (9) J AP sin e exp {^ - pt) I = L^ J.' + D' ; and so also in general. With L, = 0, and H at V, an intermediate curve is obtained, where (10) J ]\P sin 9 exp (^ - pt) i = D^, M cos J 6 = ^ {x - x{), (11) (.1/ sin h BY exp 3 (^ - pt) i = ,~-^ = {F + 6) (a; - X,) - {F + 4)3 + / v' (.7-3 - X. X - ,,,. X - X,). But with Zi = 00 , and H at I, (12) fl=^^ P = ^, (i ^J^ sin df exp 3 (»^ - jo/) z = yl, giving a sluggish motion, in the neighbourhood of the upright vertical. In the purely algebraical case of the motion of the axle, the secular term pt is cancelled by taking X + P = ; and then (13) JP = 16 (4P' + ISP' + 30P^ + -nP + ;)) = 16 (2P + 3) [2 (P + 1)^ + 1] {2,MY = 4b'[{b'' --dY + "!08j, (14) M\ = - 2 (oP^ + 12P + 6), (lo) {jiMfz, = 1 (- 56* + \W + 27), Ji^shOi = - 2 (P + 2) V(3P. 3P + 4) (16) (3M)^-, = 26 (6^-9^.-18), (3J/)^' sin 0, = 66(6 ^ 3) ^ (6 + 1. 6 - 3) * (17) {mYz, = 26 (6^ - 96 + 18), (3i¥)^sin03 = 66 (6 - 3) V (6 - 1. 6 + 3)' sin 03 ^ / 6 - 1. 6 - 3 sm^ V V+ 1. 6 + 3' 129 (18) M^ cos 6 = 8P2 - 12 + 2.« (19) i^M' sin ey = 24P (P + 1) (3P + 4) - (SP^ - 12) a; - x' (20) ^ - ^' - 9 ^^ - 9 (21) M 2 [4:bK (f' - sy + io8y~ 2 ^ {2b) [(b' - sy + lOSf' I^ ^ _ b jb' + 9) ^ _ (P + 3) (2P + 3) L x/[(6'^ - 3)3 + 108] x/[2P + 3.2(P + 1)3 + 1]' (22) M^F = - 6P2 + 12, ili^P = _ 8P- + 12 M'H = M'F - 2L' = _ 6P^ + 12 - 2 ^ (-P + -'^)' (^P '+ 3) . 2 (P + 1)3 + 1 13. For a Trisection 'figure, derived from § 11, with/ = |, 6 > 3, and H at L, (2) (3) ^''^ LQ ~ r+^ ~ c (5) P + C + P = ; QL ^ p / -£i _ 1 D ^' - ^ v^ V ^^ ^2 ^3 LV _ ^>^ + 3 ^ D LQ ^--^ - 9 ^ ' LT 2b ^ _ D LQ 6-3 P ' LP 2b D (6) OA _ 1 _ 3(6-1)= LQ znf /r (6 - 3) (6 + 3)i OB 12 >/ 6 OV ' 6 + 1 LQ W - 9' QL 6'^- 9' and so on. With LQ = S, the polhode of L is the intersection of the rolling surface ^''^ 6^ + 3 26(6 + 3) 26(6 - 3) ~ 6^ - 9' hyperboloid of two sheets, with the ellipsoid x^ y e _ 6^ (^) (62 + 3)=^ "^ 46^(6 + 3)2 + 462(6 - 3)^ ' (b' - 9)^ ' so that the projections of the polhode are ,^, x' (6 + 1)2 , (6 -i 1)2 3 _ 482 (y) 0" + 462(6 + 3)2^ + 462(6 - 3)2^ "(62 - 9)2' ^ellipse; (10) (//T^^^ + '^ - ^--sy = (62-.)- ^ ' (hyperbola) (^^) (62 + 3)2'' 6(6 + 3)2 + - (62-9)2 ^ (hyperbola) and then (12) p2 -p2^9 (1^)'*'' = "' ^'^^ '^- , '■ ' p' - P2' = t if^) y' = ^' ''''' "'-^ For 3 cusps on an algebraical figure, with L at 6, ' (13) 6P=/3seco, = ^^=PP=azs(l -/), , cu/zs (1 - /) = /c ; and when (14) f = i zs(i-/) = .(^'^\ -/ = rfi' (15) (6 + 3) v/6 - 3(6 + 1) =0, (v/6 -1)3-2 =0, v/6=3v2+a, (16) cnf /i' = 3v/4 - 3^2 = cos 71°, sni/i' = 'V 2 + I - 'V 4, as shown in Mr. Dewar's stereoscopic representation, fig. 84. 2S570 "^ 130 The b employed here is connected with the c of the Proceedings oj the London Mathematical Society (L.M.S.) 1896, p. 591, by (17) ■ * = \^', so that the stereoscopic representation for (18) c = 0-52, 0-5763, 0-6725 = 1 + V2 - V 4 (cusps) •corresponds with (19) 'b = 3-167, 3-7, and (V2 + 1)^ = 5-1072 (cusps) and gives a series of looped figures with an apsidal angle 60° (fig. 84). A larger value of b, say i = 6, will give a figure of three waves. Another numerical test is worked out in the next chapter, obtained in § 9 from (20) ^=^2, k'=x/2-1, when ^> = v/ 3 + n/ 2, i^ = V 6 - 2, «, = - 2, X2Xi = 8, «3 + A-s = 4( n/ 6 •+ 1), ajg = 2 ( s/ 3 + 1) ( n/ 2 + 1). 2 ^6 + 1 a;2 = 2(s/3 - l)(s/2 - 1), costu = snfZ' =7-^ = 0-4824 = cos 61°-1. 14. When 3 > 6 > 1, ^,'1 and x.^^ change place, and /' changes to 3 ; the sign of P and Q must be changed, making (1) 6P = 9 - 6^ 6^ = 9 - QP, (2) (2 = 4(9 - b') {b' - 1) = 6P (4 - 3P) = 2 .7^2 = - i^V'i, (3) ,.3 = (S + b){b- 1), ,:,;, = i (9 - ^>2) (62 _ 1)^ ^,^ = _ (3 _ ^,) (^, + 1) ; (4) .2 ^ (3 + b)ib- ly ,, _ (3 -b){b + 1)^ ^ ^ 16^ ' " 166 ' (5) sin. = sniZ' = ^, cn|/r=|^J, dn | A" = ^J:, x/ (^^3 - X^ 12 ^/ 6' -^ 6 ' <7) zn I A' = (ijL|I^*, ,, I i^' = 4+^=. o '^ 12 v/ 6 <8) / = 7 (Z + 1 Ki) = p - -^■^' + Q JJL = icos-i:^tJ:?I^:J:l^-^ _ 1 ^.^^BP ^ (.,-...- .,) 3 ^3 _ 3 sin p . On the Trisection figure, with/ = i, 3 > 6 > 1, as in § 13^ LT ^ 3jf_62 _ Z> L(J 9 - 6^ ]^ LV 2b D (10) (11) (12) (13) (14) LQ 3 - 6 ~ .4 LP 26 Z* LQ ~ 3^rT = ■" C (13) ^ + C + X» _ 0, OA ^ 2 s/6 _ 12 v/6 QL p - 9-zr7/> AtL (15) - = zs \K sin a. = \t±± , A .y _ , 3 + /r and at P n6) - = ^* .V _ * - 1 6 _ 1 131 With H at h, and cusps, (17) cos 02 = k'^ sin w — k cos w = " a" i cos 63 = sin w = r TTiJ and in the algebraical case, with L at h, '2 • — 6 „ . ^ ^/^ - 1 (18) ^T^^n = 1' ''' = 3, '^- = sin 15°. With H at b', and cusps, (19) cos 02 = — 1^'^ sin 0) — K cos w = — , ■, cos 0,, = — sin «,. A . change of sign of L as well as P, with the interchange of Xi and «2, will make the expressions serve again in (5) and (6) §12, for the motion of the axle of the top. The algebraical case, with an apsidal angle of 30°, has been discussed in L.M.S. p. 591, 1896, and the stereoscopic representation is given in fig. 85, for (20) h = 2-77, 2-45, 2-17, 1-732 = s/ 3 (cusps), six-looped figures. For six waves, take v/ 3> b>l ; say h = \[ = 1-67. With (21) -6^ = 3, P = l, . = sinl5°, zn i/^ = g^, ,.n§K' = ^^^^, and in the algebraical case (22) Z2 = cos 60°, Z3 = cos 43° : but restoring the secular term with a general value of L, a series of figures is deduced from the diagram of fig. 85. Transcribed here in this new notation, the algebraical result for f = 3 given in L.M.S. 1896, p. 590, can be written, with .r — ^'2 = y^, (23) i^M' sin ey cos S^p = 4P (3 - 2P) [(3 - P) 3/' - 9 (1 - P) (2 - Pf y] (24) i^M' sin 0)'^ sin 3^ = [?/*- 6 (1 - P) (2 - P) ?/^ - 3P (2 - P)'] ^ [ - z/* + 6 (2 - 6P + 3P3) 2/2 + 3P (4 - spy] (25) p/- cos = ,?/2 - 6 + 12P - 5P2 (26) IM* sin^ = - ?/* + 2 (6 - 12P + bP") f + 3P (4 - 3P) (2 - Py (27) IM' = 4 (3 - 2P) [1 + 2 (1 - Py] (28) L'M' = 4P (3 - 2P) (3 - P). At the node 9(1 — P)(-^ — PY (29) cos 3^ = 0, y' = -^ ^-p ^, v/3(2P- P") tan ^ - (3 _ 3P + p3) ^ (3 _ 2P)' . Where xp is stationary (SO) L.L'. = 0, ,.^ 3(1-P)(2-P)(4- JP)^ , ^2, _ _ i^il^* _ W^ = 2 L±10:^lJDl. ^ 2L'M' iiPiS - 2P)(3 - py 3 - P At an inflexion on the P curve (31) L'z' + SLz + 2LH = 0, (32) M'F = 12 - 6 P^, M'£! = 12 - SP^, M^i? = 12 - eP^ - 2L'' ; from which z can be calculated, and an inflexion exists if z lies between z-^, and ^^3. On the equator (33) cos = 0, 2/^=6- 12P + 5P', _ (3 - 9P ^ QP" - P') V (3 + - 3P^ + P') tan 3 i/. - p^y _ 3P + p=i) ^ (3 _ 2P. 6 - 12P + 5P')" When P is negative, the result is obtained for/ = |. 38570 R ^ 132 QUINQUISECTION. 9 4 15. For Quinquisection, with./ = ^^, the results in the Annah of Math., 1904, p. 78, are modified so that (1) 0!^= - (c + 1)3 (c, - 1) (c' - 4c - 1), X3 + 002 = 8c (c' + c^ - 3c - 1), A-3 «2 = - 16c' x„ Xs- ai2 = 8cV (c^ + c2 - c), X, - x^. X2 - «i = (c^ - 1)' (c' - 4c - 1) ; (2) Pi = 5P = (c + 3) (c^ -Ac- 1), Pa = 4 (c + 1)^ (c^ -Ac- l)^ (3) (2 = - 4c.^i, Qi = - 4(2c' + .5c + 1) (c' - 4c - 1), t^s = - 4cP2 (4) x^ exp b Ii = Rs/{x - Xi) + 2»S' s/ («3 - ■'"• a; - a^a) = A, (5) E = x' + Qix + Q2, S= Pix + P2 ; and with Zj measured from V, instead of L from Q, X = Zi + 4c, (6) |i¥2 sin 6 exp ^2 = Xj + 4c (x - Xi) + i -y (X3 - X. X - X2. X .;r . « - n/« X r ^ , Ac (x - Xy) + i V (xs - - a^a. *' — ^1)1^ ~~~ Ji ( 7) (i M^, sin ey exp 5 (>^ - /^O « [R v/ (« - «i) + iS .,/ (a'3 - a; . x - ajg) ] = Z\ 4 + 5L\ B + lOZ^ C + 10 A D+ oLyE + F, arranged in powers of Li, and also oi x — x^ = y^, and r = - ;(/* + 2 (c^ + 1) (c* - 2 c^ - 6 c2 + 2 c + 1) 2/' - (c^ - 1 )5 (c^ - 4c - 1), (8) ^ = 3/5 - 2 (c^ - 4c - 1) (c* + 2 + 4 + 8 + 1) 2/^ + (c + 1)^ (c^ - 4c - 1)^ (Pi + 20c) y + i [Py - (c + \y {c' - 4c ~ 1)'] v^ Y B= {Pi + Ac) 2/5 - 2 (c + 1)2 (c^ - 4c - 1) (c^ - 1 - 6 + - 3 + 1) y^ + (c+iy{c-iy{c'-Ac- iyy + i[y'-ic+ iy{c' + S)(c''-Ac-l)y']./Y C = y' -2 (c + iy(c' -A + 2-2-1) y' + (c + ly (c - ly (c^ - 4c - 1) (Pi + 12c) y' + /[(Pi + 8c) y* - (c + 1)* (c - 1)^ (c^ - 4c - 1) f^] ^ Y D= {P,+ 12c) f - 2 (c + 1)3 (c - 1)' (c" - 4 + 2 - 2 - 1) ?/ + (c + 1)« (c - 1)^ (c2 _ 4c - 1) 2/' + I [y' - (c + 1) (c - 1)^ (Pi + 8c) y'] sj Y i; = 2/' - 2 (c - 1) (c^ - 1 - 6 + - 3 + 1) 2/' + (c + 1)3 (c - 1)8 (Pi + 4c) y^ + ^■[(Pl + 16c)/- (c + l)^(c- l)«2/'l^/^^ F= {P, + 20c) / - 2 (c - 1)5 (c" + 2 + 4 + 8 + 1) 2/' + (c + 1)' (c - l)i"2/' + '■[/- (c- !)'('• + 3)/]s/>^. Here again the substitution (9) ■ ,„. - .,,. ^^-f^^ : or f, (^'-l)'(.^-4.^-l) ^ will convert A into F, B into P, and C into /> ; and the norm of ^4, P, . . . P is (10) x\ a^ [(c + ly (c - 1)^ - {x - X,)] {x - ,.1), . . . [(c+ 1)2 (c- 1)*- {x - .r,)f {x - X,)'. Also AF = BE = CD, B^ = A^JD ; so that in the condensed form (11) J.li' sin exp (^ - pt)i = L^ A' + FK With Pi = 0, and H at V, the intermediate curve is obtained, as before ; also the upright sluggish motion, with H at I. 1 3 When/= -1-, c^ - 4c - 1 is negative ; x-^ and x.. change place, as given in the Annals of Mathematics, 1904, p. 78. 133 (12) An illustrative numerical case can be constructed with 1 c + l-- = 5 + 2v'5, making Z(2)^'' = ( v/5 - 2)*, c K K Wr= s/ ^ "S") K = 0-,^56 = sin 19°. ' ' Other algebraical states of motion, complete in five or ten cusps are shown in Dewar's stereoscopic diagram, fig. 86. 1,3, can be rewritten in this The algebraical result of L.M.S. 1896, p. 601, / notation, (13) i^3P sin ey COS 5^ = P,f + P,f + P,f + P,y, (JJPsin ey sin 5^ = {f + Q^« + Q# + %' + Qi) V Y, (14) Pi = L'M^ = ^Ig (c' - 4c -1) (c' + c- 1) (c« + + 20c' + + 5c - 2) and the Qalculation of the P's and Q's may be left as an exercise. 16. The general case of 1 for / 2r has been discussed in the Phil. Trans., 2n + 1 ' 1904, The Third EiUpticr Integral and the Ellipsotomic Problem ; and the method is described there for the determination of (1) • /= i{v) = liK+fK'i) = r^p-^^9. ^, Jx X ^/ X when it is pseudo-elliptic ; and, putting (2n + 1) P = Pi, is given by (2) / = 1 2n + 1 1 cos sin _i.*" + Qix''-'^ + + Qn Y,n+h -J {x — Xi) _lPlX"-'^ + P2X''-^ + . + Pr, «ra+4 v/ {x^ X . X ^2)) 2n + 1 X" where Pi, Qi, . . . are to be determined as functions of a single parameter. 2r + 1 When/= -, ,ij and x^ change place ; working in the sequel with this form of/. (3) Jx \ xl J 1 X I \J X. cos .1 iS v/ (« — ^2) 1 ««+i 2n + 1 sin .X^ -J {X^ — X . X — A'l) P«+i + P 2n + 1 (4) R = X" + Qx£'-^+ . . . + Q„, S = Pi«»-i + PaA-"-' + (5) P,= {2n + 1) P, P„ = Q„, and Q = x,, with the homogeneity factor of (12) below. The variable s of Chap. Ill § 6 was employed for / in the Phil. Trans., and to agree with the notation of Abel and Halphen ; we take the S there in the form (6) S = 4s {s + ^'xy - [N(y + l)s + N'xyf introducing a homogeneity factor iV, at disposal. In the Second Stage of p. 65, Chap. Ill, § 16, where the resolution is required of S into factors, we take, as in Phil. Trans, p. 247. (7) (8) X = s — ma (m - a)2' N'm'a^ (1 - 2m)a m — a ^ y + . _ (1 - 2a)m 1 — , m — a s — (m - „)^J and to change to our new variable x, we take here ('9 iVW(l -2m)a y_ NVs \ \r' + (m - aY J (m - ay\ ' making (10) (11) (m - ay — S = (T — s = x. (m - a)' — o — So ~~ S — OJ ~~^ X = (x — X2) iWa Ax^ (m — a)' (4a - 1) X - N'm'a (m — aY (m - a) ' iVW (4a - 1) {m — uY Xo X N* m*a ■f. ■' 1 — 4 (»t - ay ' 134 A convenient homogeneity factor to take is then m — a (12) ^^ = making (13) X2 = 4a (m - a) = - X^Wi, % + A'l = 4a - 1 ; (14) (x, - x^y = 1 - 8 (1 - 2m) a, :v^ - x^ . x^-x^ = IGa^ (m - a) (1 - m + a), . 2 ,2 _ 64a^ (m — a) (1 — TO + a) . 1 - 8 (1 - 2m)a ' and in /in (1), writing /for /A", Q = ., = 4^^ = m^c) snVdn^/ P = NP (v) = V (.r, - ■n) zn / = ■' •' (15) (16) and the relation between a and m is obtained by putting i\^2«+i = in PA?7. Trans., p. 247. Write (17) /=z.~j.^, x = >A--, Z = X2- 7=^ = ^- 1, measuring L2 from T on the focal ellipse, instead of L from Q, 1 (18) exp (^ - pt)i = [^VC^^--^-2)+^>gvC^'B-.'^--.'^'--^i)] 2«+l and from p. 60 (12), §13, Chapter III, (19) JM^ sin exp (^ - zT)i = L ^x + ^^ + '' ^^ (i-^ L,- X — X2 — i V (.I'a — X . X — X2 . X — Xi) and then, by multiplication, thfe motion of the axle is obtained, (20) JM^ sin e exp (^ - j}t)i = [i? V (,r - x^) -^ iS n/ (.% - x . x - .i-i)]^^^n i, \/ (a^^ — X . X — Xo ■ X — Xi ) ' ■J X ; L., - X — Xo so that, expanded in powers of L2, and x — .cj, (21) (ilf2 sin 0)2»+i exp (2n + 1) (^ - pt)i = Z.^'^+i 4 + (2n + 1) L2'- B + + (2» + l)L2l+ J in which, as before for 2n + 1 = 3 und 5, with ' (22) X - ., = y„ A = y"' + '+ . . . . + i(P^y-^"-2^ _ _ _ B={P,- l).y2« + i+ + /(_V2«+ . )v/r C=.y2"+' + wr + ^-[(A-2)//^" ]v}- ,, te + l 7 = 2/ + /= (Pi-2 ?!-l) _y4» + i with the substitution + i[{P,-2x)y 4»-2 . ]v/r (23) y-, as before, to convert A into /, 5 into J, AJ = B1= . . . ,52« + i = ^2« j^ • +«'CV"' )s/Y; ''3 — ■•'-a ■ ■'?'2 — ■'i 'l • . . ; and the relations involve • . , leading to the condensed form (24) iJP sine exp (^ - 2)t)i = U^A^''~+i + J2,m i_ Here L2 = 0, with H at T, will give the upper rosette ; and with H near I and L, very large, a sluggish motion is obtained, in the neighbourhood of the upward vertical ' 135 Seven Section. 13 5 17. The next case oi; / = -^-y — can be attempted, utilising ^he results of Phih Trans., §31, and the Annals of Mathematics, 1904, p. 83, so as to obtain the P and Q coefficients in (1) «^ exp 7 li = Rv (x — ^2) + iS ■>/ {a;^ — x . x — x^) = A, (2) R^= x^ - Qix^ + Q^x - Q3, S = P^x' - P,x + P, ; starting with iV^ = 0, making m(l - m)(2 — 3m + s/ M) (3) « = -^ 2(1 - 2my ^ ^^ = 4m - llm^ + 8mS /4^ (2 = ^2 = ~ *'3 Xi = 4a{m — «) = 2w^(l - m) TQ _ 5^ ^ j7^2 _ 17^3 r 4:m' + (- I + m + m^)^M] (1 — 2m)* \ / J - 1 + 8m — 14m^ + 6m^ + 2m(l — m) s/ M (5) ,;, + «i = 4«-l= {\-2wY • The work has been carried out by Mr. T. G. Creak, and he finds, using detached coefficients of powers of m, .„. 7p_ p _ 2-12 + 14 ^ 1 + (Q-)J + 1)VJ/ (^) 7/^ - A (1 _ 2„j)2 ^ ^ _ - 2 + 24 - lUO + 163 - 78 - 20 + (0 + 4 - 25 + 38 - 12) ^/ M (7) ^1 (1 - 2m)* (8) P,= — , ^"^! ,. [0 + 10 - 91 + 303 - 469 + 342 - 106 + 16 ^ ' - (1 - 2m,)'' + (2 _ 13 + 21 + 5 - 28 + 10) n/tV/] (9) q, = , ^"i ,, [0 + 10 - 105 + 377 - 613 + 466 - 152 + 24 (1 - 2m)« + (2 - 15 + 23 + 17 - 48 + 16)N/i¥] (10) P, = -Q.= ,, ^^"t ,. [2 + 1 - 68 + 252 - 407 + 331 - 132 + 22 ^ ^ • (1 - 2m)^ +(5-26+48-37 + 13-6 + 2)^/ir]. (11) nK^fK'i)=\J- P.^)^=^co.-l^^i^ = ^ sin-5^-_|^-,). With this parameter K + fK'i, the result can be employed for a corresponding w, j3, y. But for a, 8, a parameter (1 - /) K'i can be taken, where /(I - /) Z'« is derived from (11) by the substitution (18) ^(l-/)^'=J(^'-4^)^ _, 7 P' a;^ + P9' x'^ + P,' .« + Pi 1 .™-i C'*^ + Qi' ■ ^ + Q'2) ■^ {xj -x.x- x^. X - Xi) = \ cos ( 4, _ ^)i • = -^ ^^" — (iTIT^jl (14) i^ = P-l= Z5±l^z:^^ii±0^^ Q'=-.2-4a = -4„(l-m + «). So also the parameter 2 fK'i can be employed, as in §21. The calculation of B, C, D, E, F, G, H, in the associated motion of the top, can then be carried out if required. When / = ^^, X2 above would become negative, and change place with x^. The form of the result for nine, eleven, .... section can be inferred by analogy, but the complexity of the algebra increases rapidly. 136 Halving of the Degree. (1) (2) (B) (4) 18. The degree can be halved of the expression for / in (3) §16 by the substitutioa X 1 --, /. 2r + 1 y\ - = '. -V ^'hen / 3u ""^ 3b' X Q 2 _ y 1 X!i — X^ f 1 = yJ_zX 1 — y' ,2 ' y/ = ^±^L^ = dn^l - /) K\ y,' ^ ^ iin + 1' •'' - ■■''I _ yi - y 1-/ 1 it/Q ^^ tAj m \Aj ■^~ (A/1 - .ri - sayZ' ' r C^A' dy -«,.ri (1-/)" x/^ (5) F= (1 - f) Y\ Y' = 4a (a + 1 - m) - (i« + 1)/ + y = (C„+ 2/')'- C^^', « ~ 00 a,'i a-g X «3 00 y 1 ,Vi + ^ - y ^3 Trr !- 1 ' :.: 1 (6) / 2n +1 cos" Ql Qs Q^ 2^ cos-1 (1 + ffi.v + .¥,/ . . . ) s/ (i . 1 + 2/ . 1 + ^^.y ± 1^ 2n +1 sin"^ (1 4- /7i?/ + //g/ CiN dx )./(i.l-3/.l-^.y±f a '-'0 <^y_ and the number of the ff co-efficients is only a quarter of the number of the P's and Q's. This form of / is suitable for the expression of the curve of H by (8) e! = £L = ^ P .... r^ _ ,./^,• = ^xp // P2^ X2 1 - ,y^' P2 V (1 - y") It will serve also for fi and -y, with f = f^ and a lower rosette curve of §14^ Chapter III ; wjiere, referred to the downward vertical, (9) (10) ih 1 + - _ ,^,2 Ifl = P" = iL'- = '''2 ^1 2 =cos^j0=^ = ^, hif-i^) =-§«^ + 4 _ With this?/ as the variable, the expression of the associated motion of the axle is not so simple as with the previous ,V ! ' I But as at first in §16, (11) :!■ - '^'i _ ,T (12) — ?/2^ ~ f = 1 •*'i y' - 1, with f = (•'■ - ^r,) .r :r, y - ?/- ^« + 1' •. (.''s - .r) ,r ,r. (13)--- - ,v/ ■'('o — X . X — ,' X., — X-, cn^ fK 5 y% = '3 ~ •'! 1 ■X-, (14) (15) .t'3 X2 s/X CO dn^/A"' (^^2 - 1)2' -^ = ^l^v' - (C'o + y^Y (a change i)f sign in )") .fj .Ta s/ Y - CO + ,'/3 137 (1) 19. Thus with/ = f, as m §11 i^2 + iF X = •r — .lu = I cos-i s/ (i . ;(/ - 1 . ya - 2/ . ?/ + 2/3) = f sin-i n/ (i . ?/ + 1 . 3/2 + y-y - yd ; and for the motion of the axle i A (3) ^ = ^fl\y^ t(^ + 6)y - 2^^ + 2wr] ^ = ^^y^f)^'^ t(^ + 4 - 2y^) VCV^ - 1) + 2iyvY'] (f - 1) not so simple as before in §12 (4) With the degree halved for the trisection / = ^, 3 > 5 > 1, 9 - b^ ■I !) 12 + i_^)^._»__A'|^-.jyW., = # cos = # sin (5) (6) ys =^^r^ = clnf A, 2/2 ^=i-.v^f'-^^ -y ^ = dn|Z', 6 + 1 2 sniZ" For quinquisection, with/ = ^, we write, as in the Annals of Math., p. 78, •' 1/2 (c + 3) (c' - 4.C - 1) 20 c + / - 1 % n/F- (8) Y=f-1. ¥', Y' = 16 c'y^ - [ V+ (c + 1) (c - 1)^' In seven section, and with/ = ^^^, as in §17, ^ (9) _dy_ =^fcos-i(l + //i,y+^2 2/^) N/(i.l +2/-l + ^^3/ + ^) (10) (11) (12) 28570 = ^^m-'{l-H,y + H,f) ^ [^ .\ - y .1 - ^^y + ^^ Y=l-f.Y', Y' = {C, + fy - c/f C^ = 4a (1 — m + a) = G^ = 4a + 1 + 2 ^0 = (1 - m)^(m + .jMy t)^ (m + v/lf)^ ^ ^ (1 - mY(m + s/ilf) 1 - 2my ' "" (1 - 2m)2 (1 - 2m) 1 -m + ^W 2m )^ Ci = - 1 + m - y/Ji" 1 - 2to 138 <13) (14) (15) Tj + m - 2m^ - N/Ji H, - 2 + 3m + V .17 2m (1 — m) 7P = 2m (1 - m) fli ^ m + -yJ/ ^ _ 1 + Ci S2~2(1-2ot) 2 2 - 12m + 14m2 + m^ + (0 - 2m + m^) V i/ 2 5 (1 - 2m)=' v/Coexp J/z = (1 + Z^:^ + ^2/') ^ ( J • 1 + 2/ • ^o + <^iy + 2/') + z (1 - Fi2/ + E,f) ^{\.\-y.G,-Cii) + f). If this Co should be negative, the result must be rewritten in the form (16) / = f- cos-i (1 + H,y + H,f) n/ (j • 1 + -V • 1 - §2/ - -^J- (17) F = 1 - f . F', F' = (Co - .y=^)^ - CiV, with a corresponding change in Hi and ^2- 20. The form of I for the half degree can be inferred more readily from / (2^), with 2r K'i ■A^2«+i = in the Phil. Trans., p. 248, making s (2?;) = 0, 2v = ^ — -— , say /g K'i, suitable for Tla, a, S, and an upper rosette and intermediate curve. Then (1) 1 {2v) = V\- P (2r) + ^iM^ ^ .1 (2w + 1) P (2w) .s" + P2 5"-^+ . . 2n + 1 ■1 cos +p B+1 sm .is"-i + QiS«-2 + 2 s"+* • + Qn-l (2) 2w + 1 2 s"+* The degree is halved, first by putting s = f, and then r ,q s 1 _i(2ra + 1) P (2v) f« -t- i (2w) = cos ^^ ^ — ^S 2n + 1 2 2n + 1 2 2n + 1 M-l cos sm 2^«+i ■ if — O]^ f 2 t"^^ n/t; 7^1 <3) J,i = 2^ - iV(l + 2/) ^ + 2A'2a-; ±iV%, and then brought into the same form as /in (6), (7) §18 by putting (4) t = ^^ 1, .y = - ^^'~y" , 1 + y = (2jL^i>^ a — m u a — m a — m X = (5) (6) (7) (8) y, _ iV^(l-2m)m^aV 2a 2a - 1 2mu^ \ (a-myu' VI -2m+T^ri;^''~Tr^2^ + ^; (a-m)^M^ ^ ^ Vl - 2m 1 - 2m / — m" a (a - m)2 ^ _ i\^^(l-2m)m^a ^ n „^ r 2 a ^ m ,n ^^ - («-m)3«3 (1 - ^) (rr^ + r^n^ -^ ^ ) 1 2a 2a ■ / =12^ ^^^"(1 -^^^^ + ^^^"^ • • • )v/(M-u.i-«+ir2!LV) / i2v)=C\-'P{2v)/Jl- +u' •I uL ^ Si So du w ^-^<^"V^Jllir-ly^v"^'», ia9 (9) in which (10) (11) (12) 6^= 1 - ^■■' '-'-(rfk-"' u (1 - 2m)2 P (2») = ^ (», - ..) z,;-, r, P (2..) y i= ||^ i? = 1-P(2.) /-^=^^^A!^^^ ^ ^ V «i A'a cn/2 an/2 ^1 = V = ^^ ^2» sn^ (1 - /,) s S3 s S2 Sl 00 u CO 1 u sn(l-/2) cn/2 21. Thus, with A\ = 0, Phil. Trans., p. 266, as in §17, p. 135, (1) I(2v) = \\e-1 + u^)^. 2 cos"^ /I ^ j^ , zr n //l 1 , 1 « 1 — 2m ,\ ^ ' 4m- (1 — to) 1 - 2m rr ^ (1 - 2to) (to - ^/ if )^ "^^ 4to2 (1 - my ' 1 — TO (Vt 7P[2v) / ^^ -7-7PR~ ^ ~ -"-^^ "^ ^^™^ ~ ^'^"^^ + (0 - 3m + 5to^) n/J/ " ^ '^ ^ ^ V SjSj (1 — 2m)^ * Nine and Eleven Section. Nine section has been worked out in the Archiv fur Math.^u. Physik, 1900, p. 74 • the result is (4) u du I - 2m = I sin-i (1 - K^u + K^u^- K,u^) N/(i-l-M.l + £+ ^^ m^) M 1 — 2m (5) (6) f co8~^ (1 + K^u + K^v?^- K^u?) N/(J.l+M.l-ir- + 1 n a^ + a^ + — I+n/J. 1 — 2m = — u") - '1 (a + If 1 _ a'^ + 5 + ll + 6-14-31-29-17-6-l + (a^ + 4 + 5 + 6 + 6 + 4 + l)>/^ 2^= 4a* (a + 1)= (a^ + a + 1) 1 _ 2m _ a''+3 + 5 + 6-14 + 7 + l] +9 + 4+ 1 + (a« + 1 + 2-2-2-2-1) ^^ 0) — 2^^ 4a* (a + 1)2 (a^ + a + 1) (8) ^ = a" + 2 + 5 + 10 + 10 + 4 + 1 = (a^ + a^ - 2a - 1)^ + Sa^ (a + 1)^ _ a^ -3-5-6 + 24 + 01 + 57 + 15-20-5+ (l + 2 + l-2-l<' -10-5) n/^ (9) 9i? = 8^a (^ + 1) (^2 + a + 1) and Z'l, Zg, /iTs are the same as A^, A^, A^, given in Phil. Trans., p. 271. Material is at hand in the PhU. Tram., 1904, p. 273 for 2n + 1 = 11, /= ^r,ov2^ + l ^ and for the calculation o£ the co-efficients in (10) 7=^-1 cos -1(1 + ^i.V+ H,f+ H,f + H,y') V iY, = ^sin-i (1 - H,y + H,f- ff,f + ff,f) ^ \Y, (Bulletin de la Societe MatMmatique de France, XXIX, 1901), but the results are omitted here, as too complicated, and 2n + 1= 13, 15, 17, 19, ... . still await solution, when expressed in the Second Stage as required in the applications. 28570 S2 140 2r + 1 22. With u = 4n, f = — s j ^i replaces x^ or .ri, and the result is of the form (1) ^ = 2n ^ A'" ^ ^"^' ~ '*2- « - ^i) 1 . .nPx'-^ + Pa*"'^ • • • / N = -^ sin-i -„ v/ («3 - ^), and here the degree is halved, and the result made symmetrical in a Jacobian form by putting (2) x^- X ^ 2 £ ^ ^" /^ - n^ = d - v^ ^ ^ n/ (iTs — «i. ^3 — aJg) "^ ' V {x^ — x^. X^ — X^) K ^ -^ ' n/ {x^ — Xi. X3 — X2) ~ " ' V (a;3 — «i. x^ — x^ k " ' /ox n/ (a;3 - Xi) dx _ 3 % / Writing / for fK', {d - y^)^ + d V K ^ (~ + K-^-d) dy \/ K \K d I — — ^ V Jl (4) I = -(^- y')^"^ + — sn/cn/dn/ dy ^^^^ ^^ ~7Y d -y = icos-i go + gx.y+ . ... +g.-i.<" v * F, - i5 n (d - ?/ ) *" n where E^ =,(?*", (^ = _?-/ — , H„.^ = ( - l)"-i ; and the calculation of the H coefficients in terms of a single parameter has been explained in the Phil. Trans., 1904. Here -J k> y> — n/k, — > d> k, and when n is odd, we must take K ^-^^ ^' = ~k~y- ^'^ + ^ = 1 + 2 2^^y -f=\ + 2Py- y\ J-H-,. V.-y = l-2^ Ix=-^+y. ^/'c-2/ = l-25^,-^2=l_2i^_,^ as before in § 5, where n = 1, d = 1. \ But when n is even, n = 2, for instance, as in §28, <6) 1 , „, , , ., 1 , oZsiZ' ,2 F, = — + 3/. V. + .y = 1 + 25iAiL2/ + 2/2 = 1 + 2P^ + y Thus in the Quadric Transformation of §6, - e') F = . /LjT dn 2eP' (7) y = n//c sn 2eZ = c sn eP sn (1 - e) p = /I - dn 2eP <8) ^{l±2Py -f) = en eP ± en (1 - g) p. (9) s/(l ± 2 P,2/ + 2/^) = sn (1 - .) p ± gn ,^^ (10) p = ^j^A: = ^(_L_ ^ N^c; p_zii^_,/l N 1 141 23. It is convenient sometimes to disregard the Jacobian form, and to put <1) a,'3 - X = ^, « = ~.y , D = iV^a^g, with the homogeneity factor iVat disposal, and with P for NP {v) and Q for N^Q (v), (3) F= Y,Y,^iC,±fy-4Cyy, '^*>' ^ n*^*^^ (J) - y')i" 4n~n^^" {D - y'y 4n' (5) (X> - ?/2)i« exp (ni + i,r) 2 = ^ = ^ v/ i Fi + i A^ V ^ Y^, (6) 4i = ao + aij/ + flaS''^ . . . . + a„.i y"''^ (7) .42 = Oo - «iS' + «2/ • • • • + «n-i(-y)""^ (8) ao = i>*™C„, a„.i = (-l)«-, ■and Fi, J-i, changes into Fj, A^ by a change of y into — y. If n is odd, as in the series ^ = 4, 12, 20, 28, .... (9) Y,= C,-2C,y-y\ •(10) Co = iV^ s/ (a^s - «i . «3 - aJs), Ci = J-^ [ V (^.'3 — «i) — \/ {x^ — x^"] = N s/ («3 — A'l.) zn ^K', If n is even, for the series ^ = 8, 16, 24, 32 ... . (11) Y,= C,-2C,y +f ,{12) 6'o = iV^ s/ («3 - Xy.Xz - ajg), Ci = 4iVf v/ (.«3 - a;i) + s/ {x^ - x^)'] = N V (a'3 - Xy) zs^K'. Measuring L^ from P on the focal ellipse, and puttiiig ''• '\ (14) ^ M' sm fl exp )(Z = ^-^ ^— Kf lib (15) J M-' 7V^ sin e exp ^i = ^ {D - f) . (16) i M' N' sin^ Q = LiD- (V " 2 A § - §^)y' " v' 2 D ' D iU) i^^iV^cose = - Z32+ 2X3^ + S- - 2y (18) • il/*iV^= (is'-SA^-^y + 4Z3IA tan0^ = ^ ^-^ ^ -L,' + 2L,^ +7 Then, multiplying together ■(19) exp [n {7^ - pt) + ^TTJi = (x> - yy Q 2 ■ V , -nV + '2/ v^ ^\» (20) (J M^iV^ sin 0)" exp n (^^ - z^y) ^■ = (Zj + ii — ^ _ , j (i? - y'f the motion of the axle is given by <21) (J ]\P N" sin BY exp [« (^ - pO + i 'r] «' = ^" ^ + « A""' i? + . . . + / where B, E, , . . are obtained from J. by a successive multiplication by (22) = ■ f . jy' + 'y ^ ^ the denominator D - y^ cancelling each time. 142 We can write in the condensed form (23) 1 M' N' sin e exp (^xp - pt + ^] i = L,A^ + J«, and J is derivable from J. by a change of y into Cn/y ; as it is the term which gives the lower rosette when Zg = ; and so is obtained from J. by a change of / into 1 — /, aa shown in §14, Chapter III. 24. Thus with /x = 4, and N = form of Phil. Trans,, p. 276, n/ {Xz - X-i) the result of § 5 can be rewritten in the dy = cos V y ■ 1 2 . K - ,y2 ' 4 1 ^y-K.— y_Tr_ ^{l-y\,^-f) sm V 2 . /c - y^ TT 4 (2) (3) M£^ exp (ot - ^^ + ^^ z = ^ {I + y . K -y) + i sj {1 - y . k +y), M^ = s/ {k — y^), y = K sn mt, fC ^ M^ iV^ sin 6) exp (^ - zs) i = L,+ (1 -,)f + iy^ Y f -y V(k- f) ,2\2 (4) i M^ iV* sin^ e = 2Z3 ,c(l - k) + [Li- 2Z3 (1 - /c) + ^-](,c - 2/2) - (k - 2/2) (5) M^ N' cos e = - Zj^ + 2Ls (i - ^) - ^ + 2 (k - y') (6) if* iv"* = [-2:32 _ 2X3 (1 - + ^-]3 + 8 Z3/C (1 - k). 25. Next with ^ = 12, Phil. Trans., p. 281, <^' ^-rS-'^n^)^: yV n/ F (2) D = a + a' + a', P = J^ (1 _ „) (5 + 3^ + 3^2 + ^s) Q = H^ -«*)(« + «' + a') (3) (4) (5) J.J = a^l + a + «=) T HI + «)' (1 - «) y - f, Co^a'il + a+ a'), C,= 1(1 + ay{l - a), /^ = l + a + a'T il-a)y-f ^ (6) (Ml) exp (i)_^2)t -12-^sm (2) -2/2)5 -12 2 = ^ cos -^ T-^; ' ^ „:^ --^ - " J 3 (ot - ^if) + ^ A = A,s/ iY, + iA^ v 1 r„ MJ^ = (D - f)\ (7) (8) (9) (10) (11) (12) J M' N' sin exp (^ - ^) / = 4 i il/^ N^ sin2 e = Z,,2 (d - f) + L, (1 - a^) f + a^a + a^ + a^) f _ y^ Z) -72 - - s/{D - f) (13) M^A'^cos e = - L/ + L, (1 - a^) + a^a + a^ + „8) _ g,^. M^A^4 ^ ^_ 2,^, + 2:,(1 - a*) + a'^a + d^ + a')f +\ L^ d ih M^ m sin Qf exp [3 (^ - ;,<) + ^ .] / = (^1 V J T', + / ^2 ,/ i };) r T + i ll^ijO/ + iy ^ Y L /} ^ 2 = L^A + ^L^E + SL,F + G, A= {1 + a + a') - {l-a)y - y^ ^1= - a'{l + a + a') f + (i _ ^b) '^2 + ^3 F^= -a'(l^a + a')f + a {I - ^3) > + y< G, = a«(i + a + a')y^ + „3 (^ _ ^ ^ ^^4 _ ,^5 M/^ A2 sine exp (^-^,+ -) ; = ;^^ ^4, ;, , ,; 143 The norm, of ^4 should be (a + a^ + a^ — y^y ^ )) / (a + a^ + a^ - y^y (a^ + a^ + a' - y") F „ y* (^a + a^ + d - y) (a^ + a^ + a' - y-) G „ y^ la' + a' + a' - f-f. 26. In iLi = 20, n = 5, Phil. Trans., p. 289, working with (1) ^=^, C, = P{bv), C- = g, Fl = i,-22/-^/^ and the identity derived from the differentiation of i) - 2/^; V Y,Y, ± cos-i- ^^^ ,,, V LFi - ^ = :^ sin-i ,^^^ ,,, n/ iFs 5 (X> - 2/2)5 2 ^ 20 5 (Z) - yy ^ (i) Ai = ao + ajy + a^^- + a^y^ + y* ; /i^ L - r - -?-^(l + ^) M _ _ „ + a2 + a3 <5) }^^C^ ^ - (1 + gp (1 + 2 g - 6 g^ - 2 g^ + ffl^) - 8 g^ 6^ 4,c (y Ci^ 16 aH^ " ~(1 - kY ,o^ n _ c' _ g - 6 Q _ g2 + 6^ _ c' (6) ^ - ^' ^' - ^TT' V ^~' '"'• ~ r*' " D 1 5 P 00 - P (5u) 1 ^ ^ f \ 2 a 1 + 6 n _ 5 a^ - a'% + 5 a b^ - {1 + 2 a) ¥ _ ^ a/ 2g2 /ON <^' «i _ - p ^ 1 _ ' Q _ _ 5 g^ + 5 g^ & + 5 g 6^ + (1 + 2 g) 5^ _ , The work has been completed by Mr. W. E. H. Berwick, and verified, and he finds <9) «2 - - %^W~ • Then multiplying five times by Q (10) ^f+iy^Y D-f the motion of the axle is given by <11) {\ ifW2 sin 0)^ exp [5(^ -ft) + ^ ttJ i = LiA + hUE^lOL^F^ 10L,'G + 5L,H+J, ivrhere / can be obtained immediately from A by the substitution {^y,jj^j, as also fi'from E and G from P ; and the result can be written concisely {12) J l^iV^ sin e exp (4, - pt + ^) « = A ^^ + J^ (13) P' = ^* J, ps = ^^ J^, . . . . and the norm of J., P, .... ^, is <14) (P - y^)^ ./^ (P - 2/^)^ (^4 - f), . . . . y° (^ - 2/^ The successive multiplication gives ■(15) E, ^-eoy -e,y' - 62 y' - e^y^- f, = 8^ w = 2, utilising the results of §38, Phil. Trans., p. 277, normalised to the Jacobian form in (6) §22 and writing d for D, as in §22, Y2 (1) ^=r(-^+T^.)-^ ^ ' h\ d-y'J .yj\. (2) F. = 1 - (fiiji.)!^ + y=, p^iJ- mi + 1.^), a-^', y = ■^/ K sn 2eK (3) p, = - .vp(2.) = i (^ + ^,) = (i±iy ^ 1 \ V K I Ad o. 1 f_L _ ^^]^ {(> - 1) v/ (1 + 6(/ + d') V .4) d = 4 ' (d - 1) (3^ + 1) _ ^» -4- ^' Sd ~~^T' ^ 145 Rationalise by putting (5) rl a + a" (1 - a^) (1 + ay 4a (1 -d'Y Cfi^ / = LzJ" = (1 -2a - a^)(l + 2«- a^) _ 4 /c _ 4a (1 -o^ ^ ' ■ 1 + /c "(1 + ay ' "" - (1 + y (1 + a^)2 ■ In the region 1 , ^ p_ -(^^+ 25+ 3) (^.^- 26-1) + 4(^+ 1) ^B Q _ 2(b + 1) s/^ ^^^ *^ 4p ' :D 4F ' (4) .4^ = do + a^y + a^y^ + y\ ^ ^^ JoV ^^ D^'JVY^, ^ (B-fY '''''' {D - fY ' and then a differentiation of / will lead to (6) a, = A + 4P = 26 + 1 + ^^ + 1) ^/^, (7) ^^ = P, + 4P - 4^ = 26 + 1 - (6 + 1) ^B^ ynx _ T.2 _ 6^-0-4-4-2 + 4 + 6V4 + l-(6 + l)2(6'-l-l-l)v5 \"j do — J^ ■) ^l — gT4 • In the determination of E, F, G, H by the, successive multiplication by (9) %f + iy^Y,Y,, (10) E,= - Dy - e,f - e,f - y\ e, = P, - 4P + %, ^ = p^ - iP + s^ ', _^ = P, + 6P-6§, „.= P, + 6P, 147 and 02, aj can be determined by squaring and adding, giving (7) 2a3 + a,5 - 2Pi a, + 1 + 6D = 0, Proceeding with the axle motion, (8) (9) (10) ^— P — fiP^fi^ i^x= -Df-f,f I =A + 6P-4|, e, = Pj _ 6P + D i <7i = A - 6P + 3^ = g, = (ff + ] ) (a^ - Sa^ + 3a - 3) - 6Z/ = (ilf cos l^y ei/'Ce-Wi = [ J.1 ■s/ {x — Xi, or ^2) + ^'A \/ (a's — .r . x — x^, or a;i) J exp fimti zn /lA', where A^, B^ have been determined for simple values of ^u, as 2, 3, 4, 5, . . . , in the preceding series ; and the degree may be halved by the substitution of § 18. 28570 ' T 2 148 But with fo = — , or , we have •' ^ At (5) {]\f(ty = [J.2 + iB2^/ {a;.^ - .'• . ^' - .^^ • •'■ - ■^■i)] exp fimti zsfo^K', or [^2 v^ (* - .2;i • ■*■ - •^2) + «-52 V («3 - ■^') ] exp ^m^z zs /gZ' ; and the determination of A2 and B2, as an algebraical function of x, proceeds in a similar manner. Here again the degree may be halved, as in §18, 20, .ibut the substitution required will not be the same as before in (4). 32. Thus for example, as before in § 3, with <1) /i = h /2 - J, / = 0, / = 1 (cusps), P comes to A and P' to B on the focal ellipse and HS. HS' = P = 2 OA. OB ; so that changing to the upward vertical, we take (2) cos 6 = y^, cos ft, = 0, cos H2 = k, ch Cj = — , K (3) a= cosJ^expH .j> + .p)i = ilvil + 2P^y +f)-i^ il-2P,y + f)] exp [J (1 + k) mt + ^tt] i (4) y = ?• sin J e exp i ( ^ - i.)i = ^ i\_^ {I + 2P y - f) - i ^ {I - 2P y - /)] exp [J {\ — k) mt + ^ tt] i (5) jS = / sin J e exp 1 ( - .^ + ;/,) 2 = 1 z [ v/ (1 + 2P y - f-) + i V (1 - 2P y - y^)] exp [ — J (1 — k) m^ — J tt] 2 (6) aS = cos^ J e = i (1 + .y2)^ 37=- sin^ i = - ^ {1 - f) -and the modular angle is Jtt — flg- Then, as before in §3 ; (7) a (3 = ^i sin e"^' = i ■/ [ ^ (1 - /c?/^) + «j, ^ («: - y^-jj ^^.p J^f^ (8) sin 61 exp {^ - ht) i = n/ (1 - cos flg cos 0) + i V (cos 9 . cos 63 - cos 0) S (7 , / 1 „ N Km G 7 and (10) ay = 4 »• sin ef' = ^i [2/ V (^ - 2/^) - W (l - ■^')1 exp {h't + i^)i (11) sin 6» exp (.^ - h't - Jtt) / = s/ (cos 0. sec 63 - cos. 6) - w (1 - sec 63 cos fl), representing a state of motion in which G and (r' are interchanged. Since f - pt cannot pass ^w, but oscillates, the apsidal angle, in a pseudo-reeular precession, is (12) ^ = IhT = ^KmT = kK, and the mean precessional velocity • (!