US i-i! ;'! ir i S7fFM***»>»!<«?>**^. C?9 (fendl Itttumtig Htbrarg BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF fteitrg M+ Sage 1891 I'M.^.^bA,. I93°6 The date shows when this volume was taken. To renew this book copy the call No. and give to the librarian. AU0& 1946 fEC 18 1953 *\ p r- ^ v*? ,,__ FEB10T35H" *ftfc NO All Books subject to Recall All borrowers must regis- ter in the library to borrow books for home use. All books must be re- turned at end of college year for inspection and repairs. Limited books must be re- turned within the four week limit and not renewed. Students must return all books before leaving town. < Officers should arrange for the return of books wanted during their., absence from AY -1-Q4958 to ™ Volumes of periodicals and of pamphlets are held in the library as much as possible. For special pur- poses they are given but for a limited time. Borrowers should not use their library privileges for the benefit of other persons. Books of special value and gift books, when the giver wishes it, are not allowed to circulate. Readers are asked to re- port all cases of books marked or 'mutilated. ' Do not deface books by marks and writing. his \Kjs. ■ QA 86 2.c79 rne " Universi,yybrar >' The gyroscope, 3 1924 004 047 860 H || Cornell University WB Library 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/cu31924004047860 PUBLICATIONS BY THE AUTHOR SPON & CHAMBERLAIN, NEW YORK BAROMETRICAL DETERMINATION OF HEIGHTS. fo.50 THE ATMOSPHERE. A Manual of Meteorology $1.50 THE GYROSCOPE. Theory and Applications $1.50 THE GYROSCOPE BY F. J. B. CORDEIRO AUTHOR OF "THE ATMOSPHERE," "BAROMETRICAL HEIGHTS," ETC. Ca>$=A$ NEW YORK: SPON & CHAMBERLAIN, 123 Liberty Street LONDON: E. & F. N. SPON, Limited, 57 Haymarket, S.W. 7 .< V>H A- 3 fa 3 Z U r> Copyright, 1913, BY F. J. B. CORDEIRO Stanbope jpress F- H.GIISOS COMPANY BOSTON, U.S.A. PREFACE. Ous exact knowledge of Rotary Motion, as of Dynamics in general, dates from the time of Newton. Euler, La- place, Lagrange, Poisson and Poinsot are illustrious names in the development of the theory. Foucault, in 1855, demonstrated the rotation of the earth by means of the gyroscope, and gave it its name. Its practical applica- tions date from yesterday. These began with the GrifEn Grinding Mill, and have been followed by the Howell and Obry devices for keeping a torpedo on a straight course, the Schlick Stabilsator for ships, the Brennan Gyro- Monorail, the Anschuetz-Kaempfe Gyro-compass, and the end is not yet. The theory of rotary motion is not simple, nor is it yet complete. Not all inventors have understood the reason for their devices, and not all mathematicians have had a clear conception of the theory, as evidenced by the un- necessary complication of their treatments. Attempts have been made to explain gyroscopic action without mathematics, or at least without the Calculus. It is hardly necessary to say that all such attempts are futile. It is impossible to explain the actions of a gyroscope with- out mathematics, and it is impossible to understand them without such knowledge. Many students are afraid of what is called the higher mathematics, and are permitted to avoid them in our higher institutions of learning. Mathematics, in its broadest sense, is the science of time, space, mass and force, and the VI PREFACE relations existing between these four quantities. It is the foundation upon which all the exact sciences are built, as it is the foundation of the universe. Everything else may and does change, but the principles of mathematics alone remain eternal. There is no doubt that mathematics are difficult: all other forms of intellectual effort are mere child's play in comparison. Hence many who are scientifically inclined, seek a field in the inexact sciences, or in the pseudo-sciences, where these difficulties may be shirked. It is noteworthy, however, that even here, as these branches become de- veloped, they are found to reach down to the solid bed- rock of mathematics, where their cultivators, who have neglected the fundamental science of all, find themselves in an unenviable position. This is notably the case with meteorology. The day will undoubtedly come (and the sooner, the better) when mathematics will be made the foundation of every education, and no man (or woman) can be considered educated who does not know the Calculus. A virile mind will not quail before its difficulties, but will experience a joy in surmounting its obstacles — • the gaudium certami- nis — such as can be found in no other intellectual field. The student with an elementary knowledge of mathe- matics, who attempts to understand gyroscopics from a study of its scattered parts in standard treatises, and from the few monographs as yet written, will find the task tedious — probably repulsive. For this reason, it has seemed advisable to the author to write a monograph which may be easily understood by anybody possessing an elementary knowledge of mechanics and the calculus. The book is divided into two parts — the development of the theory from the Fundamental Gyroscopic Principle, and a discussion of its modern practical applications. The PREFACE vii motions of the heavenly bodies, where gyroscopics are ex- hibited in their grandest and freest (frictionless) form, have been fully explained, but the engineer and cursory student, who care only for the elementary theory and an explanation of its applications, may omit the astronomical discussion without loss of continuity. F. J. B. C. Newton Centre, Mass., June 16, 1913. , PART I. THEORY. THE GYROSCOPE PART I. THEORY. i. The Name. We shall define any rotating mass as a gyroscope. Fou- cault gave this name to the instrument, consisting of a rapidly rotating flywheel, by which he demonstrated the rotation of the earth. We shall extend this name to any rotating mass. It is eminently suitable. The word gyro- scope means simply any body exhibiting, or showing ((TKoiireiv), gyration, or rotation. The term gyrostat, often used for gyroscope, is particu- larly objectionable. There is no such thing as a gyrostat, or instrument which maintains its plane of rotation. We shall see that no gyroscope can maintain its plane when acted upon by an outside force, although it is true that, provided it possesses a high rotational moment, it changes its plane much more slowly than a non-rotating body under similar conditions, thus exhibiting a high degree of what we might call rotational inertia. Here again it changes its plane in a totally different manner from a non-rotating body, but the slightest couple, acting for a sufficiently long time, will change its plane of rotation to any desired extent. Certain German writers employ the word " top " (Kreisel), as a generic name for the gyroscope, although, happily, an increasing number are now using the word gyroscope. This seems to be an overstraining towards simplicity, such 3 4 THE GYROSCOPE as leads a man to call his palace a cottage. But a cyclone is a gyroscope, i.e., a rotating mass, and so is a child's hoop, and so is the earth; but none of these are tops, although it is true that a top is a gyroscope. Furthermore a top is a top only while a child is playing with it. Immediately a philosopher begins to experiment with it, it ceases to be a top and becomes a gyroscope. The interest of gyroscopes for us, apart from certain recent applications, is manifold. We live in a universe of gyroscopes; everywhere there is rotary motion. We live on a gyroscope, and we ride on gyroscopes, be it carriage, bicycle, train or aeroplane. Our factories are full of gyro- scopes, where occasionally ignorance of gyroscopic laws leads to disaster. The axle of a rapidly-rotating grindstone works loose and the stone flies apart, killing a workman. It is ascribed to simple centrifugal force, but the stone would have withstood this stress if preserving the same plane. It could not, however, withstand the enormous gyroscopic couple, acting at right angles to its plane when the axle moved. An aviator makes too sharp a turn, thereby setting up a rotation of the whole machine about an axis which, in general, is not a principal axis. It becomes a gyroscope, even if the other gyroscopes he is carrying, viz., the rotary motor and propeller, are not turning. So that, even with the motor dead, and a fortiori with the motor turning, gyro- scopic couples are set up in a direction which no human in- stinct can foresee, and he is capsized. It is ascribed to a "hole in the air," or some other "theory." 2. Mathematical Definitions. If a body is rotating about some axis, the integral, / p 2 dm, where dm is an infinitesimal component of its THEORY 5 mass, and p its distance from the axis, is called its moment of inertia about that axis. If I p 2 dm = Mk 2 , where M is the total mass of the body, then k is called the radius of gyration about that axis, and the equation means that, if the total mass were concentrated into a mathematical point at the extremity of the radius of gyration, the mo- ment of inertia would be the same. Through any point in a mass, three mutually perpen- dicular axes can be found, which are called the principal axes of inertia at that point. These axes possess the property that the moment of inertia about them is greater or less than that; about any other axis in their immediate vicinity, and one of these principal axes is a maximum for the body, while another is a minimum. The principal axes at the center of inertia (or center of gravity) are called the Principal Axes of the body. In general, for an unsymmetrical body, the moments of inertia about these axes are unequal, and we shall define a body with three unequal principal axes as a Tri-axial body. If the body is symmetrically homogeneous about one axis, as in a homo- geneous solid of revolution, two of these axes become equal, and we shall define such a body as a Bi-axial body. If the body is homogeneously symmetrical about all axes, as in a homogeneous sphere, we shall define it as Uni-axial* A couple is defined as two equal forces acting in opposite directions, and at equal distances, about some. axis. The moment of a force about an axis is defined as the product of the force into its distance from the, axis. The moment of a couple or, as it is usually called, simply the Couple, is the product of one of the forces into the distance between them. * In works on mechanics and optics, what we have defined as a bi-axial body, is called uni-axial. The above definition is more consistent. 