«'>) 12^ = ^ = n s/ (^ cos ftj) and this is the precession when the axle is moving horizontally with the same spin (14) R = \^n^{2^iic%). At half time "(15) t = \T, ch 01 - cos = ^ (ch 01 - cos 02. ch 01 - cos 0^) = tan 0, cos = sec 02 - tan 02 = tan {\^ - ^0^), tan^ 40 = tan ^0^, and then dQ 149 The curve described by a point P on the axle is rectifiable, because the axle describes a series of cusps ; putting OP = ^k, the arc s described by P is given by (17) s = k cos'^ s/ (1 — cos 02 cos 0) = k cos"^ sin cos {ip — ht) and the length of the arc from cusp to cusp is (18) 2k cos-i sin 6»2 = 2k {^tt - 6*2). Relatively to space revolving about the vertical with angular velocity A, the projection of the path of P on a horizontal plane is given by (19) 7^ = sin cos ;^' = ^/ (1 — cos ^2 cos 9), \ = sin sin i-' = ^/ (cos 0, cos — cos^ 0) k (20) cos e = sec 02 (1 - j'), p = (1 - p) (psec^ ^2 " *»»' ^'2)' a C4, composed of two equal closed loops. To realise this motion, sj^in the gyroscope wheel in fig. 3, and hold the axle up at an angle 62 with the upward vertical, given by An \^ A Mh /An\ \cr) (21) cos 02= 2 \CRJ - C CR' ; Mh denoting as at first the preponderance, and CE^/2g the K.E. of rotation, both measured in g-cm, or, kg-m, or ft-lb. When the axle is let fall it will start from a cusp, and reach the horizontal position rising again to a series of cusps ; and the motion of the axle is given as in (8) by (22) sin exp {ip — ht) = -J {1 — cos 02 cos 0) + / n/ (cos 02 cos — cos ^0), (23) ^ ^ Ta^ 9-T ^^^ ^2 = »^ n/ (i cos 02) = fcm, h' = m. The value of can also be expressed by equation (11) and a, /3, -y, S can be written down, so that the naotion of any point of the wheel is determined completely. In the continuation of further algebraical cases, it is convenient to take /j = 2/1 sometimes, and then with x (/i) = 0, we find by elliptic function theory, (24) . (/2) = . (2/0 = ., + .2 + -3 + ^^^-^^^If^--^- = 4™a. Or sometimes, with /2 = 1 - A, /i = "^ ^ n + 1 ' (25) X (/a) = x^ + x-^ _fifi^ Qj. ^^ ^ ^^ — 'h—i^ or 4a, X2 Xi with JV as in (12) § 16. 33. For another series of algebraical a, /3, y, S, take (1) fi = h h = h f=h /' = 1 (cusps) as in the Annals of Mathematics, 1S04, p. 97 ; and then in § 14, from the downward vertical, (2) x = l M'^ (1 + cos 0), M' - X = 1 M' (1 - cos 0), J M' = W - I, - U' + b 2 _ 2 (3) 22 - I : ^^- b + V ^1 ~ h - \ <4) (i3 ^2,7 n/2)3= (1 + cos 0)^ exp [± | (^ - >^ - Pit)^i = (1 + ^ + ^2 _ i j n/ (2 - ^2) + 'iVTF'^nj ^/ (^3 - ^- ^ - ^1) ;(5) (a >/2,8 ^2)='= (1 - cos0)»exp[± | (^ + ^ -/?20]« eZ>2 + 3) (1 - 2) - ^.2 + 1 = -^ 2 ^ (h'- - 1) + / V (^5 - 2. ^ - .^a- ^ - ^1) 150 (6) (7) (8) ■ v: ^9 - b' m ■' 12 x/ ^' m = zs I A" = n~ 2n ~ % s/ 2 J {¥ -ly G h b' - 5 n~3f'~6^2^(b^- 1)' S + b' p2_ S + b' n ~ 6 V 2 V (b' - ly P^ _ />2 + Pi 1 12 V y - S + b' n 2 n G' h' 2An~7i~2v2s/{b'^-V 2An~n G _h b' - 5 V 2 ^ (i' - 1) ' n/ 2 V (b' - 1)' G' h' = Zo Then by multiplication, (10) (2j3g, 2ayy = sin^ exp 3 (<^ - jft) == [ v/ ( - ^■g - ^'i ) V (2: - ^'s) + '/ V (^3 - But from §11, with (11) (12) pP = ?>2 _ 1^ 2 (13) / — 2 ,/l ~ 3> /2 — 3) /=o, /' = f. 0, =^ 2.-, 2 1 + ^ - ^- 9 52 3 ^ + 1' ^- ^, _ 1' (1 + cos 6)^ exp I (^ — i/. — p^t) i = 2^. 2 Z^ = ^1)]^ *2 + 5 (14) (15) ^) V (2 - .0 + Z-^-^-^^^i-^ ^ (,3 _ ,. , _ ,^) (1 — cos ^y exp # (<^ + 'A — J92O 2 = jb' + 3) (1 - 2) - *' + !.,. , 2 7iF™r) + w (^B - ^. ^ - ^2. ^- - z,) Pi -.,na/r - {b-2,) ^ {b + 3) ™~ ""^^^ 3 {b - 1)^ • £2 _ 2g 27^' _ h^ + 2, m ^ 8{b - ly V (b + 3)* Here, by multipUcation (16) {2a(iy = sin^ e exp 3 (1/. - ;)0 i = [V (^ ?3 - ^2) s/ (^ - ^1) + / ^ {2^ - Z. Z - Z2)Y (1) 34. Take, as in Phil. Trans., p. 277, i>..¥.6'. 1893, p. 229, /. = i /2 = 1, ,/ = h /' = 1 (c _ (1 - ay 4«2 ' /' = 1 (cusps) 1 + 4a^ + a^ (3) (4) 1 + ^1^3 2^1 + ^3 ' 4a^ ,2\2 ^1 = - 2a 1 - a 7iy -1 ' ~i -III (1 + cos ey exp 2 {^ - ^ - p^t) i = {z - A,)s/{z - z,.z - z,) + / (2& z + 5i) V (2 . 03 - 0), (1 - cos 0)2 exp 2 (^ + ^ - p^t) = (.- + ^2) ^/ (^ - ^2 . 2 - ^^ ) + ,: (2^:|? c + ^2) s/ (2 . ^3 - 0), where 5i B.^ are left for calculation as an exercise ; 2l= /-!f^ (6) sin^ exp 4 (^ - pt) i = (^ - «) ^ (c - ^2 . c - 2^) + 4/?^ (. _ ct + a:'^ ^ -,2 (7) P = (i+4!)il_:Li«)^ C^ a + «« 4a5 (1 - a^) ' P = 1 _ 3„r (5) sin* H exp 4 () derived from (4) (1 + g) (1 + g^) {a + d' + a') -t=^M\\ + z), t = ^ j¥^(1 - ^), M^ = {I + a) (1 + g^) {a + a^ + a^) »i , „, (1 - g) (5 + 3a + ?>a^ + a^) ^ — = zn A A = V K To — 77 — ; — 2 — ; — 3\ 1 -^ ° 12a n/ (a + a' + a'') ' pi (1 - a) (5 + 3a + 3a^ + a^) n (5) m 12n/ (1 + a. 1 + a^ a + a + a^) /^^ ^2 1 ZT' _ / (1 + a + ay £2 (1 + a + a^)^ (6) rn^'^^^^ ~ "^ ''?>a^{a + a^ + ay n ~ 3 V (1 + a. 1 + a^. g + a^ + a'). (7) (1 + cos 0)^ exp 3 (<^ - t/- - j9i^) «■ = {z^ - QiZ + Q2) s/{z - z^. z - Z2) + i (6'& z'- - P^z + P3) s/ (2. ^3 - z) <8) (1 — cos ^y exp 3 (^ + ^ — P2O * r 3^-V (2. z - z^. z - Z2) + i (z - 1 + a + 2a^ + a^ a, ) n/ (^3 - ^)] (9) A = 2 152 5 + 6 + 12 + 16+16 + 11 + 5+1 a (1 + a) (a + a^ + «3) (5 + 3 + 3 + i)' 4 + 2 + 11 + 11 + 13+14 + 11 + 5 + 1 a (1 + a) (a + a^ + a^) (5 + 3 + 3 + 1) ' nm ^ o 3-1+3+1+3+2 4-2+7-1+3+2+5+5+3 i,W) ^i- ^ (^i + a)(l + a')(a + a'+ ay ^' a (1 + a)' (1 + a') (a + a'+ a') ^ (11) sm« e exp 6 (<^ - p7) ?: = r/ . 2 + a + a"\ , \ , A -p't , 1 + + al 2 + a\ , ,,) ^ .f L\ a + a^ / ^ '^ n\ ?,a + 2a^ + a^ ' J ng^ £ ^ j9i + Pa ^ 1 + + al 3 + 2a + a^ n 2n ~ S V {I + a. I + aK a + a^ + a^) ' h' + h / 1 + a. a + a^ + a^ h' — h 1 — a (13) ±-±A = / n V I + a^ ' n n/ (1 + a. 1 + al a + a^ + a'), So also, with (14) /i = 1, .A = f, ,/■ = f, /' = 1 (cusps), (15) /; = !, f, = h f=o, /' = |. (1^) .A = i, f2 = h f=i, /' = t (17) (1 - cos 0)' exp 3 (^ + ^ - p^t) = \ ^ {z - z,. z - z,) + i^l ^ (2. z, - z) (18) P^ = z,lK'= ^k(^ +a^)ty (i + 4a + a^) m 2 V 4a v/ (a + a^ + ^3) • (19^ ^ - 1 - ^' ^ a. . -1 + + 1-4-3-4-1 ^ ^ '' - TT^^' '^ "■ '^ == 2 (a + a^ + a3) ' -1-2+1+2+1+4+1 Zy ^2 — 2 (a + a^ + a^) 7:2' (20) v' (.3 -z,.z,-z,)=^ which arise in Z. M. S., 1893, p. 251, by taking (21) r (/O -t = iM^(l + z), t-t{f,)=i M-^ (1 - z), (22) ^ (/i) = (1 + a) (1 + a') {a + a' + a'), i (A) = h- -J {k- k-h- h) = a (1 + a^) (a + a^ + a^) (2-'^) ^' = t (/O - ; (Z^) = (1 + a') {a + a' + d') (24) £? = (L+_^)^^(l_ti^±_^ £i = (1 - g) (5 + 3a + 3a ^ + a^) n 2 v/ (a + a^ + a^) ' „ 6 V (1 + a^ . a + a'-^ + d') ' (25) f-f^^y -M-...l-...l-.3 = ^-^(i-±4^i±^, a + a^ Jr a^ ' (26) (^)'= - J- 1 + .1 . 1 + ., . 1 + .3 = _- (1 - c^y 36. With/i = ^,,/^ = |,/= >,/^i^ (1) '^ (/i) = 0> ''^ C/i) = 4c (r + 1):^ (r - 1) = ,1/-; (2) . = 1 M^ (1 + cos e), 4c (c + 1)3 (. _ 1) _ ,,, = ^ J/. (1 _ ^^g y). (3) 5 = _^L _ 1 , _ - f'^ + + 1 For example, c - -= ^ o + 1, z, = cos 144", y^: = A" . or c 1 - v' 5 - 1, .^, = cos lOS", K/K' = ^ 5. 153 - c" + + 2c= - 4c - 1 ± 4c v/ C *3) *1 — H ^1 — (c + l)«(c-l) c* - 2cB + - 6c - 1 (c + Vf (c - 1) r5) £i = zn i A" = (c + 3) (-c^ + 4c + 1) p^ _ (c + 3) (-c^ + 4c + l) ^ ' m 20cVC' ' n ~ 5iW Ce) £2 ^ zs 2 ^' _ 3cB + 7c^ + c + 1 £2 ^ 3cB + 7c8 + c + 1 ' ' ^ M' + A\2^ c (c^ + 4c - 1) M - 4 \ ^ (-c^ + 4c + 1)^ ^ ^ \ n / 2 (c + I)-'' ' \ »i / 2c (c + 1)B Xc - 1)' (8) (1 + cos 0)5 exp f (^ - ^ - pi^) 2 = [(1 + zY + Qi (1 + ^) + a,] n/ (^ - ^3) + z Fs £i(l + ^) + P2I V (^3 - ^. ^ - 2i) (9) (1 - cos 0)* exp f(^ + ^ - P2O 2 = 5 ^ [(1 - ^)2 + i?^ (i _ ^) + E^\ + 'L c (c + 1)^ Thence a, j3, y, 8, and ^, ^. v/ ^3 — Z. Z — Zi^. Z — 2^1) With/=i,/ = 1,/, = 1/3 = 1, (10) ^ = \M\\ + cos 0), M^ - X = \M^{\ - cos 0) M' = 8c(c + l)^(c - 1) (11) X^ = (r + l)B(c - 1)(- c^ + 4c + 1) . . -, 2a;3 _ (c + l)(- c^ + 4c + 1) ,,„. — cB + 3c^ + c + 1 (13) z-i, = (14) ^3 + ^: 4r - 4c 1 (, + l)2-(c - 1)' — 4c^ (1^) "■'"' = (c + l)^(c-l) - (16) 1 + 1 = 1^ 1 + z^z^ = ^2(^3 + 2^1), the cusp condition ; (\1\ ^^ - zs ^ r - "' - c' + 7c ^ P2 = cB - c^ + 7c - 3 A.8 a verification for cusps, we find (18) sin exp ( — Z9. Zo — Z, . , 1 + ^3 ^1 (19) ^ - -3, ^ ,;_; \ w^th ., = -^^. The method is general, and can be applied as an exercise to seven section in § 19, with (20) A = h h=h f=h /' = i, (21) X = i liP (1 + cos 0), yP - :v = J M' (1 - cos 0) (22) i¥2 = X (/2) = ^2 + ^- V^ ^ = .r, + ,ri - -^S or 4 a m § 32. 28570 ^ .154 Curvature of the Motion in Plan. 37. The radius of curvature is of assistance in drawing the curves of the next Chapter. The curve of P is considered, plan of the motion of a point fixed in the axle of the top, at a distance from denoted by a, so that a is the radius of the spherical curve described by the point. ■ Then with coordinates {x, y) in the plan, (l^l X + yi = a sin (i eM = —^ (p exp m) = — f ^ + {^] ^ ndt ^^ ^ ^ n\dt dtl if 5, y\ denote the co-ordinates of H and we take a = \lc ; (2) , , X -^ yi — a (cos 6 + z sin !^)e>/'« (3) (« — yi) (x + yi) = x x + y y + i {xy — xy) = a' (sin dcosO + i sin^ 9 xj/). Writing, as before in (11) §1, Chapter III, p. 43, G = 2Ah, G' = 2Ah', {4) sin^ e ^ = 2(h + h' cos 6) (5) l¥ = n'{H^ cos 0) - 2 (^ + ^' ^"^ ^)'. \ sin / Then if v denotes the velocity of P, and p the perpendicular from on the tangent at P, '(6) fv = xy - xy = a^ sin^ d^p = 2a^ (h + h' cos B) (7) ?/ ■= ? -rf = d (cos^ % 02 + sin^ %^^) 'ir -2 = 2^2 (// + cos 9) cos\9 + 4:{h + h' cos 9y a ^ 9«2 2«2 {H + cos 9) - sin^ 9 ff. f^\ c^ ^ W (H + cos 9) cos^ 9 , / (h + h' cos 0)2 + ■^' from which the p, r relation is inferred of the curve of P, with r = a sin 0. Or otherwise, with z = cos 9, and s denoting the arc of the curve of P, (10) (*;= aX'L±J _ ,), giving 5 by a hyper-elliptic integral. Then R denoting the radius of curvature, (11) R = r±, ^=.^1P_ . dp R sm9cos9d9 ' and differentiating (8), (12) _2_ci^dp^ ln'{- 2Hh - 3A cos 6 - A' cos^ ) cos sin P' d9 (A + h' cos 0)3 (13) « jn' (2 Hh + U cos 9 + h' cos^ 0) ^ l.i^' (^ + cos 0) cos^ t) + (h + h' cos 0)2]t' 4-^(//..),2i^^^l^^ 2 (fi +z)z^+4 (-^ y^'^)7 The radius r of curvature of H has been given already in (61) § I, Chapter III, p. 47. Thus at an inflexion of P, where /? — on fV,^ ^„ i x- , . spherical curve, ' ' *^' ^s^^lating plane is vertical of the (14) 2Hh + 3hz + A' .2 ^0, or 2 Z (// + ,) + l, + ^;. ^ 0^ (15) , cos sin2 9xp + in-j- (H + ^) = Q, so that, with L negative, cos and ^ must have the same sign It :.n inflexion. J 55 On the equator, where z — (16) > R ^n '■ a Mm' At an inflexion of H, where )^ is stationary, R (2. H + zfz 3 n (17) ^ L + L'z = Q, - = ^-^ — -^ ' = ^ cos' 6V (J. H + cos «), If G,h,L = 0, R ^ (2. E + ^)% _ ^ ;o' cos e ' (18) a gZ' - * Fg' ' so that R = 0, ip = ^ when cos = 0, and the P curve has aTramphoid cusp on the' equator. In a spherical pendulum, G', h', L' = 0, QQ^ ^ _ [jn^ (g + zy + h' |i ; ^ ^ a - l«% (^fi- + 3z) and at an inflexion of P, 2H + Sz = 0, as is evident dynamically ; the tension of the' thread vanishes here and changes sign. 38. Interpreted geometrically on fig. 88 of the focal ellipse, which may also be made to serve for any intermediate position of the articulated hyperboloid, as the tangent plane at H, with 0' the centre of the hyperboloid above at a height 00' = HK, L + L'z ^ g + g' cos e ^ HQ + Eg ' ^Q± ^ OQ' sin 6 M ~ k k k k (1) /9^ P7 ^ 9 0' Q'^ ■ fl 2 OQ'. HP' (2) , H + z = 2 — ^i sm = ^ , (3) 2 {H + z) z'- + a{^^ ^M^) = ^ ^T~''°^' ^ ^ ^ T "'"'' ^ (4) _ , OQ'^ + 00^^ cos^ e 44(g+.) + 2^Jt^'f. = 8HQ.O:Q! + 2^sin0cos9 M^ ^ M k ¥ k . HaO:Q;i ^ 4 HP:iM! oos fl ^ o HQ (OQ'^ + 00'^ g H c'. OQ'^ = 8 p + 4 p cos « - » p + 5 p . Q c'. OQ'^ ^ . HQ. 00'^ = 8 p— + 8 p , to be substituted in (13) §37. In the plane of the focal ellipse 00'- = 0, , , a _ Q c' _ 9 c R _ y if OQ cuts HQ' in q. Or analytically, //.N '7 f\ TT , <.i fL + L' cos 0\^ Af L + Z'cose \g /'L^'/'^^ « sin L L + L' cos 6 „ ^ ^ L d^p ^°^^ + ^ J/ if sin^ " cos + 2 ^-^^- Thus if Z2 = 0, ov z, — 0. R = a. and the P curve osculates the equator. 28570 ^ 2 156 The same figure 88 will represent the focal hyperbola, with VHQPT, V'QT'HT' in the position V2BP2T2, Vg'Qg'Pg'HTg', passing through S and S', and the geometry is the same as before in the focal ellipse, corresponding letters being distinguished by tJie suflBx 2. So draw O2Q2 cutting HQ2' in q^.^ and Rg ^2 parallel to HQg meeting O2Q2 in U ; tten a sin flg _ /2 ^2 39, On fig. 88, with H at a or a on the tangent at A or A', <1) H = - z^, , L + U z^= (Z + Lz) z + '■2L{n + z) = L' {z- z^) {z -2 z^) ^ y (2.^-^1) ''^^W <2) a = T The P curve has cusps when H is at 6 or V on the tangent at B or B' ; (3) E = - ^2, L + Z'z2 = 0, ^ V (2. 2 - ^2) [^' + 2 yg (^ - Zg) J Thus with s'j < SiTj, an inflexion comes in ; and when z^ = 2 Z2, z^ = k. In a lower rosette, H is at P in fig. 89, and c, 7 coincident with P, f5^ ^B _ OQ _ OT. OY _ OT. QV _ OV ^ [ a QP QP. TV 0S2 ~ OG ' as in Chapter IV, §14 (19) p. 95 ; while rg = 0, and there is a cusp at H,. But A_ = 22O2 R2 _ O2 Q; [ ^^ a sin 02 " 72T ' ^ ~ "^ and r2 = rK2. In the upper rosette, H is at T in fig. 90, and R2 parallel to TQ meeting OQ in /; -n-3 yJq a qc Here c^, !' = 1 ^^ (4) tan f), = §^ = tan 12°-2, sin B, = 0-21 ; ) (6) ' ^^~l: ^'=^r89' ^ = -^V-89, J? _ Ifit* 3'> ^ + - - 9-71^' ^'^ + -. =y^,. 159 On the representative figure of the foca^l ellipse, fig. 89, begin by taking QL = LP = 1 (cm), and then ^ , , ,(7) QV = 8, QT = 10, OQ = 4 V 5, OA = 10, OB = 8, OS = 6, OV = 12, OT = |, OQ = 3, QW, OW = 10, \b = 6, OK = 1 OT = 3 v/ 5, OV = i OV = 6, ¥ = LS^ LS'2 = 9 (13 - 4 V 5) 9 (13 + 4 v/ 5) = 3*. 89, /I- = 3 V 89. ' Where \p is stationary and H at an inflexion, ,„, fl ^ ^/ 89 g V 10 v(8) cost* = - g, = —ff-, sinl= — ^y^-, tan X = ^^J-^ = 68°. 4, >!'__= t^ + ^ = 74°. For the rolling quadrics and line of curvature, § 17 Chap. IV, § 10, Chap. V, ,m) ^ 4, B, C , _ 1, - 1, - 9 ' A',B ', G\ _ 33, - 21, 243 <10) D 9 ' D' 89 X, V _ - 83 ± 3 v/ 89 S' r _ o^ „„2 „„,. ^' <11) ,-. , -p. = 45 dn^ ra^, 1^ = 36 cn^ mt, "^ = f sn^ mt. (. From § 40, Chapter V, ^ ^ a sin 03 79' rg ' a sin Og 31' r^ ~ 9 ' •and on the equator, R/a = 0.032. At the inflexion on the P curve,, (13) cos e = ^^ ^lo'/^to^^^^ = - 0"365 = cos lll°-4, sin 6 = 0-931: At half time, (14) f = ^-^=i, sniZ = s/|, cnJZ^vf, dnii:=^|, 2Mzr_ V 40 + V 15 y/ 10 (2 s/ 2 + V 3) ( y/ 5 - v/ 3) ^^ 4 ^ ~ s/ 40 + n/ 24 ~ -8 <15) cos 4 t;y = yfc = I = C0S4 36°.8, tan 2 w = i = tan 18°.4, tr = 9°.2 ; O O ( < cos = ,r^-^, sin = 1^^, tan = 20 V 2 = tan 88° , , o V 89 o v o9 tan X = 2 = tan 64°, ^ = ^ + ^^ = 73°-2, 1 2 ' ^°^ '^ " V5' ^^"'^ ^75' *^^ (i '^ - 2 x) = v^ 5 - 2. Where i/- = Jtt, y = P = 4v/ 5 (16) cos = ^^, sin = ^^, tan = i^ ^ ^^^^ ^^o.^ ^ ^ tan (tr + I :r) = — ^= tan 45°-7, ar = 0°-7, x = 44°-3, ^p = 45°. These configurations are read off on the plan of fig. 89, and can be represented in elevation, as in fig. 91 ; the results were incorporated in fig. 82. 3. For a diagram of the motion of the axle of a top, kicking higher still than in fig. 89, select as leading to results of greater analytical interest, and shown in a figure, (1) ./ = J, /c' = n/ 2 - 1 = cos 65°-5, ' making K^ K' V 2, (2) /c = y/ (2 v/ 2 - 2) = 2V 2sin22°J = 0-91, tn 1 A" = ^ = y ^4t^ = tan 45°-5. (3) 4 P2= /c + i - 2 = i [( v 2 + 1)5 + (2 - 1)5 - 4], ^ = 4(V 2 + 1)^[(V 2 + 1)^ + (v 2 - 1)« + 4] = 451. 160 4 (4) sin 03 = / -T ^-n = 0"094 = sin 5°-3 ^ 1 + 1 1 + 9 (5) sin 03 = TXI^^i" »B = 0-589 = sin 144°, ^^ = jj^e^!'^!!,. = I'l- On the equator (6) ^ = 0, cos 2;A = ^ ^^ ^ ' ^' -^ = sin 74°-8, ^ = 45° ± 37°-4. Where i/- is stationary (7) J^ = 0, sin0 = g-^^^|^= 0-94 = sin70°-3, sin2^ = 3 s/ 3^^ = cos 76°, ^ = 45° ± ;i8°. At half time (8) .nlK = ^, en ^ /^ = y ^^1=_\ dn J ^ = ^/ (s/2 - 1), /= v/(v/2- 1), (9) sin 2^ = 2Py p. ^ (r_^^)^J. _ y^ = ^^^ 1^2°, ^ = 86°. Motion still more extensive would be given in another figure by taking (10) /=! ,c' = (./2-l)^ K=2K'. Another curve of intermediate interest is given in a diagram by (11) K= K, K = /c' = J v/ 2 = 0-707 = sin 45° ; (12) tan 63 = 4 '^^^^/^ ~ 5) = tan 17°'7, sin 63 = 0-30, tan 02 = - ^ i^^^ ~q^^ = tan 103°-8, sin 02 = 0-97 Where the curve crosses the equator (14) z = 0, cos 2^ = ^ ^^ +.^^^- ^^ ^^-^) = sin 36°, ^ = 45° ± 18°. Where the azimuth is stationary (15) 1=0, ,in» = ^_2^..(2_^2)V2=sin68°-2 (16) ,.i...au-.([4^5)'. ta.(i.-2*) = J^^^QL,^=.,„43", 1^ = 45° ± 21°-5. 4. In the original numerical example of the Annals of Math. 1904, we took (1) /=! , = 3 P = -J-, ^ = ^=_ L L' _%' _ ^6 5 v/15 M ^ 2V24' J»f~^~2V24» I'.-ili' 3 = ^4- •*=^''. ^ = ||. ^ = ^^. ^=^^ '^'^^ "8 ' si°^3=-g-, ^2 = 0, sm02=l, .^ - _ ^ n/ 6 ^ « 2 ' /,2 ^' ^^ini;^- ^' P2 = 111, p, _ S/10V24 ,'3 1 n. 128 a P3 16' a sin 03 ~ X' Pa = l'-^- 161 At the inflexion of H, (3) cose=-|=-^ = _i_, .^e = yl P - V6 _^ /3_ , P_'/25, p_i .,.^' F - ^ ^ ^ - 76 + ^ = 276' ^ = 3' ^^^ ^ = ? = 72 = ^'^ '■' In this case the inflexion of H is at half time, because (4)- sniZ=f^, cnJ/f=27A' ^^ *^^ = 7 f" In the rolling quadrics, as in §17, Chapter IV, p. 99 , A,B,C 1, -1,-4 ^',^',g' 3, - 1, 8 ^^^ i) ~ 4 ' D' ~ 4 ' with S'^= 6S^ ; and a model was made, with S = 2 cm. The polhode lies on the cone (6) ' ' - 3^2 + 5^2 + 32^2 ^ 0, and it is a Une of curvature on the confocal quadrics ^'^^ 16 + 4 V6 "^ 476 "*" - 9 + 4 v/6 " ^'' ^^> 16-4^/6■^ -4^6+ _9_4v6~' an ellipsoid and hyperboloid o£ two sheets, confocal with the elliptic plate 3? v^ 2^ ^ (9) 7^ + ^ + = 1' (10) « = 5g, i3 = 3S, ^.= -9±4s/6, and a?, ?/, 2^ are given in § 17, p. 99, Chapter IV. The rolling polhode may be made in stiff wire, or cut out on the cone (6), as in Schilling's (Grassmann's) model in fig. 63, according to the dimensions in fig. 91. The design of Kirchhofi^s associated Elastica has been given in §17, Chapter IV ; and the deformable hyperboloid was made of four pair of rods, set out to the design in fig. 74. We find also that H is on the pedal if placed at Qi, where LQj = 2LP, and so a representation is obtained of *the motion of a Spherical Pendulum, as on p. 103. 5. For a moderate extent of movement of the axle, select a smaller value of k, say (1) K = V2 - 1, making K' = K^% p = J^^^ = V 2 sin 22°i, tan % = ^^^^^^/"^^^ = tan 29°-3, tan B, = 2*^^(3 v/ 2 + ^) ^ ^^n 64°-3, sin Q, = 0-49, sin B^ = 0-9. . \ Or else take (2) K = {s/2 - 1)2, K' = 2K, P =1, R3 = 00, tan 03 = 4 = tan 26°-5, tan 63 = | V (5 V 2 + 1) = tan 35°-2, 2 2 \/ 2 — I cos 03 = — ^, cos 02 = -^ — , sin 03 = 0-45, sin 02 = 0-58, (3) sin2 exp 2^ i = (cos - -^) ^ (cos^ + A cos - ^) ■+^'75V5v/(75-"°'^)' 28570—3. X 162 (4) ^^^^1"'= ^(-2^^^^ + T'"'^-5V5)' sin'' 9 ^ = ' — .A v/2 2^'^ 3V2 n~ V5' n ^/5V5■ IK j-1, j-„i.^ ^ / — 1 P — ^ P, = T ; there are no inflexions For auotner k, take t> = v k = 2« -^ — 4=5 -^1 4 > "" of the H curve, and the axle curves are not looped ; (5) tan0B=|| = tan29°-2, sin03-O-49, tan 63 = ^- tan 44°- 7, sin 0^ = 0-70. For the representative focal ellipse, take OB = QV = PT = 20 (cm), OA = QT = PV = 80, OQ = 40, QL = 30, OL = 50, OV = 20 V 5, OT = 40 V 5, tan a, = 2. Then at half time, (6) sniZ = N/10 -v/6, cnJi: = Vlo ^ ' dn J Z = '-^, tan 2 <. = y I = tan 52°-2, and in the herpolhode, e, r? (4 v/ 3 ± 3V2) (s/5 - n/3) (7) ^r- = 8 • With K smaller still, take = n/k = 3, P = f, Pi = k? (8) tan % = Y^ = tan 22°-8, tan Q, = ^ = tan 27°-2, giving a featureless motion of four waves ; and so on. 6. The formulas of § 28 Chapter V for /n = 8 can be applied here too, for the construction of a figure, utilising the arithmetical results of fi = 4 for bisection, as well as the algebraical results of Phil. Trans. § 38. Thus for K = K' s/ 2, 1 / 1\ 1 V n/ 2 +1 . = ^[^i^2-^l)+^^^] + '^i^^ + l) = 2-497 > ^ 2 + 1. Or with K=2 K', and in this region of a, (3) rfiX' = j(<. + i)[4(<.-i) + l]i, nc=i/r=i(a + l)[4(a-l) + l]a, cn IK' = a en I K (4) —^ lg^2 , -^ — =(a2-2a + 3) (a^ + 2a-l), from which the motion may be calculated. The algebraical curves for/ = | and |, Z = 2 K\ are shown in figures. As an exercise in the geometrical interpretation, design a diagram of the focal ellipse, taking OB = 400 mm, OA = 441, making J- =^ = x{a-l\ a- •' 163 Teisection. 7. For a numerical case of /= f, as represented in fig. 93, take b in §11, p. 127, Chapter V, a little greater than 3, so as to have a large k, say (1) * = T = ^"2^' ^ = 3^ = i 5 and taking LQ as the unit, say LQ = 100 mm, (2) LV = 868, LT = 2600, LP = 104, OQ = 1440 OA = 1620, OB = 384^13, OS = 204 v/ 17 2 ^, 8 . 15 ^ 8 cos w = en o A = Y^, sm w = ^, A w = q, tan a> = ^ = 1-875 = tan 62°, o K = ^^^ = sin 58°-7, k' = ^-^^ 5^ = al 01307 = 1-351, ioO loo ^ TT 9 4vl3 4V13 sm = jj, cos <^ = TT^ ^ * = ~1I~- The co-ordinates of L and P are r^\ mr ^^Q^Q m ^Q^OO 14580 _.^ 19968 Move H from L to 0', so as to make n = 0, ^ = tt ; then T n' 1 9 i TO' 1 W or = f = !? = »■«' e| = 8. f = al 1. 8693 = 0-74. Here _ 225 _ 17 _ 17 X 225 (5) Xs — -j^g , X2 — 2g) ssi — — 1024 ' and with s = I6x, . - -o-s + 17 X 225 , (6) -/^ ^,' 5 = 4 (225 - s) (s - 17) (64s + 17 x 225) (7) ^=^,,,-. (s-25)vi64s^+U.225) - 1 sin-i 2^ ^/[(225-s)(s-17)j _^sm g-j , and the figure of the algebraical herpolhode of L can be drawn to scale, with P = 96s/s, p3= 1440, p2 = 96x/17. In the associated motion of the axle of the top, (8) ^M'co.9 = ^-^^^ - 6 + . = - ,^^+ . 13199 225 7 X 13 X 211 W ii/^ cos 03 = 9-^^256 + 16-= 9x256 (10) ^4_ 13^ X 433 X 5113 ^^ ~ 3* X 21* 7 X 13 X 211 , „ 180 (11) cos 03 = 13 ^ (433 X 5113). tan 63 = ^^^^ = tan ^ , sm 63 = 0-12, 827 HOO / 1 7 (12) cos 62= - ^(433x5ii3) > *^^ ^2 = - g^^ = tan 123°-8, sin 0^ = 0-83. The axle is horizontal when (13) *' = r^' i^^cos3st = 5v(.^-^'0, which leads to .^ = 60° ± 19°-6. 88570 X 2 164 The precession is zero when L n/ (433 X 5113) ,,.9c-7 ^^ cQo.f; (14) cos e = - ^ =-^ ^ = ^^^""^ ^ ^^ ' ^ = 60° ± 20°-7, X = 77°, tr = 3°-7. 70321 When i/- = ^TT, cos 3 t/- = 0, ■''' ~ y^ x 313' 461149 , . 300 X 7 X 31 ««^ ^ = 13 ^ (433 X 5113 -)' *^^ ^ = ~ 461149 " *^^^ ' At the inflexion of P we find (15) z^ - 1-09742 - 0-4608 = 0, z = - 0-3241 = cos 109° ; also i?3 . . 9^ 2929 gg f..79 P2 2279 8. A more interesting value of h to take for/ = f would have been 6 = n/ 3 + v/2 = 3-146, as this makes k = V 2 - 1, iT = ^' V 2, as before in §3, and so a combina- tion is obtained of/ = J and / = |, With this value of 6, (1) 6 = s/ 3 + s/ 2, i^ = 3P = v/ 6 - 2,^ a'l = — 2, ^3 a;2 =8, a^g + «2 = 4 n/ 6 + 4, ^3 = 2 ( V 3 + 1) ( v/ 2 + 1), ^2 = 2 ( n/ 3 - 1) ( v/ 2 - 1), (2) cos(. = cn|Z' = ^A^=l -^^^^-^1 =cos61°-2, sn^Z' = ^^i^ = 2sinl5°, + 1 \/ Z \rii ■ QQ_dn^r- ^^ + 1 -1 OA"'^''^^ V2 Taking LQ as the unit on the focal ellipse, say LQ = 100 mm, (3) LV^5^2^,6, g^ = 3v/2 V(V3- V2), ^ = 3(y3+s/2)v/(v/3 + l.v2 + l), and so on, as in §13, Chapter V. We find also (4) sin 63 = 0-077 = sin 4°-4, sin 63 = 0-687 = sin 136°-6. For / = i^, and the same modulus, we should have to take (5) ft = ^ 6 + s/ 2 - 1, 5^2 = - 1 a;3 «i = 2 ( v/ 3 + 1)2 (2 v 2 - V 3 - 1), «3 + ^'i = 2 62_6=12 + 8s/3-4s/6-4v2, a,'3 = 4(s/3 + 1), a;i = -2 (V3 + 1) (2 x/2 - ^3-1). The same modulus will serve also in FUl. Trans., p. 277, for making (6) / = L^ with a = s/ (2 + s/ 2) + V ( v' 2 + 1) [= 3-215 > s/ 2 + 1 = 2*414) (7) tn i Z = tan 30°-6, tn | Z = tan 74°-5 ; But we must take (8) a = V[V(^^2 + 1W2UV(^2^ ^^ j„^ ,„ j^, ^, For a tri -section of the time, take (9) b'= ^S- ^/2, sn^/f = L+1' = 0-65. en ^ Z = LzA' == o-35 • ^ - " 2 ' and then calculate ^. Or else take, for another representative figure ^ = T2 (^^^ + '^'^ ) = 3-023, making K = 2K', f = I 165 9. Another interesting numerical case for / = f would be given by (1) b = 2'^2-l + 2v'(V4 - 1) = 2V2 - 1 + -l^^l— = 3-0526, b + 1 = [v/(V2 + 1) + s/(V2- 1)P, P = f [2V4 - 1 + V2 - 3 + (2 - V2 + V2 + 1)^3] as this makes (2) /c = sin75°, ^ = v 3, ^=1-762, (3) cos w = en f /r = ^-- =V2-v/(V4-l)= 0-4935 = cos 60°-4 ; U) OQ = dn#Z'= 2 _ >/(V2 + 1) - V/CV2-1) (5) sniK=VS-l, en f /sT = 2 - v/ 3. (f.. LT _ 2V2 - 1 + 2v(V4 - 1) ^ ^ LQ V(V4 - 1) [1 -^v/2v(V4- if] _ 3 + (2V4 - V2 + 2)v/(V4 - 1) (V2 - 1)W(V4 - 1) T p LQ = 3 (V2 - 1)2 (3^2 + V (V 4-1) LV ^ 2V2 - 1 _ 2V4 + V2 - 1 LQ _i + 3v(V4-l) 3v/3-V2-l LQ_ v(V2 + 1) - v/(V2 - 1) VV2 + 1 „_ OS ~ " 2 V ^72"^^: = ^"^ ^^'^• The algebraical case of three cusps at 120° in Dewar's .stereoscopic diagram, fig. 84, has been discussed already in §13, Chapter V. 10. In the numerical case of six cusps in (20) §14, p. 131, Chapter V, and in the stereoscopic diagram of fig. 85, where L and b are coincident on the focal ellipse, (1) f=h b=^S, P = 1, /c = sinl5°, J^ = 1-0174, (2) sin w = sn iK' = j-^ = s/3 - 1 = 0-732 = sin 47°-l, (3) cos <|) = en |Z' = 2 - V 3 = 0-268 = cos 74°-5 = ^, K (4) ^3=^3-1= cos 42°-9, ^2 = i = cos 60°, ^i = - v/3 - 1, /»% 6r o Li j^ 6r o Jj , (6) F=l i:=\ H=-h (7) -t-=n/3+1 ^= — 1, _= — v'3 + 1, (rolling hyperboloid of two sheets), (8) B' = 0, A' = C = D'. The motion of the axle is given by (9) sin^ cos 3i^ = ( - 1 + 2 cos Q)\ .' , _, ' •, sin^ sin 3i^ = (1 - cos + cos^ 0) V (2 - 2 cos - cos^ 0), (10) M =2, cos = - 1 + i«, X =A cos2 10, .% = 2 V 3, X2 = 3, ,i'i = — 2 n/ 3. The time from cusp to cusp is, as on p. 48, the beat of a pendulum through 60°, of length _ . /in T - -J—- I 1- -^ ^\ R^r=^~V3' '"JTK 166 Take cos Si/- = J, ^ = -^tt = §^, (12) i {1 -cos'ey = (-1 + 2cosey, (cose + V4)'^ = (V2 + i)^, cos = - V4 + V2 + 1 = cos47°-8, cos Oo — cos 6 (13) sTd^ mt cos 6I3 — cos ft V2 - 1 + V4 - V2 - 1 2 -[(2+ V3) (V4- V2) -1] V 3 - 1 - * n/ (V4 - 1)^ = 2 (V4 - V2) - 2 ^3^2 ^ ^^ ^ (^4-1") = 2(V4- V2) - 2(V2 - 1) V(V4- 1) sn m^ = s/ (V 4 - 1) - V 2 + 1 = 1 - en I ^ = sn -^ i:, because of (3) § 9, so that, in one-third of the time from apse to cusp, two-thirds of the apsidal angle has been described. Similar applications may be cited here of the elliptic function analysis for the same modular angle, 15° or 75° ; thus a clock hand pendulum, starting from I o'clock, will go to III, from III to IX, and from IX to rest at XI, in the same time as the beat of a pendulum of threefold length swinging between V and VII o'clock. Or, with three such pendulums, ticking regularly, one will be starting from I o'clock while the other two are crossing at IX. Or, a step ladder, an equilateral triangle in end elevation, if the cord breaks, will slip down on a smooth floor, and fall flat in the same time as a plummet at the end of a thread of the same length as the side of the triangle will swing through 60°. 10. Numerical applications of § 25, Chapter V, to ju = 12, may be considered here, by a Quadric Transformation of ;u = 6 on p. 127, selecting Complex Multiplication cases. With b the equivalent of y, in Klein-Fricke's Modulfunciionen, and a as in § 25, p. 142, (1) ^ /I _ \' Cl_ ('^"'^Vi) \'"'~'ji) _ (» + 1)' (h - 8) ^» "'' '^' 16(<.+ 1 + 1) "» ' Thus from Weber's Elliptic Functions, p. 499, (3) f=V6, ^,-o'={^2 + \)\ «^=.= (V3 - ^2)(-^^lz_iy, 1 (1 -t- 0^= l(2^/3-|-^/6-|-l), 6=v/6+n/3. (4) J^ = 2^/3, i(l, - 0^)= v2(s/3-f- 1)S j(^2-t- o2) = 15 + 8v/3 (^-o) = 2+^/3, j(i + 0) = V6 + s/2, 5 = 3-h2N/3, 1 2 ..:^(...V8)=y^-3 V3" (5) K' = 2K, 0=^/2-1, i(i-o)'=l, 6^-66^-246-3 = 0, (62 + 3)2_i2(6 + l)2 = o, 6=v/3-^^/2V3, ^_1 + V (n/2 V3- ^3) 1-n/(v'2V3- v/3)' (6) I' = 2^2, i(l,-.^) = 8(./2 + l)3, i(^\+ ^^) = (4s/2 + 5)^ i(l+ 0) =V2-M, i(i- 0) = ^/(2^/2 + 2), 6 = ^6-^^2+1. (7) ^ = 3^2, i(J^- o2)=(^/3^-v2)^o2=(72-l)3(2-./3)^6 = 3(^/3-^^2). (8) ^ = 2./6, I (1, - 0^) = 8 (v2-t-l)^ (2 + ^3)' ( ./ 3 + v'2)l (9) |^ = 3V3, 4..' = (B^2-l)^ -ilz^.^: = illV 4_ (10) r = 4v3, 1(13- o2)'=32v2(v2 -H 1)°(^3 + 1)»(^3 .Hv/2)«. 167 QuiNQUISECTlON. 11. The period rectangle of the (c, V C) relation in §15, p. 132, Chapter V, will show the character of the algebraical motion when the secular term is pt is cancelled by taking Z + P = 0. There are eight regions of c to be considered, with the associated / = ^' ^•^' * , and apsidal angle ^, and the substitution (c, — -j will change/ = f- into f, and/= ^ into f. Writing a for ^ ic j, Kiepert's function L (2) is given in Phil. Trans, p. 262 by (1) Z(2) 24 64c* (7 2a + 1 64 {C^ - lf{c^ - 4c - 1) ~ a\a - 2)' 64 (g^ + l)(a^ - g - 1)^ a\a - 2) Region. I. II. III. IV. V. VI. VII. VIII. L{2r 64 4U' " / 1 \M - Wk'-' 4V/v' / 1 ~ 4A'" 1/1 V - 4/c=k'» * / c imaginary c / ^ \ imaginary v/5-1 2 1 « = 1 IV - = III V « = K = l - 0-108 - x/5 + 2 5 waves 5 cusps h loops 2 36° 18° i 10 waves 10 cusps 10 loops 2 v/5 + 2 II VI K = l K =0 - 3 s/ 10 + 1 3 -1 10 loops 10 cusps 10 waves 1 54° 72° i 5 loops 5 cusps 5 waves 7.4 00 I VII K = VIII K = 1 - n/5-1 2 imaginary * f c imaginary c / * For Quinquisection, in Region I, / = f , c > v^ .5 + 2, and in the algebraical case Withj9 = 0, n = * = |7r(72°), i^\ QL _ zn f K' _ p / - ^1 _ (c + 3) (c^ - 4c - 1) ^ ^ QV" coswAw V 5^2 «3 20c OB _ /c _ / ajg — a;2. — a?! _ 4 , ^ PT QV cos w A w v «2 *3 ' QV ^ '^ ^' tan o) = n/ JJs and so the focal ellipse is constructed for a given c. 168 For example, with (3) c = 5, « = J#, k' = 0-3609 = cos 69°, u, = 74°-5, and we take QL = 64, QV = 200, QT = 2540, OQ = 713, OB = 694, QP = 136, OA = 925, OS = 610, T& = 2570, LQ' = 150. At an inflexion of H, cos =-r^/ = ^^t: = cos 64°' 7. LQ loO To obtain more character in the figure with a larger modulus k, a better value of c is closer to the limit V 5 + 2 = 4*236, say c = 4*25 = V, making k = sin 83°-8 ; and then on the focal ellipse QL _ 29 PT V (11 X 17 X 31) OB _ ^ V 11 x 17 x 31 ^^ QV ~ 5440' QV~ 8 ' OV ~ ^ ^ ~ 64 The same focal ellipse will serve for the associated state in Region V of A more interesting case is obtained from K = 2 K', k = sin 80^"1, as before ; then ... Z(2)2* 1 l±v/(v/5-2V5 + 2) c = 4-272 (I), or - 0-234 (V), ^ = 72° or 36°, cn|-Z' = J- :^cos 36°-2. -+v/(2a + l) But a = - —J: — - makes K' = 2Z, k = sin 9°-9. V 5 + 1 12. In Region II on the focal ellipse, rn ^ OL zn jK' en ^^' ^' '■ OT - sniK' dniK' " ^^ PV _ OW _ /OP /.t-2 Q^-s/0, OT-v/OT=v/^' For a representative figure take c = 2 ; this makes b and L coincide, so that the algebraical figure has cusps, as in the stereoscopic diagram, fig. 86 ; and then .ON 27 40 ± 32 V 10 (^) ^2 = 20 ' *3' ^1 = 100 , 8 - 2v/10 sm w = g = 0-5585 = sin 34°, cos 02 = —3" = cos 66°-8,' cos 63 = sin w = cos 56°, /o^ '2 2 64 ± 19 s/ 10 ' g ,, 35 . „ Z (3) ic V = 128 ' ^"^ = yli, -v = sin 10°, ~ = 1-01 On the focal ellipse take QT as the unit of measurement, making QT = 200 mm • (4) QL = 125, QO = 297, QV = 441, QP = 191, OA = 200 V 10 - 356, OB = 62. Move H from L to P, O-, = 0,0^= 113°, n = l°-2 ; or from L to T 0, = 180° (),= - 68°, n = - P-2 ; checked on fig. 87. ' ' ' With the same focal ellipse, and = - J, / = ;^, H = S- = ■; , (54°) ^nd the curves will make three circuits to close. But the ten-cusp stereoscopic figure with apsidal angle 54° is obtained from X2 (c + S)(- - c' + 4c + 1) + OA OT = vc. 20c Tb TQ- (o- ■ mo 4c 1) 169 For a smaller modular angle still, and the motion between closer limits, take p = 1-5 = I in Region II, with f = \, and the associated c. = - § in Region VI, / - s • Then with c = 1'5, k- = sin 3°, ;^ = 1, ^^, . QL _ 57 QP _ -' ^^ ^' QT ~ 80' QT ~ ■"• Take QT = 240 mm, QL = 171, LT = 69, LP = 67-7, OQ = 244, QP = 239, QV = 249, OA =342, OB = 17-3, OS = 341-6 ; as in fig. . At 0' where R = 0, TO' = 6, O'P = 0-7 mm. Moving H on the tangent at P will give a series of curves within close limits, and when these are looped we see the pseudo-regular precession magnified. 