6 THE GYROSCOPE The plane of the forces is the plane of the couple, and the axis about which they act, which is perpendicular to the plane of the couple, is called the axis of the couple. The word " Torque" is engineering "slang" for couple. It should never be used. A single name or symbol should be strictly preserved for every mathematical entity. It would be as logical for engineers to use a separate and different symbol for ir, or for a man to be known to one set of acquaintances as Smith, and to another set as Brown. A couple acting about an axis is subject to the same laws as a simple force acting along a straight line. That is, the couple is equal and opposite (Newton's third law) to the moment of inertia of the body about its axis into its angular acceleration about that axis, just as a simple force is equal and opposite to the, mass of a body into its linear accelera- tion. The moment of momentum of a body about an axis is defined as its moment of inertia into the angular velocity about the axis, and this is evidently the time integral of the couple acting about the axis. The kinetic energy about any axis is denned as half the moment of inertia, into the square of the angular velocity, and is evidently the angle or space integral of the couple about that axis; or it is the equivalent of the work done by the couple. It is evident that all these quantities, depending upon an axis having a certain definite direction, are directed quantities and, therefore, by the parallelogram principle, we can resolve them into components having other direc- tions about other axes. Thus, when a body is turning about some axis, it can be considered to have a component of this turn about any other axis, and the amount of this component is found by multiplying the original turn into the cosine of the angle between the two axes. There are two directions in which a body may turn about an axis — right and left rotation. If we adopt one THEORY 7 direction as positive, then the other is mathematically negative. The symbol plus applies to one, the symbol minus to the other. It is important that what is regarded as positive, standard or normal rotation should be fixed by convention. The lack of such a convention has resulted in much confusion in mathematical figures and demonstrations. A priori there appears to be no reason for choosing one direction over the other. It might be argued for right rotation that most people are right-handed, that clocks turn to the right, and that most civilized nations make screws with a right twist.* All this is probably accidental. The more convincing argu- ment is that in the Northern Hemisphere, where there is the most land, and the greatest population and the highest civili- zation exist, practically all the motions of nature are to the left — cyclones, heavenly bodies, etc. We shall therefore define as positive rotation that to the left, or against a clock. It would be a great advantage if a uniform notation could prevail everywhere regarding gyroscopic motion. The fol- lowing notation will be used throughout this book. A , B, C, represent the principal moments of inertia in ascending or- der. Velocities and accelerations, or first and second deriv- atives with respect to the time, will be represented by super-dots, and double super-dots, respectively. Thus, represents the angular velocity of the angle 0, and its angular acceleration. The signs, J. and |] will be used to designate perpendicularity and parallelism, respectively. Motion about an axis through the center of gravity of the body will be called rotation, while the motion of the center of gravity about an external axis will be called revolution. The symbol a> will be used generally to designate the * This is not universal. Germans and Americans rifle their guns to the right; English and Italians, to the left. 8 THE GYROSCOPE angular velocity about the principal axis C, or the axis of greatest moment of inertia. In the case of a tri-axial body, o>i, w 2 , o) 3 will be used to designate the angular velocities about the axes A, B, C, respectively. 3. Gyroscopic Action. In Fig. 1 let us suppose a circular disc with its center of gravity fixed, and capable of turning about this center in any direction. Let us suppose it is rotating about the Fig. i. axis OD with angular velocity &>, in a positive direction. In other words, it is rotating about the axis of the disc, an axis J. to its plane through the center of gravity, in a direction to the left when viewed from D, as indicated by the arrow. The moment of inertia about this axis is a maximum for the disc, and will be designated by C. Let the line OD represent the moment of momentum about this axis, both in direction and magnitude. By the parallelo- gram principle we can resolve this moment of momentum OD into two component moments OE and OF, the line OE representing a moment of momentum, both in magnitude and direction about an axis making an angle Ad with OD, and the line OF representing a simultaneous moment about a -L axis both in magnitude and direction. Let us now start to turn the rotating disc about an axis through 0, -L to the page, in a positive direction, i.e., in the direction indicated by the arrow. If the disc were THEORY 9 not rotating we should have no difficulty in turning the axis OD into coincidence with OE, but with the disc rota- ting it is a different matter. We do not yet know what would happen. Let us examine what occurs at the very first instant of turning. We can consider the angle A0 to become as small as we please, and write it dd. Let us sup- pose that in the infinitesimally small interval of time dt, in which we turn the axis OD through the infinitesimal angle d8, this axis will remain in the plane of the page, and that the component moments of momentum OE and OF will not have had time to change. In other words, the moment of momentum about the direction OE will remain what it was, although the axis of the disc has changed from the direction OD into the direction OE. Now such a supposition cannot be strictly true, although it will approach more and more to the truth as dt and dd become smaller, and at the limiting value of zero for these quantities, it will be strictly true. The rate at which the moment of momentum about an axis changes, measures the couple acting about that axis (Art. 2). The couple about the axis of the disc willtherefore , OD cos dd — OD , , , , 1 , , . . . , be - , and the couple about a _L axis, in the dt plane of the page, is — . At the limit the first Q/L couple becomes zero, and the second couple becomes OD ~ = CJ. dt We find therefore that, if we turn the axis of a rotating body about a ± axis, the moment of momentum about the original axis will be unaffected. Further, although the body will start to turn in obedience to the turning couple, there will, however, immediately be set up a couple acting about an axis -L to the axis of the turning couple IO THE GYROSCOPE and the original axis of rotation. This couple is called the gyroscopic couple, and its amount is Cud, or it is equal to the original moment of momentum into the angular velocity of the J_ turn. The axis of this couple will be in the direction OF, i.e., it tends to turn the body positively about this axis, or to raise the axis of the disc into coinci- dence with the axis of turning. And if we continue turning about a fixed axis, the original axis of rotation will, in fact, be brought into coincidence with the turning axis. By the same reasoning, if we turn negatively, 8 is negative, and the gyroscopic couple Cwd becomes negative, and the axis of the disc will be brought down -L to the page into coinci- dence with the negative axis of turning. This is the Fundamental Gyroscopic Principle, which may be stated thus: If we turn a rotation axis about a ± axis, a couple will be set up tending to bring the axes into coin- cidence. If o> is the angular velocity about the original axis of rotation, 6 that about the -L turning axis and ^ is the angular acceleration about the gyroscopic axis, which is JL to the other two, then the amount of the gyroscopic couple is ^4^- = Cwd, where A is the moment of inertia about the gyroscopic axis. This follows from Newton's third law that action is equal to reaction. Due regard must be had to signs. If w and are not of the same sign, the couple will be negative. Ccod = A\f/ is the Fundamental Gyroscopic Equation, from which we can derive all the properties and motions of a gyroscope. To recapitulate: If we turn a rotating body about an axis J. to its rotation axis, a couple will be immediately set up having an axis i. to the two former axes, and the amount of this couple will be the product of the rotational moment by the angular velocity of the turn, its sense being such as to tend to bring the rotational and turning axes into coincidence, both in sense and direction. THEORY IX 4. Gyroscopic Action the Result of Centrifugal Forces. The gyroscopic couple we have just investigated is essentially a centrifugal force, or rather a compound cen- trifugal force. We are familiar with the centrifugal force due to a simple rotation which acts in the plane of rotation, and always away from the axis of rotation. Its amount is Mrj/ 2 , where M is the mass being carried about the axis (supposed to be concentrated into a point), r its distance from the axis and ^ the angular velocity. We have now to deal with a compound centrifugal force due to the com- pounding of two simultaneous rotations. We shall see that this compound centrifugal force does not act in the plane of either of the rotations, but in a plane -L to both. In Fig. 2, a bi-axial body, here a bar, is pivoted about a horizontal axis HE', through its center of inertia. This axis is held in a ver- tical frame which can turn about a vertical axis W. We shall sup- pose throughout this book that the supporting mechanism is without weight, and that there is no fric- tion, unless otherwise expressly stated. In practice we can approximate these conditions to any desired extent. Let the inclination of the axis of the bar to the vertical be 6, A the moment of inertia about this axis and C that about a J_ axis. We now turn the mechanism about the vertical axis with an angular velocity ^. We can resolve this velocity into a component $ cos about the axis of the bar, and a component $ sin 8 about a i- 12 THE GYROSCOPE axis. It is evident that gyroscopic couples will be set up since the body has rotations about two -L axes. One gyroscopic couple, Aty cos 6, • \p sin 0, will tend to bring the axis into coincidence with the axis of \p ; the other gyroscopic couple will act in the opposite direction, and is equal to C\p sin • ^ cos 0. The resulting couple is (C — A)j, 2 sin cos 0. But this is the well-known expression for the centrifugal couple acting in a vertical plane through the axis of the bar. The bar will therefore oscillate above and below the horizontal plane by an amount equal to the original inclination. By writing the result down at once from gyroscopic principles, we have saved ourselves a tedious integration. As another example, let us suppose a sphere, Fig. 3, rotating about an axis OC with angular velocity w, this Fig. 3. axis being held in a ring CC, which is pivoted about a horizontal axis HH' held in a fork SHH'. The ring can turn about the horizontal axis, and the whole is set revolving about the fixed point S, in a horizontal plane, with angular velocity \p. Let R be the radius of the sphere, D the dis- tance SO, the inclination of the rotation axis to the verti- cal and the rotation and revolution in the same sense. We shall calculate the centrifugal forces and show that their sum is simply the gyroscopic couple we have previously obtained. THEORY 13 We can resolve trie rotation w into w cos about a vertical axis, and co sin 9 about SO. Let us first consider a ring of radius r rotating about SO with angular velocity co sin 0. The velocity of any particle in the ring is D^/ — no sin cos #, where <£ is the angle any radius makes with the vertical. The centrifugal force outward from S 1 dtfi will be — [D\p — no sin cos ] 2 , where dm is an elementary mass and equals rd • 8, 8 being the density of the sub- stance. It is evident that there will be a couple about HH' since the velocity, and consequently the centrifugal force, in the lower half of the ring is greater than in the upper half. The centrifugal moment of a particle about the axis HH' is — 7: — [Dxf/ — no sin cos #] 2 r cos <£. 1 D The sum of these moments is 2*- /r8 — [Dip — roi sin cos ] 2 r cos d = 2 ur5 • r 2 a) sin 6\p. D But 2 irr8 is the mass of the ring = m, and mr 2 w sin is the moment of momentum of the ring about the axis SO. Hence the sum of the moments of the centrifugal forces about HH' is the moment of momentum about SO into the angular velocity, \p. This is at once seen to be nothing else than the gyroscopic couple tending to bring the rota- tional axis SO of the ring into coincidence with the axis of \j/. For a disc the centrifugal resultant would be V 2 irr z 8 • sin 0^. T 2 R 2 But w R 2 8 is the mass of the disc and — is k 2 , where k is the 2 radius of gyration. Hence the moment of momentum of 14 THE GYROSCOPE the disc about SO into the angular velocity ^ will still be the resultant of the centrifugal moments and this is what we know to be the gyroscopic couple. For the whole sphere the centrifugal resultant is E V-7rr 4 5 • w sin 0f dh, o where h is the distance of any disc, along SO, from the center of the sphere. We have the relation, r 2 = R 2 — h 2 , and substituting the value of dh from this equation, we have as the resultant centrifugal moment R 2)nr*5 -w sin 6 tydh = i t R s d ■ f R 2 u sin Bj,. o But f ttR s 8 is the mass of the sphere and f R 2 is k 2 , k being the radius of gyration. Hence, the moment of momentum of the sphere about SO into the angular velocity ^ repre- sents the resultant moment of the centrifugal forces about HH', or the gyroscopic couple, indifferently. As the apparatus turns about the point S with the angular velocity ^, it is evident that the sphere is actually turning bodily about an axis J. to CC with an angular velocity ^ sin 8. It is further evident that the component ^ cos 6 of this turning does not in any way influence the rotation &> about the axis CC, since we have stipulated that there is no fric- tion. The original rotational velocity remains constant, although the axes HH' and the axis _L to CC and HH', which we shall hereafter call the jp sin axis, both turn about CC with an angular velocity f cos 6. In the fore- going discussion we considered only the component w sin 6 about the axis SO. This was for the sake of simplicity. It will be seen that, resolving the rotation about the ^ sin axis on the SO axis, the total rotation about the SO axis THEORY 15 is w sin — ^ sin cos 0, since these components are in opposite directions. We shall obtain, as before, that the sum of the centrifugal moments about BE' is M k 2 (w sin — $ sin cos 0) f = Ca> sin 6 ^ -C^ 2 sin cos 0, where C is the moment of inertia of the sphere. But we can write the same result at once from gyroscopic prin- ciples, thus saving ourselves a tedious inquiry into centrif- ugal forces. For the sphere is turning about the ^ sin axis with angular velocity ^ sin 0, and this axis is being turned about the J. axis CC with angular velocity ^ cos 0. Hence there is set up a gyroscopic couple Cty sin • i/* cos 0. There is also a gyroscopic couple Ceo • ^ sin acting in the opposite direction. Hence the resultant couple is C\j/ 2 sin cos — Cwj/ sin = C0. The rotational component about the vertical axis, or a cos + ^ sin 2 0, has no gyroscopic effect, since its axis already coincides with that of i/-. It can be easily shown that the resultant centrifugal force due to compounding the revolutional velocity with a rotational velocity about a vertical axis is M D^ 2 and is the same whether there is a rotation about a vertical axis or not. This is the expression for the revolutional centrifugal force and represents the stress on the rod SO, or the pres- sure on the point 5. Nevertheless, these combined centrif- ugal forces in the horizontal plane produce a stress in the body, which it would not have if it were not rotating about a vertical axis. For it is clear that the actual velocities of the particles about 5 are, in the outer half of the sphere, the sums of the velocities due to rotation and revolution, while in the inner half they are the differences. We could thus, by integration, find a point in the outer half which we could call the center of centrifugal effort. On the other hand, the center of gravitational effort always lies in the 16 THE GYROSCOPE inner half, so that generally, in all planetary bodies, forces are developed which tend to pull the body apart in opposite directions from the center and along the line SO. An elastic sphere will therefore be deformed into an ellipsoid, with the long axis always pointing, towards the attracting center. In the case of the earth it would seem that these solid tides are not inappreciable and may possibly amount to half a foot. The effect of the tides of the shallow, mobile oceans in slowing the earth's rotation is insignifi- cant in comparison. Where a body rotates in an opposite direction to its revolution these solid tides would be greatly reduced and might be obliterated. 5- We shall now take the general case of a tri-axial body having a rotation about some axis. We shall always sup- pose that the center of inertia is fixed. Such a rotation cannot in general be stable, for we can resolve the angular velocity about the momentary axis into its three compo- nents about the principal axes. Let coi be the angular velocity at any instant about the axis A, C02 that about B and w 3 that about C. The turning about B will set up a gyroscopic couple with the rotational moment about C, and so on reciprocally, so that there will be six gyroscopic couples — a pair about each axis. Taking the sum (alge- braic) of each pair, and with due regard to signs, we can write A^f = (B - C) 0*0*, at B^ = (C-A)^, C — = (A — B) W1W2. at THEORY 17 These are Euler's celebrated dynamical equations, which we have been able to write at once from the fundamental gyroscopic principle.* It is evident that a rotation about any axis not a principal axis will set up gyroscopic couples which will cause the body to move away from that axis. But a rotation about a principal axis will be stable. Referring to Fig. 2, we see that the rotation was not stable because it was not about a principal axis, thereby causing the bar to oscillate about the horizontal plane. In the case of a sphere, where all axes are principal axes, any rotation will be stable. 6. Let us suppose a sphere, Fig. 4, rotating about an axis OC with angular velocity a. We now give it an impulsive angular velocity $ about OA -L to OC. Let us say that we strike the end of the axis OC a sharp blow downward through the page. To find the motion. There will be an instantaneous rotation about a new axis, compounded of the other two rotations. If 1 is the FlG inclination of the new axis to OC then tan 1 = — , and the angular velocity about the 03 hew axis is "^ co 2 + 2 - Now this first instantaneous rotation, being about a principal axis, will persist, and the motion will be stable. * The usual demonstration is somewhat complicated. Some mathe- maticians (Staeckel) have expressed the opinion that the reasoning in the orthodox derivation is not wholly convincing. Euler himself (Berlin Memoirs, 1762) stated somewhat diffidently that he believed the equations were correct, but that he had no idea what the motion would be. l8 THE GYROSCOPE Our impulse, therefore, simply shifts the rotational axis to a new position between the other two, where it persists with a rotational velocity compounded of the other two velocities. Let us suppose that the body is bi-axial, C and A having the usual significance. There will be an instantaneous new axis, as before, determined by the same conditions, viz., its angular velocity will be "^ u> 2 + 2 , and tan t = - » i being the angle between the instantaneous axis and OC. But the new axis is not a principal axis, and the motion cannot be stable. Consequently the instantaneous axis will move, and the plane COI must move also. Let us see if it is possible to find some line in this plane which does not move. Let OL be a line in this plane making an angle 7 with OC. Let the plane turn about this line with angular velocity ^. Let us impose the conditions Aty cos y = Co> and $ sin 7 = . These two equations are sufficient to determine the two quantities f and y, and only one definite value for each can satisfy the equations. Multiplying the first by ^ sin y, we have Af 2 sin y cos y = Cco^ sin y. Now at the beginning of the impulse the conditions are these. The body is rotating about the axis OA with angu- lar velocity = f sin y. The plane CI LA is turning about OL with angular velocity ^. Hence the axis OA is turning about OC with angular velocity tp cos y. The original rotation &> about OC is uninfluenced. We have thus two gyroscopic couples, Aj/ sin y ■ ^ cos y and Cw • ^ sin 7. But the conditions imposed make these equal, and as they are in opposite directions, there will be equilibrium about an axis ± to the plane. As the plane COI turns about OL, the angle 7 must remain constant and likewise the angle t. From the equations Cu = Aty cos 7 and ^ = tan t, it THEORY 19 follows that A tan t = C tan 7. Hence the motion is fully determined. It is the same as if a cone having OC for an axis and a half angle 4, fixed in the body, were rolling on a cone, fixed in space, having OL for its axis and a half angle y — <.. The axis OL is called the invariable line. A body moving in this manner, under the influence of no external forces, is said to perform a Poinsot motion. The motion of the plane COL about OL is called the pre- cession. The velocity of the axes 01 and OL relatively the body, in which they describe cones about OC, is evi- dently \p cos 7 — co. From the equation Cw = A\j/ cos y we have Ceo cos y + Af sin 2 y = Aj/, which means that the moment of momentum of the body about the invariable line remains constant and equal to A^. If instead of starting with a bi-axial body rotating about OC and then imparting an impulsive moment about OA, we start with the body at rest and then give it an impulsive moment about a line OL, equal to G, we may decompose this into two simultaneous impulsive moments about OC and OA, equal respectively to G cos 7, and G sin 7, where 7 is the angle COL as before. If u and are the impulsive angular velocities about these axes, then tan 1 = — , and, CO A d> A from the above equations, tan 7 = -^ - = 7; tan 1, and C 03 L the case is the same as before. Hence we see that, in a bi-axial body, an impulse about any axis, not a principal axis, imparts an instantaneous rotation, not about that axis, but about another, found by the relation A tan t = C tan 7. The axis of the impulse will be the invariable fine, and the moment of momentum about this line will remain constant and equal to G. If the body be tri-axial and receive an impulsive moment about a line OL, not a principal axis, let a, /3, 7 be the angles 20 THE GYROSCOPE which the principal axes OA, OB, OC, respectively, make with OL, and let a*, a*, w$ be the angular velocities about these axes at any instant. We can decompose our impulse, G, into the three partial impulses, Gcosa = Awi (i), Gcos/3 = |3co2 (2), Gcosy = Cw 3 (3). Multiplying Euler's dynamical equations by «i, C02 and w 3 we have Awi 2 , But 2 , Cco 3 2 rr, . . — - -| H = T = a constant. 