13. For a larger modulus in Region II or VI, select K = K' , k = sin 45° ; and then (1) a = i(N/5 + l),c = 2cos36±2cosl8° = 3-52(II),or-0-28(VI), i' = 18''or,54°. : Other values of a are J ( - s/ 5 + 1) in IV and VIII, a = - V 5 + 1 in III and VII, and imaginary results ; but a = V 5 + 1 will give real results in I and V. To a larger modulus still in Region II, take Z = Z' v/ 5, /c = sin 83°, and we find (2) 2a + 1 = n/ 5, c = 2 sin 18° ± 2 sin 36°, in II and V- Expressed as a section value of w the period of the elliptic integral f C'Mc, this ioY K = K' ^ h IS, G (i w), while c (^ a>) is the c ior K = K ; and c (i w), c (f w) a boundary values of two Regions. Another numerical illustration is obtained from (3) Z=^' s/10,orr v/f, ,c' = (s/2 - l)2(^/10 - 3) =sinl°-6, or ( v/ 2 + 1)M n/ 10 - 3) = sin 71°, a = V 5 + 2 ; (4) K = K' s/lh, or K ^ |, /ck' = (sin 18°)* or (sin 54)*, 2 a = (±^-±l)\ c are (5) = / V5 -1^ '■ = ( 12 /x/5 - 1\^ 2 / ' \ 2 Other numerical values can be t^ken from Weber's list, and thrown into a different shape ; useful too in the construction of a numerical case of u = 20, in § 26, Chapter V; where in Phil. Trans., § 42, we put (6) 1 ^-, = a + 1 -i = 2/3+ 1, a -± = 2(3, a a' a _ Co _ - (/3'- P - 1) V (i3^ -^ 1) - V ( 2/3 f 1) C C\ 1 J. 2 v/(2/3 + 1) 4k —4k or ('') C (1 - kY "' (1 + kY and a, c for /i = 10 in § 11 are connected with this |3, a by (8) 2a + i = ^^ + ^--^y + ^"^^ + ^) (9) (10) c = a-h'' v/(2|3 + l)-l /3 1 - a ~ /3 + 1 - V;,i3^+ 1) In Kiepert's notation, Math. Ann. 32, p. 107. <3 + 1 + N/(j3^+ 1) n/(2/3 + 1) + 1 ■ a = - 2 ^3, + 1 = v/(/3^ + 1)„+ N/(2i3 + 1) V(/3^ + 1) -n/(2j3 + 1) T l3 - 2 J' (11) £i = 1 - 4 ?3 = 2 a + 1, ^1 = 2/3 + 1, (12) Z (2)2* = 16 (^ - k)'= 256 (7^ ((7^ + 1), ^(2)^* + 64 = 64 (^C^ + 1)2^ (13) (14) (15) (16) L (5)« = (2« + lHa-2)- ^ ^(lO)«=(i^tJ^^-i), _ 16V(2)3 + 1) [-(/3^ - /3 - l)v/(/3^ + 1) - s/(2^ +-1)1 iv (.4; - ■ jS^ ()3 - 2) ' Z (20)« ^ 4 (2/3 + 1)^' ()3 - 2) L (2)« /33 \ 28670 170 Thus from Weber's list <17) ^ = 2./ 10, h{-j-^+ VK) = (v2 + ir(s/10 + 3), ^M _ ^k)2=2(2 + ly(^/5 +2)2(^10 + 3), (i= ^5 + 2. s/ 5 + 1\2 e--) = (^^^^7(--)' <3=i(^)- 1 <19) r = ^^' * (-. -^ ^'^) =- (V5.1)HV5-l) i/l ,\ /V5 + 1V 1 /I \ __ 128 v/2 3_ / v/5,+ I v 2a + 1 = (V5 + 1) (Vo - 1)'' and so on, so that a for K'/K = Vn appears so far the same as /3 for K'/K = 2Vn. - Thus from n = 5, 2 a + 1 = v/ 5 for Z' = K-^ 5 we deduce 2/3 + 1 = v/ 5. So also forn = 1, Z = if' and 2K', a = ^ (^ o + 1), jS = V5 + 1. Seven Section. 14. A numerical case can be constructed by taking 2m = v/ 2 in §17, p. 135, Chap. V, making «2 = 0"359, x^ = 0"454, Xi = — 0'791, 2„'2 _ 4. Xs-X2. X2-X1 _ 64a^ (m-a){a-m + l) _ [ ■^in+ s/2)-v'(n/2- 1)"|* (1) 4kV^ = {x^-x^f 1-8(1 -2m) a (2) 2kk = ^i|±^|l_^i^|_Al, = sin 20, tan(i.+ e) = (4^/2 + 5)*= tan 61°, Q = 16°, the modular angle (3) tn fK = /^^' = tan 56° = tn | K', f = f, N' X2 on comparison with Legendre's Table IX., at modular angle 74° ; and the algebraical curve of the axle will make three circuits to close, with an apsidal angle 38° *. A conjecture was not verified that with this modulus, tf> = /s-, as the modular angle would then have to be about 12°. For a numerical application of /n = 28 in §27, Chap. V, take K= 2K', or Weber's results (4) ^ = 2v/7, J(^-k) = 2v/2(3 + V7)3 *(-7-.-^0 = *(^+ ^')^ H-7-.^ V.) = 2^2(3+./7), K= (2^2 - v/7)2(^2- 1)* (5) - = 2^14, ^{-j-K^ s/k) = [2.72 + 2+ ^(8v'2 + 11)] [^(8^/2 + 11)+ ^(8^2 + 10)] = {^2 + 1)2[2 + v(2./2 + 1)] [./(2./2 + 1) + ^2./(2./2 - 1)] not given by Weber, but derived from (6) I = s/ 14, i (1 - k) = [^.^^iL ± s/2)+ ./(^2-l) p^ j(l + ^)= ^2(^2 + 1)2[>/2^(2n/2 + 1) +./7]. 171 Nutation in the Axle of the Earth. 15. A numerical example, on as large a scale as we knoAV, of a slight tremor super- posed on a state of steady gyroscopic motion, is given in the Precession and Nutation of the axle of the Earth. The principal part of Precession is found to be due to the Moon, Lunar Precession being more than double the Solar, or 34" to 16" in the 50" of total annual Precession. This is in the same ratio as their disturbing power on terrestrial gravity, or as the Tide Producing Power ; that is as the (density of the Moon) (sine D 's semi-diameter)^ to the (density of the Sun) (sine 0's semi-diameter)^ ; practically in the ratio of the density of the Moon and Sun, 0*58 and 0:25 of the mean density of the Earth, their angular diameter being so nearly the same, 31' and 32', about half a degree {Principia, 1713, p. 430, 438). So too the Nutation is due chiefly to the disturbance of the Moon, of which we suppose the matter distributed in a circular orbit round the Earth, or else concentrated into two quarters, at each pole of the Moon's orbit at the mean distance. Employing Poinsot's method of treating the axis of resultant A.M. of the Earth as practically coincident mth the axis of figure {Connaissance des temps, 1858; Quarterly Journal of Mathematics, Q.J.M. 1877) the components of velocity of the vector of A.M.,' in precession and nutation, are (1) Ciesin/J, andC'sf; and these are to be equated to the corhponents of the couple. Lunar and Solar. The gravity potential V of the Earth at a point P a long way off on the prolongation of its axis OCZ, at a distance OP = r from the centre 0, is given by the volume integral 2 TT y p X d a; d z (2) V = \\ PQ where jO denotes the density, in g/cm^, and -y the gravitation constant, y = 666 x 10 ^* m C.Gr.S. units ; and (3) PQ^ ={r-zy + cc' = r'{l-2^ + ^^) To the desired approximation, w m'i'-'^-^y-'^T-^^ so that (5) V.li-y^, A,- C denoting the M.I. of the Earth about an equatorial diameter, and the polar axis, (6) C = f f 2npxdxdz . x^, A = ff 2Trpxdxdz{^x^ f 2^), ff ^irpxdxdz . z = 0, being the C.G. of the Earth. But when OP makes an angle Q with the polar axis of symmetry, where Q^ denotes the zonal harmonic of the second order, ^2=1 cos^ ^ — i- The potential at a distant point P is the same then as if the Earth was condensed in a ring of radius (8) V ^E ' about a^o of the radius of the Earth, treated as homogeneous ; and on the usual estimate (Thomson and Tait, §§ 108, 828) derived also from the Figure of the Earth, (9) 1 - ^ = 0-00327, making ^^ = 306, ^j4rj = 305, to three significant figures, against 305 and 304, the numbers adopted in Kreisel Theorie, p. 663. K.. S. Ball's Elements of Astronomy, and C. A. Young's Manual of Astronorny have been consulted for other numerical constants. 28570 ^ 2 172 If OP is directed to the Sun, yjQ\ COS = sin w sin i/-, ^ denoting the Sun's R.A. from ^, and . the inclination obliquity of th^ ecliptic ; and in one year, while ^ increases by 2., the average^ value of sin ^ ^"^^''^J.^^ and"sun /tn is and * so that the average mutual potential energy W of the Earth ii and bun ^, in which the matter of the Sun may be supposed distributed uniformly as a circular ring of radius r, is given by (U) ^ = 7|:?_I^^i^^(|sin^.-J), and the gravitation couple about Ot, tending to decrease w, is n9^ _dW^S_yS_{C^^ sin 0, COS.. This couple, changed in sign, is the same as if the Sun was supposed cut in two, and half of one half placed at each pole of the ecliptic, at the mean distance r of the Sun. The angular velocity ^l of the mean Solar Precession is then given by (13) Ci? ^ sin (u = 2 ^—^z sin w cos w, m - g Er^ C ' and by Kepler's Third Law, if n denotes the mean angular velocity of the Sun round the Earth, (14) nV = y {S + E) =j S [1 + g), in which E/S may be neglected as insensible, since S/E = 324,000. Thus in (13) B n^ C - A r? cos <^ (15) m = 2:R~C~ ^^'''^ ^04^- Denoting by iV the number of seconds of angle in the mean annual Solar Precession (16) 360x6Ux60 = n = 2^-^^"^-' where - = SbGj, the number of sidereal days in the year (Evdoxus) ; and working to three significant figures, with w = 23°-5, sin w = 0*4, cos w = 0*917, , (17) N= 1,296,000 X I X 3e6i°x'^306 = ^^'^^ ^os w = 15-9, nearly 16". A Slide Rule should be employed in these calculations as rejecting automatically more than three significant figures ; this is the extent of the accuracy of the data of the problem, and to go beyond would lead into the dishonest decimal, so called. A four figure logarithm would go into the first dishonest decimal, which must therefore be rejected in the result ; although calculated as a last figure, or it would be difiicult to know where to stop. 16. So al-so in Lunar Precession, the Moon may be supposed distributed in matter over a circular orbit of one lunation, inclined at an angle i with the ecliptic ; or else condensed into two particles, ^ M, at each pole of the lunar orbit at the mean distance ri ; and then if ^ denotes the longitude of the Moon's ascending node, (1) cos 6 = cos w cos z" + sin (o sin i sin ii , and the mean value of cos^ 9 in one revolution of the node, during which Si increases by 2 TT, is (2) cos^ 0) cos^ i + J sin^ w sin^ i = (| cos^ i - J) cos^ w + J sin^ i. The mean potential energy of the Earth and Moon is then (3) ^1 = ^ - 7^^-^ (I [(-I cos^ i - J) cos« . + J sin'' i] - Jj and (4) "^7 " ^" "^ y3 (I cos^ I - J) COS o) sin to 173 and the Mean Lunar Precession is given, as before in (15) (16), by (5) M = _SyM C - A 2 Rr ' C ^ ^°^" * ^ cos w = 2 7e • ;e ■ — c~ ^^ * ~ ^^ ^°^ '^' employing Kepler's Third Law in the form (6) n,'r,^ = jiE + M)=yM[^+l). Thus the ratio X of Lunar and Solar Precession is given by (7^ X = — 1 = ^ = >^i ^ J cos^ i - 1 . iV ^ n'' ^ + 1 M in which we may take, from Young or Ball's Astronomy, (8) ^ = 81-5 ; I cos^ « - i = 0-988, with ? = 5" 9 ; w 366- J = -g^- = 13"4, the average sidereal revolutions in one year ; and then (9) ^ = (l-'^'g)' X 0'988 _ 2.J25 j^r = 34" ^i^h N = 16", ^ ^ N 82-0 ' making iV^j + iN" = 50" ; and this agrees closely with observation, the number in the Nautical Almanack being 50"'4. But working backward from the observed value of the Mean Annual Precession, and assuming -=^ = 81'5 about, we arrive very nearly at the estimate in (9) §15 of The weight of the Moon is the uncertain factor still, and it was not known with any accuracy in Newton's day, as he takes £J/M = 40 in round numbers, 39*371 in the Principia, p. 430, and so doubles the weight of the Moon, and the ratio of gravity disturbing power of the Moon and Sun, which he takes as X = 4*48 15 to 1. He is obliged then to share the total Precession in this ratio, giving 9" 07"' 20'^ to the Sun and 40" 52"* 52'^ to the Moon, to obtain the result 50" 00*" 12", which agrees with observation (Principia, p. 438). 17. Arrest the diurnal rotation R, and the axle of the Earth would oscillate, under the influence of the solar and lunar rings of mean attraction, according to the law of a quadrantal pendulum as a magnetic Compass needle, (1) A^^ + ^a + -i) y '-'^ (^ - ^) sin cos = 0, or ^ + / sin cos 6 =0, ,„, o 3 /^ , ^\ y S (C — A}) C Ru R^ A R ^ ^ '^ 2 Ar^ A cos w p^ C fJL (d f)\^ _ j = p^ (sin^ w — sin^ 6), sin 6 = sin w sn pt, cos = dn pt. Treating the oscillation as of small extent, the period is 27r //A2ir 27r \ in which we may take A = 0, and cos w = 1 ; and with the year as unit of time, (6) 2^_ /26,000_..42 D V 3664^ 366i 17-4 The Free Eulerian Precession, and Variation of Latitude. 18. Left entirely to itself in space as a rigid body of revolution, the axis OK oi resultant A.M. of the Earth would be fixed in direction, and the axis OC of figure and 01 of instantaneous angular velocity would lie in a plane through OK (fig. 94), revolving with a precessional angular velocity n, such that, if IOC, KO(j are denoted by a, 6, the component angular velocity about the diameter OA. is (1) M tan a = fi sin 6. The component A.M. about OA is (2) AE tan a = CR tan 0, tan a = ^ tan B ; and OK describes a cone- round OC through the Earth with relative angular velocity (3) i?-,cose = i?-i?*^; = i?(l-|)= -3I; thus the period of revolution of K in the Earth is 305 sidereal days, say 304 solar days or 305 10 months ; but K revolves round C in -^ttt^ of a day. (Thomson and Tait, Nat. Phil., § 108 ; Maxwell, Scientific Papers, I., p. 260.). But Chandler's observations have shown that the period of the variation of latitude' is more like 14 months ; the discrepancy is attributed to the elastic yielding of the Earth.^ This can be illustrated experimentally with the Maxwell top, adjusting the screws SO- as to bring the C. G. to 0, and to make C and A very nearly equal, C slightly greater than A. Treated analytically by Euler's equations of motion of §36, Chapter I., and denoting the components of angular velocity about principal axes fixed in the body by P, Q, i?, {C-A)PR = Q, C~=0;. (4) "^ dt + (C - A) QR -- = 0, dt so that R is constant, and (5) 1 d^P p de " 1 d^-Q Q df = -( dt = - (2 - 1) i^' = - p'R' (6) P = i? tan |3 cos {pt + t), Q = R tan (3 sin (pt + t), and the period is 27r 2,r J A on. •-, 1 n (7) — = -^ c - A *^"^ C - A ^ ^^ sidereal days. And generally, a body of revolution makes (C — A)/A wobbles per turn, or A/{C — A) turns per wobble ; so that with C = A, the body has no sort of dynamical backbone as required for instance in Diabolo. Lunar Nutation. . 19. Let z in fig.. 95 denote the pole of the ecliptic, and Z the pole of MN, the Moon's orbit at inclination i with the ecliptic, N the ascending node on the equator, M on the ecliptic ; the figure being looked at as the concave side of the celestial sphere,' as a star chart. The Moon is supposed distributed over a mean circular orbit MN, in its mean sidereal revolution of 27-3 days ; or else replaced by two quarter moons at the pole K and the antipole at the mean distance r, ■ and then, according to the preceding explanation, the Earth s axis 00 will follow the ascending node N on the equator with angular velooity /.N 3 n^ 1 C - A ^^1 2 ^ ■ E ■ — C~~ ^^^ ^^^ ^°® ^^' M "^ ^ and the components in j^recession and nutation ^^.ill'be /ON • fl^i' i^ _ 3 V 1 C - A (2) sm « ^^ ^, dt - 2R J : '~C~ ^"-"^ ^I^ cos CK (cos z CK, sin z CK). il/^ 175 In our approximation to the result we may put (3) 'tM = pt, where p is the angular velocity of regression of the Moon's node, CzK= tt — pt ; and in the spherical triangle C^K (4) Gz = 9, CK = /, zK = i, C^K = TT - pt (5) cos / = cos i cos 6 — sin i sin 6 cos pt /^N „^^ T „^r, ^n\r cos i — cos /cos 6 ••«••/> v (6) sm 1 cos ^CK = -. — = cos t sm 6 + sm i cos 6 cos p<, sin sin / sin ^■CK = sin i sin pt, (7) sin / cos / cos ^CK = (cos i cos d — sin / sin^0 cos pt) (cos i sin d + sin z cos 6 cos |?^) = sin 6 cos (cos^ i — sin^ f cos^ pt) + sin « cos i cos 20 cos ^z = sin cos (I cos^ i - J) + J sin 2z cos 20 cos p^ - J sin cos sin^ i cos 2/)f (8) sin / cos / sin ^^CK = sin i cos i cos sin p^ — J sin^ i sin sin 2p^. Then if ^j denotes the mean Lunar Precession, the constant term in — -^ in (2) (7), as before in (5) § 16, with replaced by its mean value w ; and A0, A\P denoting the radians of Lunar Nutation in celestial latitude and longitude, (10) — 5 — = — ^i =- (sin 2 cos i cos w sin pt — h sin^ i sin w sin 2pt) at I cos'' t — i 1 i r J (11) sin w / " = - .... (^ sin 22 cos 2^ cos p^ — J ^^^ *' ^i^ '^ <^os w sin 2^^) c^if _ Ml 1 (12) A0 = — — ^ g-; y (— Jsin2z cos w cos pt + |; sin^ i sin w cos 2^?^) 7/ "h" COS 6 "^ n (13) sin w Ai^ = — .... (i sin 2/ cos 2w sin pt — \ sin^ z sin w cos w sin 2^^). If Z denotes the Sun's celestial longitude, a the longitude of the Moon's ascending node M, we put L = nt, pt = 27r — q, (14) A0 = a cos a — fi cos 2si + y cos 2L, (15) A^ = — a' sin Si + /3' sin 2 g^ — y' sin 2Z ; (16) ^ = i^lllll^i^J^ = 1 tan i tan w = 0-0097, say O'Ol ; ^ ^ a J sin 2? cos w * -^ /I T\ i3' i sin^ 2 sin w cos OJ ,, ., „ nnn^ A.m (17) e, = ^ ^ ^.^ ^ ■ ^^^ ^^ = i tan z tan 2. = 0-0116, say O'Ol ; ,io\ /^ i sin 2? , u A sin 2? cos 2w o (18) a. = " ■ ^ o . 1, ". = - -3-^ r r- -■ = i^^ o- p 2 cos^ i — 2 '■ pi COS t — ^ sm w cos u) 2^ tan 2a» in which 2irtii/p is the Lunar Precession in a period of revolution of the Moon's node, say 18^ years ; so that, with N^ = 34", i = 5° 9', 2i = 10°-3, 2w = 47°, (1^) . « = 360 f 60^x 60 - pte-|?l = ^^" ^""'^l' «' = ^^^ ^^"'^- in practical agreement with the Nautical Almanack. The terms y cos 2Z, y' sin 2L are the components of annual Solar Nutation ; they are obtained from the components o£ the velocity of C, which follows a point Q on the equator, 90° behind S the Sun, with angular velocity (20) 3 J ^~ ^ sin CS cos CS, so that in the spherical triangle tSN of fig. 96, where CSN is perpendicular to the equator Q'Y'N, (21) ^, sin ^ = - 3 J^^^ cos SN sin SN (cos Nt, sin Nt) 176 in which (22) cos SN sin SN cos N-v = sin B sin S-v cos Sif, cos SN sin SN sin N-v = sin 9 cos sin^ Sir, with Sif = L = nt; and then (23) ^ = - 3^^ ^7-,^ sin 6 sin nt cos ra/, ^ dt R C (24) sin ^'^ = - 3^' -^^- sin cos sin^ nt. cut tx o The mean Solar Precession ^ is the constant term in - -^, the same as before in (13) § 15, with = (0, the mean value ; and for the components of annual Solar Nutation (25) ^ = - 2/x sin w sin nt cos 'nt, A0 = ^ sin a> cos 2Z = 7 cos 2Z, dt ^n (26) sin w -T^= - ^ cos w cos 2?2r, Ai/- = - iL cot &> sin 2i = - -y'sin 2Z, (27) 7 = -T- sin w = sin 0". 508, y = A cot w = sin 1". 17, \ • ' 2n ' zn in fair agreement with the Nautical Almanack. Gyeoscopic Motion of the Moon's Node. 20. The regression p oE M, the Moon's ascending node on the ecliptic, may be calculated as due to the influence of the Sun, when the mean Moon is replaced by a rigid' ring of its matter and radius the mean distance rj, revolving in the same sidereal period, 27-^ days, with angular velocity n^. Here for the ring, (7 = 2 J., C'/ (C - .4) = 2, and (1) Cwi p sin « = g '^L-^{C — A) sin i cos i, fa\ 3 n^ . n 4 ni . 4 X 13'4 , „ (2) ^ = _ cos «, - = ^ —1 sec ^ = -^ ^ = 18, -^ 4 Ml ' p 3 n 3 cos ^ ' the number of years in a sidereal revolution of the node M, on these assumptions, and the Sun replaced by a ring. But the time observed is about half a year longer, and the discrepancy was explained in the third edition of the Principia, lib. Ill, prop. XXXIII in a note by Machin and Pemberton. Although the Moon moves so fast that it may be replaced by a ring, the relative" motion of the Sun and node must be taken into account ; and it is not accurate enough to substitute a ring for the mean Sun. -, The mean relative motion of the Sun and node, n + p', must be taken as the G.M. of its maximum n + 2p in their quadrature and minimum n is Syzygy, not the A.M." n + p, ao that (3) n+p'= V{n^ + 2np), ^, = ||^v(l + f) p' ~ 2p V , X ^ - , + 1 n P = p^ *-£••• =18-5. This follows from the equation for the Sun S, moving uniformly, actine on the lunar ring, as in (24) §1J^, with MS = Z - Si, and dL/dt = n, -^ & . .dSi 3n2 . (4) sm « -^ = - 2n ^"^ * ^°^ * ^^"^ MS = - 2p sin i sin^ MS, d (5) -j^{L - Si) = n + p - p cos 2(Z- - Si)^ similar to;the relation connecting true anomaly, 2(Z - Si), with excentric anomaly (6) tan s/ (n' + 2np)t = y~^ tan (L - Si), making the time taken by the Sun to meet the ascending node again (7^ ?I or ^ n „ ^ ' ^ {n' + ^npY vwv2^ = ,rTy ""^^ y^^'- about^26,00a''^°"'^'"^' '^''' '' """^ ^^""'"^ consideration in Precession, as n/p here is 177 Then if A$^ denotes the periodic term in Si, (8) , L - Si = {n + p') t - Aii, and from (6) (d) tanA« = j)'tan (Z - Si) ^ p' sin 2 (Z - Si) ^ ^ n + {n + p') tan 2 (Z - S) 2n + p - p cos 2 (Z - a)' or practically, to the approximation required /,•^^ A r, ^sin2(Z-s) sin 2 (Z- Si) . ^, . ^ , ^ (10) AS = 2^ ^ ^ = ^p ^ = sin 90' sin 2 (Z - Si). Y For the simultaneous variation Az of the inclination i of the Moon's orbit, (11) %= - 1 - sin i sin 2 (Z - R), M = % - f ^^^ \ cos 2 (Z - ft) = sin 9' cos 2 (Z — a), and with a mean i of 5° 9' in the octant, the inclination oscillates between 5° in quadrature to 5° 18' in Syzygy of Sun and node. An equation similar to (5), where R is replaced by ct, is investigated in the Lunar Theory, which gives ts the longitude of the Moon's perigee. Observation shows that the, apse line advances over 3° in one lunation, and makes the circuit in a little less than nine years ; and Newton pointed out, in the Principia, p. 126, the effect of a small central disturbing force on an orbit nearly circular in making a rapid progression of the apse, as is seen with a Spherical Pendulum, a plummet at the end of a thread. But the transversal distributing force must not be ignored in the case of the Moon,, as it is found to produce nearly half the motion of the apse line ; and so it does not seem possible to calculate the motion of the apse in the manner above, where the mean Sun is replaced by a ring, which disturbs the elliptic orbit of the Moon round the Earth. 21. In Newton's treatment of Precession in the Principia, p. 432, the equatorial belt of matter round the sphere of the Earth is assimilated to a chain or stream of free particles, and the regression of the node is investigated in the same manner as for the Moon ; thence the Precession is inferred when the spherical part is compelled to follow the equatorial belt. A similar method is followed by Maxwell in his investigation on Saturn's rings- {Collected Works I, p. 286) ; the movement of a ring as a whole is the same as if rigid, but stability requires the ring to be shattered into fragments as small independent satelhtes ; a solid ring would be unstable, and come into contact with the body of Saturn. 22. Following Newton in working with the density of the Sun and Moon, the idea may be employed in the interpretation by Mechanical Similitude of Kepler's Third Law, which may be restated in the Problem of Two Bodies, the Sun and Earth, (1) n' = j^= y ^3 = -I '^ 7 [(density of Sun) (sine Sun's semi-diameter)^ + (density of Earth) (sine Sun's parallax)'] Thus a model of this system of Two Bodies, moving in free space, an Orrery of the Sun and Earth, made to a geometrical scale would preserve the similitude if made of the same material and density ; or if the material was altered, preserving the density ratio, the period would be inversely as the square root of the density. The statement is true equally for a Planetary System, of three or more bodies, provided the density ratio is adhered to in the model (Principia, p. 294, lib. 11. , prop. XXXIL) 28570 178 CHAPTER VII. The Spherical Pendulum. 1. When H is placed at Q on the focal ellipse, G,h,L,l = ; and in a spherical top, (1) siA^ ^^ = 2A', sin^ ^ § = 2^' *=°' ^' (2) p2^ = JFA' = W|) = n. CP. p, ?; = in^sinfl = w. CP, so that H describes a central orbit, a Lame curve of the first order, and P describes the polar reciprocal, not a central orbit. But if H is at another point where PQ meets the pedal of the focal ellipse with respect to 0, and HP' is the other tangent, as in figs. 97 and 98, S' = 0, G\ h', L' = 0, equivalent to an interchange of (j> and \p, h and h' ; (3) sin^ ep = 2h cos e, sin^ 6^ = 2h, sin4e^H^ + ^) = _ eos^ie^i-%^ = h, ^ dt ''at (4) p^^ = p2 A - 1 El% so that P describes a central orbit round C ; but the (p, ct) curve of _H is no longer a central orbit, or is made so if viewed stroboscopically by conversion into a Lame curve of the second order {p, zr — ht). With G' = CR = 0, either C = and the body reduces to a rod, or else R = Q, and there is no spin of the wheel, and the motion is the same as of a Spherical Pendulum, a plummet Pi at the end of a thread OPi, whirled round in space about a fixed point 0, shown in plan and elevation in fig. 99, A and B. The thread OPi will move in the same way if attached at Pi, the C.G. of any body, and the body can have a Poinsot motion about the C.G. at the same time. The (j> motion when h = is thus a Spherical Pendulum \p motion when h' = ; and it is convenient to consider these two states of motion as associated ; and denoting AOH on fig. 97, 98 by w' or am fK', and OH the perpendicular on the tangent at P', to draw the other tangent HP to the focal ellipse, and OQ the perpendicular on it, to denote AOQ by w or am/K', thus giving precedence to the / of the Spherical Pendulum, but deriving it on the diagram from/' and H, where S' = 0. On the focal ellipse in fig. 97, 98, (5) 03 = (o' ~ w, OQ = aAw, OH or OQ' = aAco', cos(. ^.) = -Q_ = _„ cose3 = 5^; and similarly we find, with SHS' = tt — 02) HS/HS' = «"■, (6) cos 03 = - S2i^, = - ^J^-, ch 01 = $L!1 = ^WL. cos w cn/" sin 0) sn/' In the general case, where H is not restricted to lie on the pedal (7) -^ = 2 ch J01 sin 103 sin ^6,, ^-I^ = 2 sh J0i cos ^0^ cos jOg, as in (1) §19, Chapter IV ; so that in the Spherical Pendulum, with 8' = 0, (8) th 101 = tan J03 tan 103, exp 0i = ^5Li(^2_- ^3) . cos ^ (02 + 03) ' and drawing the perpendiculars SZ, S'Z' on HP', HZ = HZ'. SHZ = i (0^ + OA. S'HZ' = U^2- 63), ,1 r0„ - fl.^ = __, . ,,,^ ^„^ ^ „^^ ^ __^^ (9) COS 1 (02 - 03) = ||, . cos J (02 + 03) = HZ 179 showing the geometrical interpretation of (8) ; where S, 9i, 9^ are positive on fig. 97, and ^2 < Jtt ; on fig. 98, B, flj, flg are negative, and 62 > Jtt. The relation in (8) is equivalent to I — Zl i. + Z2 i^ + Z^ Zg + Z2 ^2^ ^'2 ^ ^3^ - ^2^ 1 + 2Z2Zi + ^2^ 1 + 222^3 + Zs^ With A' = for a Spherical Pendulum, (11) ^= (1 ^- z^) (H + z)~2^2 = a- ^') {E + z) -2^,z' ft ft, = - z^ - Hz^ + z + E (12) F=H = JS +2^= - z,-Z2-z„ E = z, z^ z, ; (13) sn^/' = ~^~/^ = ^^J^, .'2 snV = ^^^-^^ •3^2 - •S^l Z2 - Zi' J Z^- Z^ .'^snV' + 'c^ = -^, .'2snV-'c^= -^^, .'^sinV^'-l-'c'^^-^^; (14) ^3, ^2, -2^1 = {x!^ sn^/' + k2, k'2 sn2 /' - k-^, ,.-'2 snV' - 1 - /c'^) - A and substituting in (10), (15) 2)2 ^ ;c* + 2 (1 + k'2) k'2 sn^ /' - 3 k' sn*/' = r^ t ^y » ^2 *]. *2 — ^1 snV (1 + .'2 - .'2 sn^/•T ^'^ -^ ~ >c* ^ 2 (1 + ,c'2) ,c'^ snV - -6 k' sn* /'' '2 ~ *]. . . as in (6) ; so that (17) ..2 , ^ sny (1 + .'2 - .'2 sn^/•T "^ ^ en / - 1 + 2 (2 k'2 - 1) /c''' cnV' - 3 k' cnV'' nq^ dnV (2-/2-dn^/7 ^ ^ an / - ^'4 + 2 (2 - yc'^) dnV' - 3 dn*/" On fig. 97, 98, with K the midpoint of PP' on OH, sin ID sin J . . , Aw , —: 1- — - — r sin (D + sin w — r (20) tan w = , ^, = K^ ; = K^ — 5 ' a; + X cos w cos w , Aw — r 1 -— r cos w + COS w ; Aw , Aw Aw /oi \ Aw K^ sin w COS w' — COS w sin w' „ / / n i^i) -r-7 = To— = ; y = cos 6, = cos (w -w w) Aw k'' sin w cos w = \ / ^QQx tan w _ 1 + k'^cos^w' _ 1 + K-'^ + tan^ w' _^ 2 — k^ + tan^ w' • ^tan w' ~ K^ - /c'2 sin^ w' ~ 1 - /c'^ + (1 - 2^'^) tan^ w' ~ ^^ + (2^^ - l)tan'^w" leading back to the relations in (17) (18) (19). Writing s, c, d for (sn, en, dn) f K' , the co-ordinates of H and P are (23) a cd, a sd, and a -^-^ = " t -5 ^-jj « k^ — ~ = "'' — — UK dn/ rf K^ + k' s^' dn/ 5 1 - /c'2 c^' Ll TTS2 HS'2 (24) ~ = ',, =k' + 2(1 + k'') >c'V - 3/c'* 5^ a (J A. ES' . HS'2 = OS* + 2 (0A2 + 0B)2 OQ^ - 30Q* (25) cos 03, cos 02, Ch0i = p ((c'^S^ + k% k' S^ ~ k\ 1 + /c'2 - k'^s^) (26) sin 03, sin 02, sh 0i = ^ ( - 2/c'2 sc, 2k' sd, 2k cd). 28570 Z 2 180 2 Given 63 and 6^ of a Spherical Pendulum, as in figs. 97, 98, make SHS' = tt - 63, SHP = S'HP' = i (^3 - ^2)' P^P' = ^3 ; draw HO at right angles to HP', and take an. arbitrary to give the scale of the diagram of the focal ellipse. Since the areas POH, P'OH are equal, (1) OQ . HP = OH . HP', 1^' = ^ = cos 03, and, similarly, HT' HV (2) ^ = cos 02, HV " '^^ ^'' Reinstating /i and /a, where/ = /, - /i,/' = 2 - /^ - /i, in § 9, Chapter III, p. 56, n^ , - dn (.A - /i) , _ en (f, - /O _ sn(A-/i) (^^ '' - dn if, + /O' "^-cn(A+/x)' ^ sn(y2+/0' (4) ch 01 = 1 + cos 02 cos 03 g^ ^-^^^ ^^g g^ ^ ^.Qg Q^ -g positive, 02 + 03 < t, ^ ^ ' cos 02 + cos 03 ,.^ / sin0tZ^ \^_ sin^ 02 sin^ 03 ^ '' \ de ) (cos 03 - cos 0) (cos - cos 02) [ (cos 02 + COS 03) COS + 1 + cos 02 COS 03]. • «^ ^^"^ " dt ^/ (2Z) dm hz z-z n m that X (revolves ) ^^^^o^ding as z, is ( P^gf^^^) ; that is according as H is above Q, and G, h, 8 are positive ; or between P and T, when the H curve is looped, and G, h, 8 are negative ; where the thread is horizontal, z = 0, and ro- is stationary. With H on the pedal, as in the Spherical Pendulum, the relation (8) §1, with (35) (36) (37) §15, Chapter IV, p. 97, becomes (7) si Ci dy = -^3, snf.K'cnf.K' dn f^K' = sn {f,K' + Ki) en {f^K' + Ki) dn {f,K' + Ki) the condition which occurs in Lame's equation of the second order ; and writing kz k''= k, V= y, ^2 = Y, (8) .V(2/ - 1)(% - 1) - z{z - \){kz - 1) = 0, (2/ - ^){Kf + y^ + z") - (1 + k){y + ^) + 1] = 0. /q^ , _ „ or 1 + ^ - % + V [(1 - lcy+ 2(1 + k)ky - SFy^ ,^^ v^i'c'V + {d, + k'cY] - n/[/c'V + {d, - k'cY] ^ 2/c'ci(a?i Draw the H and P curve to a scale such that k is twice the length a of the thread OPi ; (10) 0H^ = 2anH + .), CH^ = 2a^ (E + ^), ^ = sin^ - 0^^ CP. OK = OC. OPi ; and if V denotes the velocity of P, (11) V^ = n' [2z' (H + z) + 4|] = n^ (QW cos^ + OC^) = n' (OK^ + HK'^ cos^" 0) (12) V3 = n. OH3, V2 = ». OH2, V3 sin 03 = V, sin 0. 181 If CY is the perpendicular on the tangent at P, cos -p . ^„ (13) tan PC Y = — = — = sm 6 cos 9 y^^ = cos'' 6 ^^^ = cos- tan x ; • nClip An at \J\j biv Sill (/ — = — and in (14) §6, Chapter IV, page 86, (14) A' = 0, Qh = 2R, ^ + \ + h = 2, A + B + C = 2D, ^ a c . for the rolling quadric which generates the herpolhode of H, while, with 8' = 0, the associated rolling quadric is a cone ; and then the condition in (14) may be written, with proper attention to sign on the diagram. . A B C _ HQHQHQ_ (15) D^D^D-"^' HV + HT"^HP~''' and in fig. 97, (16) HQ = 8, HV = S - QV, HT = g + QT, HP = g + QP (17) S3 + (- QT . QP + QP . QV + QV . QT) 8 + 2 QV . QP . QT = 0, which, with the relation, in (15) §14, Chapter IV, p. 95, reduces to (18) |! + ^-L|^±^ I + S^^i^^Lflfa) ^ 0, D^AL^m = ^, as before in (9) §13, Chapter III, p. 59 ; or, in the notation of Weierstrass, (19) U + SZp?' + i^'v = 0, so that a cubic equation is required for the determination of L when v, w, and / is given. But ill terms of/', w', /OuN JL C KJLL sin t/o " f> '2 ^' £' J -0 (20) -^= ^ = ^^ = - p2.^ sn/ en/ dn/. Solving the cubic (19) (21) Z = [ v' (Q^ + ir A') + QY -1^{Q + « A') - Q]J, so that there are two equal roots if (22) Q' + J^ A' ,r= 0, p^v = 1 p'^v, ^2 pw + //3 = 0. The construction of a numerical case is restricted by this cubic equation, which implies that H must be taken on the pedal of the focal ellipse ; the extra constant is no longer at disposal for making p = by bringing H and L into coincidence, and the motion cannot be made purely algebraical. 3. The Spherical Pendulum for / = 1 has been considered already in §2 Chapter V, p, 117, in which the bob is projected horizontally in the equator with angular velocity 2A, and then reaches an extreme inclination 63 with the downward vertical ; (1) E = 0, F = H = 2-„ = secO^ - cos 6, ; or cos Os^vll + K) n \ n I K V? k' T H (2) ^=n = j7r + ^Z=j7r+v' (/c'2 - H?) K^liT + Kk sin 03. Here OH = OS, and H is to the left of B, at K^ in fig. 97 ; if at K^ to the right, the H curve is looped, and (3) n = J ,r - V (/c'2 -l^)K, ^=W-7r = -ln -s/ (k'2 - K^) K, and the motion is reversed. In the associated ^ motion, the axle starts from a cusp on the equator, and sinks to the extreme 62 as before ; and the apsidal angle (4) ^ = h t/' + KiufK', sn/' = cos Q^ = ~, k < sin 45°. K Thus with K = K = sin 45°, h — 0, giving plane oscillation through 180°. 182 n Test by other numerical cases, such as -;^' ^^ = ^^°' 5 = 2^'^ = ;^' ^' = ^^°' 1 = K = sinl5°, ^3 = tanl5° = 2 -v/3, /' = f , ^ = '^^5 and ultimately for k = or very small, z^ = 0, Sg = 90°, - = oo, "^^ = tt, or to a higher approximation, "^' = tt (1 — f /c^). When^'-i ::'- ^^-^ ,/2_ v^5 + 1 2./ = A, the modular angle is J tan '^2, say 32° ; and we find /' = i, cos 03 = sn/' = ^^~^ 2 sin 18°, * = 155°; the representation is given in fig. 99 A ; and is associated with the motion of a particle sliding on a smooth vertical cone {Proc. L.M.S. 1896, p. 654). In a Spherical Pendulum, when P, Q, L, H are in coincidence with B on the focal ellipse, / = 1, 03 = 0, and the motion degenerates into a plane pendulum oscillation through an angle 202- Give the bob a slight tap horizontally which imparts velocity v when stationary for an instant in the highest position ; the bob will miss the lowest point by a small anglq 03, such that (5) V sin 02 = a sin^ ^ ~T ~ ^^^^ ~ ~ ^^^ 2^i ^^^ i^2 ^i^i J03, in which we may put ch J0i = 1, sin J02 = k, the value at B ; so that (6) sin 403 = ^ cos i02. On the focal ellipse, H will move a short distance from B along the pedal in fig. 97, and the tangent KiBKg will turn through a small angle into an adjacent position HiHQHs, so that AOQ = am/K' is a little less than a right angle, AOH = amf'K' is a little greater ; and / a little less and /' greater than 1 ; and (7) QL ^ z n/g' ^ ^ LP zn(] -f)K" which assumes an indeterminate form when /= 1 ; zn/= 0, zn (1 — /) =0. But (8) ^zn/Z' = /r dnV - H' = K' .^ - H', |zn (1 - /) ^' = - K + H\ CQN QI^^ H' - K'k ' QL _ H' - K K^ ^ ' LP K' - R ' QP K k'^ ' (10) Q^ = K^^nfcnf ^ K'^sn(l -/)cn(l - /) _ ^ ' OA dn/ dn (1 - /') '^ sn(l.