222 Or the kinetic energy remains constant throughout the motion. From (1), (2) and (3), G = Aaicosa. + Buncos fi + Cco3Cosy (4), and G 2 = ,4 V + BV + CW (5) Let us suppose that the preponderating rotation is about OC, so that, as the body rotates about this axis, OA and OB become successively _L to OL. Since, from (4) the moment of momentum about OL remains constant and equal to G, it is evident that the moment about any other fixed line must remain constant, and the moment about any line ± to OL must be zero. Hence, whenever OA or OB becomes _L to OL, wi or 02 becomes zero. If OB is _L to OL we have, from (4) and (5), Awi 2 CW3 2 _ j, 2 2 Ao>i sin 7 + C03 cos 7 = G, AW + CW = G 2 . These three equations determine o> 1} 03 and 7. In the same manner we derive three other values for a>2, « 3 and 7 when OA is JL to OL, and it is evident that at these in- stants the motion would coincide with that of a bi-axial THEORY 21 body under the same impulse. It will be seen, therefore, that a tri-axial body, subjected to an impulse about a line OL, will perform an irregular precession about that line, with the axis OC approaching and receding from that line within definite limits, being nearest to it when OB is _L to OL and farthest from it when OA is ± to OL. To recapitulate: When we impart an impulsive couple to a uni-axial body it will, from ^the first, rotate about the axis of the couple with a constant velocity. The axis of the couple, the instantaneous axis and the invariable line are one. If the body be bi-axial, the instantaneous axis does not coincide with the impulse axis (unless this be a principal axis) and the motion consists of a constant rotation about OC, which is carried about the impulse axis (or in- variable line) with a constant precessional velocity at a constant distance from the invariable line. The instan- taneous axis always remains in the precessional plane at a constant distance from the invariable line. If the body be tri-axial, the motion consists of a rotation about a principal axis, which is not constant, but varies within fixed limits, while this axis is carried around the invariable line with a precessional velocity which varies within fixed limits, being greatest when OB is ± to OL and least when OA is _L to OL. This axis, further, does not preserve a constant distance from the invariable line (or impulse axis), but moves in the precessional plane within fixed limits. The instantaneous axis does not remain in the precessional plane (COL), but moves to either side of it, within fixed limits, being in this plane only when OA or OB is ± to OL. We may describe the motion, therefore, as a varying rotation about a principal axis, while this axis performs an irregular or wobbly precession about the invariable line. None of the elements of motion are constant, but vary within fixed limits. In all three 22 THE GYROSCOPE cases, however, the axis of the impulse couple is the in- variable line. Such motions of a body about a fixed point, started by an impulsive couple and without the action of any other forces, are called Poinsot motions* For a tri- axial body, the path of C, the extremity of the main rota- tion axis, will be a symmetrical wavy curve about the invariable line, like that represented in Fig. 13. The mere translational motion of a body, or motion in which the axes remain ||, can set up no gyroscopic action. Hence the translational motion of the center of inertia, and rotations about that center, are independent. So that, generally, whenever an irregular body is struck, we can resolve the impulse into an impulsive velocity of the center of gravity and an impulsive couple about that center, and the center of inertia will describe its path just the same as if the body were not rotating, while the body will execute a Poinsot motion about that center, just the same as if it were fixed. Motion of the first kind we are familiar with and can easily anticipate, while motion of the second kind, or gyroscopic motion, is contrary to our everyday experi- ence and cannot be foreseen. It can only be brought out by analysis.f We shall refer once more to the popular superstition that a gyroscope "tends to maintain its plane of rotation," or, in other words, that it is a "gyrostat." If it is understood by this that a gyroscope opposes to a change of plane what we have called rotational inertia, just as a simple body opposes ordinary inertia to a change of position, then no great harm is done. But if it is supposed that a gyroscope is a gyrostat, viz., that it maintains its plane, then grievous * Poinsot. Theorie Nouvelle de la Rotation des Corps, 1834. t Gilbert, in one of his operas, humorously refers to the terrible punish- ment meted out to a billiard fiend, who was condemned to play with "elliptical billiard balls." THEORY 23 harm may result. A gyroscope spinning with an infinite velocity — a mathematical fiction — cannot have its plane changed by any finite force. In this case, it is a gyrostat; but finitely spinning gyroscopes do not and cannot maintain their planes when acted upon by an outside couple. A misunderstanding of this fact has led to many strange mechanical attempts. The late Sir Henry Bessemer, of steel fame, misled by the name gyrostat, actually con- structed a cabin on a ship, swung on fore and aft trunnions, to which was rigidly attached a heavy rotating flywheel. The idea was that the "gyrostat " would maintain its plane and with it the cabin, so that the ship might roll, but not the cabin. Of course the cabin swung just the same as if there had been no rotating flywheel attached to it. The only result was that at each swing a heavy couple was brought to bear on the bearings of the wheel, tending to stop its rotation by the increased friction. If he had allowed the axle to move relatively to the cabin — to pre- cess — then a slight swing would have set up a precessional reaction — our rotational inertia would have come into play — and he might have anticipated Schlick's Stabilisator. As another example of the prevalent misconception that a gyroscope is a gyrostat, we find the following in Prof. Sylvanus Thompson's classical work on "Dynamo-electric Machinery." "Another point, which arises only in the case of dynamos used on shipboard and motors running round a curve on a track, is the gyrostatic action of the revolving armature, which always tends to keep its axis pointing in the same direction." He then states that this gives rise to a force "On each bearing, alternately acting up and down at each roll, if the axis of the dynamo lies athwart the ship." There is, of course, no such thing as gyrostatic action. There is, however, in this case, a gyro- scopic couple, which does not act up and down, but back- 24 THE GYROSCOPE wards and forwards, i.e., about an axis J. to the deck. From this we see that it is advisable, as far as possible, to place the axes of all rotating masses on a ship in a fore- and-aft direction, since, in such a position, the only gyro- scopic couples set up will be those due to the pitching of the ship. 7. External Forces. We shall now investigate the motions of gyroscopes under the action of external forces. Let Fig. 5 represent Fig. 6. a bi-axial ellipsoid rotating with angular velocity w about its axis of greatest moment OC. This axis is held in a ring which is pivoted about a horizontal axis HH', in a vertical frame W. This vertical frame can turn about the verti- cal axis W. To the lower extremity of the rotation axis, at a distance I from 0, is attached a small mass m. We shall suppose this small mass and the supporting frame to be negligible in comparison with the mass of the gyroscope. Fig. 6 represents a bi-axial top, its point fixed at 0, and its THEORY 25 center of gravity at G, OG being represented by I. To rind the motion. The two problems are identical, with the exception that, in one case, the axis OC is urged upward, while in the other case it is urged downward. Both gyro- scopes start from rest and are influenced by gravity. The rotations are both positive viewed from above. The gravitational couple is ± mgl sin 0. It is clear that any velocity will set up a gyroscopic couple about an axis _L to OC and in the vertical plane COV. We have called this the ^ sin axis. The horizontal axis HH' we shall call the axis. ty represents the angular velocity about the vertical OV — or the precessional velocity. It will be noted that the \p sin axis and the axis both turn about OC with angular velocity f cos 0, while the rotations about the three mutually ± axes we have taken are u, ty sin and 0. We can therefore write the equations of motion at once, and they are (Fig. 6) mgl sin — Coji/- sin + A^ 2 sin cos = Ad, (i) Cad - Aj, cos 06 i = AD t (t sin 0). (2) Multiplying (2) by sin 0, C03 sin 06 i — A f sin cos 66 i = A sin 0D t (j, sin 0) . (3) Integrating, Coj(cos O — cos 0) — A I if/ sin cos 00 = Aj/ sin 2 -'/ ^ sin cos 00, or, Aj, sin 2 + Cw cos = Co> cos O , (4) where 0o is the initial inclination. Equation (4) informs us that the moment of momentum about the vertical remains constant, as was a priori evident, 26 THE GYROSCOPE and we might have written the equation at once without integrating. It is evident that there is no couple acting about the vertical and therefore no increase (or decrease) of the original moment of momentum about that axis can occur. And, generally, it is evident that the time integral of any outside couple acting about a fixed axis must equal the increase (or decrease) of the moment of momentum about that axis. Gyroscopic couples, which are centrif- ugal forces and which we may call internal forces, can perform no work on the body. For action is always equal to reaction, and to perform work it is necessary to have an external "point d'appui." Work can only be done by external forces, and in our subsequent work we shall see that gyroscopic forces never perform work on the body. Gyroscopic couples react mutually on each other and can no more perform work than a man can raise himself by his boot straps. But gyroscopic forces change the direction of the motion caused by external forces and, as we shall see, in a most peculiar manner. Multiplying equation (i) by and equation (2) by ^ sin 0, and then adding and integrating, we have if n m A ' ei , ^ 2 sin 2 , . mgl (cos O — cos 0) = 1 — - (5) 2 2 That is, the kinetic energy imparted is strictly equal to the external work done by gravitation. This was also a. priori evident, and might have been written at once. From equation (4) we have Ca> (co s O — cos 0) c , ... ... •• / \ ^ = . . — -. Substituting m equation (5) If a a \ Ad 2 , [Coo (COS O - COS 0)] 2 /e . •rtCcos 0o - cos 0) = — + \ Ashi2d " • (6) THEORY 27 Whence , sin 8 dd y/^(cos0o-cose)(i-cos 2 0)-r^(cose o -cos0)l 2 This is an elliptic integral which can be readily solved, though the transformation is somewhat tedious. We can thus find the inclination of the axis at any instant, and from equation (4) the value of ^ at that instant. The motion is completely determined. We are, however, not so much concerned with the exact quantitative determination of the path described by the] axis, as with its quality, or what kind of motion the axis undergoes. Let us see if the axis falls continuously, or if there is a limit below which it cannot go. Putting = in equation (6), we have ^ (1 - cos 2 0) = %^ (cos 0„ - cos 8). (7) Whence COS = (Co>) 2 ± J\ (C*) 2 cos 9o , (Co,)* 4mglA * 2mglA 16 m 2 gH 2 A 2 The positive sign leads to a value for cos 0, greater than unity, and is therefore inadmissible. Hence the axis will cease falling (or rising) at the point where cose = i^_^_(^!£^ + -^-. (8) qmglA Y 2tnglA 16 m l g 1 V A* The nature of the motion is now clear. At a certain point, the axis stops falling, and all the kinetic energy is converted into horizontal motion. This carries it up to the inclination from which it fell, when, all the kinetic energy being used up, it comes momentarily to rest again, only to repeat the process over and over again. From 28 THE GYROSCOPE equation (4), the horizontal motion or precession must always be positive. Hence the axis keeps moving about the vertical in one direction, executing meanwhile a series of symmetrical dips. If co is very large, we see from equa- tion (7) that the dips, or (cos0 o - cos 0), must be very small, and from equation (4) that the horizontal velocity, or <£-, must also be very small. is accordingly very small, and the squares and products of these very small quantities can be neglected in comparison with the quantities them- selves. If co is very large, we can, therefore, write equations (1) and (2), mgl sin — Ccoi/- sin = Ad. (9) Cwd = AD t {i> sin 6) . (10) Now the very small portion of a sphere included by one of the dips, we can regard as practically plane, and the quantities \p sin and become rectangular coordinates, \f/ sin being represented by x, by y. The origin of co- ordinates is where the motion began, viz., from rest. We can further regard sin as practically constant in this small area. Hence equations (9) and (10) become mglsva.0 — Coox = Ay. (n) Co>y = Ax. (12) Integrating, mgl sin Bt — Cwx = Ay. (13) I Ccoy = Ax. (14) Writing the equations mgl sin 6 A VCoil . (Cu ,\ "1 , s x= ^v^hr- sin (xvJ (IS) mgl sin 6 A r /Cu \1 , .. we see that they are the integrals of (14) and (13) respec- THEORY 29 tively. For differentiating (15) and (16) and substituting the values of x and y, we obtain (14) and (13). Equations (15) and (16) represent a cycloid with its base horizontal, convex downward, and having a cusp at the origin of coordinates. The equations of a cycloid are usually written x = a ( — sin <£), y — a (1 — cos ), where a is the radius of the generating circle and is the angle which a radius makes at any time with its initial position. Hence our cycloid is generated by a circle of radius „, , — , and the angle which any radius makes with its C 2 ar initial position is proportional to the time and equal to /~* — t. If, therefore, we supposed our axis to be attached A to the circumference of a very small wheel, of radius ** — , rolling along the under side of a parallel of C co latitude with the uniform angular velocity — , it would describe the path taken by the extremity of the axis, both in time and position. The axis will thus describe in space a cycloidally fluted cone about the vertical as an axis. The time of falling through one of these minute cycloids is 2ttA Ceo The rise and fall is so minute and rapid that the eye cannot follow it, or at most detects only a slight blurring. The ear, however, can detect the humming caused by these minute vibrations. When a gyroscope is turning about its point of support, a distinct note is usually heard, and the frequency of the vibrations can be very accurately determined by comparing it with the note of a tuning fork. Unless the gyroscope is driven by electricity or some other constant source, and thus kept up to pitch, it will be noticed 30 .THE GYROSCOPE that the note constantly gets lower, corresponding to the slowing down of the rotation due to friction. The axis progresses along the parallel of latitude a dis- tance 2 x mg J^"„ — , or through one cycloid, in the time 2 ttA -pr- • Hence the time of a complete revolution about the Geo vertical will be , . The times of a single vibration mgl and of a complete revolution are, therefore, independent of the original inclination, and while the time of a single vibration is independent of the outside force, that of the revolution is inversely proportional to it. The horizontal motion, or motion in longitude, is called the precession, while the variations of inclination, or lati- tude, are called the nutations. The nutations we have just investigated are called free nutations. We shall shortly come upon another class known as forced nutations. If the gyroscope (Fig. 6) swings about the point 0, be- low the horizontal plane through 0, it becomes a gyroscopic pendulum. If co is small, it will execute a series of festoons about the vertical, but it will never reach the nadir. If co becomes zero, the body becomes a simple pendulum, and, from equation (6), we have j deVI I v 2 mgl (cos do — cos 0) the law of the pendulum. If we constrain the end of the axis to move in a frictionless groove in a vertical plane, the gyroscope falls like a simple pendulum, for, in this case, the gyroscopic couple called into play exhibits itself solely in pressure against the constraint, and the only accelera- tion is that due to gravity. THEORY 31 8. Case of Constant Precessional Velocity. Let us suppose, Fig. 7, a ring which can turn about a horizontal axis EE'. Within this ring is pivoted another ring capable of turning about an axis PP', _L to EE'. This inner ring carries a rotating disc with its axis OC _L to PP'. A constant angular velocity, ^, is given to the outer ring about EE' . Rotations are positive and the angle EOC is 0. It is evident that the axis OC will move to set itself into coincidence with EE', that it will go be- yond to an equal distance and then come back, thus oscil- lating about OE. It is in fact a horizontal pendulum, with the directive couple Coif sin — Aj/ 2 sin cos 0, tending to set the rotational axis along EE'. We have at once the equation — Co^ sin + A j/ 2 sin cos = A6, where A is the moment of inertia about an axis ± to OC. Let the initial inclination be 9q. Hence, integrating, /-. •/ „ /. \ 1 a -o /sin 2 6 sin 2 o \ Ad 2 C = sin -, where is an auxiliary angle. 2 2 It will be seen that when = o, 4> = o; and when aking this substituti i djj> = d , = -. Making this substitution, 2 dt V« . / . ,1/ , vi- sin a — sin 2 rf> T 2 The radical can be developed by the binomial theorem, and i + - snr - sin 1 ' 4> I . > - -, I'll • 2 y i — sm^-sm^ 2 ,2 3in 2 c6 2 + --2.5 sin 40O sin 4^ + I.3.S sin 6^ sin 6^ _ _ _ etc 246 2 246 2 Now / sm 2n d = ■ sin 2 " A/iU = — 2 -4-6 ... 2« 2 Hence :| T +• I ■ ! sin ■■- !• I — ~- I sirv' " 2 Val + ( lL3L Hfysin'r etc l \2 • 4 • 6/ !2 J THEORY 33 where t is the time taken by the axis OC to swing from O to coincidence with OH. The time of a complete swing is T = 2 xy//^i + (1) rin«* + (^3) sin^. . . . etc.l » Caj^L W 2 \2 -4/ 2 J If we attach the outer ring firmly to any point of the earth's surface with the axis PP' vertical, it is evident that the horizontal component of the earth's rotation, about an axis in the meridian, viz. ^ cos X, where X is the latitude of the place, will cause the positive end of the axis of the gyroscope to oscillate about the true North. The vertical component, f sin X, will, of course, have no effect on the gyroscope. The gyroscope can thus be used as a compass to indicate the true North. Or if the axis PP' were set horizontally, due East and West, the axis of the gyroscope would oscillate about a direction || to the earth's axis and, coming to rest in this position by damping, would indicate, by its elevation above the horizon, the latitude of the place. The rotation ^ cos X in the case of the earth is small enough for us to neglect its square in comparison with a high value of gyroscopic rotation. Hence, for small oscillations, we can use the formula = 2 -Vi Ceo \j, cos X A i Since for any disc, - = - . if the gyroscope has a rotation C 2' of 20,000 turns per minute, a value which is reached in practice, the period would be about n seconds, at the equator. 2 7T j/ = = 0.0000720211. w 86,164.1 sec. ' co = 2tt X 333-33 = 2094 radians per second. 