-/) when en (1 -/) and dn (1 -/) are replaced by 1, (11) OX ^ ^ '/ '' ^" (^ -f^ ^^^ "^ (^' - ^^-'^ (1 -/)• In equation (18), (19) §1. with 1 - /■ and /' - 1 small, and introducing the approximation. (12) (1 - /) K = sn (1 -/) = sn (/' - 1) ^^ - ^^ - ^^J sin 45° ; and H then crosses to the other side of Q. 5. When / = and H is at A, the pendulum makes plane revolutions and a tap anywhere except at the highest or lowest position will give Spherical Pendulum motion, grazing the highest and lowest point. Then H is on the pedal close to A and /, / are small, (1) (2) (3) (4) QL K' - H' QL_/-, ^\ ,j., QH k'^ sn /' en/' g^ = - tan e.dnf = ^'2^^/^' ^'^2 dn/ = /c - sn/ , sn/ 1 + r^ f *n/ 7" n = ^irf- LH OA K /[hK + H'K - KK' - Y~Y2KK' + (l - J) KK = .fK'{H-^2K) = f'K H - {K - H) k'' ^ = n TT. 184 When the pendulum is whirled round fast, ^ is small, but /,,/' need not be small, and this approximation fails, but (5) tan 03 = ,7/1'^ = - ^' s^/' (^Bneg.), tan 0b = - | ^n/ (9, > J :r), tan 6*3 = K tan 02) (6) n = ^-;^72 '— tan gg = ,r + *. At the other extreme, where k' = 0, K' = ^tt, k = I, K = co,hut Kic =0, (7) n=/r = j7r/. Eealise with a plummet whirled round at the end of a thread. 6. Next in bisection,/ = l, discussed in § 5, Chapter V, p. 119, and taking (1) .3, -. .. = l,l-'c, l-\; 4 = 3-^-. = l-4i-, Q = 2P, the cubic (18) § 2 required for the Spherical Pendulum, on putting L = 2P c, becomes (2) 4P^c«+ (1 -4F)c + 2 = 0, 4P2 = ^^, A = - ~ J J ■ ^ ( 1^2c)(2.e-2 _cl) z^.2^^^, ^' - ^ = - I 2z:p = ^^:t|^, 5=-8P^ _ 4 (1 4 c) (1 + 2c) , ^ A = QH ^^ c^ (1 - c) ' 2P QP' in which c may be taken as the arbitrary parameter for the construction of a numerical case ; and in § 5, p. 120, Chapter V, (4) ii»f^cose = i^-,S (i.lPsin0)^=(L±^ii-^ + 2i-^V-- (5) ^^ = Ji -^ ^, V2 -- -' -J - c (1 - c) ■ - c ^ y 1 - c^ _ c n/ (1 + 2c) _ 1 Tl^' ^2 + ^> - - (1 + c)^ (1 - cY '"''- iT~c' _ n/ (2 + c. 2 + - 2 c^) '' '' (1 + cf (1 - cf ' (6) p_ L + P _ (2 + c)^(l + 2c)^ h = k = c (2 + cY n M 2V2(1 + c)*(l -c)^' n M V 2 (1 + c)Hl - c)* (1 + 2c)*' (7) jM^ sin e exp (^ - pt + i^)/ = [y (Lj_^_i^^) - 2/]y [i . i-^^y^^-^^ and l>c>0> — l>c> — 2, for a real case of a Spherical Pendulum. Thus in the example in (16) §19, page 103, (Q\ ^ ^ ,1/^ = M 71/2 _ 4V10 pa _ 1 r2 _ 3 T /3 W '- 2' 45' ^^ -T^' ^-15' ^-5' ^=~v/5' and the cubic equation (18) §2 becomes giving no other real value of L. Then (10) z, = y g = cos 37°-8, :, = - y | = cos 129°-2, z^ = - s/ 10, P = - J- ^ = _ 3 » VIO' /; 2V10' and ■^^ = — 175°, as shown in fig. 99B. 185 Measured on fig. 68 of the focal ellipse, (11) OA = QT = 10, OB = QV = 6, QL - LP = 2, LH or LQi = 4, A RQ , B HD 3 C _ HQ _ „ (12) ^ - HV and the rolling surface is (13) 1 2) D - HT 2) 2>" HP \x' - if + 2>z' = S^ hyperboloid of one sheet ; and the polhode is the intersection with (14) i^^ + If + Qz' = g^ so that the polhode cone is (15) x^ - Ihf - 24.z^ = 0; the projections of the polhode on the principal planes are (16) ~ + 6f + 15z' = S' (ellipse), f^^ + ^ + ^z'^ = S' (ellipse), g«' - ^f + J = ^" (hyperbola). en-/ 8) 2 /' _ i snV n/15 sn u. (17) dnV' = l, and along the polhode (18) |=yidn., |=--Lcn., The motion of the thread is then given by (19) sin « exp » - pt + J.) i - S/ |(-lj + y) y (i . 1 - A _ ,') When c = 1, P = 00, k = 0, z^ = Z2 = 0, h = co, and the thread swings round horizontal with infinite velocity, and "^ = n. Take c = 1 i = P = Vf, Pi = x/f, Vk = ^"^f j/^ k = sin 6°-7, if* = 96, 3V3 A2 o2 n' 12^. -a = >/ f = COS 52-2, ., = ^^^ n/2 cos 54° ; and for the same P, v/53 4n/5' 20c^ = 17c^ + 6 = (2c - l)(10c2 + 5c - 6) = 0, so that we can take c= -^ ± as well as c = J, to determine the points where PQ cuts the pedal. Finally when c is very nearly zero, the motion reduces to a small conical revolution about the downward vertical, and a comparison can be made of the apsidal angle with the Lagrange-Bravais approximation ; but as the ratio of sin 62 to sin 9^ approaches equality, the motion is nearly circular. With c negative, — 1 > c> — 2, the associated values of c are imaginary for the same P ; and in a tabular form. c 1 1 1 -tV± Ay'f 1 /2647 ' - 1 ~f 3 -2 p CO ,44 v/21 A y¥ 7^5 8 ^v/7 n/33 ao !» 2 n/15 1 n/15 P' 00 ,65 y2i yi y¥ n/309 8 4v'13 v/33 -Jj 00 n/19 n/IS 4 n/15 1 K sin 5°*6 sin 6°-7 sin 5°-0 sin 3°-7 sin 2°-8 sin 21°-8 -sin 37° 1 03 90° 65°-3 52°-2 37°-5 29°-3 24°-7 90° 52°-2 37°-8 e. 90° 66°-4 54°'0 39°-0 ■ 29°-7 25°.0 90° 110° 129°-2 180° h? 00 99 5 12n/6 3 5 7 CO 25n/6 24 9 7v/10 n' 28n/70 20v/10 64N/2i 44 n/ 330 * 180° 157°-5 135° 112°-5 103° 99° 90° -180° -178° -l?.-)" -90° 28570 2 A 186 7. In the Trisection o£ §11 Chapter V, p. 127, (1) QL^P' QV=*-^ = "12"' QV-^^' and from the relation i (2) L' + AL + 2Q = 0, with ^ = - 9P^ + 12, 2Q = 12P(3P + 4), (3) 9(i - 4)P2 + 48P - L' - 12L = 0, cubic in X, but quadratic in P ; a G„ not unicursal, in the co-ordinates L and P. When K is small and Q near B, QH > 2QV, L > 4 ; and with x large, and (4) z = 4^,Pi-y, (a;.v - 2xy = 4(.« + 2) (a?' + 5x^ + 5x + 2), Thus X = 7 makes Z = ^, y = 3P = ^|^, « = cos 16°, sin 63 = 0-4225 = sin 25°,. sin e, = 0-5078 = sin 30-5. ^^ = 1-2, ^ = ^. + nX^ = ^"^^^ ^ ^'^' '^ sin 63 '-'■^ ao:ainst (1 + § sin 02 s™ 63)2'^ = 1'08 x Jtt of Bravais. Other numerical cases can be constructed by taking L= 1,^P = - l,b'=7, K = sin 50°-5,/ = i ; T 1^,2 29 ±2 s/ 129 2 1./-- 7A2_22i.'_a r_ 6qp_inv — 2J_2 /,2 _ 5_3 qj. 2 1 iv — — 3, ox^ — f or g^-, f — 5 Ul e-5-. Steady Motion and Small Nutation. 8. In the Steady Motion of a Spherical Pendulum of length I, at inclination with the downward vertical, represented in ficr. 72, if L denotes the pendulum length which beats time with the small invisible nutation, and X the height of the equivalent conical pendulum with precession ft, we have found in § 4, Chapter III, p. 50, /i\ m^ I 1 o , q a X / cos , o .2 a (1) -_- = = i sec f f cos 0, -^ = —J — = 1 - f sm 0, V(l - |sin2 0)' and when is small (3) ^P = i. (1 + § sin^ 0), agreeing with Bravais, and verification enough to have warned Lagrange of his error, if used in conjunction with a figure ; but a diagram was excluded from the scope of the Mecanique analytique. In the case considered by Lagrange of the small movement of a Spherical Pendulum round the neighbourhood of the lowest position, k may be replaced by zero ; A, P, T, S coalesce ; the focal ellipse reduces to the straight line SS', and the pedal of the focal ellipse becomes the two circles on the diameter OS and OS', as in § 21, Chapter IV ; the two figures, 100 A and B, representing the same motion in opposite direction, with G and h positive and negative, and C above or below 0. The geometrical meaning of Bravais's corrected result of the apsidal angle, given in general by (4) § 16, Chapter IV, p. 98, is shown in fig. 100, A and B, with the approximation of § 21, Chapter IV, p. 106 ; so that making SR = SO, when R is ultimately the midpoint of QV, and V of QH, >P HS ^ RH ^ „QH ,8 <^^ I^ = OS = 1 + OS = ^ + 5 (jy = 1 + |^= 1 + i ch J01 sin |02 sin ^%, ni. ~ n m ""va-vZ VX TT > ^ > Att, 2' when S' = in (1) §19, Chapter IV, p. 102, in which we may put 0i = 0, as at V. Then if a, h denotes the apsidal distance from the lowest point, and I the leno-th of the thread, a, 6 = 21 sin (J02, ^3), *= /gA Jl = X + H ^^ ^ ]^ ^ i area of the spherical ellipse a'^ '-" ^ „ „ hemisphere which is the correct result of Bravais. 187 Otherwise to the same approximation in fig. 101, A and B, dropping the perpen- diculars HA, Qq, Rr on SS', HS - HS' _ AS - AS' OA hq ^^^ *^ 2«i - HS + HS' ~ AS + AS' - OS " 20S' (7) ^ = 1 + I ^ = 1 + I th 101 = 1 + t tan 102 tan pg. 2' OS The experiment with a plummet and thread is useful in showing the progression of the apse line, and illustrating the motion of the Moon's perigee. The motion is the same for a ball rolling inside a spherical surface in the neighbour- hood of the lowest position. In the more general case of a surface whose equation near the lowest point may be written (8) x'- + f' = 2lz + mz^ + ... it may be proved as an exercise (Problem xiv, June 8, 1899, St. John's College Cambridge) that for a smooth particle sliding about in a small orbit, the apsidal angle. (9) ^ = J.(l + i±^.^^)' and the time through the double apsidal angle is reducing to the case of the spherical pendulum when m = — 1. Move H up to Hj in fig. 100 A, where OH is nearly vertical, and OPj flies round nearly horizontal ; and ,-,,. * 1 , HiR , OS - HiR ^^'^ i;;='+W ^ = '^--208^'^' and in the nutation the thread never reaches the horizontal position. But move H down to S, and another motion can be obtained, where the bob flies round with great velocity, in a curve, very nearly in a sloping plane, and crosses the equator ; and here the apsidal angle cannot be distinguished from w, without a closer approximation. 9. In the general case of the motion of a top in the neighbourhood of the downward vertical shown in fig. 101, A and B, where G' is not zero, move H towards V so as to make G' positive ; and then in (7) §1, p. 178, (1) thi 01 = ^~ g, tan J 02 tani- 03, ri)\ +1,1 a — HS — HS' _ AS — AS' _ OA ^^^ t^2 «i - HS + HS' ~ AS"+"1S"~ 06- Since (3) It ^^ - It ^Q' - 1 we may take as the midpoint of q q to the same approximation as before, and iA\ G - G' _ hq - hq' _ OA . ,^ _ OA ^*^ G+ G- hq + hq' ~ Oq ' ^ ' ~ OS' (5) ± = 1 + % - 1 + Oh + jOr = 1 + /^l + iO£\ OA ^ + l^ + 2 OAJ OS = 1 + (l + J 1^) thl 01 = 1 + 2 (Q + 'Q') ^" 2^2 tan J 03 = 1 + g .^ + ^S ®^^ ^2 sin 03, to our approximation ; and agreeing with (7) § 8 for (r' = 0. 28570 2 A 2 188 Thus for example, when H is at £, (6) ^ = !:i = 3, and ^I- = i,r. ^ (jT rq In a cusped motion, in the neighbourhood of the downward vertical, is small, — - — = (1 — cos 6 k 8' _ CR _ sin le. (7) ^ = I = - cos (^2, ?-tl' = (1 - cos e^) ^ = 2 ch 101 sin JSa sin Jfl,, (jr rC K ^^^ /fc 2 An sini0o' 2 "2 /n\ "^ 1 — Scosflo— Ij- ifl J. ifl 1 sin J03 , Ci2 _ , S' Limits of the Apsidal Angle. 10. On fig. 99A where h is positive, and h' zero, and in Chap. Ill §12 (1) (2), p. 58, and in (9) (10), p. 59, h h ,„,,_/•*! n dz I r n '^^ _ n i ^''^ ^^ ~ J.„- 1 -^ VW) = 2'^ ~ "i' "^ = J r^="2 '7X2Z) = "a - 5,r, 2^ (.^) ^ = n=.-n, +n/ = .-J_^-^-^3;^p^<-; and if' = TT when z^ = — co, z^ + z^ =' 0, A = c», and the velocity is infinite. On fig. 99 B where h is negative, replace A by — Ai ; so that (4) ^ = - ^^1 == _ h _ h dt 1-^2 I + z \ - z' (5) *=-n,-n,_-, + n',-n,--,.f;_-^,^, and ^ = - TT when 2^^ = — oo ; and the motion is reversed. Following Puiseux in Liouville, 1842, since (• ^(1-^3.1-^3) ^g _ ^ ^ ^{i.z,-z.z- z,)- ^"' . - \ f sin 03 sin 62 <^2 r / / ^'^ J (l-^3)V(^3-^.^-^3) = f^(l- +^3.1 +^2) + n/ (1-^3.1-^2)]^ = ttCOS 1 (02 - 63), <^; and ^Tj - ^Tj > - ^■j > ^2 - 2^1 between the limits of integration, o i ^ ^2 — Z-^ J *= n. + nt"*" ^'="'^"1"'"' "•">' "-e condition (10) §1, and with h positive a.d V 1,1 + 5^3 + /^3^2J 2 n/(1 + 2;r3 J, + -2') (11) .<-:?: ^l_(^!£!A_i:jiBlM ^ - Jt cos J (02 - 03) ^ ^'- ^3) 189 Limits can be assigned otherwise to the apsidal angle * of the Spherical Pendulum given in (31) § 2, Chapter III, p. 49, by the inequalities (12) 1^> Z3> Z> Z2> - 1, (13) A> V[{z^ + Z2) z + ^ + ZsZ2]> B where (14) A= n/ ( ^3 + ^2 + 1 + ^3 ^2) = ^ (1 + -^3 ■ 1 + ^2) = 2 cos i 02 cos 4 03 (15) B = ^/(-Zi- Z2 + 1 + z,Z2) = V (1 - % . 1 - ^^2) = 2 sin 1 02 sin i (16) A ± B = 2 cos 1 (02 + 03), AB = sin 02 sin 03, ^ = th J 0i, making f^3 Adz ^ f Bdz <^^^^ J., (-1 ~.z') s/{z,-z .T-'z,) ^ . ^ i "(1 - z-') ^{z,-z.z- z,) where as in (6), r Mz^ ^ f Bdz^ ^ ^ (18) J (1 + z)^/ (23 - Z. ^ - ^2) J (1 - ^) v/ (^3 - ^. ^ - ^^2) /■ sin 02 sin Q, dz ^ r a -d\ ra a \ (19) \ il-z^)AH-^.z-z.^ = i'^ (^ + ^) = - COS (02 - 03) and then, as in Webster's Dynamics, p. 53, or Gray's Dynamics, p. 198, ' ,+ 1-) > ^> *7r 1 + B^^ ^ }^ x^2. y. . ^^. B (20) i - (^ When 01 is-small, these limits are too far apart ; 'for closer limits, take (21) J.'2 = (^2 + ^3)^3 + 1 + ^2^3 = -42 - (1 - -^3) {H + h) > {Z2 + z^) z + I + z^z^ >B'^ = (^2 + Z3) ^2 + 1 + ^2^3 = 52 + (1 + ^2) (2-2 + ^3) 5' / 1 + 2 ^2^3 + ^2^ _ ' j'_ n/ (sin 02 - sin 03) (22) 2--v/ 1 + 2^2^3 + ^3''""' with ^2 + ^3 always positive in the Spherical Pendulum ; , sin 02 sin 0; r dz sin 02 sin 03 r dz {^^) j;' J(i_^2)^(^^_^_^_^^) ^ ^ 5' j^i^z')^ (z.-z.z-z^) (25) cosm-h) ^^^^ <=0'i{h-h) ,^ closer limits than before. (A. de St. Germain and Hadamard, Darboux Bulletin, 1895, p. 282 ; 1896, p. 114.) 11. If the wheel of fig. 3, swinging as a plane pendulum, receives a small twist about the axle at the end of a swing, communicating A.M. 2 Ah' , or if the wheel has this A.M., but with precession arrested by holding the vertical spindle, and then releasing it at the end of a swing, the motion will start from a cusp, and H will move a short distance from B along the tangent at B, such that ,. A' _ BH _ OQ - OB sin (i) _ dn//- ^ sn/ _ 1 - en (1 - /) _ sn (1 - /) ^' n OA OAcosw en/ sn (1 - /) 1 + en (1 - /) or isn (1 -/) when 1 — / is small ; and (^\ ^ = QQ siQ w - 0^ _ sn/dn/- k ^ en (1 - /) _ dn (1 - /) ^ ) OA OA cos w en/ sn (1 - /) dn (1 - /) sn (1 r- /) = [l-cn(l-/)][.^-/^cn(l-/)] ^ .sn(l-/) .^ - k'^ cn,(l - /) ■ ■ ■:, sn (1 - /) dn (1 - /) >^ 1 ]+ en (1 - ./V)> ' dn (1, - /') orH'c^--'^)sn(l-/), , 190 (3) (4) QL _ H' - K' k' QP _ /c'- sn / en /■ QP ~ K' k' ' OS ■ "dn/ "' H K — .1 TT + A' K ' K At = i- - {2H-K)^' n («) (7) For an intermediate orbit, with H at V close to B, and 1 — / small, h = h , 6*1 = 0, - = sm * 02 8in,| 63 = ^ = — -^ .-^ K^ sn (1 - /) cn(l-/)dn(l -./) OA or^^sn (1 -/), sn/ QL /H' 2\ n /■^ ^L H' ., .^ fH'K + HK - KK' H'K\ , .-, ^ K' -^)'^'(^-f) *--(- = Jtt + (Z- F) sn"(l -/) = i^ + l—^E ^ = Jtt + £-_?sin J03. (8) (9) In a lower rosette, with H at P near B, 1 — / small, n n 03 = 0, h h sin J 62 = '^j — - = — = sh J 01 cos J 02 = '^' sh J 0i . QP , .^snyW^ ,.. ,„ (1 _ /), LP ^ (j _ |;^ ^„ (^ _ ^^_ OA dn/ i' = i,r-i^(l -/)- (l -|J)Zsn(l -/) = J,r-fi^sn(l -/■) K ' n In a lower rosette, with H at P near A, and / small, (10) (11) (12) QP OA H' LP = '"-^' g^ = (-§)-/. ^ = (f-«')-/ LP * = ^/-gi/^=(fi^-^K'^)sn/ ^ = QP _ QP _ K^ . n k OB ~V '''•'• In an upper rosette, where H is at T near A, and / is small, CIS) QT _ sn/dn/ _ QL _ /. ^'\ . LT fl" . (14) (15) LT -^ + ^ = n = ^.f-^K= -{K-H)^nf ^ = ^- = 01 = sri/- n « OB K ■ With H at a on the tangent at A,/' = 0, PH = J PT, and near A, /'is small, (1^) *+ . = n = (i/_ ^f H- jAV)sn/ 191 12. In the ordinary treatment of a Spherical Pendulum, a plummet at the end of a thread of length I, and T the tension of the thread in gravitation measure, gT absolute, referred to fixed rectangular axes Ox, Oy, Oz, with the Oz vertically downward ; (1) (2) (4) (5) (Pz T z _ W ^ ^Ml = ''' dl^ ^ M I - '^ x^ + y"^ + z^ = P, XX + yy + zz = 0, XX + yy + zz = gz, v^ = x^ + y^ + z^ = 2g {H + z) XX + yy + zz = — x^ — T -1/ = 2H + 2,z. '-^^ = -n' {2H + 8^), f = n^ Writing //' for the vector x + yi, (^^ w d? = Proceeding as before for the top, (7) z = z^ cn^ u + Z2 sn^ u = z^ — (% — z^ sn^ u, u = mt, - = ^ "^^ — \ 1 d?w rfi 2 = fi/c^ sn^ u — h, ■ ^ ^ 2(2g ^- 3..) _ - 4fa -..,)+ 2.. ^ ^ .'.K^fK ^ ' z^ — Zi ^3 — Zx sor V ' •' -4-. = dnV"/^', sn' V ■' ' and (8) is the Lame differential equation of the second order, in Jacobi's notation. « Special solutions of (8), analogous to those of the first order on p. 70, are v' K'i K + K'i K h 1 + ^ 1 + 4k^ 4 + K' w cd ,s^ K-'sc Wi fdu fdu rdu The general solution of (8) is (10) w = A(t>2(u,v) + B(j>2{u, — v), d^ H (m + V) (X _ zn «)« d_ 9(m + V) (X _ za «)» , or J TT e ' an 9m V ^ K + fK'i (11) *2(«,^)=^ ~-e^— ^ and V or /is given in terms of v' or f by the relations in (17) (18) (19) §1 ; 2 - 4k'' sn y en J' dn y ^ 4 en ^«' dn ^t;' (12) X - ^4 + 2(1 + k")k" sn Y - 3k'' sn y k' sn V[k'V + 2(1 + k')^ + 3] The three special solutions above make X = 0. (13) In the Weierstrass notation of §18, Chapter III., p. 68, 1 — cos 6 _ pwi — Tpu 1 + CO'' _ p'/, — p/'i 2 pui - pua' 2 pui- pwg' cos = 2pz^ - pi;i - pi;, pi'l - pW2 ' i¥^ = pwi — pwa, /, .X , ^ cos 00 — cos 6 pw — e, 2 ^ (14) sn^mt= --—^ ::3:;7-^ = -^ j^, en'' to« cos Oo — cos 6*., - eg' ^^ - P^^ dn^ mt = ^l^l-P" ei - 63 ' 192 so that (6) assumes the form (lo) po = 6 -rjr. DM — 3n" *— ^ Li _ '•An'H, with -=- = -^r^, (16) — ^^ = 6pM — A, A = SiDWi + 3pi'2 + 2 (pi'i — pwg) S Zt/ etc* Equation (11) §23, Chapter III, p. 75, points out a solution of (16) in the form (17) ' w = {u, Wi)

'\ = 0, \p Vi/ the relation discussed in Halphen's F. E. I, p. 106, of which the geometrical interpretation is given here in §17. We have to take w^ = wj — /'wj, V2 = /2W3 = (2 — 2/') W3, so as to make (2) V = z^i + ^2 = wi + (2 — 3/') 0)3 = wj + /(W3, W' = t^l — W2 = ^1 — (2 — /') 0)3 = Wj — 2w3 + f'b)3, The condition ^"vi = arises in the revolving plane catenary, discussed in (11) § 25, Chapter III, p. 78, and by Terradas in the Proc Math. Congress, 1912, II, p. 250. In the notation of § 6, Chapter III, p. 53, and with S in the form employed in the Phil. Trans., 1904, (4) /S = 4s (s + m^ x) — [(1 + y) ms + re? xyf, 2 = — m^ x^y^, and the identification of S with P on p. 78 is made by , . p^ _ H _ s — s (wx) _ s + m^ X ^^^ 1? J n^ wF W W ' s {vi) = - "*' ^ = p- |^^^> s («^2) = s (2 wi) = 0, (6) 1 + ?/ = 0, making ^"vy = x (I + 1/) = 0, 28570 2 B 194 as also in the motion of a pflrticle sliding on a cone {Proe. L. M. S., 1896, p. 649) (7) m« ,.^ = ^; (^-^^\ r = - ^ f V , with y=-l; and the Phil. Trans, may be consulted for the construction of a pseudo-elliptixj case. Writing r for p, and 6 for w, and with w = re^' the vector of the curve, + ^ du, with -— T-, = an, S ^ 5 w rfM^ \iodu) "^ ds\^ s . I . (8) w r r (9) \ dw _ \ ds w du 2s du (10) Id'w /I div\' 2 - 5 + s""^ d, a + * S' = 1 . ± = 2 (s + m^^) - im' (1 + ,y)^ (11) 2 (s + m^«) - im^ (1 + j?)^ = 2p!< + pVg, so that z /3 > r > y > a, <13) r' - y' = i(i' - y2) sn^ 2eK, jS' - r' = {(i' - y') cn^ 2eK, „^ -r' = («2 - y')dn'2eK, (14) 2.Z = f^ ^(^'-y')dr'^ (1-^) e = J fe|! = 2eK zs 2/7l ' - / {2fK'i) (16) dn 2/'Z' = ^L_, en 2/'Z' = ^-L, dn 2f K' + en 2/"^' = 1, P + 7 P + 7 and this is the condition resulting from (17) p"r, = 0, 2pt;i + p2ui = 0, puj - e^ + \)i\ - e-, - (ej - p2i'i) = ; and in §6, Chapter III, p. o3, with i\ = w, + /'w,, (19) {e, - e.) cny + (e^ - e^) dn^ - 'h^, = 0, .-'^ cnY' + dwV' - -A^ = ( sn" // ■' ,sir 2/ and writing S, C, T) for (sn, en, dn) 2fK, (20) .„v" = ^; = ^44^, dtf r = f t ? . ^---- ';^ir__^ (21) (Z> + C) (1 - 7)) + (Z) + C) (1 - f) -1 = 0, (Z) + C - 1)2= 0. Otherwise in terms of s = sn /", (22) ;,'2 (1 _ ^2) + 1 _ ^'2 ,,2 (1 ->^''s'Y _ [1 - 2(1 + k')s'^ ;v^s•'p=o. If iJi had been a fraction /Z of the real jieriod, and s = sn / K. (23) p.i - .3 = fLJ;^«, 2 _^ ^ _- -'^ '= 1 + ,2 (24) (1 - .^sy + S (J - s^) (1 - ,^,3) _ ,i (1 ^ /.) ^,^j _ ^,^ ^^ _ ^^, ^^,^ ^ ^^ [3 - 2 (1 + k') s' + k'sJ = 0, s= /I t_'^ ^/3 + K'^ _ /i _ ,. ,/3 ^ ^ 195 In the associated motion of a particle sliding on a smooth vertical cone, we take a a radial distance r, such that (25) L= ^ + »^''' in the plane development or projection, and'so obtain an orbit described under a constant central force, or else with an additional term varying inx^ersely as rl In the associated Spherical Pendulum, with [I = 1, A' = 0, as on p. 56, §10, Chapter III, and with r^ for Q, — n/2 (26) ^ = ^-2 sin-> /f,^ =36-2 cos-^ ..^ \-yj. . , {I + z) s/ {I - z) (1 + 2) V (1 - 2) ('27') 1 + z __ s + n?a' 1 + ^ _ — s — n?x 1 — ^ _ ^ C9S> ;7 = §l(fjt "*^^ _ 9 ^^ = 2.S' _ IR A^ _ _ 1 _ 2mVM? am When |3 = y, or nearly so, k = 0, k = 1, dn 2/7v" = en 2/7i:' = \ = tn\f'K\ 2/'A ' = lir, K' = (»,/' = ; and the apsidal angle /29> e=f 2^Vr^ _ ip^ df^ ^jr_: ^ ' J j3' v/ (4 . 3/3'^ . /32 - r' . r^ - y'^) v/ 3 J , n/ (jS'^ - r^ . r'^ - y'O n/ 3 ' an angle 103° 55' 23", as in the Prijicijjia, p. 127, for an orbit nearly circular, with a constant central force. Hence if the endless chain is a loose fit on the cylinder, as in fig. 102, -^ V 3 or 0*577 of the circumference, 208°, will not be in contact, but loose and showing daylight, when the chain is in revolution ; so that 0*433 or 152° will be in contact. As the length of the chain is increased, O decreases gradually to ^tt, so that (26) ^r. > e > 1^. 15. Pseudo-elliptic cases can be constructed of the catenary, and the extension o£ the analysis to the associated ^ Spherical Pendulum is easily carried out. iV Thus with f ~ §, H = 0, and with j = 2 a^ 2 — — ■" " ) (1) e = -'. sin-1 ^ = J- cos-' i., r' cos 3 = cv\ a catenary for a central attraction vai*ying as the distance r. Withf = I, and r^ = 2*', (2) I=i Bin- ^ = i cos- -•^, (3) i\'2 = r» - 1 i? = 4 AV" - 4 b-^ r + a% a square ii b = a ; 2r r. /3 — r. r + y _ a ,• -1 /« + r, /3 + r. r (4) 7=4 cos- / (^ + « ) (^' + « ^ - «') = 4 sin- / {r_ - a) {r' - ar - a-^ f cos ' = isin- y^ ^ /.x dl i ar do >i ra , t\ ['2(i dr^- 4e K a s/b ± 1 (6) >7=— T~' a=l' «- 2"' «-^ + 7' ■'2 /c^ _ ' " 2V5 ' r., ^/5 - ] „-,„ 3 -n/5 ^5 - (7) dn ir = — ^ — , ' en 4rK = — ^ — , sn ^K = j as before in §3 in the associated Sphelical Pendulum, with /" = i,/= 1. 28570 2 B 2 196 This integral is discussed by Abel, (Euvres I, p. 143, where his a = 0, 6 = J, and ; and our W = b for trisection : and with Te = — ^5 ^3 = ~ 117) 1 — V 10 r2adr^ r dx (8) r' = a' (1 - a;), j-VR = } ^ (x' + x' - x) ^ an integral associated with Quinquisection, in Chapter VI, p. 167. With/' = ^,x = y = - lin Phil. Trans, p. 229 and with ^ = ^' I^ ~ ' Inho - "" , , , , , 4ar* - 3aV + a' , ■ i i^' - «') ^ [4^'(^' " «')' " «'] (9) 1=1 cos-i 2? = i ^^^' ~2? , (r + a)-y (2r' - 2aV + a') „ . . , (r - at) n/ (2r^ - 2aV - a^) = I cos- 1 ^ ^ ^ 2ri = * ''" 2^^^ (10) . dl = 1-^' - d9, introducing che relation c^ + c^ - 5c - 1 = 0, or 4a(a - 1)^ = 0, as in the Trans- foj"mation of the Eleventh Order, and Abel (Euvres I., p. 142. So also for /' = f, 3/ + 1 = 0, 2^ - 2 - 1 = 0, 2 = J ( n/ 5 + 1), in Phil. Trans., p. 230, (11) i^ ^^^ •' -'•"'^'' - '^ - "' '^^U'"' -^'''•- ^ "'^^ 2f^ f sin-i (*"' + «^ ^(^^^ -2) -a' '"/ )y2r^ - 2aV - a^y. K f = ^^^,^ + 1 = 0, (1 + c)(l + - 4c2 - c^ + 2c* + c^) = 0. And/' = f, 2/ + 1 = 0,/ - / + - 1 = 0, giving;) = (16Ac/c')-'^for-^ =7 31 With (u = 8, /' = i, and (Z.iHr.6\ 1893, p. 226 ; PhU. Trans., p. 277) 1 - 2^2 1 1 (12) ^ + l=-^-^- = 0, 2 = ;72, «4 = J, A,. = - 1 +2 = _ 1 +— . (13) . = .(.-:q^7-(:^)' = (5- J) [4s2-2(n/2- 1)^5+ (V2- 1)2] and putting, 2 (14) « - i = i^ = i (^ - 1), 6^ = 1^ [(^ + 1)2 _ ( V2 - 1)¥] = K5-0E-(-^-i)^^(-^-i)l = (15) 8a«* J .,„,-. (-/2-l)V(2.-l ) i - , cos ^^ = J sin - ^^IifL^l(^!:2^^i)!ijM:^ - 1)^) , .i(n/2-1); = Jcos 1 ^^-^2 ^ + 1 V 2 ^ ^-J '2 - 1 a catenary under a central attraction, an open branch stretching to infinity s/2 - 1 radr" n/2 ]r~^'^ 0c'V f /c'V ^ ay/ r=^T¥ ~ 6y/' sng/' 2scd' e' en 2/' _ k'^^c T' (2) sn 2/' (3) sn 2/' Interpreted on the focal ellipse of fig. lOo, -j^ log scd = J- = U, df ^ s c d or HV - Pn - HP' = 0, from (17) (18) (19) § 14, Chapter IV, p. 95 ; a,ijd, as in § 15, if X, ./ are the elliptic co-ordinates of H, Tjrp sd dn 2/" 11 i = a — = <-/ C dn 2/' (4) HV = «'?^ = -^, = V(X +«^), s sn 2/ sn 2/' n/ (A + i32). HP' = « K ^sc n/X ; t/ sn 2/' (5) OH = adllf = V ( - ,.), HZ = a,c' sn/" = ^/ (v + a^), OP' = n/ (X - v) = /■ ; (6) HS' 4- HS = 2HV, HS' - HS = 2flZ, HS' = VZ, HS = VZ' ; and the asymptote of the hyperbola v and OH make equal angles v^'ith OV and OT. Making HL = HP', L will be the Landen point on the contrafocal toucliing TV at L. To construGt fig. 103, select an angle AOH = am f'/C, and draw TH\' at right angles ; then from (1) ..X ,, _ I - 2s' _ 1 - tn^ f 2 _ 2 tiT f - 1 4 /" 2s'^ - 3s'* 2tn2/'-tnV 1 2 tn' f - tir f" (8) 1 > K- > 0, 1 > tn /' > ^ = tan-i ;55°-3, tt > 2 am / > cos"' J. Make HP' = H V - HT, and draw the normal P'G ; then OS is the G.M. of 0(j and OT ; and so on for the rest of the geometry. By solution of (1) (9) (10) fO 'ill K '■ sn^ / 1 + ^-'2 _ V(l - kV^) dn^ / ' = _ 1 + ^^ + ^ (1 - A-'2) *■■' s'^/' = ^ 1 + k'n/3 + k' ■1 - /c'x/3 + k' 6 y - ' dn/'= /^±J^^±J ^ dn(i-/') = y 1 + ^V3 6 z^" "v/ 3 4 a:^ ^ cn (i-/') = y--V^-y'-'^-^' 0/ cn 2/ 2 '1 4- KK OA ~ ■/ ~'2 ~" "^ 2 '1 - /I - z.^- v 2 ' 1 - n/(1 - A-'-) dn2/' = ^ILzu^"''^-'') (U) (12) cos 03 sin (^9 = 2^■'i■^ n/2 = t.n^/", k' + a: 4- s/(l- of the Spherical Pendulum motion by an accent, W ^-«-^g = |-f =M = i-^(2-^-os.), (21) -<\>' =p\^{2.H + z)ndt = p\ y ^^-±J. dz, the elliptic integral of §3, Chapter III, p. 50, which gives the length of the arc described by the centre of gravity. 201 CHAPTER VIII. Motion referred to a Moving Origin and Axes. 1. Take a frame of reference o£ three rectangular axes OX, OY, OZ, moving in space so that Vi, V2, % are the components of the velocity of along the axes, and 0,, 62, 6^ the components of angular velocity of the frame about the axes. If I, m, n denotes the direction cosines with respect to the frame of a straight line OP in a fixed direction in space, then (1) ^^ = m^a - «02, 'Jt ^^^^~ ^^^' "JJ ^ ^^^ ~ ™^'' dt For if the points of intersection on the unit sphere are P, X, Y, Z in fig. 105, — is (J/v the rate at which the plane YOZ separates from P, and this is the velocity of X along XP on the sphere multiplied by sin XP. But the velocity of X due to the components 0i, 621 ^3 of the A.V. of the frame is the direction XP is 0, — 63 cos PXZ, % cos PXY ; so that dl (2) J= - ^2 ®i^ ^P <^°^ P^2 + ^3 ^"^ ^^ ^o^ P^Y ij/i = — O2 cos PZ + 03 COS PY = —7162 + m(>2- Denote the components of linear momentum of a body or system by Xi, j:-., Xg, and the component in the fixed direction OP by i7 ; (3) H = Ixi + mx2 + nx^ , , ^ dH , dx, dxo dx« dl dm dn ^ ^ dt dt dt dt dt dt dt = l{^ - ^'s 03 + % %) + ™ (^ - ^'3 ^1 + X, e,) + ni^^-x,d, + X2 9i) = IXi + mXg + nXg for all values of I, m, n ; X^, X2, -Xg denoting the components of the resultant applied force from the exterior, the interior system forming a system in equilibrium, by D'Alembert's principle or the Third Law of Motion. Next denoting the components of angular momentum of the system by y^, 3/3, ^3, and taking a fixed origin and axes through it parallel to the axes of the moving frame ; and denoting the coordinates of by x, y, z, and the component A.M. about the fixed line through the fixed origin in the fixed direction by G, (5) G = l{yi - X2Z + xsy) + m (ys - x^ x + x^ z) + n (y^ - x^ y + x^ x) by § 28, p. 24, Chapter I ; and differentiating with respect to the time t, and afterwards moving the fixed origin up to the moving origin 0, as if the moving origin and frame of reference was passing through the fixed origin and frame, (6) ,=y=, = 0, but-| = ., 1 = ., I = .3, ^'^^ ^ " K^ ~ ^' ^' + 2/3 03 - *2 V, + X, V2) + m ( ) + n ( ) = lYi + mFg + nYs for all values of I, m, n, ; Fj, Fg, F3 denoting the components of the impressed couple about the axes of the frame, is their instantaneous position. Thus the six equations of motion of the system are proved, as required in a dynamical problem, 28670 2 C 202 Motion of a Pendulum Body and Gyroscope about an Axle on a Whirling Arm. 2. Apply these equations of motion to a single body, moving like a pendulum about an axle, when the axle is fixed in any position to a vertical axle, and the system is made to revolve about the vertical axle with constant angular velocity fi. With the origin anywhere on the pendulum axle, say at a distance r from the foot of the shortest distance a between the two axles, and with the frame of reference XYZ of fig 58, fixed in the pendulum body, while the frame xyz is carried round, preserving a fixed direction in space, with Oz vertical, then, with Routh's order of displacement of p. 73, (1) 0j = P = - ^i sin e cos (j>, 6-2 = Q = M sin 61 sin ^, Bz = R =^ fi cos + -^ ^ (2) Uj = an cos (^ cos — y sin <^) R^ cos 6 — Mr {x sin (j> + y cos + Mry) cos (j) + (E + Mrx) sin 1^] jj? sin cos + M (i cos f — y sin ^) o ^t^ — K^ ^i sin sin ^ — K2H sin cos .^, from which -^ and J/i has disappeared ; and (9) ^3=— ,^+ Mry) sm i> - {E + Mrx) cos ,j>] ^.'-^ sin cos 6 — 2 M (x sm(i> + y cos (j>) an^ + 2 (gMx - K,f,) sin 6 cos .^ - 2 (yMy - K,^) sin (^ sin ^ + H, of the form ^^^^ i'it) " " ''°' ^'^ "^ * ''° 2^ + /cos + ,(7 sin ,/, + /i, is the most general case ; and this is reduced by the substitution tan J,/, = c to the form (12) (;^^'^n^^z,,y^,,z, ^,^^J^ the general Elliptic Integral of the First Kind. 203 3. The reaction of the axle on the pendulum body, given by the forces A'l, Xg, X3, and couples Fj, Fg is found from the remaining dynamical equations of (4) and (7) §1. The reactions are reversed on the whirling arm ; and in this way the theory may be tested of a suggestion of gyroscopic control of stability, as in a flying machine. In these cases of motion on a whirling arm, the vertical axle is forced to rotate with constant A.V. ^, and the couple of constraint can be investigated required for the vertical axle._ If this constraint is absent, u will fluctuate so that the A.M. about the vertical remains constant ; but the solution becomes hyperelliptic and is abandoned as intract- able, except in some cases of a small oscillation above a position of relative stable equilibrium, with the system left free to fluctuate in A.V. about the vertical. If the outer ring is held fixed in fig. 57, or the vertical spindle in fig. 3, the body swings like a pendulum according to the law (1) tan J0 = tan Ja en nt, k = sin Ja, even when the wheel is in rotation ; but the extra reaction due to K or CB, the axial A.M. do of the wheel, is a couple CR-j, about an axis perpendicular to the inner ring in fig. 57. And when the outer ring is forced to rotate about the vertical axle with constant A.V. ;u, the modified motion and the extra reaction is calculated by the preceding equations. 4. When the axles intersect, a = 0, and the origin can be placed at the intersection, making r = 0. Further if OZ is a principal axis, ^ = ; and D, E can be made to disappear by turning the axes OX, OY in the body ; the equation then reduces to (1) C (^)' = ^{A- B) fi^ sin^ d cos 2^ + 2 {gMx — Kifi) sin 6 cos (j> — 2 {gMy — /iTj^) sin (> sin <^ + H. With cos

+ h, with a simple elliptic function solution. An application is shown in fig. 106, where a flywheel has its axle OC held at a constant ^angle y with OZ ; we must replace A, B, C, D, E. F in (10) § 2 by J. cos- y, A, A siv? J, 0, A sin y cos y, 0, where A denotes the equatorial M.I. of the wheel at ; and put a = 0, r = 0, i = A sin y, y = 0, Ki = K sin -y, K^ = ; and then (4) A sin^ y ( -^J = — i Afi^ sin^ y sin^ d cos 2^ + 2Afx^ sin y cos y sin d cos 8 cos f + 2 {gMh — K/j.) sin y sin cos

— {gMh — K^i) sin ^ sin ^. The arrangement is equivalent to the Gilbert Barygyroscope and other apparatus such as the Gyro-Compass, and the original Foucault experiment with a long free simple pendulum intended to detect the rotation of the Earth ; but here the axle Oz is held parallel to the polar axis, and ft denotes the A.V. of the Earth, K being made large enough for Kfi to be appreciable, as described in the next chapter. The position of relative equilibrium is given by ,„.._„ J , Afi^ cos y cos -I- qMh — Ku (6) sm d) = 0, d) = or w, and cos ^ = -^ ^-r—. f—r: " For a small value of ^, such as the A.V. of the Earth, where fi^ can be neglected, equation (5) gives simple pendulum motion, of equivalent length (7) /=_A.^!!^I; Mh-^ sin and the gyroscopic pendulum would reverse if we could make KiJ.>gMh. 28570 2 C 2 204 Differentiating (5) again with respect to (p, putting d/dt = q, d^d^ Id^^, ^.^3 eos 2^ - ^^' ^'^^ ^ ^°; ^. + ^^^ - ^^ sin cos ^ = - f, ^ ^d