34 THE GYROSCOPE 9. Case of Constant Couple about a Fixed Axis. Let a gyroscope be subjected to a constant couple about an axis which we shall take as the vertical. All symbols have the usual significance. Let H be the constant couple about the vertical. We shall suppose E to be negative, so that, if the angle be less than - , a, the original rotational velocity about OC, 2 will be retarded. The equations of motion are -flcos0 = CZW (1) — Cco 3 i/< sin 6 + A^ 2 sin cos = Ad. (2) — H sin 6 + Cw 3 d - Af cos 00 = AD t (f sin 0). (3) Multiplying (1) by cos and (3) by sin 0, and adding, — H = cos 6D t o3 3 — Cw3 sin 60 — A ■j/ sin. 6 cos 66 + Asm6D t (j / sm6). Integrating, — Ht = Co> 3 cos 6 + ^ sin 2 — Cu cos O - (4) This is the momental equation, which states that the time integral about the vertical axis is the increase of the moment of momentum about that axis. Multiplying (2) by 6, and (3) by ^ sin 6, and adding, - E$ sin 2 6 = Ad 6+ AD t (4, sin 6) ■ ^ sin 6. Integrating, — ] H sm6 • smQ d\}/ = — ■ -\ — *■ = T. (5) This equation states the fact that the work done by the couple about the ^ sin 6 axis is equal to the kinetic energy about the ^ sin 6 and 6 axes, or to the kinetic energy out- side of that about the rotational axis, which, of course, it cannot influence. THEORY 35 Substituting from (i), cos = — —D f o>3, in (4), a. c 2 — Et = — — u 3 D t 033 — Cw cos 6o+ A j> sin 2 0. (6) H Integrating, and substituting the value of / ^ sin 2 from (5), we have (co 3 2 - a, 2 ) + '-cosOot+T. (7) «3 ^ X 2 A 2 A Let us suppose that the couple H is small relatively to the initial rotation a, and, there- fore, o>3, for a short time, does not differ greatly from a). Eliminating t from equation (7) by the aid of equation (4), we find that the equa- tion contains only the square of 0, while it has the first and second powers of ^ sin 0. Hence when f sin becomes zero, the two values of are equal but of opposite sign, while, when becomes zero, we have two unequal values for ip sin 0. It is evident that the rotation axis will at first move in obedience to the couple, but will shortly be deflected downward, executing a series of loops, as in Fig. 8. Let us take the case where the axis, at the beginning of motion, was ± to the axis of the couple, and, as before, the couple is so small, relatively to «, that we may consider &> 3 as reasonably constant for a short time, and equal to o>. Fig. 8. Equation (7) becomes HH 2 _A0 2 A+ 2 sm 2 2 A 2 2 36 THE GYROSCOPE — Ht — Ceo COS Substituting from (4), tp sin — A sin HH 2 AO 2 , 1 (Ht + Cw cose) 2 m we nave — — = 7 • r-rz \o) 2 A 2 2 A sin 2 Let us now determine the locus where becomes zero. We do this by putting = o in equation (8). (Ht + Co> cos 0) Hence Ht=± L ^— , or sm0 Ht (sin =F 1) = ± Ceo cos 0. (9) Measuring the angle from the equator instead of from the pole, cos becomes — sin $1, and sin 6 becomes cos 0i, where 0i is a very small angle. Hence equation (9) becomes Ht (cos 0i T 1) = =F C« sin 0i. Since the equation must vanish, identically, when X = o, the lower signs are in- admissible. Putting sin 0i = y, we have y = — (1 -cos0j). (10) This is the familiar equation of a cycloid with a genera- Tft ting circle of radius — . That is, the radius of the genera- ting circle increases proportionately to the time. The points where becomes zero all lie, therefore, on this cycloid as a locus. The cycloids are represented by the dotted curve in Fig. 8, and the looped curve of the axis touches these cycloids at the points 3,1, while the points 2, 4, indi- cate where ^ sin becomes zero, having equal but opposite (in sign) values at these points. The proportions are, of course, very much exaggerated in the figure. Actually, with a very large value for w, the loops would be very minute and the axis would move along close to the equator for an appreciable time. It would, THEORY 37 of course, finally get away, for, as the excursions become larger, the value of a> begins to increase and our equation no longer holds. The general motion would be that, while executing a looped path, it would gradually spiral down- ward, pass through the nadir, rise again, though not to its original level, and thus, by a series of swings, finally come to rest at the nadir, where the couple would be expended solely in increasing the rotational energy about OC. 10. Attraction of a Distant Body on a Tri-axial Body. Let 0, Fig. 9, be the center of inertia of the attracted tri- axial body, OC the axis of greatest moment, OA that of a A<*J Fig. 9. least moment and OB that of middle moment. Let OS be the direction of the attracting body, cos a, cos /3, cos 7 the direction cosines of the axes OA, OB, OC, respectively with respect to OS. Let

i. (1) (A — C) [K cos a cos 7 — a^] = BDt&i. (2) (B — A) [K cos a cos /3 — coia^] = CD ( w 3 . (3) The terms containing K express the attractional couples about the respective axes, while the terms containing the to's express the gyroscopic couples about these axes. Multiplying equation (1) by «i, equation (2) by «2, and equation (3) by w 3 , and adding, K [(C - B) cos cos 7C01 + (.4 - C) cos a cos 702 + (B — A) cos a cos /3co 3 ] = A(aiD t ui + Bc^Dtw + Cco3Z>,6) 3 . Or, * Routh. Advanced Dynamics, Art. 5519. THEORY 39 K / [(C — B) cos /3 cos 7coi + {A — C) cos a cos y«2 + (5 — A) cos a cos /3a> 3 ] = ^ + i? ^ +C ("3 2 -C0 2 ) , () 2 2 2 The left member is the work done by the gravitational couples, while the right member is the imparted kinetic energy. It will be seen that the work done by the gyro- scopic couples is zero. (See Art. 7.) From a simple geometrical consideration, it is readily ,, . , cos j3 j • , cos a , ., , seen mat cos = — — , and sin^ = - — , and that sin 7 sin 7 7 = 01 cos 4> — c^2 sin 3 cos a — 0)1 cos 7, sin aa=aj cos 7 — 033 cos /3. Multiplying equation (1) by cos a, equation (2) by cos /S, and equation (3) by cos 7, and adding, — (C — B) «2C03 cos a, 1 — {A — B) C01W3 cos p, \ = A cos a Z)(&)i + B cos @D,W2 — (B — A) a)i02 cos 7. J + C cos yD t w 3 . Or, — .4o>io)2 cos 7 + Awio>3 cos /3 + A cos aD^i | — 5o)20) 3 cos a + 5o>io)2 cos y + B cos p\D ,012 1 = o. (6) — Coiio)3 cos B + Coi20>3 cos a + C cos yD t u>s j Now the first line is by (5) the derivative of Awi cos a, the second line is the derivative of Bo>2 cos /3, and the third line is the derivative of Cu s cos 7. Hence, integrating equation (6), we have Aui cos a + Bui cos /3 + C013 cos 7 = Co> cos 70. (7) This expresses the fact that the moment of momentum about OS remains constant, as was a priori evident. 40 THE GYROSCOPE From (5), we have cos /3 cos ?o!i = sin 7 cos 77 + cos a cos 7012. COS a COS j(3co3 = — sin a cos ace + COS a COS 7CO2. Multiplying (1), (2), (3) by wi, an and co 3 , and substituting these values, we have K (C — B) sin 7 cos 77 — K (B — A) sin a cos ao = ^4a)lZ)(0)l + Bo32D t Ol2 + CwzD t wz. Integrating, iT [(C - 5) (sin 2 7 - sin 2 7o ) - (B - -4) (sin 2 a - sin 2 «„)] = il CO! 2 + 5C02 2 + C (c0 3 2 - CO 2 ) , (8) which we can write in the symmetrical form 2r[C(sin 2 7-sin 2 7o) + ^(sin 2 ^-sin 2 /3 ) + ^(sin 2 Q ! -sin 2 a)] = ,4 CO! 2 + #C0 2 2 + CC0 3 2 (9) The work done is therefore expressed by half the left member. We have further by combining equations (1), (2) and (3) in pairs, A B r _ B <°i 2 + Q - A °* 2 = K ( sin2 7 ~~ sin2 T °)' ( IO ) B , , C -C02 (co 3 2 — co 2 ) = K (sin 2 a — sin 2 a), (n) C-A ' B-A ^4^ <* 2 + jztj («* ~ "* 2 ) = * ( sin2 0° - sin2 /»)• ( l2 ) These are the kinetic equations. The earth being a tri-axial body, although only slightly deviating from bi- axiality, its rotation, a> 3 , is not constant, but variable. It is possible that this irregularity, slight as it is, might be determined astronomically, and the positions of the axes A and B, as well as their ratio, approximated. This would be a great improvement over the laborious methods of THEORY 41 direct measurement, employed by Col. Clarke and others. Col. Clarke's first results were that the longer equatorial axis exceeded the shorter by about a mile, and that this axis was situated in longitude 15 34' E. Later, he changed this to 8° 15' W. The uncertainty shows the futility of using minutes in the results. The general solution of the motion of a tri-axial body under external forces, has not hitherto been effected. [See Note at end.] 11. Attraction of Distant Body on a Bi-axial Body. In the case of a bi-axial body, we have as the equations of motion, ^ (C—A) sin cos 0— Coij, sin 8 + Aj, 2 sin cos =A 0. (1) Cud — A I cos 00 = AD t ($ sin 0) , (2) where the symbols have the usual significance. Put ^ (C — A) = K, and multiply (1) by 8, and (2) by ^ sin 0, and add, and integrate. K / sin 2 8 _ sin 2 O \ _ ^10 2 Atf sin 2 , , This energy equation states that the work done by the gravitational couple is equal to the kinetic energy imparted. We have also the momental equation, which is Cos (cos O — cos 0) = A'ty sin 2 8. (4) This is derived from (2) by multiplying it by sin and inte- grating, and states that the moment of momentum about OS remains constant. Substituting the value of 4, sin 8 from (4) in (3), /cos O — cos 0\ 2 2 = f(cos 2 0o-cos 2 0)-(^)^ sin0 \ > limiting v \^ it descrit 42 THE GYROSCOPE / AK, „, /CoY/cos Oo- cose s'] = (cos O - cos 0) ^- (cos O + cos 0) - {^-J ^ ^r^ JJ. Hence, besides the initial value, cos = cos 0o, will become zero when K , „ , „ N /Cw\ Vcos O - cos 0\ , . I (cos0o + COS0) - ( T ) (— J^— j- (S) This is a cubic equation in cos 0. If co is large compared with K, it is evident that for a very slight increase of 0, equation (5) will be satisfied, and that for a value beyond this, will become imaginary. The axis will, therefore, oscillate between O and this other ; value. We thus see that describes a fluted cone about OS as an axis, in a positive di- rection. The path is indicated by the left hand side of Fig. 10. If the force is repulsive, instead of attractive, the precession will be jr IG IO negative, or in a retrograde di- rection, as indicated in the right half of Fig. 10, and the nutations will be inside the pre- cessional circle. If co is very large, ^ and become very small, so that we can neglect the squares and products of these small quanti- ties, in comparison with their first powers, without appreci- able error. We may also consider K sin cos as practically constant during the minute motion. Hence the equations of motion (1), (2), become K sin cos — C o)^ sin = Ad. CtJ =AD t (j, sin0). » THEORY 43 Taking \j/smd as x, and 6 as y, and the origin of coordinates at the beginning of motion from rest, we have K sin cos — Cwx = Ay. (6) Cuy = Ax. (7) Integrating, K sin cos Bt — Cwx = .4y. (8) Coiy = -4a;. (9) Writing the equations K sine cos BA /Cut . /Coit\\ , N * = cv lX- sm (x)> (lo) K sin cos BA I /Ccot\\ , N ' y= ~^cv — ^-^(TJJ' (II) we see that they are the integrals of (9) and (8) respectively. Equations (10) and (11) represent a cycloid having a generating circle with radius ~r-z , and the rate of /~* rolling of the circle is — . The problem is very similar to that in Art. 7. The time of describing a cycloid is — — . C01 The time of a complete rotation of the body about its axis is — . Hence the ratio of the time of describing a cycloid to CO A a complete rotation is — . In the case of the earth, where C — A — — — = 0.0032, the time would be 0.9968 of a sidereal day. The time of the complete precessional period is, under the conditions of the problem, — If the earth and .K cos O sun were the bodies considered in our problem, the nuta- tions would be far too minute to be measured, 0.1" being 44 THE GYROSCOPE the smallest angle which by the greatest refinements it is possible to measure by astronomical instruments. This 'angle corresponds to about 12 feet on the surface of the earth. 