= ^u sin sin ^, and the body makes sin sin ^ beats for one revolution. But at (^ = or TT, i^ cos (0 + 7) + ^ -. (9) ;)^ = or ,= COS («-,)+ iM^S sin y' sin sin -y' and a negative value oi f' implies that the position of equilibrium is unstable. Another application is made in Watt's Grovernor (fig. 107), where B — A must be positive, (10) ( T^) = — p^ cos^ = P^ cos sin ^ — n^ sin (^, so that in a sidelong position of equilibrium, at inclination ^ = y, cos y = n^/p^, and y is an acute angle, and the equilibrium is stable. We write then (11) f-^ j = p^ (cos f3 — cos <^) (cos

< a, when the arms oscillate between inclination a and /3, above and below the equilibrium position, to an equal vertical extent, up and down, since (12) cos a + cos j3 = 2 cos y. The elliptic function solution is given by (13) tan J + 2n^ cos (j) + h, -t| = — p^ cos (j> sia (j> — n^ sin ^, a rotation of the ball being represented by the term Ki in n- ; so that in the position of ■equilibrium cos y = — n^jp^, and the angle y is obtuse, and the equilibrium is unstable. Then when (2) f -J- j = p^ (cos (jt — cos a) (cos (p — COS j3), > a> j3 > f, (3) cot J<^ = cot Jo sn mt, or tan J<^ = tan Jj3 sn mt ; and as a or /3 goes out of existence, cos a is replaced by — ch a, or cos /3 by ch j3 ;- (4) (— j = p^ (cos ^ + ch a) (cos (jt — cos (3), tan J«^ = tan Jj3 en mi ■(5) f -^j = p^ (cos a — cos (j)) (ch (3 — cos '^ ~ UD " n~K" ~ rrr ~ la' (3) LR = LAdnAZ, RS = 2QR = ADdn2AZ, QL = OL cos ;A = X. OD en 2AZ. ,,. 1 2 1 rr K dnhK , LR , RQ + QL <^) ^" ^^ = dn (1 - h)K =''m='' RQ-QL _ 1 - A Ai^ + Xcos^// _ 1 - X 1 + X sn (1 - 2h)L ~ 1 + X Axp - \ cos iP ~ 1 +X 1-Xsn(l- 2h)L /KN .2 4. 2 z.z^ tn AZ FS <5) can^. = tn^hK = ^, ^^ ^^ _ ^^^ = ^-.-^ _ 1 + X RQ - QF ^ 1 + X Ai^ - cos ^ ^ 1 + X 1 - sn (1 - 2A)Z ~ rz^ RQ + QF 1 - X A;/- + cos ^ 1 - X 1 + sn (1 - 2h)L 206 ,„, ,,,. 1 + X 1 -sn(l - ^h)L . • ,,2;,/r-j_Jli 1 + sn (1 - 2A) L (6) sn^ AA = -2 - . 1 _ X sn (1 - 2A) Z ' '^ ^^^ ' ^ ' 1 - A su (1 - 2A) X^ 2n /.^ r 1 + X 1 + sn (1 - ^A) ^ (^) ^^ (1 - ^) A = -2- i+Xsn(l-2A)£ an n/.^ 1 - A 1 - sn (1 - 2A) Z cn^ (1 - h) K = -g- 1 + X sn (1 - 2A) Z Thus in Watt's Governor, the quadric transformation gives .21 . 1 .- 1 /al + Xsn (1 - 2A) Z (8) : tan^i = tmia tan J ^ j -Xsn(l-2A)Z (9) 1 - cos ^ = 4 sin J (a + j3) sin J a sin J i3 [1 + X sn (1 - 2h) Z] -h Z>, . ^1 + cos(^ = 4 sin^ (a + /3) cos J o cos J /3 [1 — X sn (1 — 2A) Z] -=- Z>, (10) sin<^ = 2sini(a + j3) >/(sinasin/3) dn(l - 2A) Z -^ Z>, D = sin a + sin j3 — (sin a — sin j3) sn (1 — 2h) Z, (11) cos /3 - cos <^ = (cos j3 - cos o) sin /3 [1 + sn (1 - 2A) Z] -^ />, , ; . '-'-^ '■■ cos ^ — cos a = (cos ^ — cos a) sin a [1 — sn (1 — 2A) Z] -f- Z> (12) X = ^"^"-^"^f , A = |. ^ ' sm (a + /3) 1 • Draw the tangent AVRj to the circle ac? in fig. 108 ; under gravity with velocity due to the level of E the midpoint of LL', the time along the circle ARD from A to Rg is twice the time from A to R ; so that ADRj = am 2hK, with ADR = am hK ; and ^iTz^ LR2 _ VRa _ oD _ OD - Oo _ OD - PL sin V _ 1 - X sn^ 2AZ which is Landen's Second Quadric Transformation ; and then ., ,, o, r^ (1 + X) sn 2AZ 01 7^- en 2AZ dn 2hL ^ '^ 1 + X sn^ 2AZ 1 + X sn^ 2AZ Moving Axes for the Motion of a Top. 7, Resume the treatment of Chapter III, §1, for the motion of a top, referred to the fixed coordinate frame xyz^ and moving frame x^y-^Zi, of §20 Chapter III, p. 70, and fig. 56^ Denoting the components of A.M. by h„ h^, A^., and hi, h.^, h^, (1) K + ilhi = (^1 + ih) '?'^', ^ ' dt dt dt dt ^ ^ ^' dt and at the instant when i/- = 0. , . (iA^. (iAi 7 d\ij ,, c?A,, rfA, , J d\L . . „ The components of A.M. are given in Chapter III, §1 by (5) Ai = 4 sin e cos e^+Ci? sin f^, h, = - A^, h, = A BinH^ - CR cos 6, measuring 9 fi-om the downward vertical ; and (6) -j^ ^ -^ = 0, and A3 is the constant denoted by ; Ji bin n g^^ and the first equation of (4) is satisfied identically ; while the second equation giAes ,., - , ^0 jGco^e + CE) {G+ C RcobO) i^) ~ ^ df^ A sin^ fl - ^l" sin « = 207 a,nd integrating (9) (I)" = 2," iH . cos ,) - i^^^^p^)' = S„. iH . cos e) - 4 (^i^f^-')', (10) (f/+sin»()(gy_2„^(ff + cos«), the equation of Energy, as before in Chapter III., p. 44. Denoting by X, Y, Z the components in gravitation measure of the reaction at with respect to the frame xyx, but by qX, gY, gZ absolute, and by ^ = x + yi = a sin 0e'/'» the vector of the projection of the C.G. mi ^^ X + Yi _ dK Z d'aeose ^^^^ ■'-ir--d?' ni = ^--d^^ and taking ^ = at the instant considered, so that X acts in the vertical plane through = - n' (2H + 3 cos d) sin 6» + 4 (A + h' cos 6) h' cot - 2ih' '$ at yi Qx Z g , ■ lid^O , „ /d6\'^ 9 _ n- a + n^ (2H + 3 cos e) cos 6/ + 4 (A + h' cos 0) h'. Thus at the reaction in gravitation measure is composed of y (i) an upward vertical force ( 1 — — ) M ; an? (iv) a force 2 — — -j, at right angles to this vertical plane; (ii) a force along GO, — {2H + 3 cos 6) 31 ; (i) and (ii) combining into a single vector given by a limaqon, as in the Dygogram of a plane pendulum ; (iii) a force 4a (h + h' cos 0) — ^—- = 2aMh' sin ~, ^ ^ gsmQ dt acting at right angles to the axle, in the vertical plane through the axle ; aMh' de g and (iii) (iv) combine into a single vector 2aMh' -, along the vector of A. V. w. The expression for R the radius of curvature of the projection of the C.G. could be •obtained by resolving normally, and will be found to agree with that given in Chapter V § 37, p. 153. In a Spherical Pendulum of length a, g — an^, h' = ; and the reaction at reduces to the single component (ii). Writing Q for dO/dt, and differentiating (8) with respect to B, ,^,. d d?9 _ld'Q _ d (G cose + CR){G + CE Gosd) ^^^^ ddd?~Qd? -^9 A' sm' e -n cosO {G' + €'£') (1 + 2 cos^ 0) + GCR (5 cos 9 + cos^ ) ^ ^ - A^ sin* 9 "" ''°' ^ - ~ -P ' isuppose : and from (8), in a state of Steady Motion, with dip/dt — ^, ,,,, 2 (G cos 9 + CR) {G+ CR cos 9) ^ a r ^ m? a a -2. <15) n^ = -!^ A" ■ i a ^5 ^^'^ G + CR cos 9 = Au sm^ 9, G cos9 + CR = {Ai-L cos 9 + OR) sin^ 0, CRft = A{n' - fi' cos 0), G^i = A{fi' - n' cos 0) ,, -- 2 _ (G' + C'R') (1 + 3 cos^ 0) + 2 GCR (3 cos + cos^ 0) (i*>) P ^^8in*0 ; -and the axle make p/fi beats for one revolution round the vertical, while the top makes R/fi revolutions, as before in §21, Chapter IV, p. 106. 208 But spinning upright, with 6 = it, G = CE, (18) A^= , ^^ . , ^ dt 1 - cos ff and the motion cannot be treated as steady ; but in (14), with cos = — 1, (19) f^<^ tP-^^ + »' -» » - ^f - < ^ A^ {1 — cos Qy 4:A^ as before in (24) § 21, Chapter IV, p. 108. 8. The motion may also be referred to the moving frame Wi y^ Z of fig. 56 with the axle of the top along OZ. This system of moving axes is useful for the discussion of the motion of a spherical ball, spinning and rolling on a spherical surface (a bicycle ball in a cup or wine glass), and the motion is comparable directly with the motion of the gyroscope wheel of fig. 3. Calling h the radius of the ball, and a + Z> of the spherical surface in which the ball rolls ; gX, gY, gZ the components of the reaction, absolute, at the point of contact, and Ai, Ag, Ag of A.M. of the ball about parallel axes G.«, Gy, Gz through Gr the C.G, With the coordinate system of fig. 109, the geometrical relations are (1) U=a~, V=asme^, W = 0, U - bQ = 0, V - bP = 0, (2) .. = sine|, e.= -f, .3=oos4j h, = MP F, h, = Mk^ Q, h= Mk^ B, and the dynamical equations are (4) «^_F«,+ w,=^-.,rine, ;!|-we,+ OT,=^, In the discussion of the dynamical equations, take the third equation of (3) first ; it reduces to (5) ~-j- = 0, E Sk constant. Next eliminate F between the first of (3) and the second of (4), (6) ~^^ + ^^^' - ^^^^ + F ( W ~ ^^'^^ + ^^^0 = ^ (7) f . ue. - bEs. . ^li^^ . ^,) . (1 . ^ (^ , ^eos e I) . .4« wt-:)«H§-^-os.f^)..4^=o (8)_ _ (l.|:)a|;(sin^.f).,^.i.ef = 0, and integrating (9) (l + p) « sin^ e^ -bE cos e = G^ei constant Lastly, eliminate X between the second of (3) and the first of (4), (10) b^- bEO, + bPe, + i; (^ - Ve, + TT^03 + 9 sin 9) = (11) (l + p) (^ - V6,) - hE6, + ^^^sine =0 209 = - oA'sinfl + jGi + bRcoBe) {G,coBe + bR), as before in (8) §7 with a f 1 + -p j sin^ 9 2=£_Z_ 7 = .ri.^'-A 9A_ (^i ^v - bR (1^) ^^- ^./. ,. , ^ = al4-^, 2A=_- ^IL^ 2h' a 1 + ^, ' a I 1 + A.S an example, spin the ball with angular velocity R, such that (15) R^ = -p- (l + pj) that is 35 ^ for a homogenous sphere, about the common normal at an angular distance of 60° from the lowest point ; when the ball is released it will start from a cusp, and proceed fco describe the motion discussed already in Chapter VI, § 10, p. 165, given by (16) sin^ exp Si/^z = ( - 1 + 2 cos Qf + i{l- cos + cos^ 0) n/ (2 - 2 cos - cos^ Q) ; and starting from a cusp the ball will describe one -third of the apsidal angle 30° in two-thirds of the time down to the apse, as before on p. 166. With no spin and ^ = 0, the motion of the ball is comparable directly with a Spherical Pendulum. The sign of h must be changed, and of cos 9 as well, in the discussion of the motion of the ball, spinning and rolling on the top of a sphere of radius a — b. The equations may be modified, without material alteration, by making the spherical surface kinetically symmetrical and capable of moving freely about the centre ; a term —j^ must be added to 1 + y^, / denoting the M.I. of the sphere about the centre ; the proof may be left as an exercise (^Quarterly Journal of Mathematics^ XV, 1878, p. 191). Equation (4) determines Z, the thrust between the ball and the surface ; (17) -^ = (/ cos e - Z702 + F^i = (/ cos + ^I!±J1 = ^ COS + a(|y+ a (sinewy ,2 V = g COS + 2g j^ ir(^ ■•■ ^^ ^) 2H + (S + j-,)cosd SO that Z is proportional to the depth below a certain horizontal plane. And (19) . .^ = ^sine + ^ - F03 = ^^{g.in9- bRe,) (20) -^ = ^ + f^^3 = -j^ (9 -OS 9- bR9,) 9, Applying this frame of reference to the gyroscope wheel of fig. 3, with U, T, W the components of velocity of the C. Gr. (1) U = a^=a9„ V= -amn9'^ = -a9„ W=0; (2) g±.= g COS 9 + U92- ¥9^ = g cos 9 + ^ = ,,cos0 + a(^) +«(Bm0^) = g Gos9 + 2an^ (ff + cos 9) = (g - an^) cos 9 + an^ {211 + 3 cos 0), introducing the principle of Energy ; 28570 2 D 210 /ON X . a , dU T/fl Y dV ^ jja (3) g-^=g.me+^-VK 9.M=Tt^^''^ and taking moments about the C.G. (4) ^1 - hA + ^2 = gYa, ^' - hA + hA - - gXa, with (5) h, = A'P = A' sine ^, h, = A'Q = A'f^, h=GB, A' = A- Ma'; and the elimination of X, Y is found to lead to the same equations as before. 10. When a sphere rolls on any other surface it is convenient to consider the parallel surface described by the centre of the sphere. For the sphere, cylinder, and cone, this parallel surface is of the same character ; and so no limitation need be made in b the radius of the ball. But for other surfaces we shall take the sphere as small, a little bicycle ball, so that the two parallel surfaces are practically undistinguishable, and limit our consideration to a surface of revolution. With the same dynamical equations as before in (3) (4) § 8, the geometrical relations for any surface of revolution about a vertical axis are replaced in (1) by (1) U = pf, F = .sin9^, where p denotes the radius of curvature of the meridian curve of the surface described by the centre of the sphere, and a- the normal to the axis. Then the third equation of (3) § 8 becomes intractable in general ; as well as the other dynamical equations. But the equation of conservation of Energy can be written down, and of A.M. about the vertical axis. 11. For a sphere in a cone, as at Roulette, = J tt - a, if o denotes the semi-vertical angle, and if r denotes the distance of the centre from the vertex of the cone on which it lies, ^= U, with p = ^, p Tj , -i 3 (18) y( 6^+ F \ H + ^nr^ b'sin'a + ky ^X (19) c^(^ - pt) ^ /{_^±JL\ dr V W sin^ a + F/ V + f) // + n-;)(l + P X= 2 f;; F 1 + sm" a r\/X g cos a (Ji r^ — r"^) k' b"^ ¥ k^ 1 ' // + ,n . ,, ^Tp- (-^ — »^ sin a) &r cos a — ;=- nr^ b^ sm^ a + k^ ^ ' 2 a quartic in r. 28570 2 D 2 212 "When the sphere rolls on the outside of the cone, the sign of g and b or r must be changed. Then, for instance, if A = 0, X is equivalent to the form employed by Abel, CEuvres I., p. 140 ; and assuming a factor a + r oi X, ^ ■' a~ Pm - PSv ~ s + X - y ' y a ■[■ r' as in Levy's Elastica under normal pressure, discussed in Math. Ann. 52, 1898. But it a + r a. factor of X in the general case of (19) (21) X = (a + r) {- — i- Ar + Br' - i nV), and putting /r,g\ r _ 1 - z dr _ 2dz a 1+2;' r \ — z 1 H- z \1 + zJ * W + z/ 2Z (1 + zY' (23) X=--^- ^ ^ 1 + z suppose, and putting (34) B = ^■., i»' = ^, the value of V ( — ^) when z = ± 1, ^ VX V{2Z) L + L'z as before in the motion of a top ; and so the algebraical results of Chap. V can be utilised, while the extra constant at disposal can be made to cancel the secular term pt. Put 11=0, then X has a factor r', and the integrals are non-elliptic ; ,q„s dip ^ k^ J ( D , /^^ sin^ a + F <^^) rff^i'^FTF' andr = .l + i?cos^y ^, ^ ^, , a curve which can be developed into a limagon ; this motion is realised if the sphere is placed gently on the surface. On the plane whirling table a = Jtt, and we write (28). X = P'r' - (H - ^nr'Y = X, X,, X„ X, = ± ^nr' + Fr T B, (29) ^ = 2 cos-^ y ^ = 2 sin-^y ^^, 2 Pr cos ^ = X, - X, = nr' - 2H, P / fP' H\ a circle in space, centre C, where OC = -, and radius /(— „ + 2—), described with n w \n^ uJ k'P constant velocity p p. Thus if the ball is dropped gently on the whirling table, it will proceed to describe a circle passing through 0. The method may be applied where the axle is not vertical but an angle j3 out of plumb ; and then the ball describes a trochoid in space, made by a point on a circle of radius — ^^-yj '- 4 sin /3, rolling with velocity „ ■'- sin |3. Similar results will be found to hold when the axle is vertical and the table askew • the ball will describe a bicircloid on the plane, a circle if tan /3 = . A,' The cone is at rest with n = 0, and the projection of the path (illustrate with bicycle ball in a conical lamp shade) is a central orbit, described under a force of the form „ + vr-^ and v can be cancelled by a suitable change of ^ into^^i/,, equivalent to the applica- tion of the surface of the cone on another cone of vertical angle a', sin a = p sin a. 213 Pat ^ = to obtain the motion of a particle sliding on a smooth cone ; the equations are then similar to those in the plane revolving catenary of Terradas, §14, ChajD. VII. For any other surface, except the, sphere cone, and cylinder, the equations appear intractable ; Routh's Advanced Rigid Dynamics, Chapter V, may be consulted, and F. Noether's Diss. Munich, 1909, '■'■RoUende Bewegung einer Kugel auf RotationsMchen.'''' 12, For the motion of a sphere on a cylindrical surface, as a bicycle ball visible inside a glass tube or jug, return to the moving axes of §7, with Oy along' the axis of the cylinder ; and then in fig, 109 (1) 01 = 0, 02 = -f, 03 = 0; (2) U= a'^, F = ^, W=0, U - bQ = 0, V + bP = 0, and with the axis of the cylinder at an angle |3 with the horizon, /o\ dU qX r> ■ n dV qY ■ n dlV j-ra 7^ „ „n .^^a . (3) ^ = -L^~ gcosfi.rn9, _ = 2^ - ,9 sm /3, _ - f7e, = ^^ - ^cos/3 cos0 ; Eliminating X, ... F dbQ P dU dU ^ . „ /, P\ d^6 , ^ . „ ^ as in plane pendulum oscillation, of length (6) / = M + ^Jasec/3. (7) 1^ = 2nN/ (cos^ 10 - cos^ Ja) or 2?2V (cos^ JO + sh^ Ja), n^ = |, according as the pendulam oscillates, or makes a complete revolution. Eliminating Y, rQ\ dV . r, k' fdbP , ,p. N /n ^F\dF , . ^ , k\j.de „ (8) - + g sm /3 - ^-.(^ + bRO,) = (1 + -) _ + ^ sm/3 + ^,bR^^ = and from the third equation in (4), (10) (Wf:)f + fiJ.fsinp| = 0. Then,putting((=yg+l)* = ^, / ( COS^ ^ — COS^ Jaj (12) ^ COS ^ + 72 sin ^ = 4 ~ i'^ | (p ^ ^) *^" '^ m r COS xfj dip J (13) ^siu^p - Rcosip = B - ^n^ (|J + l) tan (3 y(cOS^^-COS^Ja) sin ;^ dip 7 ■^ COS^ — — COS^ Ao m 2' And (8) may be written <") (i + iVf ^^'^f *^''"'3 = »' the equation of Energy. 214 These equations are intractable in general, but in the special case of k^ = \h\ m = I, ^T? , A ^ -ifsinhe ^^ sin ^e \ (15) 2§cosie + Rsm^e = A + Cco. ^[^^, or-^^j (,e) 2 f sin je - i. cos je = ^ - C (ch- ^£f, or sh- ^-^). With the axis of the cylinder horizontal, (3 = 0; and the radial plane through the centre of the sphere will oscillate like a pendulum of length (17) '"(l + p)" (18) (i^^\v' + lfS? = i'jV", a constant, and F = S ^, (19) (fc. + F) (f y = F (A-' - i<'), R^NUn^^^y For example, with k^ = ^b^, (20) R = ^Vsin H^ - «)' ^^= i ^^<^°^ (^ ~ ") ' , , . , , , iV rcos ia cos id + sin Ja sin J0 ,^ (21) 2/ = i &iV./cOS J (9 - a) rf^ = iZ- - J V (Cos'ie - COS^a) ^'^ (22) ^L^ = J ^^ (cos ia COS- '^ - Sin Ja ch- ^ v^'^* 6 ^ n \ sm Ja COS ^a A/" / , , sin i0 -1 i__i cos ifi t - ( cos y cos-i^^^ - sm Ja sh ^-T-^ ^ n \ ^ ch. Ac. -^ sh *a With the axis of the cylinder vertical, /3 = Jtt, (23) b? ^ ^' di^ ^'^ constant, (24) 6f = .J, 5i^ = .(,-/■) (25) (l + p)^ + pM^(2/-^)+.9sin0 = O (26) y -Jc + ^^^ sin /3 = ^ sin (m^ ^ ^-^t^F + ') so that the level of the centre of the sphere oscillates, in the period -^/(l + pj- A rotation of the cylinder about its axis with constant angular velocity n does not alter the character o£ the motion. Sphere containing a Flywheel, rolling on a Table. 13. In a dynamical problem considered in a Senate House problem, Jan. 20, 1869, also by Bobyleff, and Kolosoff {Moscow Math. Collcetions, 1892), the motion is investigated when a flywheel is mounted with its axle fixed as a diameter inside a sphere, which rolls on a table. When the centre of gravity is not coincident with the centre of the sphere, the result will apply equally to the top with a spherical point which rolls on the ground, and the flywheel is mounted on the prolongation of the diameter. When the sphere is floated on water or mercury, the motion A\-ould be the same as on a smooth table. But the motion is very intractable, except when the C.G. coincides witli the centre of the sphere, and there is uniaxial symmetry about the flvwheel. Begin by supposing the flywheel is at rest, and measure M, A, ( \ of the previous, notation ; M and A mcludmg the flywheel, but C iunoring it, as a rotation of the sphere about the axle does not set the flywheel in motion when the pivots are smooth. PA5 Combining the two frames of reference of § 7 and 8, in fig. 110, (1) U = U^ — bb)y, V = Vy — — Jwj:, zc = (2) P=ei=-^sin0, Q=fl2 = f, i2=^+^ cos = ^+03, 03 = «cos0, (3) h = AP, h, = AQ, h^ = CR + K, if K denotes the A. M. of rotation of the flywheel, constant during the motion, (4) K = K cos d + hi sin 0, hy = A2, K = - h siii Q + h cos 0, (5) w^ = P cos + i? sin 0, uy = Qi w, = - P sin + P cos 0. The equations of motion are, in absolute measure, (6) f-M.-!-*. f + A=-X«, f=0, Hence ^^ is a constant, say 6^^ ; and (8) - h sin + A3 cos = A^i sin^ + (CP + iT) cos Q = G. Eliminating X and F, (10) ^ (A, + 1/6^0,,) - (A, + If^H) M = 0, |: (/i. + Mb'u^y) + (A, + im^o,,) /x = 0, so that we may put (11) hy + Mb\ = - Hsin ^, or (A + Mb^) ~= - H dn 4. (12) h^ + Mbhv^ = H cos ^, or {CB + K+Mb^R) sin 0- (4 + iW) ^l sin cos = i?cos i/, (13) (^ + ilf62-)2 ^^y + [((7P + ^ + ili^^P) sin - (4 + il/52) ^ sin cos 0]^ =3-. Then (14) {CR + K + Mb^R) sin 6 = {A + Mb^) fi sin cos + H cos ;A (15) ^ [(CP + X + Mb'R) sin 0] = (4 + ilP»s) (2 cos^ ^^ + sin cos ^) (16) sin e~[{CR + K+ Mb'R) sin 0] = (J. + Mb') cos 0(sin= ^ + 2 sin cos 0^ ^) = cos 0|-[(4 + Mb')fi sin^ 0] = cos ^ "^/-^^ '^ [(? - ((7P + Z) cos0] (17) A sin ^^ l(CR + K+ Mb'R) sin 0] + (^ 4 Mb') cos 9 ~[{CR + K) cos 0J = (18) A{C i- Mb') sin e J [{CR + K-AR) sine] + C (A + Mb') cos 6^^ [(CR + K- AR) cos 0] = (19) [AC + Mb' {A sin^ + C cos^' e)}~- (CR + K - AR) Mb' sin cos 0^^ = and integrating (20) (CP + Z- 4P)2[4(7 + Mb' {Asm'& + {7 008^0)] = a constant, say A{C +Mb')E' ; so that, with (.21) 2 = COS 0, A = -. — jz ^rpToJ 1 — /C = -^ ^-—5 ^ ^ ' A.C + Mb" A. C + Mb' (22) AC + Mb' (A sin^ + C cos^ 0) = 4 (C + Mb') sin^ + C (.4 + lf^>2) cos^ = A{C + Mb') [sin^ + (1 - ^) cos^ e] = A{C + Mb') (1 - kz') ^ (23) 6-P + A--^P= ,,/,,o P= ^ ^ 1 s/ (1 - >t2=^)' A - C A- C V{1 - kz') (24) ZT sin cos ^ = (6'P + /^ + il/^^P) sin^ 6 - (A + Mb') ^ sin^ cos = {CR + K+ Mb'R) cos20 - ^ ■^'^^ [ <5 cos - ( CP + Z) cos^ 0] A + il/6^ .^ A + MP ,-, a C ^ Mb' „ .. , 2N = ^^ _ ^ A ^^ (j cos - ^ _ ^ A V u - -?;^') so that the motion of the axle is algebraical ; and 216 (25) dz\^ -add diJ \ dt H H \' A + Mb' A + MbV (1 - i-) sin^ Q &m^ \P — ^ -)'(sm'^ - sin^Scos^^) K G A-C A cos b A + Mb C+Mb' E A-^ Mb' A-C -,v^(l -hz') = H'' (1 - z') - [K' - G'z - E' V (1 - kz')f = Z, (26) n={A + Mb'^)H', K={A-C)K', G = AG', E=^±^, {A-C)E, (27) (28) U /-, dQ rj, ■ , V = — Wa; = u sm COS 9 — ^ sm = — // cos i/- + ^ =rjy^ — sin t/ — it sin 8 5 . J. + Mb^ D' , Ci? + /I - .4i? . „ „, . ^ A - C E sin = — H cos i/- + J — ; — ^^^,g- sm = — H cos i/- + -p^ w ^ + Mb' With respect to fixed axes (29) (30) 1 dx u , V ■ , y -=- = ^ cos i/- — 7 Sin i// = 6 f/i? b ^ b A-C E' C + Alb' V (1 - /5:^2)- l4 — C E sin e sin ^^ C + Mb' ^{\-kz') sin e «(/ 6' + ii6^ ' H' V (1 - /fcs') (/^ I dy u . , , V , o, J. — C ^' sin cos li b dt b b C + if6^ s/ (1 - /(---) = - H A- C E^ ^ A- C E' K - G"' C + Mb' H' C ^ Mb' ' H" v^ (1 -kz'Y 14. Bobyleff considers the case oi A — C and ^ positive, so that as 1 — A; is always positive, put k = sin^ a. But with A — C and )?; negative, put k = — sh'a, I — k = ch^a ; and then in parallel columns : — k = sin^ a ^ sm a = 2q 1 + q' -I 2 rf^ sin a = 2 -y- i-„-^ da (1 + q-y ^ c?a; _ 2dq ^ = -^ - C E' b C + Mb' H' sin a ^ = tan J -, 2 sin a = sin - c c Q = (1 + q'fZ= H'' (1 + q'Y - i:?V sm^a K'{\ +q')-2-^-E'{l-q^) sin n ^ ' / G'q sina VZ I + g'- ^Q dy . b + "- H'dt = c c sin a K (1 + ?^) - 2 ^''/- - i;' (1 _ 02) sm o ^ i ' 1 + q' 2_dq k = — sh^ a 1 - /..^ = fL±x U - 9=^ C sh a 2y \- q' dz sh a = 2 -^ + t 2^9 -,rf^ b d^ c ~ 1 - ?■' C - A E C + Ji/r' R sh a .y = th J :^, r sh a = sh - Q = (1 - 9^)2 Z = Z?'2 (1 - 0^)2 _ 4g'^y ^ ' ' sh^ a 6 n/Z \ -~a^' s/ n 1 i^- + * E'dt ""'^'-^^-'^-^(^^r). so that ^ and y are given by elliptic integrals of the third kind. sh o r+^ q' Jq 2ir A special illustrative case can be constructed, in the manner of §2, p. 116, Chapter V, by adjusting the constants so as to make (1) ^ Sh-l 1 ± o2 2 pj^_i ]^ ^ ^2 _ cos"^ / . ~ ch-1 V r I ± q^ ^iQi _ sin"^ ^ch-i sh- / ^2Q V 1 ± Q1Q2 = Q ; as, for instance, with E' = 0, and R constant, (2) /= cos- sm , /[" h - a 2{h + a)g + c(l + q^) ' V L 2cA ■ 1 + ^2 ■1 /\ -h - a 2{h -a) q - c(l + q^) ' V L 2cA " I + q^ ,„. dl ^ 1 - q' V (g^ - A^j ^^ (^j 1 + ^2" v/ Q ' as developed later in § 29. Put C = 0, k = 1, when the casing is light compared with the flywheel ; and then (4) cos 9 = sin -, Q = H'\l - q') ' - [K'(l + q') - 2G'q - E' (1 - q')]' = QiQ^r (5) (6) Q„ Q, = {H' T E') (1 - q') ± K {I + /) T 2G'q ^ A-t Tj'fif = Qi ~ ^2 dq c c 1 + q' ^{QiQ^y Putting K' = as well as C = 0, the flywheel has no rotation, or is reduced to a diametral rod, fixed in a light spherical case, rolling on a table. By putting Z^ = in the general equations the motion is obtained of a solid sphere with uniaxial symmetry, rolling on a table, say a homogeneous ball loaded symmetrically with equal weights on a diameter, and no bias. Then (7) dt. = H'' (1 - ^^) - [^V (1 - kz') + G'zf, and, as in (11) § 2, ^ and y are given by integrals of the form r (8) Z sin- + ilfcos- / (a cos 2-+^ sin 2- + h V \ c G or Zsh- + ilf ch^ ^(ach2- + bsh2-+ h and by a change of origin in x, these integrals are reduced to the form in (15) (16) § 12, and are non-elliptic, but expressible by the functions X X sm . X sm - sm cos -^_^ sh-i„, _^ ch-i sin a sh a ' cos a £, cos-^ X COS - c ch o _ A ^ With {? = 0, -^ is constant ; and with yl — C = 0, the ball rolls uniformly in a straight line. (1) (2) (3) 15. In Jukovsky's exceptional case, J.-C=0, ^ = 0, and (19) § 13 reduces to C (C + Mb') ^ - Mb'K sin cos e ^ = 0, ^ dt dt C {C + Mb^) R + ^MU^Kz' = CF, a constant = — W sin !/■, ?) Til ' 1 K sin d -^=-n cos ^ + -,^-^„ 1 clx bit K sin sini// _ K ■ a dO C + Mb' H^'^'^'dt' — + —rZ — constant, b // ' 28570 2 K 218 (4) hdf~ ■ C + Mb' (0) H' sin e cos ^ = gj^ + ^ty^-i^ ,i,3 , _ ^' eos . + ^^^4^cos^ « C + Mb' C + Mb' C. C + Mb'' C (6) (7) H' sin sin i/- = tt j (i) = ^" ^' - '^ - ■ F + K _ ^, , IMb'K^ y_ r, C + Mb' C.C \ Mb'\ ' and Z can be thrown into Abel's form (8) Z ^ E \ Fz - {a ^ ^z ^ -yz'f. Then in the notation of Math. Ann. 52, p. 468, (^) ^-^-^^?^=^^(^-7f.)' '=M^'' (10) a + |3^ + 7^2= Jf 1 - + I + y 2(s +'x) ^ 2(s + a;)2 M d X du \_s + X l^S — s (11) /(a + ^Z + yz') dt = 2M'^ (i^ - lu) = M'y-{-l{u + V) -1{U-V)] so that 2/ will be given by the Second Elliptic Integral. 16. From equations (11) (12) § 13 (1) dQ {A + Mb')'^= -F/iCos^ = - {CR + K + Mb'^R) ;u sin + (4 + Mb') fx' sin Ocosd = F (0), suppose ; and differentiating with respect to 9, (2) (^^^^^^^)^^ = ^'(^)' (3) F' (6) = - (CE + K+ Mb'R) fx cos 6 -{CR + K + MPR) sin i-^ Qdt -{C + Mb') ^ ^ sin + {A + Mb') fj? (cos^ 9 - sin^ Q) +2^i sin cos ^4' Q dt Differentiating (8) § 13 with respect to 0, A sin- i^ + 24 ;u sin cos + C dR (4) 4sin=0^^ + 24;usin0cos0 + ^^^'0080 - {CR + K) sin = 0, and in a state of Steady Motion, (5) Q = 0, H= 0, Ci2 + Z + .M^>2i2 = {A + .1///^) ^^ cos ; and then we find, after reduction, (6) F (0) = - (4 + Mb') ,', ^^+,,2^0, so that in the small oscillation superposed on the steady motion the axis makes one beat up and down m one circuit, and the apsidal angle is w. The radius c of the circle is given by (7) c^t = v= -bw^= -b{- ,.i sin cos + i^ sin 0), with ,. = ^^^ ^ K + Mb'R (8) so that table in c__ CR + K - AR . {A + ili62)cos0' if in fig. 110 A, OC, OA cut the vertical through the centre of the circle on the JJ, Hi, 219 (9) ^ ^ CR + K - AR DE {A + Mb')R ^ ' b CE + K + Mb^R' ^~ b - CR + K + Mb'R' h R b - BE cos 6' Thus with K = - (C + Mb^)R, c = oo, and the ball rolls in a straight line ; while K = {A — C) R makes c = 0, and the centre of the sphere is at rest. But the formula for the beats does not apply when = Jtt, and the ball rolls in a straight line with the axle horizontal. Here ) = {K sin e + CRsmd - C^ cos 0] ^ dv do (20) u = b ~^, V — bfi sin cos 9 — bR sin 9, h, = - A^ sin 9 cos + {CR + K) sin 9, K = A ^, (21) ^ - ^ic = b^ + bRfi sin 9 - b^' sin cos = ^ (22) ^ + uh, ^ A^^ +{- A ^i cos 9+ CR + K) ^ sin = - Xb, 28570 2 E a 220 in which fi^ may be neglected ; so that (23) K^ -r -^^ . ^ (A + Mb') ^+ (CR + K+ Mb'R) /x sin = 0, dR (24) A{A^MW)^^^ -h {CR + K^Mb'R) {CR + K - C ^ co. 0) = ^ df ,,.- 2 CR+ K+ Mb'R CR + K- C (CR + K - AR) Mb' cob' e Tic + Mb') sin'' d + C (A + Mb^ cos^ 9) [K+ (C + Mb') R] \rC+ _Mb' sin^ 6)K+ (C + Mb') CR] - A. A + jW WIA {C + Mb') sin^ e + C{A + Mb') cos^ 0] reducing to the previous value in (15) when 6 = Jtt. Thus with C = 0, when the inertia o£ the case is ignored, (26) / = {K + Mb'R) K 'A. A + Mb-' K - Mb -I sin 6 K A. A + Mb' so that the ball will not roll straight if bowled underhand with (27) V > iTsine or 72 > K MT' Mb-'' And generally / is negative and the ball cannot roll straight with (28) [K + {C + Mb') R'JliG + Mb' Bin' 6) K + {C + Mb') CR] negative or ■k-^,±J€v b sin d \C + Mb-' sin^ Q)K- ^ + ^^^'- ^ V (29) Kb sin -,, (X-Z, = (7) (X - ^) (X - B) (A -C){X-D) + ^Xi-X- B) (X - C) + -^X (X - C) (X - ^) + J X (X - ^) (X - 5) = 0, a quartic for X, the roots of which interlace with A, B, C, D. The polar plane of a quadric (g>) uF + G = T / ah bh ch \ ■ ,i with respect to the pole (^ .^ _ ^ , x^^T^' T^"Cl reducing, when X is a root of the quartic (7), to ^.,(APa ^ BQb , CRc ni) - q and is therefore the same for all the family of quadric surfaces like (8), passing through the line of intersection of (1) and (2). The four polar planes for the four values of X in (7) thus form a tetrahedron self conjugate to the family of surfaces ; and referred to quadriplanar co-ordinates x^, ajg, x^, x^ of this tetrahedron of references, (1) and (2) may be written in the form (11) 2F = a^Xi^ + a^X^ + a^X^ + a>^X^ = Sop^p^ (12) 2G = Xiaia?!^ + Xaoaica^ + X^a^x^^ + \^a^x^ = ^XpOpajp^, with the suffix p, q, r, s in any order, instead of 1, 2, 3, 4 ; and here APa , BQb ^ CRc ^, (13) -p = x7^^ ^ x;^^ ^ x-^ - ^^' and \ F — G = gives the four quadric cones through the single curve of intersection. The identity of the expression in (11) and (12) „„ ^ / APa , BQb , CRc r)j^\^ requires the co-eflBcients of Pa, Qb, Re to be zero, from which (16) Da^ = - \- ^- K- B . X^- C ^ Xp — Xq . Xp — Xr . Xp — Xg The identity is assured at the -same time of the co-efficients of AP^, BQ^, CR^ ; for the co-efficient of AF^^ in (14) is /i7\ A ,.-i Y °p = — -—^ Xp — -B . Xp — C (1^'' {\-Ay 2) "Xp- A. Xp-X,. Xp-X, . Xp-X, _ J^2 A- B . A - C _ D ■ Xi - ^ . X2 - 4 . X;, - ^ . X, - ^ ~ ' by the theory of the partial fractions of A- B . A- c • .-. . ^ a; C181 -T =;^ -A =i^ r ^ -A ^ in the form 2 - — H— , ^^°^ .4 - Xi . A - X2 . .1 - Xg . ^ - X4 A - X,,' and also fi-om the quartic equation (7) (19) Xi - A . X2 - A . X3 - .4 . X4 - .4 = 1!"' . .4 - B . A - C. 223 So also with (20) .., = {AP + ah) — -« + (BQ + bh)-±-^ + {CR + ch) -^, - ^, -^p — -1 Ap — iJ A„ — L Ap in the comparison of (12) and (16), the co-efficient of Z>%= in (12) is (21) 2^= - i-S 1 ^ X„ - .4 . X. - 5 . Xp - C Xp U Xp . Xp — Xq , Xp — Xj . Xp — Xg But resolved into partial fractions (22) X — A . X — B . X - C ^ ^ Xp - ^ . Xp - ^ ■ Xp - C ;v - Xj . X - Xg . .« - Xg . a; - X4 « - Xp . Xp - Xq . Xp - Xj . Xp — X/ so that putting ;i; = (2S) V Xp — J. . Xp — ^ ■ Xp — (7 _ _ ABC _ _ j^ ^P • ^P "~ ^q • ^p ~ ^r • ^p ~ ^9 ^1^2^3^4 ^ as required, since XjXgXjXi = ABC D in (7). The coefficient of (J.P + ahf in (12) is (24) V . \«P«^ = _ ^ 2 ^p • ^p - -S . Xp - C (Xp -AY D \- A.\-\.\-\.\-K = ^ A- A- B . A- C ^ , D \- A.\- A. \- A.\- A ' as required, and the coefficient of AP + ah vanishes by reason of equation (14) ; so that the identification is verified completely. 19. Employing a variable X, we can put (11 «...'= ^^•'-'" *p '"p along the polhode curve of intersection of (21) and (22) ; and then (2) DM . X„ - X ■p ^ A.\-B.\-a /M_ ^ {\ - A . \ - B . \ - C . \ -\) , "^ ''^ V x» • Xp - Xq . Xp - X, . Xp - X, and by the solution of (20) §18 we find ^^pap«p_ (3) P = ah -4HJ=_ii, AP^ah = ah ^--^ ^_Xpap^ (4) a = hh ;r ~^ , bq + hh = ih \^~ ^ Xp«p.a;p (5)- E= ch t'~ ^ , CR + ch = ch i^~^ With i^ and G in the homogeneous form in (11) (12), (6) jj-f^^ = B (BQ + hh) CR - C{CR + ch) BQ = BC ih R-h u) = sc^= ^^^^w (7) dl^iiD = la {AP ^ ah) + b {BQ + bh) + c {CR + ch) - D'h] AP vt V ''1 -^ / + A {AP + ah) Dh P= ah ^ "p^'p -Xp - A SapiCp Q =bh ^ «p«p -Xp-5 2«pa;p ' V «P*P R = ch ^Ap- C 224 and eliminatinp- Z>, » (8) %-t\^'p} = [«^ i^P + «^) + *^ (^^ + ^^) + ^-^ (^^ + ^^) - (^P + ahf - (BQ + bhy- {CR + chy] P + (AP + ah) (AP" + BQ' + CR') = (AP + ah) (BQ' f CR') - BQ (BQ + bh) P - CR (CR + ch) P = BQ [(AP + ah) Q - (BQ + bh) P] - CR [(CR + ch) P - (AP -r ah) R] = bc(q^- R^-^ \ dt dt These equations maj- be written symmetrically ,. ^rlR " dQ h d (G,F) ^^ ^ dt ~ ^ dt - ABC d(h, P) ' ^ ^ dt ~ -^ dt' ABC d(Q, R) ' with four more obtained by a cyclic change ; and by the linear transformation of (3) (4) (5) they are equivalent to (") ■'• t - -. It - ^&} = ^' (*' - *■) "'■■"■'••'■ .,o\ 1 dx 1 dx„ _ 1 f 1 1 \ dX _ j^r /. n, ^ a^a^x^x^ x^ dt x^ dt \X — X^ X — Xp/ dt reducing to (13) '^ = 2iV n/ (a,aoa,a,) V (X - X^ . X - X^ . X - X3 . X - X4), nf = J^, A = n (X - Xp), so that X is given by an elliptic function of the time. 20. The linear substitution which leads to the original Weierstress variable s in § 6, Chapter III, is obtained from the correspondence Xi, Xg, X3, Xj, 00 (1) X — X„ _ s — Sg A ^ S K — Xj _ " — s dX _ ds X - x^ ^ - sa' (x-x,)4 2' X - X, o^^6^' r^rT~] s Si — .? Si — S3 Ss — s Ss — S3 ■S — S3 X3-X3. X - X, - cn^ M - ^'' ~ '^^• X2 - X X2 — X3. X - X4 - sn2 u - ^2 - X4. X - X3 /ON X„- \.dX ^ n/S ds 1 V (Xj _ X3. Xj - X,)tA _ v (sj _ S3) ds , ^ ^ VA Si > a- > ^2 > s > S3 > — 00 , (4) [' - '_ . = dn'u = ^' ~ ^^- ^1 - ^ (5) (6) ... _. S2 ~ S3 X2 — X3. X — Xj (7) . sn^/Zr =''-""= ^i- ^^, Si — S2 Xj — X^ (8) cn^/A" = ^~-l2= ^J - '\ Si — Sj Xj — X^ (9) dnVA" = "-^^ = ^-""-^ (10) .