12. Forced Nutations. In Fig. 11, the circle BAVN represents the orbit of the earth, ON the line of nodes, or the intersection of the earth's equatorial plane with the plane of the ecliptic. E is the pole of the ecliptic, P that of the equator. The angle EOP is the inclination of the earth's axis to the pole of the ecliptic and is 6. The plane EOB is thus 1 to the ecliptic and equatorial planes. In studying the attraction of a body on the earth, it is indifferent whether we consider the earth revolving about the body, or the body revolving about the earth in the same orbit, with the earth's center fixed. We shall suppose the earth's center fixed and the body (sun or moon) revolving in a corresponding orbit about the earth. Let us suppose the initial position to be at V (the vernal equinox). That is, the attracting body is at V, and the line of nodes, ON, likewise at V. Let the angular velocity of the attracting body in its orbit, which is supposed circular, be constant and equal to a. Let the angle made by the line THEORY 45 of nodes with the initial position be \p = VON. This angle ^ is the precession. Let us suppose that at any instant the attracting body is at some point A in its orbit, measured by the angle VOA . Then the angle between the body and the line of nodes is at — \p. The gravitational couple is always about an equatorial axis _L to the plane A OP, containing the attracting body and the earth's axis, OP- If S be the dec- lination of the body at any time, then this couple is K sin S cos 5, where K = =±— (C — A). We can resolve this couple into a component about an equatorial axis J_ to the plane EOB — the component — and a component about an equatorial axis ± to the former — the i^ sin component or axis. Since PBA is a right spherical triangle, cos PA = cos PB, cos BA, or sin 8 = sin 6 sin (at — \f). (i) Hence the gravitational couple is - K sin 6 sin (at — f)Vi - sin 2 sin 2 (at — ip). We use the minus sign because its tendency is to decrease the angle 6. We must resolve this couple into the 6 and ^ sin components. Let A be the angle APB, the angle between the planes AOP and BOP Then the 6 component is the couple into cos A, and the ^ sin 6 component is the couple into sin A. By spherical trigonometry, ctn (at — \p) _ cos d tan A Hence ctn (at — \p) sini = cos4 = Vcos 2 6 + ctn 2 (at — ip) cos0 Vcos 2 d + ctn 2 (at - 4>) 46 THE GYROSCOPE The couple about the line of nodes, or the component, is thus — K sin sin (at — \p) Vi — sin 2 sin 2 (at — tp) COS0 V cos 2 + ctri 2 (at -yf) But / — j-t— — - , , -r Vi — sin 2 sin 2 (at — \j/) Vcos 2 + ctn 2 (at — \p) = ■ ^^ *r "■• v Y ' sin (at -$) Hence the couple is — K sin cos sin 2 (at — (cosO— cos0 o )+^ sin , will be — Ceo ^ sin , and the gyroscopic couple about the ^ sin axis will be Cud. But from equa- tions (11) and (12), these couples are K sin 2 at sin cos and — K sin = — K sin at cos at sin 6. These are ex- 2 actly equal and opposite to the gravitational couples. Hence the axis will need no restraints, but will move of itself in the path we have prescribed. We have thus re- duced a dynamical to a simple kinematical problem. It will be seen that by equating the gravitational and gyro- scopic forces and integrating, we shall obtain the equations of the curve traced. The motion of the axis, therefore, under the attraction of a revolving (performing a revolution) THEORY 51 body, is a harmonic motion in a positive direction in an ellipse having a major semi-axis — — - sin 0, and a minor 4aGct> zr semi-axis — — sin cos 0. It moves in this ellipse harmon- 4aCco ically with a constant angular velocity 2 a, while the ellipse itself moves with the constant horizontal angular velocity 37- — sin cos 6. The long axis of the ellipse always 2 Geo points towards the pole of the orbit. The constant retro- grade precessional velocity of the center of the ellipse is K 2 Ci - cos O , where O is its constant inclination to the pole, CO or f = — cos O . 2 Ceo The constant, or regular, precession due to the sun is about 15" a year: that due to the moon is nearly i\ greater, so that the regular annual precession of the earth's axis is about 50". The semi-axis of the ellipse OB is for the sun about 0.53", while the semi-axis OA is about 0.48". The axes of the ellipse due to the moon are about § as large, or barely within the range of measurability.* The ellipse always maintains its position with respect to the line of nodes. This is necessarily so, as it is a forced nutation. When the attracting body is at a node (vernal or autumnal equinox), the axis is at the lowest point of the ellipse and at -rest, for here the constant precessional C — A * In computing K, we must, of course, use astronomical units. ■ — -. — = .0032. For the sun, the attraction -=- must equal the centrifugal force of the earth, which is M a?D, M being the earth's mass, and a her orbital angu- lar velocity. Hence 2)j = ° 2- 52 THE GYROSCOPE velocity of the ellipse as a whole is exactly equal and opposite to the velocity of the axis in the ellipse (see Equa- tions (10) and (n)). If, therefore, we started our rotating body with its axis at rest, it would immediately fall into motion in that part of the harmonic ellipse necessary to bring it to rest at the node. It is for this reason that the axis of the moon always lies in a plane J. to the intersection of her orbit with the ecliptic, or, stated otherwise, the plane of the moon's orbit, the plane of the moon's equator, and the plane through the center of the moon 1 1 to the ecliptic always intersect in the same line. For, we can consider the moon's center to be fixed, and the sun to revolve about it in its corresponding orbit in the plane of the ecliptic, while the earth revolves about it in the plane of the moon's orbit. Let us suppose" that the moon's equatorial plane intersects these two orbital planes in two different lines. The forced nutations we have just considered will strive to set the moon's axis ± to each of these lines, and this can only be accomplished when the two lines of nodes coincide. The result is that the moon's axis moves as if it were rigidly attached to the plane of her orbit. This is known as Cassini's theorem, and was discovered by Cassini from observation. The extreme range of variation of the inclination of the earth's axis, from these forced nutations, is about i.i", while the extreme range of variation of celestial longitude from the mean is about i". Since the velocity in the ellipse is 2 a, while the orbital velocity is a, it will be seen that two complete ellipses are described in every complete revolution to the same node. The axis is nearest to the pole of the ecliptic at the solstices, while it is farthest away at the equinoxes. The axis would make a complete preces- 2 IT sional circle in — - = 26,000 years, about. THEORY 53 For a single attracting body, the path of the axis is like that represented on the right side of Fig. 10. Two of these curves are described during every orbital period, while in the free nutations one of the cycloids is described in a period somewhat less than the rotational period. The actual forced nutations of the earth are relatively large, while if there were free nutations, which, however, do not exist, they would be extremely minute. 13. Gyroscopic Motion of the Moon's Orbit. The plane of the moon's orbit makes an angle of nearly 5 with the plane of the ecliptic. If the mass of the moon were uniformly distributed over her orbit into a thin ring of matter, and this ring were rotating in its plane, with the same angular velocity as the moon in her orbit, the attrac- tion of the sun upon this ring would set up gyroscopic couples. Now the moon, by her motion, practically dis- tributes her mass over her orbit, so that, the motion being rather rapid, the attraction of the sun is practically the same as if it were acting upon such a rotating ring. It is, therefore, a simple matter to calculate the motion of such a ring. C, the maximum moment of inertia, would be MD 2 , where M is the mass of the moon and D its distance MD 2 from the earth. A — . It is clear that the rotational 2 axis of this ring, i.e., the pole of the moon's orbit, will describe, in a retrograde direction, a small precessional circle about the pole of the ecliptic, at a distance of 5 . It can easily be calculated that the time of describing one complete precessional circle will be about i8f years. The rapidity of the precession, compared with that of the earth, is due to the very much greater moment of momentum of the ring about its axis, and to the greater gravitational couple. It is further clear that the motion of the ring will 54 THE GYROSCOPE be precisely like that of the earth's axis, just discussed in the previous article. The axis will move in an ellipse with harmonic motion, in a positive direction, and the major axis of the ellipse will always be directed towards the pole of the ecliptic. The center of the ellipse will move with a constant retrograde precession about the pole of the ecliptic. The motion in the ellipse is due to the forced nutations arising from the \p sin and components of the gravitational couple. These forced nutations, it appears, have not been considered in the theory of the moon's motion: at least, the author has been unable to find such corrections in lunar tables. In any case, they must be applied, and it is believed that they may in part, at least, if not wholly, explain certain anomalies in her motion. At present, the position of the moon, as given by the best tables, may be " out " 3" or 4", corresponding to 3 or 4 miles in her orbit. These deviations are now on one side, now on the other, of her calculated position. Two of these ellipses will be described in every synodical period. 14. Effect of the Earth's Equatorial Protuberance on the Motion of the Moon's Orbital Plane. Regarding the moon as a uniform ring of matter rotating about the earth, it will be seen that, as the plane of this ring and the plane of the earth's equator are not coincident, but are inclined at an angle varying from 18 to 28 , the earth's equatorial mass will set up a gyroscopic motion in this ring, precisely similar to the motions we have just considered. It will be very slight, however. Taking the average inclination as 23 , the complete precessional period would be about 159,000 years. This is at its present rate, but the factors are continually changing, so that long before this period is ended, the rate will have changed. This THEORY 55 precession, of course, takes place about the pole of the earth and is retrograde. 