- = r^T^~ ^, -^'^ = ^ ^'- ^^ ~~^' \ - X3. \ - K \ - X,. X., - x\ n n "1 *'i^ - X4 - Xg. X3 - x^. X, _ X x^ - x, s, Xi - X^. X, - X3. X - X4 Xi - X, s,~- ,, n.2 „ „ . 2 ^' S, — (T, 1 — T. ^1 Si - S„. Ni - .;/ Si - So. 52 - S3' „, a;' 52 - S3. Si- S3 225 But the previous a and fK' of § 6, p. 54, Chapter III, corresponds to A. = 0. When a return is made to the original Poinsot motion by taking a, b. c zero, the four roots of (17), § 17 are A, B, C, D ; and if the variable X is employed in a symmetrical treatment, (12) A]-= ^ /•/-/. . , W= ''-'-^ A- B. A- C. A- D' ^ B - A. B - C. B - D' r-R^ = M.\-.C _ ^,2 _ M. \- D C - A. C - B. C - D' ~ D-A. D - B. D-C /jgx AP" D-A. D-B. n-C. A-\ D¥ ~ A - B. A - C. A - I). X - D' 21. If a = 0,\ = A ; and if a = 0, ^ = 0, Xi = ^, Aj = 5 ; the flywheel axle is parallel to a principal axis of the body and X^, X^ are the roots of (1) (X-C)(X-Z)) + c^^-=0; (2) ^i = ^P, ^■2 = BQ, a', = ^^,-m, x,=^^-Dh, (3) ^^'~ 1 _ 1 ~ X3-X, (^4 -A3) A3 — O A4 — o (X4- C) Xi - (X3- C) x^ (4) Dh X3 - A^ (5) X3 - 6\ X4 - C = 6-2 £ X3 - i> . X4 - D = r (A/3 *"" ^64 (6) -K = ch — ^^^^ (A3 - 6) .T3 - (A4 - 6) a,'4 /™\ 1 1 — X3+C A4 — C ^^^ "^~Z' "'-W "' - (X3 - A,) i)' "^ - (A3 - X4) D- With a = 0, b = 0, in the general equation (8) AP" + BQ' + CR' - Dh' = (9) A'P' + B'Q' + {CR + chf - D'^h^ = (10) A(A - B)P' = C(B - C) R' -- 2chCR + [D {_D - B) - c'] h' (11) B{A- B)Q' = CiC - A)R' + 2 ch CR + [D {A - D) + c'] ¥ With X3 = A, = B, (12) {B-O (B-D) + c'^ = 0, D (13) A{A- B)P=%[^{B. B - C)n-^{D. D - B) h]'. But if X4 = Xi = A, (14) {A-O {A-D) +o'^=0 (15) BiA-B)Q' = ^[>y{A. C - A)R +V{D. A- D) hf And if Xg = X4, then P and Q^ have a common factor i?\/C + An/Z>, making X3 = — n/ (C1>) ; so that in these three cases, the dynamical equation (16) C'^^'={A-B)PQ determines R, and P, Q, by the circular or hyperbolic function of the time t. Thus when Xg = X4, putting R^ C + h'^ D = x, (17) ABC{^^^=x'[{B-C)x-2{B + VCD)hVD] [2 (A + ^CD) h^D - {A - C) x], and the motion is expressible by the circular function. 28570 2 ^ 228 The motion of the axes in space is the same as if A, B, C were constant and the body was acted on by a couple X times the A.M., and about the same axis, where X = — ^tM; because /(O (7) A^- {B- C)QE = L= -\AP, dt (8) ^^ [APfm - (B- C) QRfit) = 0, or a ^ - (& - c) ?r = 0, (9) p = P.m, a = A, dt Thus with X constant, f(t) = e-'>'* = —,\r = e^* - 1. ClT and with/(() - 1 - 4 ,'-tl>r. 'O ^0 *0 as in Sundmann's transformation employed in the Problem of Three Bodies. The Liquid Gyrostat. 25. This is an instrument invented by Lord Kelvin {Nature, Feb. 1st, 1877), where the body is a shell, mounted centrically in gimbals, and filled with liquid, which may be supposed frozen at first and moving bodily, then melted ; and finally frozen again, or else moving bodily with the original A.M. unchanged, when internal motion in the liquid has been destroyed by viscosity. The motion will be the same as if the flywheel in §17 was first clamped, and then merely released. The behaviour of an itg^ spun on a table will illustrate the same difference, according as the egg is raw and liquid inside, or hard boiled and solid as a china or chalk Qgg. So also with a sparklet, according as it is empty, or full of liquid carbonic acid compressed. Consider then the motion of gravitating liquid filling a case of thin sheet metal in the shape of the ellipsoid (1) fi+lJ f £^ = 1. a^ ¥ c^ and mounted in Cardan ring gimbals, permitting general motion about the centre 0. First suppose the liquid frozen, and the ellipsoid to have components of angular velocity ^, v, t ; then u, u, to the velocity components are given by (2) u = - yt + zr,, V = - zS, + xl, w = - XV + yS,; and the body has a Poinsot movement. Next suppose the hquid to be melted suddenly, and additional components to be communicated to the case of angular velocity Qi, Q^, O3 ; the additional velocity components of the liquid are to be derived from a velocity function as may be verified by considering one term at a time. Then denoting by «', v', w' the velocity components relative to the case which has components of angular velocity P, Q, R, ' ' ' (4) P = i\ + E, Q = Q,+ „^ R = Q, + I (5) «' = „.,ii-.-a = -^0^_^^0,. (7) »=» + .Q-,P_,j£La,.-^-^a^. 229 Then since (8) ^ -a + ^' T^ + "^ -2 = 0, a liquid particle remains always on a similar ellipsoid, a stream sheet. The hydrodynamical equations of motion with moving axes, taking into account the mutual gravitation of the liquid, become (9) ^-f 4 ^ f> ax . J du T, , ^ , , du , , du , , du ^ i-n-yp Ax + — vM + wQ+U-^+V -3 ^ W -^ = V, ctt cix ciy ctz (10) .4,5,C = j^ aha %, P2 = 4. X + a^ X + &^ A + c^ the form (11) '0 X + a^X + h\\ + c^ P' With the values above of m, w, 10^ u , v', iv', the hydrodynamical equations reduce to Idp p dx + iwyp Ax + ax + hy + gz = 0, (12) (13) -J + 47ryp By + hx -^ ^y + fz = 0, - ^ + ■^^yp Cz + gx + fy + yz = ; and integrating (14) - 4- 27r-yp (4«^ + P^/^ + C^:^) + J (aa;^ + jS?/^ + yz^ + 2/92r + 2gzx + 2^«t/) = constant ; P so that the surfaces of equal pressure are similar quadric surfaces, which must be co-axial with the ellipsoidal case from symmetry and dynamical considerations ; thus /, g, h vanish, as can be shown also by a long algebraical reduction ; and (15) ^c'(c'-a') c,2 2\2 Q.: (c' + a') {a' + b'f ' c^ — a^ c' + a Q,- 3 "2 6^ .N^ \a? + b^ with similar expressions for jS and y. If we can make (16) {4.irypA + a)a^ = {4:7rypB + (3) b^ = (i^rypC + y) c\ the surfaces of equal pressure are similar to the external case, which can be removed without affecting the motion, provided a, f3, y remains constant. This is so when the axis of revolution is a principal axis, say Ox, when- (17) Qi = 0, Qs = 0, t = 0, ., = 0. If Qg = or P = 2i in addition, we obtain the solution of Jacobi's ellipsoid of liquid of three unequal axes, rotating bodily about the least axis ; and putting a = b, Maclaurin's solution is obtained of the rotating spheroid. The reduction to the form (11) (12) (13) is obtained by the introduction of the -dynamical equations (18) where (19) ^- hR + hQ = 0, ^-h^P + h,E = 0, dt " at dk^ 'dt - hQ + h^P = 0, hi = 2m {yw — zv) = Qi ^1^; 2m {f - z') + ?2m (t/^ + ..^) .(20) (21) and <22) 5 h = ^M i^^ Q, + {b^ + c^) ( (c' - a') Qa + (c^ + a^) V C + a' a^ + h' hi 4- h2 + h^ = G^, a constant 228 The motion of the axes in space is the same as if A, B, C were constant and the body was acted on by a couple X times the A.M., and about the same axis, where X = — ■ ., J ; because (7) j^dP _ (^B - C)QR = L= -\AP, Cto (8) I [APfit)] -{B-C) QRM = 0, ovaf^-ib-c)gr = 0, (9) P = P.m, a = A, dt Thus with X constant, f{t) = e-^* = _, Xr = e^« - 1. But with /(O =f=l-'-, ;- = log^, / = 1 - exp (- ;-) ; ■p, - - tn , '0 ''0 *o as in Sundmann's transformation employed in the Problem of Three Bodies. and with/(0 = 1 - .-3, r = th -, In t-n '0 The Liquid Gyeostat. 25. This is an instrument invented by Lord Kelvin {Nature, Feb. 1st, 1877), where the body is a shell, mounted centrically in gimbals, and filled with. Uquid, which may be supposed frozen at first and moving bodily, then melted ; and finally frozen again, or else moving bodily with the original A.M. unchanged, when internal motion in the liquid has been destroyed by viscosity. The motion will be the same as if the flywheel in §17 was first clamped, and then merel}'^ released. The behaviour of an egg spun on a table will illustrate the same difference, according as the egg is raw and liquid inside, or hard boiled and solid as a china or chalk egg. So also with a sparklet, according as it is empty, or full of liquid carbonic acid compressed. Consider then the motion of gravitating liquid filling a case of thin sheet metal in the shape of the ellipsoid o 9 9 and mounted in Cardan ring gimbals, permitting general motion about the centre 0. First suppose the liquid frozen, and the ellipsoid to have components of angular velocity S„ v, t ; then u, v, lo the velocity components are given by (2) u = - yl + zri, V = - zS, + xt, w = - xn + y^ ; and the body has a Poinsot movement. Next suppose the hquid to be melted suddenly, and additional components to be communicated to the case of angular velocity Q.^, Q^, Q3 ; the additional velocity components of the liquid are to be derived from a velocity function as may be verified by considering one term at a time. Then denoting by w' «', z«' the velocity components relative to the case, which has components 01 angular velocity P, Q,R, ' (4) P = i\ + ^, Q = Q, + ,^ R = Q^ + i (5) u=u.yR-.a = ^^^--^^^^ (6) .•=».. P -...= 5^... _^^,.„ (7) »' = »-y X^- X^l ^^ 'iJ 3 \ J JS JP2 _ ^2 ' requiring the third elliptic integral ; thence the expression of Xy + x^i and y^ + y^i. Introducing Euler's angles 0, ^, ^ as in fig. 56, (14) Xi_ = F sin d sin ^, x^ = F sin cos ^, x^, = F cos 9, (15) ^'1 + ^^i ^ _ g_2^i ^ I / Xi + x^i ^ df Xi — x^i ' dti ^ V xi — x^i dt'' given by the rigbj-hand side of (13) ; (16) v,= U = -^^= px, + qy,, J. dT ^^2 = ^=J^= P'^'2 + ?2/2, axa^ (17) e,= P='^^ = q^,^,ry,, dT O2 = Q = -J- = qx2 + ry2, ^3 = -K = — = q'x^ + r'y^, (18) sin -^ = P sin ^ + Q cos = (qxi+ ry^) sin ^ + (qx2+ ry^) cos .^ = qF smQ + r (y^ sin * + V2 cos ) = qF sin d + r ^i-Vj + ^a.V2 •^ r ^3 r; ^ P sin «3,y3 Fdx^ given by elliptic integrals of the third kind. 28570 2 G 232 And conversely, as in § 30, Chapter I., if the kinetic energy T is expressed as a quadratic function of the components of momentum, denoted m § 1 by Xy, x^, x^, and y„ 2/2, 2/3, the partial diiferential coefficient with respect to a momentum component will give the component of velocity to correspond. These theorems, which hold for the motion of a single rigid body, are true generally for a flexible system, as considered here of a liquid medium with one or more rigid bodies swimming in it : and they express the statement that the work done by an impulse is the product of the impulse and the arithmetic mean of the initial and final velocity ; so that the kinetic energy is the work done in starting the system from rest. Thus i£ Tis expressed as a quadratic function of U, V, W, P, Q, R, the component of momentum corresponding is dT dT ^_dT d_T dT dT ^^^ '''^W ""'^W '"'-JW' y^'dP' y^ dQ' ^' dR- But when T is expressed as a quadratic function of «i, a-g, «3, yi, y^, Vi, /IT rJT dT ^ dT r. dT n dT ^^^ ^ ~ S;' dx^' dx,' dy,' dy^' dy. The second system of expression was chosen by Clebsch, and adopted by Halphen in his Fonctions ellvptiques ; and thence the dynamical equations follow, given already in §1 of this chapter. When no external force acts, the problem we shall consider, there are three integrals of these equations of motion which can be written down immediately : (A) T = constant, (B) x^ + x^ + x^ = F^,a constant, (C) x^yi + X2y2 + x^3 = X = GF, a constant ; and the dynamical equations in (1) §1 express the fact that «i, x^, x^ are the components o£ a constant vector i^in a fixed direction ; while (7) shows that the vector resultant of 2/i, 2/2> Vi moves as if subject to a couple, of components (3) x^W - x^V, XsU - XiW, ,«ir - x^U, with U, V, W for Uj, Vj, v^ ; and the resultant couple is therefore perpendicular to the vector F, so that the component A.M. along OF is constant, as expressed by (C). If the medium was absent, these couples would vanish, and the C.G. of the body would move uniformly in a straight, while gyrating about the C.Gr. in a Poinsot movement. But the presence of the medium has the elFect of making the apparent inertia a diiferent constant in the direction of the three axes of reference in the body, while adding a constant term to the moments of inertia. A geometrical representation of the imotion has been given by F. Kotter in Crelle, 121, p. 300, analogous to Poinsot's, when the medium is absent : — " Roll a plane conic on the velarium cone of a spherical catenary, made by a disc of flexible cloth cut out by concentric circles and hung over sloping rods ; the normal of the plane and two lines fixed in the plane at right angles will move parallel to the principal axes of a body moving in infinite liquid." Kotter considers the special case where (4) 2T = %,^ + b,yi + h,yi + {- + m) x,' + [~ + m) x^ + {- + m) .v,\ and the motion can be determined by the elliptic function ; the centre of the conic is then at the vertex of the cone. 28. If a fourth integral is obtainable, the solution is reduced to an integration, but this is not possible except in a limited series of cases, investigated by H. Weber, F. Kotter R. Liouville, Caspary, Jukovsky, Liapounoff, Kolosoff, and others, chiefly Russian mathematicians ; and their solution requires the hyperelliptic inteo-ral. In the motion we shall consider which can "be solved by the elliptic function, the most general expression of the kinetic energy was shown by Clebsch to take the form' (1) T=i2) {xy' + xi) + Ip'xi + q {axvy + Xoji/^) + q'x,y, + |r {y^' + y^) + ^r'y,\ so that the fourth integral is given by (^) ~dt ~ ^^ .'/a '^ constant ; 233 (3) ~ = xi{qx2 + rys) - x^iqxi + ryj) = r {x^y^ - x^y^) 1 (dx \^ (4) -2 [-—) = W + xi) iyi' + yi) - {xiy, - x^y^f = i^i' + ^i) iyi' + yi + yi - G') -{Gx,- Fy,)\ (5) x^ + xi = F^ - .rg^ Xi iji + X2y2 = FG - x^ y^, (6) r (yi' + yi) = 2T - p {x^^ + x^) - p'x^^ - 2q {x^y^ + x^y^) - 2q' x^y^ - r' y^ = iP - P) '^i + 2(? - q) ^3 2/3 + TO] (7) m^ = 2T - pF^ - 2qFG - r' yi, a constant, and so making (8) ;i f^^)'= X = {xs, ly = {x,y, - x,y,Y, dtJ where X is a quartic function of x^, and thus t is given by an eUiptic integral of the first kind ; and by Abel's inversion, x^ is an elliptic function of the time t. Next we write (9) {x^ - x^i) {yi + y^i) = «i2/i -l- x^^ f i («i?/2 - x^{) = FG - x^y^ + i-J X QQN yi + yji ^ FG - x^y^ + i-J X Xy "r X^ X\ ~r X^ (11) ^^ {xi + x^i) = -«■[(?'- q) «3 + »"V3](«i + ^aO + «>% (2/i + 2/2O (12) ^Jog («i + x^) = - {q' - q) x^ - r'y^ + rx^ "^^^^ J'g ^ (») B ">« y i^- =-(''- ») -" - ('•' - ••) ^' - ^^ %^' requiring the third elliptic integral ; thence the expression of x^ + X2i and y^ + y^i. Introducing Euler's angles 0, ^, i// as in fig. 56, (14) Xi = F sin Q sin ^, x^ = F sin cos ^, x^ = F cos 9, (15) ^^1 + ■^'2^ ' = _ g-2^i A w /£i_±_f8! = ^ a;i — ^2*' ' dti •>/ x^ — x^i dt'' given by the rigbj-hand side of (13) ; (16) v^= U = J- = px^ + 92/1, «2 = ^ = 5^ = -P^'2 + ?2/2, (17) 0,= P = ^=^^, +.,2/,, dT 0-2 = Q = -^ = qx2 + ry2, ^^ = ^ = 5I = ^'^^ + "'^^^' (18) sin -^ = P sin ^ + Q 00s ^ = (qxi + ry{) sin (|) + {qx2 + rya) cos f di = qF smQ + r {y^ sin + y2 cos ^) = ^F sin + /■ ^^^ ^ '^j'^^ given by elliptic integrals of the third kind. 28570 2 G 234 Then, employing the formulas of p. 72, §21, Chapter III, dx dy_ (20) §£= f^cos X« + F cos Yx + PTcos Z,r, f=n cos Xy+V cos Y?/ + W cos Zy dt {21) ^ + i^ = U (cos Xx + i cos Xv) + V (cos r.'c + i cos Yy) + W (cos Za; + z cos Zy) ^ ' dt dt ^ .7/ V = (/'■^'i + 9^1 ) (cos ^ + 2 cos sin ^) e+i + (iO'^2 + qVi) (- sin (j> + i cos e cos ) e't'^ + {p'x^ + q'y^) (- «sine) e'f'^ = q {yi cos ^ — ?/2 sin .^) e"^* + i [j) cos (a'i sin f + X2 cos 0) + ^ cos (?/i sin .^ + 2/2 cos (j>) — (p'xs + q'y^) sin 0] g^j = _ ^f%-_^=|/i .J'^- + ^■ [(^ _ ^') ^3 + (^ _ 9') 2/3 j sin e"^ ^ sin a = - <] + iq i sin ft x^ + x_^y^ cos - V3 sin ^J/i e"''» + i [(j3 - /)') «3 + (5' — 9') 3/3J sin e"^* Fsind integrable and verifiable by a differentiation, as shown by Kirchhoff, in the form (22) F (x + yi) = Fy^ + Gxs + «VX^^ s/{F-'-x^') (23) iF {x + yi) = y^ (cos ^ + / cos sin <^) e+* + 2/2 ( — sin ^ + i cos Q cos '!»i + qyi) sin sin (^ + (pa^'a + qy^) sin cos <|) + (^'^'3 + q'y^) cos = p sin (.I'l sin ^ + ajg cos ) + q sin (?/i sin

^-,2 / x - m V (x + mY \-T r ^^ X + m r \x + ml - {^'^ - ^y4^)' - F'X {x + my m (12) X = 4.-x r Jl {x + my + 2 ^-y^.V. (^^'^ - m^) + (/; - /) {x - my t , \2 (G — y^ , G + y^ \^ - (■'■ + ™) ( p • ■'*'■ + — Y^ ™J ('[^\ w^rif - '^^■*3 - ^^^^' J^G - x^y^ _ V Jt {16) jirm - ^7^ - ;7^, ^x VX ^-y^x + ^^^m) (14) (15) di^ - qFdt (. + «)(i^.. + «^=») dx TX' G - 2/3 dxp - (qF + IrG) dt = 2F x' + ^'™' dx VX' before in § 11, for the ball in a cone, with x for r. 28570 3&S 236 This again can be made to depend on the associated motion o£ a top, when a factor a + X is assigned of X', and we put /-, ^x X 1 - ^ G + ys _ L ± L' G,y3 ^ L, - L' . ^ ■> a ^ T+~z' ~T S^' F M ' so that the solution for a top can be utilised again in the construction of an algebraical case of the motion of a solid in a liquid, or evolutions of an airship. 30. When the body is projected in an axial direction, i^ - ^3 is a repeated factor of X, and 2/3 = G, so that (1) X={F- xiy [P-^ {F + x,f + 2 2l^ {F + ^3) - G' (2) rdt=^ dxo _ — dy ~ y ^{A-2By + Cf) suppose, putting F — x^ = y; so that if J. is positive, ._, .1 sh-i ^A^{A - 2By + Cy') _ J_ ch-^ A - By ^^^ VA ch-i y ^(B'^ AC) VA sh'i y^{B'^ACy the motion is unstable, and the axis falls away ultimately from its original course. Thus with G = 0, A = 4- ^ ~ ^ F^, which is positive for an elongated body like the gas bag of an airship. But if A is negative /4^ rt - ^ sin- ^(-^)^(A-m + Cf) ^ 1 eos- ^— %_ ^^^ '^^- V{- A)^ y^{8'-AC) ^(-A)^^^ y^{B'-AC) and the axis makes - \/ ( — A) nutations per second. TT To secure axial stability in an elongated body, A must be made negative by a suitable increase in G ; as discussed already in Chapter I, § 22, p. 19. The Motion or a Perforated Solid in Liquid. 31. In the preceding investigation the liquid stops dead when the body is brought to rest in it ; and when the body is in motion the surrounding liquid moves in a uniform manner with respect to axes fixed in the body, and the force experienced by the body from the pressure of the liquid on the surface is the opposite of that required to change the motion of the hquid ; this has been expressed by the dynamical equations. But if the body is perforated, it is possible for the liquid to circulate through a hole in reentrant stream lines linked with the body, even while the body is at rest, and no reaction from the surface can influence the circulation which may be supposed to have been started by the application of impulsive pressure across an idealmembrane closing the hole by means of ideal mechanism connected with the body. The body is held fixed, and the reaction of the mechanism, as the resultant of the impulsive pressure on the membrane and surface, is a measure of the impulse, of linear components S, »,, t, and angular components X, ^, v, required to start the circulation. This impulse will remain of constant magnitude and fixed to the body, which thus experiences an additional reaction from the cinjulation, which is the opposite of the force required to change the position in space of the circulation impulse ; and these extra forces must be taken into account in the dynamical equations. Take the simplest illustration of a ring shaped figure with uniplanar motion, and denote by t, the resultant axial hnear momentum of the circulation, • f ' -1 ^ ^^ •?■ n' ^^/P*'' 1' '^ ^^' ''"*^"" °f *^^ "^^g i« ^^oved from to 0' in ch'^nge ta'^^ ^^ ^ '°^ ' '^ ^' '^'' ^^^^^P^"'^"*^ of liq^Hd momentum 237 (1 ) aM'u + S, and jiM'v, along Ox and Oy, to aM'u + S and (3M'v', along O'x' and O'y', the axis of the ring changing from Ox to O'x' ; and (2) u = QcosO, V = Q sin 0, m' = Q cos (0 - i?0, v' = Q sin {0 - Rt), so that the increase X^, Fj, iVi, of the components of momentum, linear and angular, are (3) Xi = {aM'u + S) cos Bt - aM'u - ^ - l5M'v' sin Rt = (a - j3) M'Q sin (0 - Ri) sin Rt - ^ (1 - cos i^O (4) Fi = {aM'u + S,) sin Ri + (iM'v' cos i?^ - (3M'v' = (a - j3) iJf'Q cos {9 - Rt) sin i?^ + ^ sin Rt (5) iVi = [- (a M'u' + £) sin (0 - Rt) + (iM'u cos {9 - i^^J 00' = [ - (a - j3) M'Q cos (0 - i?0 sin (0 - Rt) - ^ sin (0 - i2^] Q? The components of force X, Y, and couple iV acting on the liquid, and reacting on the body, when the centre is at 0, are then (6) X= It— 1= (a - j3) M'QR sin = (« - /3) M'vR (7) F = It ^ = (a - i3) M'QR cos + ^S = [(a - /3) i/z« + ^]R (8) iV = It — 1 = - (a - jS) il/' Q2 sin cos - £ Q sin = - [(„ - |3) If' m + 5]u If the body is free, the additional forces acting on the body are the components of the kinetic reaction of the liquid ^o that the equations of motion are <10) if (^ -vR)= - aM' (^ - vR) - (a - j3) M'vR R <11) M{^^ + uR) = - aif ' ( J + uR^ - [(a - 13) M'u + ^] (12) C~ = - eC ^ + [(a - jS) M'u + S,]v etc Q/b and putting as before on p. 18, (13) M + aM' = Ci, M + I3M' = C2, C + sC = C^, (14) Ci~ C2vR = 0, ^2 -T + (^1^ + ?) ^ = 0, Cg --^ (ciw + H — C2M) V = 0, showing the modification due to the circulation £. The integral of the first two equations of (14) may be written <15) ciu + ^ = F cos 0, C2V = - F sin 0, R = ~ (JjL ^(16) ^ = M cos - w sin = - cos^ + - sin^ - i cos 0, U/t 6'i Co C| (17) ^ = M sin + w cos = ( - - -) sin cos - ^ sin 0, /-,Q^ n d'9 (F^ F'\ • a a F^ ■ a V^V ^18) (7 = — — — sm cos ^ sm = i'-i ^ ' ^ d1^ \Ci 62 / Ci dt (19) (73 ^ = i^y = y\-~ cos^ - :^' sin^ + 2 :^ cos + fl^ so that cos and y is an elliptic function of the time ; and the reduction can be made ^by the substitution tan J0 = q, as before in §2. The circulation t plays a part similar to the influence of a flywheel in § 1 7 ; and in the general investigation of §28 the co-efiicients q, q may be taken to represent the influence of circulation. 238 dx When i is absent, — in (16) is always positive, and the centre of the body cannot describe a loop ; but with I present, the influence may be great enough to make ^ change sign, and so loops occur. The Motion of a Cylinder surrounded by a Vortex. 32. When a cylinder of radius a is placed across a stream of liquid flowing with velocity Q, the stream function changes from (1) Qr sin (J to ^ = Q (r- - -) sin 0, and ^^ = (2 (l + '^) sin 0, so that the liquid slips round over the cylinder with tangential velocity 2Cl sin 0. But if the liquid forms a whirlpool round the cylinder, the velocity at a distance r can be written, (i)\ m _ a^iD ^ •' T IT ' making the velocity tangential and aw at the surface ; and if there is no slipping, w is then the angular velocity of the cylinder. When these two states of motion are superposed, the tangential velocity is 2(2 sin + aw ; so that, on Bernoulli's theorem, if PP' is the ordinate across the stream in fig. 112, the pressure at P' exceeds the pressure at P in absolute measure by (3) \f> (2Q sin e + a^f - \p {2Q sin 9 - a^Y = 2pw Q . PP'. as if submerged with PP' vertical in a liquid o£ density 2pw Q/g ; and the resultant thrust on the cylinder is 27rpa^mQ, across the stream, so that if u, r denote the components of Q in the direction .rO, yO the cylinder feels a thrust component 2wpa^u)V, 2TTpa^wu, in the direction xO. Oy. Keverse the velocity Q of the stream so as to reduce it to rest and to give velocity components u, v to the cyhnder, along 0^', Oy ; the cylinder will move through the liquid carrying the vortex with it. The effective inertia of the cylinder through the liquid is increased by the amount of liquid displaced, so that with Oy vertically upward, and o- the density of the cylinder, (4) TTo-a" -■ = — irpor -J- — A-Trpa'^wv, -n-aa^ T' ~ ~ '^P'^ ~T "•" 2-Kpa^MU — (/(a- — p) ira, (5) (ff + p) '— + 2pi„r = 0, (ff + p) Li - 2pwu + g ( i, a and /3 are imaginary, and the expression for h in the form of (9) changes from , o, r 1 /I - en 2yW<^-^^ 9. rcosv/(4iV^-l)a^ to 2J. cos log , jr- dv = 2A \ —)^r^^ Sr — . J ^ \1 + en 2u/ Jo V{c]ra-x) With a sharp point, x = 0, iV = 0, and the previous equations are obtained of a spinning top, as in Chapter III. The case has been considered already in Chapter VIII, § 13, of a spherical base of radius b, with x = b sin 6, z = b cos d, p = b. The base may sometimes be considered as equivalent to an anchor ring surface with p constant, and x = a + p sin 9, z = c + p cos cos 9 ; but no simplification results in the D.E. for p. and A3 or E. When the ease is thin and light compared with the flywheel, as in the Kelvin gyrostat, or a thin eggshell full of liquid, put C = 0, and then h^ = K, a constant, Y = 0, and the A.M. about the vertical is a constant G, expressed by (12) AP sin9 + K cos 9 = Ap sin^ 9 + K cos 9 = G, and then (19) §2 becomes /I Q\ dR , T, o G — K cos 9 , . ^\ , Kz (13) ^ x^ + Rpcos9=^^^^{x-ps.n9)+^; and further integration is intractable. This may be taken to represent the motion of a top with a round ball point, spinning on a table, the ball being a light sphere, free to rotate on the axle. 4. In the Steady Motion at a constant inclination 0, Q = 0, and from (21) §2, (1) -.r-f + z'] cos 9 + xz sin 1 M ^2 P^ sin 9 CH + K {z cos9 + X sin 9) xRp — ^^ — ^ sin 6 + g {z sin 9 — x cos 9) = 0, (4) as investigated already in Chapter I. In fig. 113 of the prolate egg-shaped body, (2) (yQ^ = AB. = z cos 9 + X sin9 = p, PQ = s sin - x cos9 = q ; but the sign of q is changed in fig. 114 of an oblate body ; and if b is the radius AGr of the circle described by G, /QN h V Pz — Rx . „ Rx (o; ^ = — = = z s\n9 — — , p n a R _ zsin9 -b _ GJ - GO _ OJ P xsin9 JP JP' showing that OP is the instantaneous axis. Also to a geometrical scale (5) ^c=CR + K, ck = AP = Apsin9, kc = (CR + K) sin 9 - Ap sin 9 cos 9, so that, with g/p = \ = AB, and Fp parallel to GB, equation (1) may be written (6) {CR + K) ^ sin - V sin cos = ilV {z cos9 + x sin 0) {z sin - ^x) + gM {z sin - a; cos 0) equivalent to (2) § 13, p. 12, Chapter I, (7) M- kc = gM{^+ q^= gM. Gp, (8) ^^^^=4cos0+^^Jll_^ Mp M sine ' 243 (9) Writing (1) in the form 'A ( TTi- + ^M COS + xzsin 6 fx — (z cob 9 + x sin 9) -^ R — CR + K T M = r (2 cos e + a; sin 9) -^ R + ^^ + ^ sin 9 -4.9 {-=rj + z^ j COS 9 + xz sin If 2f sin 9 — a cos 9 sine ' the right hand side vanishes in the critical state when a wobble begins. When 9 is small, /u is large, as is seen when a coin is spinning nearly flat on the table, or a wine glass with the stem nearly upright. But spun upright, = 0, on a rounded base, like an egg-shaped body, x = 0, but x/ siD.9 = p ; and the wobble begins when (10) (A.,.C^)=^|.. .)(.-,, 5. For the determination of the small nutation of the axle, differentiate (21) § 2 with respect to 9 ; omitting terms that vanish, and putting (1) d^dQ ^ _ 1 d^ ^ 2 d9 dt Q df "* ' for m/27r beats per second of the axle, or m/fi beats for one revolution of the precession n, so that the apsidal angle is Trfi/m, (2) i^j^ + a? + z^^m^-2fj?zpB\r^^9cos9+ 2 (-^+ ^2) ^^4 sin cos + (^ + z^\ fi^ cos2 9 + 2fi^xz sin^ 9 + fl^zcos9 - x sin 9) p sin^ 9 + fjT xz sm2 9 — -^ Ex (z cos9 + x sin 9) — n -j^« {z c,o%9 + x sin 9) d9 d9 h d — fiR (z cos 9 + X sm9) p cos 9 + p.Rx (z sin — « cos 0) — ^ -^ sin ^^cos0 - __^sin0 + g {p - p) = 0, M M d9 putting (3) a; = j9 sin — g' cos 0, z = pcos 9 + q sin 0, dq _ = ^cos0, ^= -psm0, ^ = - a d9 ^' d9 p- p. (4) By a rearrangement of the terms (f. + ^' + 'O A + «2 + z']m' + '^^ d9 C , da . p.x {z cos9 + X sin 0) — -p. u sin 2 f 4 + ^^) M cos + 2 axz sm9 - R -A„ (z cos + « sin 9) - ^ \M J sm ' M. - 'ifihp sin^ cos + (-^ + /) ,x^ cos 20 + /x- {z cos - * sin 0) p sin^ + u^ ^^ sin 20 — ijlR (z cos9 + X sin 9) p cos 9 + p. Rx {z sin — a; cos 0) - M^cos0 + g(p - p) = 0; and introducing the values of -j- and -^sin from (19) (20) §2, 0/0 W0 28570 2H2 244 (5) (f .^.^.^.^)(^..^..^) + [Afi {x- p sin 0) X sin d — ARxp cos + h^xz] nr C ■jxx {z COS + « sin B) — ~n sin + r_ 2 (— + Ax^+ CzA pcosO- Cfi {x - psinO) z sind + CRzpcose + 1^-^^+ x^^ h h^ 2 f 4 + 2^) M cos e + 2tixz sine - B -^ {z cos + a; sin 0) - ^^ V M J + ^2^ ^2 COS 20 - /x2 {z co&d + X sin 0) ,o sin^ + ^2«ir sin 20 - ^i? (2^ cos + « sin 0) ^o cos + (iRx (z sin - « cos 0) M-=^cos0 + g{p - p) = 0; to be reduced. As a verification, (0) should give m = /n for a "spherical case of radius b, as m Chapter YIII, §16, p. 218, with p = b, x = b sin 6, z = b cos 9, p = b, q = 0, and h.i= (A + Mb') ft cos d - Mb'E. And spinning upright = 0, on a rounded end, x = 0, but -. — - = p, (5) should give as another verification 'A .M (6) [^ + "f "^^ ^M + ^^JM -]^-^^P A third verification can be obtained when the body rolls with the axle horizontal = Jtt. Also for a disc at any angle, rolling on a sharp edge, with z = Q, p = Q. Then, with the Steady Motion conditions of § 4, (7) (f + A^ + C.') (^ (A , sN w , M J. (a — (O sin 0) X sin -4 (2^ sin 0-5)pcos0 + ^«2r cos + M ^^ '^} xz sm . C ■ X (z cos 6 + X sin 0) — -^ sin - 2 (^ + Ax^ + Cz'\ cose - C {x - p sin 0) ^ sin + C- (z sm 6 - b)p cos + f^, + ^^) (.1 cos + If^-t+i^' Vili / \ sin . + ^'sin0)-4cos0-£^Jli^" ^ i¥ sin + (^J + Ax' + Cz'^ (^~ + z'^j cos 20 - (z cos 9 + x sin 0) p sin^ + xz sin {z cos + a; sin 0) p cos + (z sin - &) (^ sin - a; cos 0) A 2(jr- + z'j cos + 2xz sin (2^-^ — ^ ) (2 cos , ^ „.„ „ , ^^ . ^ V sm 0/ ^ ^ i¥ sin 'A 20 - — cos^ - {2^b + q\) cot + X (^ - p) A C expressed in terms of the geometrical quantities x, z or p, g, p, X, 6, , — , and ; J. rJ. IrJ. (8) (^ + Az' Cz') (4 + x' + zA ^' = (C + Mx') (9L+JP) JL ^ ^'^ ^ ^^ ^ i"^ ^ V sin / sin - C-[p {x - p sine) + bp cos 0] 4^ sm (9). - (U" + ^^' + G^') [iP - P + g cot 0) X - ^^ - p? - bq For the reaction of the table Y^ C_dE ^ _ CQdR M Mx dt Mx do = _ ^ -4 (a; - p sin 0) ^ sin - ^jgp cos + (CJ! + K) j j1/ + Ax^ + Cz^ 245 in which qQ^ may be neglected. 6. The special case of the axle vertical or horizontal may be examined separately. Spinning upright = 0, and on a rounded base « = 0, but -^- = p ; and then sin (V) ^ = n- z ? _ 7 h Rp „ ^ ^^ sm e ^' sm /u ' and we find L Vil/ ■ " / ^ "iT A* Spun about a vertical diameter, = Jtt, 2 = 0, i? = 0, ^> = ; and (8) § 4 requires A3 = 0, Z = ; and from (19) (20) § 2, and (2) § 5, Thus a solid body like a curtain ring or biscuit, spun on the table about a vertical diameter, begins to wobble when (6) ^2 = 9 (^ - P) and then the energy ^ ^ jM (o! - p) M A Rolling in a straight line with the axle horizontal, like a cask with no bias, 2 = 0, /i = 0, = Jtt, (r,s dfx _ CE + K dR _ ^ ^ ' d0 4 ' de ~ /nx M ^ 2\ 2 {CR + K)' CR + Kr,, f . ,, W (l7+ * j ™ iK4 ~ —j—Rx' + ^(^ - p) = (9) (_ + ^^j™^=__^^(_ + ^3Ji2+_J_.9(^_,) so that, to run straight with velocity V = Rx, this must be positive. Rolling as a solid body, K = Q ; so that stability requires «' = ^> 4 - -^ + X' (10) - '1?^9-Q U M A acquired in rolling down a plane and descending a vertical height 4 7^ (« — />). But as a Kelvin gyrostat, the body can hold itself upright with the axle horizontal as in fig. 33, when the case appears stationary, with i2 = 0, provided (11) K'>gMA{x - p). When the body is rolling in a circle with the axle horizontal and with bias, so that z is not zero, as in fig. 115, (12) (^ + Ax' + Cz') ^= Apx {x- p) ^ (CR + K) xz (13) (^ + Ax' + Cz') ^= -Cf.z{x-p) + (J. -^ x') (CR + K) (14) {CR + K)p = gM . GE = gM(q + ^^ and p = x,q = z, z - b on R , Rx n (IS) ^ = RP=7' * = "" 246 (16) (17) ~ m '^ W ^ M d9 M de^ ^ ^^ ^^ _ (^ + x^) fi [Af^x (x - p) + (CE + K) xz] - [Rx^- ^^xz + ^^^) [- C,z (x-p) + {^+ x^) (CR + K)] + [^ + Ax' + Cz') [- (^ + ^') A*' - Ap + M^^^ + ^ (^ - P)] = (18) (f -H^.^H-(7.^)(^+.^ + .^)^^ = (^ + cc') [Ax (x - p) + M (zX + hx) xz] + [x(z - b) - 2xz + z\ + bx'li- Cz (x - p) + (^^ + x'^M{zX + bx)] + (^ + Ax' + Cz') \^ + z' + xp- z(z -b) -\{x - p)'j = n± + xA [Ax {x - p) + M (Xz + bx) xz + (\z - xz)[- Cz {x - p) + {C + Mx') (\z + bx)] + (^ + Ax' + Cz') ^^+bz+xp-\{x-p) = C + Mx') {z\ + bx) z\ - Cz' (x - p)X - (^ + Ax' + Cz') [{x-p)X-^x'- bz^ Unsteady Motion of a Body Sliding on a Smooth Table. 7. For a smooth table, the conditions in (1) §1 would be replaced by equating to zero the vertical velocity of the point («, y, z) in the body in contact with plane, so that (2) (U - Ry + Qz) a + (VPz + Rx) ^ + {W - Qx + Py) y = -, and expressing the conditions that the resultant of X, Y, Z is the pressure R normal to the plane by (3) X = Ra, r = R(3,Z= Ry. It may be taken then that the C. G. merely rises and falls in a vertical line, as any horizontal velocity would persist unchecked and constant. Consider a solid of revolution as before in fig. 113, sliding and rolling on a smooth table, with G rising and falling in a vertical line GQ ; then with , R 9 (5) dt de dt Taking moments round Gy, dd' de df ^ dt dp _dpde _ . dp' dQ , , ^„ M (6) ^ - hA + AA = ^^ - (Ci2 + Z) ;. sin + V dQ dt sin e cos (J = - Rq= - Mq dQ (^) {m-^Tw-^p-^^^^'-'i i, m' sin cos - ?^ + ^^' P sin 9 + gq = 0. 247 and taking moments round Gcz and Gx, (8) -^ = 0, Ag = CR + K, a constant, (9) ^ ^ + ^P cos e - CE - K = 0, ^ (4P sin 0) - {CR + Z) sin = (10) ^P sin + (CR + K) cot = ^^ sin^ + (Ci? + K) cos = 6=, a constant, the component A.M. about the vertical GQ. In the Steady Motion, Q = 0, (11) Afj? sin cos - {CR + Z) ^ sin + gMq = 0, so that the body begins to wobble when (12) {CR + K)' = AgMAq cot 0. Spinning upright on a rounded end, = 0, cot = oo , ^ = 0, but 5' cot = |> — jO, (13) {CR + KY > 4gMA {p - p), or the body will wobble. Expressed in terms of G instead of /j. by (10), (U) oMa = [^ - {CR + K) COS03 {CR + K - GcobO) _ I d [G - {CR + Z)cos0p ^ d { CR + K - GcosO) ^ de A sin2 ' °^ 2 rf0 A sin^ Differentiating (7) with respect to for the small nutation of the axle, and omitting terms that vanish, (15) (^ + q^) m^ -. 2^^| sin0cos0 + ^/.^ cos 20 - ^:^^(|sin0 + ^cos0) + 9{P- p) = ^ (16) 4 ^ sin + 2^/x cos - Ci? - i:= (17) (m + ^')^'= ^^'(1 + 2cos^0)- 3^:^^MC*os0 + ^(^^^/)' -g{p-p) (18) A{A + Mq') m.' = A'fi'{l + 2 cos^ 6) - 3 {CR + K) Afi cos d + {CR + K)' - g MA {p - p) = A' a' sin^ + {CR + Kf - gMA {Sq cot d + p - p) (19) A{A + Mq') ^ = A' sin^ + (4 cos + MXqY - MAX {Sq cotO + p - p) A* 8. The investigation can be continued to the motion of a body, like a rounded stone, called a celt, when it is spun on a table, rough or smooth, about a point Q where GQ is the vertical, or nearly so, and the motion does not depart, far from the position of equilibrium. A sheet of plate glass was employed by Kelvin in his Royal Institution lecture, May 1893, in a raised frame to keep the body from flying off. With rectangular axes {x, y, z) through G with Gz to the point of contact C in the body in the position of equilibrium, and with Ga;, Gy parallel to the lines of curvature at C, and GC = A, the equation of the surface near C may be written. with p. (T the principal radii of curvature at C ; and then, to the order of approximation required, the direction cosines a, /3, y of the vertical are given by ra\ dz X r. dz y , or, if required to a closer approximation x^ y^ (3) 7= -'(l-«^-^^)=l-v-& 248 With the previous notation the geometrical relations required are (4) ^-R(i+Qy = 0, ^ - P^ + i?a = 0, ^ - Q„ + P^ = ; and, rolling on a rough table, (5) U - By + Qh = 0, V - Ph + Ex = 0, W - Qx + Py = 0, in which it is convenient to use a, /3, y, instead of x, y, z. When the axes {x, y, z) are not restricted to be principal axes, we have (6) h, = AP-FQ-ER, h^ = -FP + BQ-DR, h, = -EP-DQ + CR + K, when the axis Grz carries a flywheel, having A.M. K. In the dynamical equations, neglect quantities o£ the second order by comparison with Z7, V, W, P, Q, X, y, X, Y, of the first order ; and then with R and Z finite, but W =0, (7) !^-jiy = ,. + |, f + ijp = ,p + I, u = ,,^|, the last equation giving Z, as if the body was at rest in statical equilibrium. Taking moments round G-, (8) ^ - Rh + Qh =yZ- zY, ^ - Ph, + Rh, = zX- xZ, '^ - Qhy + Ph^ = xY - yX reducing, in the order of approximation employed, to (9) A^ + KQ- (B -C)QR- D (Q' - R^) - E {^ + PQ) -F[^-RP)+ gMy + Mh{^^+RU- g(i) = (10) B*^-^ - KP - {C - A) RP - D {^^ - PQ) - E {R' - F^) ' - ^i"^^ Q^) - Mh{^^ - RV-ga) - gMx = (11) C^-§- {A-B)PQ-I)[^ + RP)-E{^- Qr) -F{P^- Q^) = If the body is not to depart from the position of equilibrium, either I. With a finite spin R, D = 0, E = 0, so that GC is a principal axis ; otherwise the body could not spin steadily upright ; or II. the spin R must be small. 9. In the first case (1) A'§*KQ-(B-OQR-F{^-llp)+g3Iy^Mh{'^ + SU-gi3)=0, (2) £§ -/fP-(C-4)iJP-i,'(^+Qij)_,M.-.VA(f -fir-^,)=0, dR (3) C -^ = 0, P is constant, and also h^ = CR + K. (4) ^ = P^_^_Q;, dV _ j.d,i'dP. and then (5) ^ f + m -(B-OQS- F{''-§ -Rp),Uki- fi|' + iPu^BO -,3) + ()ily - 249 or (6) {A + Mh^)^-^^Ka-F^^- [(5 -C)a-FP]R- MhR^ + MhRIT , . -1 , -9^W-y) =0; and similarly (7) {B + Mh^)^-KF-F'^- [(C -A)P + FQ]R- MhR^l + MRV + gM {ha -x) = 0. From the geometrical conditions in (4) (5) § 8, with 7 = 1, and with w = pa, y = trfi, (9) U=R.(i-h[-^ + Rf3), V= -Rpa + h[f^ + Ra) and then (6) and (7) become (10) (^.M.^)( f .i^|).ir(-|.i.,)-^(-^.^|) (U) (£.«.)(- 5, ■4f)_^( |,«„),^(_J^,^|) - ilf7ii?<7 ^ + MhR (^- Rpa + h'^ + Rha'j + gM (h - p) a = 0, and (10) (11) are the two differential equations which determine a and /3 ; the condition of stability requiring o and /3 to remairl small periodic terms. With Perry's notation of and 0^ to represent the operation -^ and -^; instead of D and D^ as before, the letter D being required here already to represent a product of inertia, write equation (10) and (11) (12) F%\ - (^ + MK^) e'(3 + [(A + B - C + 2Mh' - Mhp) R - K]da + FR'a - [(B - C + Mh' - Mha) R' - KR + gM (A - ^)] |3 = (13) - (B + Mh') 9'a - Fe'(5 + [(A + B - C + 2Mh' - Mh^r) R - K]d^ + \_{A- C + Mh' - Mhp) R' - KR + gM{h -/))]«- FR' \i = and treated symbolically ri4^ 1 = - (^ + ^^') e' + (B - C + Mh' - Mh {A + Mh^) gM[^{h- \^(R'.A' + Qy[MA'{h-p)+MP{h-.)]+^{A'.B' + py[MQ{h-p) + MB'(h-rr)W i A'B' - PQ ■ ^ -\ 7,D !n^D"- l"" this second case, where R remains small, as well as P and Q, so that R\ PM, UK can be neglected in the approximation, equations (9) (10) (11) §8 become (2) B^-KP- D'l^- FdP_ dU ^ ^ dt dt dt ~dt ^ {h - p) a = (3) C^-E'^-D^ = ^ ' dt dt dt ' «o that <4) CR- EP- DQ = ^ constant. In the geometrical conditions of (5) §8 (^) ^'= - Qh, V=Ph, W= 0, and (1) (2) become 251 (6) {A + M¥)'i^ + KQ-F'^- E^- gM (h - a) jS = (7) (B + Mh') ^- KP- D^ - F^ + gM(h -)pa = or, introducing the condition of (23), (8) ( 4 + m^ - ^ )^ - (i^ + ^) ^ + /^Q _ gM (A - .) )3 = (9) [b + Mh^ -D^)da_^p^EDY_P_Kp^ ^m {h - p) a - in which we can put, in consequence of (4) §8 (10) P<= ^, Q= -^. Thence again, in the symbolical form ^ _-[a + M¥-^)9^ + gM (h - .) ^''^ ^ " {F^^)e^-Ke ^ - (b + Mh' - (12) ^^ -[B + Mh^-^)e' + gM{h-p) (i? + ^) 02 + Ke ^''^ ^^f - ^1 - ^^^^^^ = 0' ^4 ^ ^f - -^^^^^ = ^ leading to the symbolical equation for d, in which 6^ can be replaced by — ?n^ in a stable oscillation. The same treatment will serve for the analogous D.E.'s in the electrical theory of the transformer and alternating dynamo. As an illustration, resume the consideration of the stilt gyroscopic pendulum of §4, p. 30, Chapter II, or of the equivalent gyrostat on horizontal trunnion axes not at the same level, as in §22, p. 74, Chapter III ; as they may be considered representative models of the Schilowsky carriage balancing upright on a single track by the influence of an interior flywheel, with preponderance such that carriage and flywheel depart slightly from their upright position of statical instability ; the preponderance of the flywheel is found to improve the gyroscopic action. The equations of motion of the carriage and flywheel may then be written A- symbolically, ,.,. a _ A,d' - gMA j3 _ A2e' - gMJi, ^^*^ ^ ~ : M ' a~ - KB ' or with — m^ for 6^, (15) (Aiin^ + g3£A) (Am' + 9^2^) - K' m^ = - iy[£. - (y-^' * y^) so that stability requires (17) K>^ (A,A,) y{^^ + ^^) > ^ (^^^2) (-^ + -^^ where 2 tt/wi, 2 Tr/wg is the period of oscillation of the carriage and flywheel about their position of statical stability, when K = 0. With a = ao cos {mt + e), |3 = j3a sin (mt + e), a„ ^X'+ 9^A _ K^ / Am^ + g^A /A /m^_±nl ^ ^ ^0 ~ Km A^m^ + gM^h^ v A^m^ + gM^h^ ~ V ^gV ni'^ + nf with TO = mi or TOg, and TOi m^ = n^ n^. 28570 2 12 J-2 ' 252 11 On a smooth table we can take in our approximation .(1) Z = - .9 if, X= - gMa, Y= - flM(3 ; so that with x = pa, y = = 0, ^ = 0, and Grz a principal axis. II. R small, as well as P and Q. 1 2. I. —In this case the equations reduce to (5 - C)QR + FRP - gM (h - a) ^ == 0, (C - A)RP - FQR + gM {h - p) a = <1) A^f + KQ- dt j^dQ dt (2) B ^f^ - KP ■ dt dt (3) pdR _ ^ ^ dt "' Room Writing X for g/R^, and taking as in (8) § 9 (4) ^ = f + ^-. «=-5 + «'^' the differential equations reduced to two, which as before may be wi-itten symbolically « _ - AS' + IB - C + M\{h- (t)] R' - KR ^^^ ^ Fff' + [{A + B - C)R- E]e + FR' ^ _ - B9'^ + [A-C + M\ ill - p)] R^ - KR '^^> a FB' -[{A + B^ C)R-K]9 + FR^ ' Multiply these equations together, and replace 0"^ by — m', to obtain an equation equivalent to Puiseux's result, in Liouville, 1848-1852, for a solid body, K = 0, on turning the axes G,c, Gv through an angle 8 so as 16 become the principal axes, but no longer parallel to the lines of curvature at C. The U, V, W equations reduce to (8) ^ 1^ "*■ ^ W = ^' U' + VHb constant ; (9) ' U^-Vl^l \^J dt dt V — rj2 y2 = 0, tan~^ I J is constant ; 253 and _ TF is constant and zero ; so that the C.G. moves with constant velocity in a horizontal- straight line, as is evident otherwise ; and it is convenient to reduce the C.G. to move- ment m a vertical line. _ Make Puiseux's S = 0, or else take i^ = 0, to obtain the results given in the Math. Tnpos, question 8, June 4, 1897, m. 8 ; we have to discuss then the conditions required that the two values of rr? j B should be real and positive, given by the quadratic (10) [4|!+5-(7 + m(A-.)][s|; + ^-(7+MX(;.-^)' - (-4 + 5- 0=^1;= 0. These conditions are (a) 4 - C + MX (A - p) and 5 - C + M\ {h - c' n/ (c* - a*) - V (c* - ¥)' b' + d\ 2g 2{a^ + b')c " {LiouvUle, 1844 ; Routh. II., p. 202.) 13. II. — On the smooth table, with R restricted to be as small as P and Qj the •equations reduce to with (3) the same as before in §10, but with the omission of the term Mh?. « ' ■ ^-t ^-% 254 Thus for example the stilt gyroscopic pendulum, of fig. 45 in §4, p 30, Chapter III,, can hold itself upright on a smooth table when K^ > gMhA, neglecting the inertia ot the- frame ; and the result is the same if the flywheel axle is horizontal or inclined. Simplify further for a solid of revolution with 5 = ^, i^ = 0, ^ = p, on the smooth, table ; and then (4) AD'f3 + {2A - C) RDa - [A - C + MX {h - p)] R'(3 = 0, (5) - AD'a + {2A - C) RD^ + [A - C ^ M\ {h - p)] iE^a = ; and with o + j32 = y (6) - AZ)^y - i {2A - C) RDj + [A - C + MX (A - p)] R^y = ; so that, with y = exp mti + e, (7) Am' + {2A - C) Rm + [A- C + MX {h - p)] R' = 0. The discriminant (8) (2J. - CY - ^A\_{A - (7) + MX (A - p)] = (72 - ^AMX (A - p), which is positive, if as before in §7, against ^^"^ 7^ {C + MhpY ' on the rough table, in (10) §4. All these special cases may be made to serve as exercises for an independent treatment. ___ The investigation can be carried out in a similar manner of the stability of a body with a flat base, rolling and spinning near the highest point of a spherical boss, as given by Routh. GrYRO-COMPASS. 14. We have seen in §11, Chapter I, how the Serson-Fleuriais Gyroscopic Horizon feels and registers the rotation of the Earth, as distinctly as the Foucault pendulum ; and the idea of utilising this rotation to replace the magnetic field for directing the mariner's compass has often attracted attention (J. gyrostatic working model of the magnetic compass^ by Sir W. Thomson, British Association Report, Montreal 1884 ; Nature, 30, p. 524, 1884), and at last has been made a working success by Dr. Anschutz, by the aid of the improvement in technical science, enabling the high speed of rotation requisite to be attained, 20,000 revolutions per minute, 333^ per second, about double the spin of a 6-inch (15 cm) shot. So too the Flying Machine was obliged to await the construction of a suitable motor. The mechanical details are described in the Gyro-Compass, by Gc. K, B. Elphinstone, of Elliott Brothers, 1910, where the theory also is given, as well as in the Klein- Sommerfeld Kreisel Theorie, p. 851, due to Fritz Noether (fig. 116). In the dynamical explanation, take axes UA, OB, OC fixed in the compass, and suppose the body to receive small angular displacement, a, j3, y radians from the mean position 0^', 0,y, 0^^, where Ox is drawn to the north, Oy to the west, and Oc to the zenith. The diurnal rotation of the Earth, w radians per second, has components in latitude X,, (1) o) cos X, 0, w sin X, about Oa;, Oy, Oz ; (2) cos X sin y, CO sin X, about OA, OB, OC ; to the order of approximation employed. 255 As the axes OA, OB, OC are fixed in the body (3) "1 = -t = j7 + w COS X COS y, Wg = 4^ = -TT — «j COS X sm -y, B^ = R = -j2 + w sin X, dy ~dt and K denoting the A.M. of rotation of the flywheel about an axle parallel to OA, (4) Ai = ^P + K, K = BQ, hs = CR, A, B, C denoting the M.I. about OA, OB, OC. The body ikf, composed of the flywheel and compass card, is floated in mercury with a metacentre at and metacentric height h ; so that in the dynamical equations of §1 (4) (5) ^1- hA + M2 = A^- {B - C)aR= - gMha, and ignoring here, and elsewhere, insensible terms of the second order such as QR, (6) AD\ + gMha = 0, giving mere simple pendulum motion about OA, of equivalent length A/Mh, unaffected by the rotation w. Again (7) ^ _ hA + Vs = BD^(i + K(JDy + u, sin X) = - gMh(3, ,(8) ^ - hA + hA = CB'y - K (X>j3 - co cos X sin y) = 0. Writing equation (7) (9) BD'^ + KDj + gMh (/3 - ^,) = 0, ^, = - ^^^^^, «hows that the card and axle dip to the north through this angle /3q, and the minus sign shows that the north end of the card rises in north latitude, as in the Fleuriais top, in order that the couple — gMh^^ about Oy should give the precession w sin X to K^ the A.M. of the flywheel. Treating these equations (7) or (9) and (8) in the previous symbolic manner nm 13 - j3o _ - KD _ CD"^ + K,., cos X ^ ' y ~ BD'+ gMh- KD (11) {BD^+ gMh) {CD'+ Kw cos X) + K^D''= ; and, replacing D^ by — m^, the motion is oscillatory, of period 27r/m, where (12) {- Bm^ + gMh) ( -' Gm^ + Ku, cos X) - K^m^ = 0. The small root of this equation gives the slow oscillation important in practice, obtained by ignoring BC, so that . . , _ gMhKo) cos X '^^^^ ™' " K' + gJMhC + BK is affected, and the period of oscillation of y, but not its mean direction, and so no compass correction is required : except in such high latitude that v = 900 cos X, where the ship would keep up with the Sun or stars and the compass would lose directive action. But if the course is due north at a speed of v knots, and the latitude is varying, a term— must be added to Q, and 0^ in (3) § 14, and (19) § 14 must be replaced by ■ ^.o cos X sing = K'^ dt aiD cos X 900 cos X' if a denotes the radius of the Earth ; and in degrees, for a small angle such as S, (5) g° = 57°-3 - . 900 cos X Thus at the equator, for a course due north at 30 knots, g°.= 1°-91, say 2°. For any other course, v must be taken as the component to the north of the velocity or the speed northing ; and the deflection S is to the west, changing sign to the east if the course is southerl3^ o n o There is still the correction ^ to consider for a change of speed, called the BallMc Defection, as m a Balhetic Galvanometer ; here again it is the component in the meridTan clanged to' '''' ""' '"^ ^' '"' ^^*'"''^ it to y + ^ ; and then (18) § uS (6) ^^^ (Dy + D^. f „ sin X) + Mh (gB - a^) = so that (7) K%=Mhai}, I^t d{^ ' in accordance with the gyroscopic law of angular momentum ; and integrating (S) ;, = ¥l!l ^ =^ M]nt_w cos X . K dt K 257 16. W hen the length I of the thread of a pendulum bob swinffing near the surface of the Earth becomes comparable with a the Earth's radius, it is no Wer allowable to take gravity as a uniform field of parallel lines, because the convergence of the field to the centre ot the Earth becomes comparable with the angular deviation of the pendulum thread ; that is, if is the deviation of the pendulum, and ^ of the gravity corresponding, (1) /sin0 = asin,/., l§+g{9 + ^) = 0, ^^ = ] + I = ^1 + i) which does not agree with the ordinary pendulum formula when l/a is appreciable. In a straight track or tunnel (2) / = oo, -V, = i, gi' a so that (8) §15 may be written /5^ M ^ oMhw cos X T _ Ko, cos X T^ _ T ^^ 8 K -^^ T~1?~T}- At a place where T = 7;^ ^ = ;u, so that this ballistic deflection is dead beat, that is there is no sudden change in the card when full speed is reached, and the acceleration has come to an end. Consider the movement of a carriage oscillating under gravity, on a length of track made perfectly straight and not curved to sea level, say a Channel tunnel, cut straight not level. The time from rest to rest through the tunnel is given by (4) ^ = V^ and this is about 2,500 seconds, making 17 double oscillations in the 24 hours ; implying that a satellite grazing round the Earth would make 17 revolutions a day, with a velocity of 17 X 900 = 15,300 knots, in a period of about 85 minutes. If the Earth could be considered a homogeneous sphere, the same result would hold for any other tunnel cut straight through, including the diametral tunnel or well discussed in Plato's Phcedo, or the Voyage au centre de la terre, of Jules Verne. And the water in the well, sunk to the centre of the Earth, and through to the Antipodes, would be at rest in equilibrium ; but drawing a bucket of water would set up an oscillation of 17 tides a day, independent of the interior distribution of density. 17. According to the figures in Elphinstone's Cryro-Compass, p. 68, Kio = 20190 = al 4-3051 (ergs) with 86400 — 236 = 86164 mean solar seconds in the sidereal day, for greater accuracy, "> = -J^ = al 5-86287 (radians/second), - = al 4-13713 = 13713, 86164 ^ ' hj K = CR = al 8-4422 = 10^ x 2-768 (erg.seconds) where R denotes the angular velocity of the flywheel, and C its axial M.I. ; and at 20,000 revolutions per minute, E = 20000 ?^ = al 3-3210 = 2094 (radians/sec) 60 C = al 5-1212 = 132100 (g.cm^), against 136,000, the number supplied by Dr. Anschutz in Kreisel Theorie, p. 'si^'l. Again, Gyro-Compass, p. 6S, / = 10« X 4-04 = al 9-6064 (g.cm^) K^ = al 16-8844 ,,Mh = K'/I = al 7-2780 = 10^ x 1-897 (ergs), in Kreisel Theorie 10' x 1-5 ergs ; and takings = 981 cm/sec' = al 2-9917, Mh = al 4-2863 = 19330 (g.cm), so that if the wheel weighs 4 ku', h = 4-8 cm ; not A = 10 cm, or iM = 1500 g, as given in Kreisel Theorie, p. 86Ji. 28570 • 2 ^ 258 Then ^ = al 5-3013, / ~ = al 2-6506, tt / ^ = al 3-1478 = 1406 (seconds), the time of a single swing of undamped vibration at the equator. But in the Gyro- Com'pass we find the time of a double swing is given as 3,680 seconds, and TT / T^ = 1840 = al 3-2648, V K(o cos A n/ COS X = al r-8830, cos X = al 1-7660 = cos 54° 20', which is the latitude of Kiel, where the instrument is made and tested ; and so we have calculated the latitude, not stated, of this place from the data given by Elphinstone. If k denotes the radius of gyration of the flywheel, and d the diameter, given in Kreisel Theorie, p. 862, as c? = 14-8 cm. = al 1-1703, and M = 4000g, F = ^ = al 1-5191, 4! = al 0-1785 = 0-151. M d} 18. A damping arrangement is required to prevent an oscillation once set up from keeping on continuously unchecked. This is provided by a mechanical arrangement {Gyro-Compass, p. 33, fig. 18) in the ventilation of the flywheel, by which the air driven out is directed to one side or the other as the flywheel tilts through the angle |3, and a couple F^ is called up tending to increase y ; and so equation (19) §14 is changed to (1) - KD^i + Koy cos X sin y = i^/3 (2) - KD^ + Ki^ cos X (sin y - sin yo) = i^ (/3 - j3o) El (3) Kiii cos X sin yo = F^^ = — Kw sin X -^r^ meaning that the damping moment i^|3o must balance the directive moment at the angle yo ; (4) sin !.=-— tan X, showing that the compass is deflected to the east through an angle, which expressed in -degrees is (5) a = 57°-3^tanX. Combined with equation (18) §14, ^^^ ^ -^'^ = " ^^^ = ^ (/3 - /3o) - K. cos X (sin y - sin yo) (7) (/Z>^ + ^i> + Z. cos X) (y - y„) = 0, the diff'erential equation of a damped vibration, with the solution (8) y - yo = ^e -*-^ cos {pt - e), (9) f=IIL = l •' gMhl K' (10) ^2 ^ 1 ^2 _ K^ cos X j^,2_ ^2 ^2 unramped^''"'*' *' *™' ""^ ^ ''''^^' ^""'""^ according as the Vibration is damped or In a single swing the amplitude of oscillation .1 is reduced by the factor (11) e-^f^^ = e- ^, where £ = J/ri, (12) 1=1 ^T,' The damping arrangenient can be cut out, and so T, observed as well as T, in testinp- a compass m the factory ; thus, on p. 68, Gyro-Compass testing (l-'^) 27', = 4110, 2^0 = 3680, seconds ; 259 over one hour, so that the motion of the compass card is comparable with the minute hand o£ a clock ; and thence (14) 6 = 1-56, / = 0-00152 = al 3-1818 ^''^ ^h = ^f=if= ^' 2"3447 = sin 1-26, (16) a = l°-26 tan X. Thus in latitude 60°, a = 2°-2. 19. At such a high speed as 20,000 revs/minute, the whirling tension becomes important, and must not reach the yield point of the metal. For a ring the tension length h = v^g for a peripheral speed v ; and here with a diameter d = 15 cm, (1) V = TrdN = ,r X 15 X 20000 h- 60 = 5000 tt = 15708 cm/s (520 ft/s), u = al 4-1962 v^= al 8-39^4 ^ = al 2-9917 A = al 5-4007 = 251600 cm ; and with steel of S.Gr. 8, this gives a tension of 8A or two million g/cm^, or 2,000 kg/cm^, or 2 t/cm^, say 12-5 tons/inchl In an experiment by Car us Wilson, described in Nature, May 14, 1891, a steel flask 12 inches, say 30 cm in diameter, began to yield at 16,000 revs/minute, a peripheral speed of 840 ft/s ; giving a tension of over 30 tons/inch^. 20. Fixed in the meridian, the ring zTjz'VJ of fig. 106 partakes of the diurnal rotation fi of the Earth about an axis O^', inclined to the zenith OZ at an angle a = ZOzy the co-latitude ; so that in a state of steady inclination (1) Afi^ sin 9 cos + CRfi sin = gMh sin (0 - a), in which fx is the angular velocity of the Earth, taken as given in radians per second by (^' '■ = sirdhTio = 8^50 -'''^■»"« = '»'''< '•2"' so that fi^ may be neglected as insensible ; and putting 9 — a = E, and denoting the latitude by X, (3) CRfx sin (a + E) = gMh sin E, tan E = ^.P^^"^^ ^. and E the deviation of the axle from the vertical could be made appreciable if R could be made large enough, as in the Gyro-Compass. This is the theory of the Gilbert Barogyroscope, where a flywheel is mounted in i. gimbal ring provided with knife edge bearings on trunnions in the line OB, placed east and west. , A slight preponderance Mh is given to the body or its frame, so that it swings once in T seconds like a pendulum of length I, where (4) 1 = ^^, gMh = Af = A^,; and then at iV revolutions per second, ^'^ ^^ = 86400' -RfT^ NT^' and taking the wheel as disc or ring shaped, C = 2A, (6) tanj;= '^^^^ 10800 . ' sm X ■ NT" and the wheel would reverse in latitude X, if we could make 10800 (7) N = T' sin X" Thus with T = 7^ as in the 400 day clock, N = i^, or 384 in latitude 30°. sm X 28570 2 K 2 260 But when E is small and expressed in minutes by E' , fr,s , „, NT'cosX ^, NT' cos \ (^) ^'^^ ^ = ^0800-' ^=—;^- In the Gyro-Compass, at 20,000 revolutions a minute, N = 333^ ; and making 7^== 1, (9) E' = ^^cos X = 106' cos X = 1° 46' cos X, Oir which is quite appreciable ; the experiment is described by Foppl in the Math. Encydopcedia ; and if the line of the knife edges is inclined at /3 to the east-west line, a factor cos |3 is required for E in the above (3). In the Gilbert Barogyroscope of M. Koenigs, the wheel can be spun up to 55,000 revolutions a minute, over 900 a second ; but the diameter must not exceed about 5 cm, or the material will begin to yield. If we could make T = 5-7 with N = 333^, then in (6) (10) t^n E = , '^°^^ , , ^ = 45° + 1 X. ^ ^ 1 - sm X' ^ Denoting by i?o and Nq the angular velocity and revs/sec which would enable the body to reverse and rise to the upward vertical position, provided precession was not checked by the trunnion knife edges, (11) CR,^ = -igMhA, cos X cos X (12) *''" ^' = TJ-HJ^ T- = „.,.„ C F~' T- — ,- ~ji- — sui X 86400 -— r ^ — sm X 4.4 Jlfx 4:A F and with C = 2.4, p _ cos X r" -^ ^°'^ ^ '"' " 43200 f - ,in X ' '^^' P-™"'^- But in the Serson-Fleuriais top of §11, p. 10, Chapter I, E is small and given by (14) sin E, or tan E = ^\T ^ • gMh 21. Apply this theory of § 4, Chapter VIII, to Gilbert's Barogyroscope of § :.^0, when the trunnions are turned through an angle j3 with the E. and W. line : and ic is neglected [Kreisel-Theorie, pp. 731, 736 ; fig. III., p. 753]. There z is not the zenith, but the pole, and y = J tt, = jZ, cos t> = sin a sin Q where a is the co-latitude zl, I being the zenith, as in fig. 117. ' At a steady inclination, and neglecting ^i^ the couple (1) N = - h^Q + hP = - Kfji sin sin ^, and with the zenith at I, and nadir at v in fig. 117, (2) N = - gMh sin (^ + 8), where S is the angle VLz, (3) Kfi sin sin + gMh sin (^ -f g) = 0, and putting (j> + S = E, (4) tan E = A'^«L^ij«Li__ ^ ^<^M sin a cos (3 O^^Jh - A> sin 6 cos g gMh - A> cos a' since, in the spherical triangle tzZ, ('^) sin sin g = sin a cos )3, sin cos g = co-^ „ • and this agrees with § 11 Chapter I, where (3 = 0, and X denotes the latitude. If i.; denotes the value of E when /3 = 0, as given in (6) § 20, then as stated there v^-' tan E = cos (3 tan E^. rotation, or the opposite. ^ * ""* ' ■^'■'"* ""^ '"^'^ •'><' »■"= 'i"'«tio„ of 261 22. Float a flywheel in a ship at rest on water, the axle longitudinal. The couple due to the Hearth s rotation will be CRfi sin PC in the plane PC through the axle and the pole P ; and the component couple in azimuth is CRf^ sin PC sin PCZ = CR,. sin ZP sin PZC so that in latitude X, and azimuth from the N. point, ^-m + CRf. cos X sin = 0, ~ + n^ sin = 0, n^ = -^^ cos X, a pendulum oscillation, with a heat - = tt /(^^ sec X) ; and with a = tt"^ L, where L is the length of the pendulum which beats the second, about one metre or 40 inches, R^i _ 4.ir^N _ N g 86400^ 21600 X The couple to change trim is CRfi sin PC cos PCZ = CRu sin X, independent of the azimuth, corrected by moving a weight P aft a distance £ ^ sin X = ^ S^X, P g P ^loOO X as in balancing a compass card against magnetic dip. So also the couple for the flywheel on land may be investigated, due to the rotation of the Earth. Thus a flywheel, diameter 12 feet 6 inches, a steel disc 9 inches thick, weighing 20 tons, would feel the influence in an alteration of load on the bearings, placed east and west 5 feet apart, of about one lb/ton, or 20 lb, at 8 revolutions per second, 480 a minute, in a direction parallel to the polar axis. Gyeostatic Chain. 23. This can be realised practically with a number of bicycle wheels, linked together by rods, and stretched to a tension T, practically uniform when hung vertically with a heavy weight at the lower end, or else held down by a strong spring. The motion has been considered by Sir W. Thomson in Proc. L.M.S., Vol. VI., 1875, and Perry's Spinning Tops can be consulted for a diagram ; the results can be obtained by a reduction to a state of Steady Motion, considered in the elementary manner of Chapter I, where the gyrostatic chain forms a uniform polygon, wound in a helical manner round an axis. Begin by considering the simple case of a string of particles like beads, each of mass m, at equal interval 21, on a uniform helix round an axis O2, swinging bodily with angular velocity n, on the thread at tension T. Each bead is describing a circle of radius c with velocity /; ; and projected on a plane perpendicular to the axis, the beads are seen forming an equilateral polygon, a side subtending an angle 2/3 at the centre. The thread making an angle a with the axis (1) m — = 2T'sin a sin /3. Looked at sideways the string of beads is seen projected into a wave like figure, advancing with wave velocity U, such that ,„. 2/ cos a . J, J ... ^, , „„ 2c(3 ^^ _ ^ cos a __ sin /3 (2) — jj — = time of describmg the angle 2(3 = — ^, ^^ = ^-^^ ^ 3~ ' 2T . . ^ 2Tj . 2 n2 ST; 2 /sin /Sx^ (3^ w^ ^ =^ c sm a sm i3 = — I sia'a, b^ = — / cos^ a — — £- , ^ ^ m m m \ p / The wave length X is the advance at velocity U during a revolution of a bead, in time 2Trc/v ; n U o , sin i3 (A\ X = 27rC — = 27rC COt a -—J~ ^ ■' V p (5) turning couple = Tc sin a cos j3 = -^ tan a -t- . The line density is on the average lo = m/2l, and (Q) £73 = - cos^ a l^-^)\ reducing to W = -cos- «, v' = ^ sin^ a, for a continuous chain, with j3 = 0, sin /3/|3 = 1. Realise with a Dynamobil and chain, as in Fleming's experiments in Proc. Physical Soc, Nov., 1913 ; and imitate direct Reflexion and Refraction. 262 Thus in § 25, p. 78, Chapter III, when the chain forms a uniform helix, two of the roots are equal of the cubic R = in (6) ; and denoting the roots by a^, P, H^, (6) R = V (p' - T^)' - ^f\ X r = 4 (p' - «^) (p^ - n 2' (8) r cos a = Z, Tb sina = N= Zb tan a, T + ^n^wP = H and the apsidal angle is -— — — r, — r, verifying for the plane revolving catenary with V (3 + cos" a) a = Jtt, as before in § 14, p. 195, Chapter VII. 24. Replace each bead particle m by a flywheel, of mass If and axial length 2a, making an angle 6 with the axis, and spinning with A.M. K, joined up by links of length 21, making an angle a with the axis, to form the gyrostatic chain in a uniform polygon wound helically round the axis, having the appearance of a procession of waves. With link tension T and 2y the angle subtended at the centre by the projection of the flywheel axle, the central force on a flywheel is (1) 2T sin a sin (B + y) = Mn^a sin cot y = MnH - — ^ cos y, ' ' sm /3 ' . /QN Mn-l ^ sin j3 sin ((3 + y) , _ 3InH cos (3 cos (/3 + y) 2T cos y ' 2T ~ cos y The couple which acts on a flywheel to give it the precession n has its axis along Ox, and moment (3) L = 2Zy - 2Yz = 2rcos a. a sin - 2T sin a cos (/3 + y). a cos = 2Ta [cos a sin — sin a cos 9 cos (|3 + y)] and then, K being the A.M. of a flywheel, and K sia 6 - An sin 9 cos 9 the component A.M. perpendicular to the axis O^-,. (4) (K sin 9 - An sin 9 CO& 6) n = - L = 2 Ta[sin a cos 9 cos (j3 + y) - cos a sin 9] (5) - ^n^ cos + Kn + 2 Ta cos a = 2Ta sin a cot 9 cos (j3 + y) = Mn^ cl^ cos 9 cot y cot (/3 + y) (6) tan y tan (jS + y) = ^''' ^' ^"^ ^ — J.n- cos 9 -^ Kn -V 2Ta cos a (7) l + tanytan(|3 + y) = ^^i^ '■^ cos y cos (j3 + y) ^ - -in^ cos + Zn + 2 7a cos a + ii/n^ a^ cos - ^n^ cos -t- Zn + 2 Ta cos a (8) cos ^ cos (/3 + y) ^ ^ _ ifa; / cos y 2T MnH\ (9) cos= (|3 + y) = — ^ V 2T ) - An^cos 9 + Kn + 2Ta cos a + Mn' a^ cos • 9 '^^^ «^^'if^ ^' ^ (^ + 7^^ fr°™ ^'^^ %^b«el to the next, and the axial distance is Za cos + 2^ cos a, so that the apparent wave velocity iiS /1 0^ n - axial length _ ^ a cos + Z cos a ^ ^ time lag ~ " flT^^ ' /3 + y = (a cos + ^ cos a) (11) sm (a cos + Z cos a) ^ ^ + ^ ' H f y = ya cos t) + t cos a) —, = sinH/3 + y) -4n^cos0 + Zn + 2T^{acos9 + I cos a) ^ fc ' j]/ii^l ; ^ > - ^n^ COS + Kn +^Ta^^^~rM^^^^^ -JT' and denoting the line density — M. >,„ ,„ ^^ 2(acos0 + Zcosa)^y"'°'^P' 263 (a cos 9 + 1 cos a) n -.^ .sin (a cos 6 + Z cos a) -==. <12) — ^re^ cos 6 + Kn + 2Ta cos a + ifnV cos /a cos 9 + cos a - An' G(^9 + Kn + '2Tj (a cos + Z cos a) ^ ' ^ When the chain is taut, cos 9 and cos a can be replaced by unity, and the expression is obtained of the next article, where an independent investigation is given of a small oscillation in the neighbourhood of the straight axis. The heHcal motion is a mechanical model of a wave of light, circularly polarised in a magnetic field, as described in Larmor's Aether and Matter, Appendix E, To represent a wave of plane polarisation by a steady motion, the gyrostatic chain would be swung round the axis Oz as a plane polygon, and the conditions investigated in a similar manner, as given by Sir W. Thomson in Proc. L.M.S. VI, 1875, and Kouth, Advanced Rigid Dynamics, 1905, p. 289. The chain can take the form of a plane zig-zag wave, with the centre of each flywheel on the axis Oz ; and then (13) - An' sin cos + Kn sin 9 = '2Ta sin (a + 9) (14) - An' cos e + Kn = T^^ \, where X = 2 (a cos 9 + 1 cos a) is the wave length. Thus with 9 = Jtt, , K _ Ta X Tsin a '^^^^ M~ ~hlM = ~p^^- 25. Suppose each gyrostat is equivalent dynamically to a bicycle flywheel, of axial length 2a, and each link to be a light cord or steel wire of length 2Z, stretched to a tension T. Denote by x, y the components of the slight displacement of the centre of a flywheel from the central straight line 0^^ ; and let j9, g, 1 denote the direction cosines of the axle, and r, s, 1 the direction cosines of a link, distinguishing the different bodies by a suffix, k. Then in the previous notation, to the order of approximation required. (1) ^1 = - § ' ^2 = % ^1 = ^4^1' ^' = -^^2, h=K; to be employed in the dynamical equations (4) §1, in which flg/ii and 63^3 can be Omitted as insensible. For the ^th flywheel (2) ^ - hA + hA = -A^ + K^= aT,{q,- s,) + aT,., {q, - s,^,), (3) ^ - hA + hA = A'^ + K^=-aT,{p,- r,) - aT,., {p, ^ r,.,) ■ and for the motion of translation (4) ^W^^ ^'"'^ ~ ^'-' ''''-'' ^^^ '^ "^ ^*'' ~ ^'-^''-^' while the geometrical relations are (5) x^ - x„.i = a(p„ + ^A_i) + 2lr,„ y„ - ^/s-i = a {q„ + q,,,^) + 2ls„. Writing (6) X + yi = w, p + qi = zs-, r + si = V, - iKDT^, + a(r, + 7,.i) tr, - a {T,<,, + T,.,H + T,cn - an') (1 + /) ; while U" in (13) will not be positive, and the straight chain will be unstable, unless the tension exceeds ' (15) (kh + „/r) (1 + ;•), 265 27. In a gyrostatic chain hanging vertically, with the lower end free, put (1) z, = 2k {a + l)= 2ka (1 + /), T, = kgM = .j^^yVJ)' and then (2) IaD^ - iKD + gM- Zk + 2k-l ^k'^k + ^k-\<^k-\ _ ^A. - gM -^"1 - -_^-J|-^ =. or 2 (1 + /) (4) ^^- - g 't ("i 'r;;- = (5) W^. - MJ^.i - a (W;t + CT;t-l) - 2 / 1 +/ requiring the hyper-geometric series for the solution ; but reducing to the Bessel function of zero degree when a and k are zero, or else an = a:. Realise with a chain revolving round the downward vertical, 28570 2 ^ 266 28. In the discussion of the small vibration near the vertical of a single flywheel when suspended by a thread of length 21 = b, we can put r,.i = 0, T, = gM, and then ■omit the suffix k in the preceding equations ; and then (1) (AD'- - iKD) sy + gMa (^ - 4. gMaA. When the point of the axle is placed in a small smooth cup, as in fig. 3, & = 0, and the condition (7) becomes (11) - {A + Ma') n' + Kn- gMa = 0, requiring for stability (12) K'> 4. gMa {A + Ma'), :as before in §3, Chapter I, since A + Mc? is the equatorial M.I. about the point 0. Spinning upright inside a smooth spherical surface of radius h, the sign of a must be •changed, to obtain the condition of a stability, as in the gyroscopic horizon of Fleuriais in § 11, p. 10, Chapter I, and Fig. 9. For a top spinning upright on the summit of a smooth sphere of radius h, the sign of a and b must be changed, or else the sign of g, which amounts to the same thino-. The relation (7) is obtainable by elementary reasoning on a state of Steady Motion, as in § 13, p. 12, Chapter I, where a flywheel is whirled round the vertical with its axle at a constant inclination 6, at the end of a thread of length h at an inclination a so that the thread and axle are in the same vertical plane. The dynamical equations are <13) Tsin a = Mn' (a sin « + Z> sin a), Tcos a = qM (14) Kn sin % - An^ sin cos = Ta sin (a - 0) ; and thence (15) ^2 tan a = a sin e + 6 sin a it <16) - An' cos % + Kn + gM.a = Ta sin « cot = a Ma ^-"l' , , ' tanW (17) - ^n- cos + Kn + gMa cos a _ sin g _ a sin a a 9^^a cos sm K. tan a - b sin „ -^ _ j 1 • , /rr\ 1 « -> . "' "■" cos O reducing to (7), when and a is small. 267 29. Denoting by ^, r, the components of horizontal displacement of the point of the- flywheel or top, then ^ ^ (1) br = ^, bs = n, ba = k + ni, (2) w = a7s + ha. If the point is forced to take the motion (5, „, t) by components of force X, F, Z, the equations of motion become (3) - AD'q + KDp = aY- aZq, (4) AD'p + KDq= -aX+ aZp, (5) MD^w = X+ Yi, M {DK - g) = Z; (6) {AD'' - iKD + gMa) t^ + MaZJ^^t, = MazsBX (7) [(^ + iWa^) Z)2 - zZX> + ^Ma] -us + ilfa&D^^ = il/aziri^^r. Thus if the point of the top is made to take the periodic motion given by (8) L, 28570 . 2 L 2 268 The geometrical relations are, as in (3) §28, (6) as - ^ = - ap, y - r, = - aq, and the dynamical equations for the motion of the C.G. are, as in (2) §28, (8) B'a; = D'l - aD'p = - ga, D^y = D\ - glf-q = - g^ (9) aD'p = pD'a + ga, aD'q = may be resolved iSto two equal components, J = ^ about 00 and OC ; or ^ (1 - cos 9) about OC, and M sin about OA. perpendicular to 00 in the plane COC ; or expressed by Euler's 'three angles 6, ^, ^, the motion of the rod and case is the same as if J = ^, as^ in an upper Rosette gyroscopic motion of a spherical top, in § 14, p. 61, Chapter III If the angle 6 was kept constant the motion of the rod could be produced by rolling a bevil wheel on a cone of vertical angle . - 6 fixed to OC on an equal bevll wleef fixed on the vertical axis OC, as in fig. 118. ci^uai u^.\u wueei, • To allow a variation in 0, each of the bevil wheels would be required to sHd over -i sleeve, square to ensure rotation, with springs to keep the wheels in contact. ^ 269 Consider the motion of the elastic wire from another point of view, by supposing a length of it pushed down through a hole bored through the vertical spindle 00 in fig. 2, and then through the bore of the stalk OC When the wire is clamped in the stalk OC so as to partake of its motion' the ■component angular velocity is - /x cos about OC and fi sin 6 about OA, and (j> cannot vary ; but the angular velocity of the part of the wire in OC will partake of the motion of the vertical spindle, and will be fi with respect to the frame. But if the wire through OC is clamped to the frame, as in Kelvin's suspension of the Gyroscopic Pendulum, the continuation of the wire through the stalk OC will rub round inside it, when undamped, with relative angular velocity ^, and so have a component angular velocity ,ii (1 - cos 6) about OC relatively to the frame, together with the component ^ sin d about A, as before ; and if B varies, the stalk OC, and the •wire inside it has a component angular velocity — about the axis OB perpendicular to the vertical plane COC. The flywheel has an independent angular velocity R about OC, and will rub over the axle with relative angular velocity (1) -^ = R + fj. cos 0, or R = -^ — /J. cos d, -as before in §14, p. lo, Chapter I. Swing the gyrostatic chain bodily as a plane polygon round an axis Oz with angular velocity ** or n, like a skipping rope in relative equilibrium ; then with Kelvin's elastic wire connexion, the A.M. about an axis perpendicular to 0^^, through the centre of a gyrostat in the plane of the polygon, will be, with the preceding notation, (2) — -An sin 6 cos 6 + K sin 9, for a flywheel, (3) — A'n sin. 6 cos 6 + C'n (1 — cos 6) sin B, for the case, the axle making an angle B with Oz ; and n times the sum of these terms in (2) and (3) will be the couple required for the precession n, acting about an axis Ox perpendicular to the plane of the gyrostats. If a denotes the angle which a link makes with Oz in fig. 119, and Z denotes the -constant component of the tension of a link parallel to Oz, this couple will be .(4) "iaZ sin Bj, — aZ cos B^ (tan a^ + tan a,j_i) = {A + A' - C) n" sin 0, cos 0, + {Kn + C'ri^) sin 0, .(5) (.4 + A' - C) n^ sin B„ + {Kn + C'n^) tan B„ + aZ (tan aj. — 2 tan B^ + tan a^^.i) = 0. In addition (6) Mn^y,, + Z (tan a^, - tan a^.i) = ; with the geometrical relation (7) y^ - y^.i - a (sin B^, + sin 0,.i) - 21 sin a„ = 0. When the gyrostatic chain is stretched taut, so as to form a polygon nearly straight, then sin B and tan B may be replaced by 0, and sin a, tan a by a ; and the rest of the ■solution proceeds as before in §26, putting <8) Vk, h, «^ = m P, Q) exp cU. A half wave length of the curve is covered when ck = tt, so that v/c is the number of gyrostats in a stationary half wave, which is therefore of length tt —^ . A plane polarised moving wave is given when exp cki is replaced by exp (nt + ck)i ; •and a helical wave of circular polarisation when lo, ct, a of §25 replace this y, 0, a. (Proceedings of the London Math. Society, VI., 1875 ; Waves in a stretched gyrostatic chain, by Sir W. Thomson.) In a continuous chain of revolving gyrostatic links, an analytical difiiculty arises because a and B do not coincide with the tangent in the limit, and the curve would be given by a function without a differential coefiicient. The analytical thumb passed over the curve would feel an infinitesimal roughness as of a milled surface, increasing as the ■curve makes a greater angle with the axis Oz of rotation, even with an ordinary chain and K = 0, due to the transverse M.I. a. 1 a 270 Gyroscopic Pendulum with Flexible Wire Suspension. 32 With the previous notation of § HI for a single flywheel in its case with a short length of wire suspension clamped at 0, and employing Euler's angles 6, i, f, the kinetic energy is (1) T=HA + A') (i; + M^sin^e) + JCV (1 - cos 6)^+ J(7 (^ - M cos fl)^ and the A.M. about the vertical is (2) {A + A') ^ sin^ + CV (1 - cos Bf - <^' (^ - M cos s) --= G, a constant ; while (3) g-^cos0 = i?, a constant ; so that the Equation of Energy (4) T = gMh (H + cos 0), where H is a constant, will lead to (5) (A + 4') ^ + (G + CR cos ey ^ 2^^ (^ ^^^ 0) and putting cos = 2: (^^ Wi = ATA' ^^ - ^^ ^^ + ^^ (4 + ^')[U + 4')(l+^) + 6''(1-^)] ntroducing the Hyperelliptic Integral, except when C' = 0, or when C = A + A', which identifies the motion with an upper rosette of the spherical top. When the wheel is clamped on the axle in the case, and the case is supported by this elastic suspension of a short flexible wire, <^ = 4'i ^n^ (7) T=\{A + A'){^ + .^sixxH) + H^ + C")m'(1 -cos0)^ and (8) \_{A + A) sin^ e + (6' + C) (1 - cos 0)2] ^L= G (9) m - i^\. (1 - ^') is - ') - .^:i^,t.-' dt) A+~A'^ '^ ' (^ + 4')[(^ + ^')(l+^) + (C+6")(I-^)r obtainable from (6) by putting C = ; and the motion is hyperelliptic, except when C + C = ^ + J.', or when C + C" = 0, like a spherical pendulum. To realise the motion given completely by the elliptic function when the inertia of the stalk is taken into account, ignored in the previous treatment, the suspension of the stalk must be made by a smooth ball and socket, or else a Hooke universal joint. Whirling Deflection or a Shaft. . 33. The motion is intractable if the wire suspension is not restricted to be short except when the gyrostat departs slightly a distance a from the position of equilibrium in the downward vertical. Then if EI denotes the flexural rigidity of the wire, the bending movement (B.M.) at a depth z below 0, where the deflection is y, is v •- •/ where io is the B.M. at 0, and Y = Mn\ Z = gM, in absolute measure. Here Zy can be ignored when n is large, so that, integrating, (2) i?/| = A. - 1 IV, 7^/ tan = .L„A - \ )7r', if A is the length of the wire. 271 If L denotes the B.M. at G, the point of attachment in the gyrostat (4). L,= L+Yh, ETtane=Lh + lYh^ Ela = \Lh' + \Yh', and eliminating i, (5) EI (^ - tan e) = 1 F^^ = 1 Mn^ a¥ (6) tane = ^ - § .. h EI But the vector velocity of the A.M. of the gyrostat in (2) (3) § 31 (7) {A + A) v? sin cos + C'n ( L - cos 0) sin + Zn sin = Z, (8) {A + A - C) n' cos^ 9 + (C'n' + Kn) cos = ^- ^ tan 9 _Bl 5 5-* £J -2*'"* EI I"-''' EI sf ~ ^ =4^;- + tan A tan h '^ m 2i,-> h 1 ,^ „„ h Er~ EI 1 in which cos may be replaced by unity, giving a quartic equation for n, the critical speed at which the straight vertical position becomes unstable, (9) X Mn'¥ \(A + A') n' + Kn-4^] f ^^ ' - A = s ^I I h j\ EI ) h EI ^ h tan EI This is with Kelvin's suspension, where the wire is clamped at ; but in a steady ■deflection of the gyrostat when the wire is made to rotate with the case clamped to it, the vector velocity of the A.M. is (10) {A + A) n' sin cos + (iT - C" n cos 0) ?2 sin = L (11) {A + A - C) n^ cos^ + Z'n cos ^ •(12) {A + A - C) n' + Kn = 4.^ + -^ EI ^ :agreeing with the expression given by N'oether in Kreisel Theorie, p. 893. In this way the stability- is investigated of the shaft of the Laval steam turbine, and of the flywheel of the gyro-compass. The shaft is made thin and flexible, but it runs straight and steady provided the speed avoids one of the critical values of n given in (12). The extension to more than one flywheel on the shaft is required in the design of steam turbine machinery in rapid rotation, and worked out by Stodola and other writers. The inertia of the shaft has been ignored here, compared with the flywheel ; but if a straight shaft runs in bearings, the critical whirling speed of revolution is found to be that given by the musical note of the flexural vibration, as investigated in the Eroc. Inst. Mechanical Engineers, April, 1883, where thrust and torsion of the shaft is taken, into account as well. As. the speed is increased the shaft passes through a succession of ■critical values of rotation, corresponding with the fundamental note and/hartaonics of the iateral vibration. 272 Motion of a Body with Altazimuth Suspension. 34. With a ball and socket joint at in fig. 3, the wheel and stalk may be clamped together without altering the character of the motion. But with the freedom at in altitude and azimuth only, as in fig. 3, the motion becomes hyperelliptic, and so we investigate the inertia of the stalk, ignored hitherto in the previous treatment. Then for any solid body, with uniaxial symmetry, like the wheel clamped to the stalk, replace CR by - CV cos 0, and the equations of momentum and energy become (1) (.4 sin= + C cos^ 6) ft = G, (2) A (f)' + (A mn' 6 + C cos^ 6) f.' = 2gMh (H + cos 6) ; so that, with cos 6 = z, (31 A (*)= - 2,.M* (ff + .) (1 - ^) - ^ f /^ : ^> ,, . and t is given in terms of z by the hyperelliptic integral and further progress with it is abandoned. Bat , . dU _ (PQ _ _ (-iMh . . A - C G"^ sine cos 6 ^^ W ~ df ~ '^A ^'"^ A {A sin^ + (7cos2 0)^ — ■ sm d + 3 — sin d cos d ' ^ A A \dt ,.. ld^Q_d (dQ\_ gMh „ .4- C ^+ (4-3C) cos^ 0-2 (^- (7) cos* ^, Q df de\dtJ A A (A sin^ 6 + C cos^* 0)^ __9Mh ^^^^^_ A-C A + (A-$C) cosH -2 (A- C) cos' 6 fd^pY A A ' A siu^ e + C cos^ e \dt)' and so in a state of Steady Motion at inclination with the downward vertical, (6) .^^ ^^^cos0 = O, , = ^, cos^ + ^+(^-3C)cos^0-2U-g)cos*0 -| _ A- (A- C) cos^ J f" - ' ^^ Q dfi ~^~ /o^ TO^ 3 A- CA + '6{A - C) cos^ . , ^ /I' A A - (A - C) cos^ ' which shows that in the small nutation, the axis makes — and — beats per second and per circuit. The small oscillation can be analysed in the same way of the gyroscope wheel in fig. 3 in Steady Motion, given in §4,- Chapter III, p. 50. When C = 0, the body reduces to a rod, and the motion is that of a Spherical Pendulum, investigated in Chapter VII. With spherical kinetic symmetry at 0, as in plane pendulum oscillation, and ^ is constant. . T-^r *?^ inertia effect of a gimbal ring in the gyroscope of fig. 32 will not influence the elliptic function character of the motion of the flywheel, if the ring is made kinetically a_ sphere, by converting it into a cage by the addition of another equal ring in a plane a^ right angles (Koenigs, Darboux Bulletin, 1895, p. 225). ^ 35. When the restriction is removed of uniaxial symmetry, and a return is made to the equations of §2, with the B axis along the pin of tig. 3, and the BC plane Through the C.G., twice the kinetic energy is given as in (2), §29, Chapter I, p. 24, wi?h ^ (i) P = Msin0, Q = f E=-uco.e, u = ^^' «* dt^ referred to the upward vertical ; and applying the equation of energy and momentum, (2) ^T=b[-^J + (.4sin^0 + 2i;sin0cos0 + Ccos^0) .^ + 2 (Dco^O - FsinQ) .k» = 29Mh{H + cose) * 273 (3) (7 = Ai sin - lis cos 9 = {AP - FQ - EE) sin - {CR - EP - DQ) cos 9 = (A sin^ 9 + 2E sin cos + C cos^ 9) fi + (D cos 9 - F sin 9) — and eliminating ^, (4) {A sin^ 9 + 2^ sin cos + C cos^ 0) i^(^)'- {D cos 9 ~ F sin 0) (^)' + G' = 2^MA (H + cos 0) (A sin^ + 2^ sin cos + C cos^ 0) of hyperelliptic character ; of no more essential complication than before in § 32. 36. The movement will be of the same analytical character for a symmetrical top with a sharp or rounded spherical point, as in fig. 2, spinning on a smooth table. The horizontal motion of the C.G. may be ignored as uniform, and the C.Gr. taken as rising and falling in a vertical line ; the application of the momentum and energy principle gives, with respect to the downward vertical, (1) Afi sin^ 9 - CEcos9 = G (2) {A + Mh' sin^ 0) (^y + A^^ sin^ = 2gMh (H - cos 0) h denoting the distance between the C.G., and the centre of the ball point, and A referring to an axis through the C.G. ; so that (3) (A + Mh' sin^ 0) Q2 + ^^ tl^if '^T ^^' ~ ^^^ ^^ ~ ''°' ^^ " ^' and differentiating with respect to 9, twice, /^\ / /I , l,n.2 • 2Q\dQ , 1^2 • o a n'> (G+ CRC0S9')(GC0S9+ CR) art • n A (4) {A+ Mh^ sm''9) -^ + M¥ sm cos 9Q;— ^ '.^„ ^ — gMh sin = dt A. sin (5) {A + Mh? sin^ 0) ^ ^^^ + 4 Mh^ sin cos ^ + Mh^ (cos^ - sin^ 0) Q^ jG' + g^ig^)(] + 2 cos^ 9) + GCR (o cos + cos^ 0) ,^^ eos = A sin* '' In a state of Steady Motion .n. n a dQ r, ,„ (G + CR cos 9) {G cos 9 + OR) G = A^isiD.^9 - CR cos 0, Ai^i' cos + CR^ + gMh = 0, CR' > 4 gMh A cos & (7) (^ + m^sin^0)^^-^ (G' + CR') (1 + 3 cos^ 0) + 2 a CR cos (3 + cos^ 0) _ ^ ^ A sin^ (8) m' ((9^ + CR') (] + 3 cos^0) + 2 GCR cos (3 + cos^ 0) (A + MK' sin=^ 0) A sin^ 1 + 2 --^ cos + i^j— ■ ^ 1 + ■ — ^ sm- A Spinning upright with = 0, G = — CR, the motion cannot be treated as steady ;; but (8) reduces, as before in § 21, Chapter IV, p. 108, to (ON 1 ^^a , „2 _ „2 _ CR' _ flMh 4 /, _C _ 4gMh ^^ Q'd? ^P -^' P -^aF T-' ^R'~ A' AR' In the associated experiment of a rhombus ABCD of light rods of length 2a in fig. 120,. each forming the axle of flywheel of mass M, revolving with A.V. R in the same direction when the rhombus closes with the vertical diagonal AC, and supporting a weight Mi hung from C, the application of the same dynamical principles will give (10) {A + Ma') fi sin^ - Ci2 cos = G (11) (A + Ma') (^)' + {A + 9Ma' sin + ifa^ cos^ 0) (^~J + 2 {A + Ma') ^ sin^^ = 2g (4:M + 2Mi) a{H - cos 0) ; and the investigation of the Steady Motion and small oscillation proceeds as before. 28570 2 M 274 Concealed in a casing this apparatus would appear to act like a spring balance, provided it is allowed to rotate with free precession, even with R = 0. Arrest the precession by holding the rhombus, and the action would cease. But the apparatus is not found to work, from the practical difficulty of maintaining equably the rotation R. Referred to principal axes again, with D = E = F = in §35, the motion is of the same character as before in (3). The motion of the vertical axle fluctuates ; but if ic is constrained to move with constant A.V. ^, the solution is that given already in §2, p. 202, Chapter VIII. The altazimuth suspension of fig. 3 may be replaced by Hooke's joint, or else by a short length of flexible wire, as employed by Kelvin. As a dynamical experiment of gyroscopic interest, suspend the steering head of a bicycle by the pin through in fig. 3, the head carrying the fork and front wheel ; and investigate the motion, steady and unsteady, with the wheel spinning with angular velocity R, (i) when the fork can rotate on the head with angular displacement , (ii) when the fork is clamped to the head. A rod in a smooth sphere is equivalent to a rod across the stalk, which makes an unsymmetrical top, and the motion is intractable. Application of an Elliptic Function Table, 37. This Table is taken from a Report to the British Association, 1913, to use in the construction of a state of motion, of pseudo-regular precession, where the axle of the top has a small nutation with a precession backward and forward, the axle describing a series of looped curves ; these were indicated in Chapter V, §26, oO, as numerical applications of the analytical results. The table has been calculated to give (1) E{r) - znfK, F{r) = ™ (1 - f)K, f - ~ ; and then the elliptic function from ^(r) n{r) ^ ' D{r) ^k dn (1 -/)Z' In the pseudo-regular precession the modulus k is small, and the table gives the functions of the co- modulus ; so that an interchange of A' and K is made in using the table, corresponding to k = ( V 2 - 1)^ = sin 9". 9, K' = 2K. Drawing the focal ellipse, and reading off the angle 63 and 63 for a given f, and with H at L, an .algebraical curve is obtained for the corresponding / and k ; these are shown in diagrams for (4) A' = 2 A, ;•= 1 11-1^ -1- -1- ^ •' ' ./ 2) 4) ^) 8) 10) 1 2! .as representative of the analytical results of Chapter V, §24 to 30. A few more figures are added as illustrations of algebraical motion where the modular angle is large, say 80°-l, when A = 2A', and the axle oscillates between wide limits, as shown already in fig. 81, 88, 93. Also iov anothe^ modulus, corresponding say to A = A', or A = AV2. Thus A = A , K - sin 40 - U-70/,/ = i, gives a figure intermediate to fig. 81 and 88. These figures can be multiplied indefinitely, and provide an exercise in drawing on the focal elhpse. & Calculation is thus replaced by measurement on a drawing. 275 Elliptic Function Table. K = 3-163232 = 2/r, k' = (n/2 - 1)^ = cos 80°-l. r F0 •P B(r) D(r) A(r) BW 0(0 F(;-) f F;// 90-»- 0-0000000 00-00 0-0000000 1-00000 0-0000000 1-000000 2-41422 0-000000 90-00 3-1632320 90 1 2 3 0-0351470 0-0702940 0-1054411 05-33 06-37 07-78 0-0236055 0-0471266 0-0704807 1-00042 1-00168 1-00373 0145672 0291353 0437045 996072 995482 994492 2-41377 2-41248 2-41020 0-0105122 0-0210228 0-0315278 89 89 88 66 32 98 3-1280850 3-0929380 3-0577909 89 88 87 i 5 6 0-1405881 0-1757351 0-2108821 09-40 11-15 12-94 0-0935852 0-116354 0-1387305 1-00663 1-01036 1-01490 0582762 0728510 0874299 993121 991335 989170 2-40709 2-40309 2-39821 0-0420255 0-0525198 0-0629870 88 88 87 63 28 92 3-0226439 2-9874969 2-9523499 86 85 84 7 8 9 0-2460292 0-281U62 0-3163232 14-78 16-61 18-44 0-160624 0-181970 0-2027083 1-02026 1- 02643 1-03340 102012 116600 131193 986619 983674 980347 2-39246 2-38585 2-37839 0-0734471 0-0838875 0-094H086 87 87 86 56 20 84 2-9172028 2-8820558 2-8469088 83 82 81 10 11 12 0-3514702 0-3866172 0-4217643 20-27 22-09 23-90 0-222777 0-242131 0-2607148 1-04116 1-04971 ] -05903 145791 160394 175004 976646 972566 968111 2-37008 2-36094 2-35098 0-104707 0-115075 0-125414 86 86 85 47 10 72 2-8117618 2-7766148 2-7414677 80 79 78 13 14 15 0-4569113 0-4920583 0-5272053 25-69 27-47 29-23 0-278492 0-295428 0-311493 1-06912 1-07997 1-09156 189616 204239 218864 963282 958075 952541 2-34021 2-32864 2-31631 0-1357-21 0-145990 0-156220 85 84 84 34 96 57 2-7063207 2-6711737 2-6360267 77 76 75 16 17 18 0-5623524 0-5974994 0-6326464 30-96 32-66 34-33 0-326664 0-340925 0-354265 1-10388 1-11692 1-13066 233493 248125 262760 946636 940378 933773 2-30320 2-28935 2-27438 0-166406 0-176543 0-186628 84 83 83 17 77 36 2-6008796 2-5657326 2-5305856 74 73 72 19 20 21 0-6677934 0-7029404 0-7380875 35-97 37-58 39-15 0-366675 0-378154 0-388707 1-14509 1-16019 1-17594 277394 292031 306664 926839 919564 911962 2-25950 2-24353 2-22689 0-196653 0-206623 0-216524 82 82 82 95 53 10 2-4953858 2-4602916 2-4251445 71 70 69 22 53 24 0-7732345 0-8083815 0-8435285 40-69 42-19 43-68 0-396459 0-406863 0-414883 1-19233 1-20934 1-22695 321294 335917 350531 904045 895809 887274 2-20962 2-19172 2-17322 0-224474 0-236302 0-245773 81 81 80 65 19 72 2-3899975 2-3548505 2-3197035 68 67 66 25 26 27 0-8786756 0-9138226 0-9489696 45-14 46-56 47-94 0-421828 0-427914 0-433163 1-24513 1-26388 1-28317 365133 379723 394294 878439 869306 859692 2-15416 2-13455 2-11442 0-255351 0-264835 0-274217 80 79 79 24 75 25 2-2845564 2-2494094 2-2142624 65 64 63 28 29 ■30 0-9841166 1-0192636 1-0544107 49-28 50-59 51-87 0-437594 0-441236 0-444114 1-30297 1-32326 1-34403 408845 423370 437868 850196 840236 830015 2-09378 2-07268 2-05113 0-283484 0-292637 0-301661 78 78 77 74 21 67 2-1791154 2-1439684 2-1088213 62 61 60 31 32 53 1-0895577 1-1247047 1-1598517 53 12 54-34 55-53 0-446087 0-447353 0-447805 1-36524 1-38687 1-40884 452333 466761 481132 819360 808450 797312 2-02915 2-00682 1-98395 0-310381 0-318958 0-327524 77 76 75 12 55 97 2-0736743 2-0385273 2-0033803 59 58 57 34 35 36 1-1949988 1-2301458 1-2652928 56-69 57-82 58-92 0-447888 0-457214 0-445955 1-43130 1-45405 1-47711 495484 509771 523999 7859/5 774362 762563 1-96107 1-93772 1-91412 0-335654 0-343753 0-351674 75 74 74 37 75 11 1 - 9682332 1-9330862 1-8979392 56 55 54 ■37 38 39 1-3004398 1-3355868 1-3707339 59-99 61-03 62-04 0-444133 0-441637 0-438925 1-50047 1-52408 1-54794 538163 552258 566278 750574 738388 726016 1-89028 1-86619 1-84197 0-359399 0-367083 0-374230 73 72 72 46 79 10 1-8627922 1-8276452 1-7924981 53 52 51 40 41 42 1-4058809 1-4410279 1-4761749 63-02 63-97 64-89 0-435596 0-431815 0-427607 1-57200 1-59624 1-62063 580214 594062 607813 713473 700756 687893 1-81756 1-79304 1-76846 0-381307 0-388138 0-394706 71 70 69 39 66 91 1-7573511 1-7222041 1-6870571 50 49 48 43 44 45 1-5113220 1-5464690 1-5816160 65-79 66-67 67-53 0-422994 0-418011 0-414213 1-64514 1-66975 1-69441 621462 635000 648419 674873 661712 648419 1-74379 1-71911 1-69441 0-400992 0-406990 0-414213 69 68 67 14 35 53 1-6519100 1-6167630 1-5816160 47 46 45 90-?- F;// '/' FW CW BO) A 00 DOO B(r) f Ff f 276 INDEX. Chapter I.— Page Steady Gyroscopic Motion 1 Gyroscopic Couple ... ... ... ... ■•• ■■■ ••■ ••• ••• ■■• ••■ ^ Precession of the Equinox ... ... ... •■• ■•• ••■ •■■ ■•■ ••• •■■ 9' Serson-Fleuriais Gyroscopic Horizon 10 Gyroscopic effect of a wheeled carriage, paddle or screw steamer 16 Stability of an elongated body in a medium ; air ship or submarine boat 18 Appendix on General Dynamical Theory ... ... .•• ■.- • ■ ••• .■■ ■•- 22 Chapter II.— Gyroscopic Applications .. ... ... ... ... ••- ■•■ ••• ■•• ••• 28 Theory of Oscillation, free, damped, and forced 32 Schlick Sea Gyroscope ... ... ... ... •• ... ■■■ •■■ ■■■ ■■■ 35 Mechanical Similitude 36 Single rail gyroscopic carriage ... ... -.- ••• ... ■•• ... ••• ••■ 37 Bessemer Saloon ... ... ... ... ... ... ... .-• .■• ■•■ ••■, 37 Appendix on Dynamical Units .. ... ... . ••. ... ••• ••• ■•• 39' Chapter III.— General Unsteady of a Top or Gyroscope 42 The Elliptic Integral and Function 48 Steady Motion slightly tremulous ... ... .• ... ... ... ■•• ••■ ... 50 The Apsidal Angle -.. 51 Elliptic Function Notation ... ... ... ... ... ... ... ... .. ... 53 Third Elliptic Integral, complete 57 Rosette curves ... ... ... ... ... ... ... ... ... ... ... 61 Cusped figure ... ... ... ... ... ... ... ... .. ... ... ... 63 Notation of Weierstrass ... ... ... ... ... ... ... ... ... ... 65 Lame Function... ... ... ... ... ... ... ... ... ... ... ... 69 Co-ordinate Axes, fixed and moving ... ... ... ... ... ... ... ... 70' KirchhofE's Kinetic Analogue ... ... ... ... ... ... ... ... ... 76 , Whirling Chain' 78 Clifford's vector treatment of the Top ... ... ... ... ... ... ... ... 79 Synopsis of the Formulas ... 80- Chapter IV. — Geometrical Representation of the Motion of a Top ... ... ... ... ... ... 81 Deformable Hyperboloid of generating lines ... ... 81 Darboux's representation of Top Motion 85 Curvature and Inflexion of the Herpolhode ... ... ... ... ... ... ... 91 Constants of the Motion of a Top ... ... ... ... ... ... ... ... ... 94 The Apsidal Angle ' , ... 93 Confocal and Contrafocal Surfaces ... Convention of Sign Associated States of Motion ... Curvature of the Polhode 100' 101 106 Darboux's Conjugate Relations ... ... ... .. ... ... ... ... __ jQg 112 Deformation of the Hyperboloid ... ... ... ... ... ... _ __ ■ ^j^^ Chapter V.— Algebraical Cases of Top Motion __ jng Bisection of a Period ... ... ... ... ,., ... ... .._ ___ "" iiq Quadric Transformation ... ... ... ... ... ... _ _ '"' "" -iqi Trisection ... ... ... ... ... ... ... ... ___ '_] "" '" ,,,- Quinquisection ... ... ... ... ... ... ... .. ' '" "" -.Zc) Seven Section ... ... ... ... ... ... . _ "" "_"" "' "" -.^"^ Halving of the Degree ... ... ... ... _ '"" '" "" ■" -^5 Nine and Eleven Section .. ... ... ... ... ' "' '" "" .^q Division by an Even Number ... ... .. _ "' "" ■"" y^*? Euler's Third Angle, and Klein's Parametprs .".' It- Curvature of the Motion projected on a Plane ... ... ... \ T-i 158 158 Chapter VI.— Numerical Illustrations and Diagrams Bisection Trisection _ Quinquisection " ^^^-^ Seven Section _' 1|57 Nutation in the Axle of the Earth ... .]." ."'.' ■'-J^' Free Eulerian Precession and Variation of Latitude ill Lunar Nutation ••• Ij^* Gyroscopic Motion of the Moon's Node ... ^^^ 176 277 Chapter VII.— Page. The Spherical Pendulum 178 Steady Motion and Small Nutation 186 Limits of the Apsidal Angle 188 The Plane Revolving Catenary 193 Separating Polhode and Hess's Integral 199 Chapter "VIII. — Motion referred to Moving Origin and Axes 201 Pendulum or Gyroscope on a Whirling Arm 202 Quadric Transformation 205 Moving Axes for the Motion of a Top 206 Motion of a Sphere in a Cone or Cylinder 210 Sphere containing a Fly-wheel, rolling on a Table 214 Jukovsky's exceptional Case 217 Modification of Poinsot Motion by Interior Fly-wheel 220 Interior Circulation in a Body 226 Expanding or Contracting Body 227 Liquid Gyrostat ... 228 Motion of a Solid in a Liquid Medium 231 Motion of a Perforated Solid In Liquid 236 Motion of a Cylinder surrounded by a Vortex 238 Chapter IX. — Dynamical Problems of Steady Motion and Small Oscillation ... ... ... ... •'•• 239 Unsteady Motion of a Body rolling on a Table 239 Steady Motion 242 Small Oscillation 243 Unsteady Motion of a Body sliding on a Smooth Table ... ... ... ... ••• •■. 246 The Single Track Gyroscopic Car ... ... ... ... ... ... .•• ••. ••• 251 Gyro-Compass ... ... ... ... ... ... ... ... ... ... ... ... 254 Gilbert Barogyroscope ... ... ... ... ... ... ... ... ... -.• 259 Gyrostatic Chain 261 Kelvin's Flexible Wire Suspension... ... ... ... ... ... ... ..- .•• 268 Gyroscopic Pendulum with Wire Suspension 269 Stability of a Revolving Shaft 270 Motion of a Body with Alt- Azimuth Suspension " .•• 270 Elliiitic Function Table and Applications 274 Plates of Diagrams. Corrections. Page 132. §15 (1) /x = 10, x, - x^ = IGc^ V (c^ + c^ - c). Page 143 §26, u = 20 (8), -f I at end of the formula. Page 144. §28, /» = S, (^/; ^ ^ 5 <°4 2 ' (3)Pi = - NP{2v) Page 146. §29, ft = 16, here b is the reciprocal of a on p. 145 for /i = 8. Page 147. §30, m = 24, (7)2a3 + a,^ - 4A«. + 6i) = 0, 2^ + [^^ 4^ ^J -. 1 -. ^ = Page 147. §30, iu = 32 ; here c of Phil. Trans. §49, p. 301, is connected with h in fi = 16 by b'^ = c^ + c 2 N 28570