15. Effect of the Moon's Orbital Precession upon the Axis of the Earth. We have seen that the pole of the moon's orbit moves about the pole of the ecliptic in a retrograde direction in about i8| years. Let us examine what effect this has upon the motion of the earth's axis. We have seen that a revolv- ing bi-axial body executes, under the influence of the attracting body, a steady retrograde precessional move- ment, on which are superposed certain forced nutations. We have also seen that, if the attracting body is placed at the pole of the orbit and supposed to exert a repulsional instead of an attractional force, we shall have the same steady retrograde precession, on which are superposed certain free nutations. Disregarding the nutations for the time being, the steady precessional motion will be the same in either case, if we choose a suitable mass for the repelling body. We shall, therefore, regard the center of the earth fixed,. and the moon placed at the pole of her orbit, at B in Fig. 13, and moving in a retrograde direction about the pole of the ecliptic C. Let A be the position of the earth's axis, and the angle CA = 6, its inclination to the pole of the ecliptic. The angle CB = a is actually about 5 . The angle BA = c is, therefore, the inclination of the earth's axis to the repelling body. Let the angular velocity of the point B in the small circle be —b, and the angle CAB be designated by A, while the angle ACB is designated by C. The gravitational couple is, therefore, — K sin c cos c. We can resolve this into the two com- ponent couples, — K sin c cos c cos A, which is the couple, and K sin c cos c sin A , which is the 4> sin couple. We are 56 THE GYROSCOPE careful to write the signs thus, because the first always tends to decrease the value of 0, while the second, the ^ sin 6 component, acts in a positive or negative direction according to the sign of sin A. In Fig. 13, A is a negative angle and in this position the couple will act in a negative direction, while, when B is on the other side of CA , it will act in a positive direction. Taking a fixed vertical plane through C as the standard of reference, the angle ACB = C will be (— bt — i/-), where, as before, ^ is the precession of A about the pole of the ecliptic. t> t- • 1 ^ • , sini sina , ■, By spherical trigonometry, - — - = - — , (1) sin C sine and cos c = cos a cos 9 + sin a sin 8 cos C. (2) Hence the 6 couple is — K cos c Vsin 2 c — sin 2 a sin 2 C. THEORY 57 Now, since sin 2 c — sin 2 a sin 2 C = i — [cos 2 a cos 2 + 2 sin a cos a sin cos cos C + sin 2 a sin 2 cos 2 C] — sin 2 a sin 2 C = i — cos 2 a cos 2 — 2 sin a cos a sin cos cos C — sin 2 a[cos 2 C + sin 2 C] + sin 2 a cos 2 cos 2 C = cos 2 a sin 2 — 2 sin a cos a sin cos cos C + sin 2 a cos 2 cos 2 C = (cos a sin — sin a cos cos C) 2 , therefore v sin 2 c — sin 2 a sin 2 C = cos a sin — sin a cos cos C. The couple is, then, — K cos c (cos a sin — sin a cos cos C). Substituting the value of cos c from (2), this is — IC(cos a cos + sin a sin cos C) (cos a sin — sin a cos cos C) = — K [cos 2 a sin cos — sin a cos a cos 2 cos C + sin a cos a sin 2 cos C — sin 2 a sin cos cos 2 C] = — 2T cos 2 a sin cos + K sin a cos a cos C cos 2 + if sin 2 a sin cos cos 2 C. (3) The i/- sin couple is K sin c cos c sin ^4 = X cos c sin a sin C =K sin a sin C (cos a cos + sin a sin cos C) = iT sin a cos a sin C cos d -\- K sin 2 a sin C cos C sin 0. (4) 58 THE GYROSCOPE Our equations of motions are, therefore, K sin 2 a cos 2 (— bt — 4) sin cos + K sin a cos a cos ( — if — <£-) cos 2 —K cos 2 a sia 6 cos 8— Caj, sm9 + Aj/ 2 sia.0 cosd =A6 (5) Jf sin 2 a sin (— if — $) cos (— bt — $) sin + K sin a cos a sin (— if — \f/) cos + C«0* - Ai, cos 00 = 4Z> 4 (^ sin 0) . (6) [Note. We have previously used C to denote an angle, but now it has its usual significance. There will be no confusion.] Multiply (6) by sin (- b — ^), K sin 2 a sin (- bt - f) cos (- bt - f) sin 2 (- J - f) + if sin a cos a sin ( — bt — ^) sin cos ( — b — \f/) + C)(— b — ^) +K sin a cos a sin cos cos ( — bt — ^) + if sinacos a / sin cos sin ( — if — ^)(— i — ^) sm 2 /*. • /* • fl 2 - if cos 2 a ^^^ -Co / ^sin00+^ / ^ 2 sin 0.cos 6d=A-- (8) Integrating (7), K sin 2 a I sin (— bt — \f) cos (— if — ^) sin 2 (— i — ^) + if sin a cos a / sin cos sin (— if — f){— b — <{,) + iCco cos — Ceo 1 ^ sin 00 + i/1 / ^ sin cos 00 THEORY 59 + A J ^ 2 sin cos 00 = — bA\psm. 2 d + bA ft sin0 cos At 2 sin 2 (9) 2 Subtracting (9) from (8), sin 2 K sin 2 a co& 2 {bt+\f) |- A" sin a cos a sin cos cos(&+^) 2 - A" cos 2 a SLi _ & [c u cos 9 + ^.^ sin 2 0] = T + Const. 2 Const. = .ST sin 2 a + K sin a cos a sin O cos O 2 — X cos 2 a - — 6Ca> cos O . 2 Hence we have the equation of energy, „ . 2 „ ,,. . A sin 2 „ . 2 sin 2 o K sm 2 o cos 2 lot + M A sui 2 a 2 2 + if sin a cos a sin cos cos (bt + ^) ^ • • n v 2 sul2 1 c 2 s i ne o —A sin a cos a sin0 o cos0 o — Acos^a \-Kcos*a 2 2 - b [Cu (cos - cos O ) + AJ, sin 2 0] = T. (10) The terms not in the bracket express the work done in the plane BA, for this work is „ C . , „ /sin 2 c sin 2 c\ , — A I sin c cos cdc = A I ), and sin 2 c = sin 2 a + cos 2 a sin 2 — 2 sin o cos a sin cos cos C (mn c ciTi c \ — " ) „ . - sin 2 2 /-< 1^-2 sui2 9 o = A sm 2 a cos^ C — K sin 2 a - 2 2 + K sin a cos a sin cos cos C — K sin a cos a sin O cos O + K (sin 2 O — sin 2 0). 2 60 THE GYROSCOPE As in Art. 12, the bracket expresses the work done about an axis ± to the plane of the ecliptic, and is equal to the increase (or decrease) of the moment of momentum about this axis, multiplied by the precessional velocity of the moon's orbit. The average of this work for a complete motion of the repelling body around the small circle, from conjunction to conjunction, is zero. When w is large, as in the case of the earth, f and 6 must be very small, and we shall neglect their squares and products, bt + 4> will be very nearly bt for a short time, and will remain practically constant. Hence we can use as the equations of motion to a close degree of approxima- tion, putting \p sin 6 = x, 6 i = y, K sin 2 a sin d cos 9 cos 2 bt + K sin a cos a cos 2 6 cos bt — K cos 2 a sin cos 6 — Cwi = Ay (1 1) — K sin 2 a sin 6 sin bt cos bt — K sin a cos a cos sin bt + Co>y = .43c. (12) Now if we use only the first terms of the left members of Equations (11) and (12), we have precisely the case treated in Art. 12, and the motion due to these first terms is a har- monic motion in an ellipse with the elhpse itself moving with a constant retrograde precessional velocity. Taking the second terms only of (n) and (12), it is evident that the motion due to them will be simply an elliptic harmonic motion in an elhpse having axes K sin a cos a cos 2 d and K sin a cos a cos 6. The motion in this elhpse will be retrograde and the constant angular velocity — b, while in the other elhpse the velocity is 2 b. If we take the third terms of (n) and (12), which are respectively — K cos 2 a sin 6 cos 6 and zero, the result will be a constant retrograde precessional velocity. The actual motion of the pole of the earth will, therefore, be the resultant or these three separate motions. It is clear that there will be a continual, though THEORY 6 1 varying, precession in a retrograde direction, while the inclination to the pole of the ecliptic will pass through periodically-recurring values. Integrating (12), _ n . . „ . „ . n sin 2 bt Lay — Ax = K sin-' a sin — 20 „ . . /cos bt — i\ , -. — K sin a cos 62 THE GYROSCOPE where y is the inclination of the axis to the invariable line. The instantaneous axis executes a small circle about the axis of the body, and relatively to the body moves in a positive direction with the angular velocity ^ cos y — a>. (See Art. 5.) Since ^ cos y = — , this velocity, which is relative to IC — A\ some fixed point on the body, is co I — - — I. With a high value of co, the instantaneous axis makes a very small angle "with the axis. Now if we supposed the earth to be executing a Poinsot motion about some invariable line (which it must be dis- C — A tinctly remembered it does not do), since — - — =0.0032+ , the instantaneous axis would make a complete circuit, relatively to some fixed point on the earth's surface, in — — sidereal days. This is 312 sidereal days or about 10 months. Hence there would be a ten-monthly period of variation of latitude, as determined by celestial objects, with a maximum difference of 2 t, where 1 is the constant angle between the instantaneous axis and the axis of maxi- mum moment, i.e., the axis of the earth. Places 180 apart in longitude would experience opposite variations, a maxi- mum in one place corresponding to a minimum in the other. This imaginary variation, on the supposition that the earth executed a Poinsot motion about some invariable line, has been called the Eulerian nutation, and its period has been called the Eulerian ten-monthly period. From the previous discussion we have seen that the earth does not execute any Poinsot movements, and the consideration of what variations of latitude would result in case it should execute a Poinsot motion is purely academic. There is no THEORY 63 such thing as an Eulerian nutation, or an Eulerian ten- monthly period.* It has, however, been found by observation that there are actually small periodic variations of latitude, and that places on opposite meridians experience opposite variations. Chandler discovered a 14-monthly period in these varia- tions, but there is absolutely no trace of a 10-monthly period. The variations are extremely small with maximum differences of about half a second. It is a coincidence, perhaps more, that the periods of the forced solar and lunar nutations have a least common multiple of 14 months. Meteorological phenomena are, in all probability, re- sponsible for a part of these changes. A cyclone always rotates in the same direction as the surface of the earth under it. The tendency of these great rotating masses of air would be to shift the rotation axis slightly to their side. They occur alternately in the northern and southern hemi- spheres every six months. The heaviest are on opposite sides of the world, viz., the north Atlantic and the south Pacific. These might give a yearly period in the shift of the rotation axis, and, in fact, traces of a yearly period have been discovered, but the most distinct period seems to be the 14-monthly one. The exact determination of the causes of these changes awaits solution. Among possible factors are the forced nutations of the sun and moon, and the elasticity of the earth. * The impact of some external body, or a shifting of the principal axes by a geological convulsion, would produce a. slight Poinsot motion, but this would quickly be obliterated by the forced nutations. PART II. APPLICATIONS. \ PART II. 17. Applications. Children were the first to apply the gyroscope. They did this in their toys, the rolling hoop and the top. Some years ago they used to play with a form of gyroscope called Diablo. The fascination here is that these inanimate objects contradict their every-day experience and seem to be alive. And, in a certain sense, they are alive, for it is probable that all life consists only of more complicated forms of motion. The hoop and the top, when set in rotation, refuse to fall, to the child's great delight. As soon as they begin to fall and]acquire the slightest velocity about a horizontal axis, immediately a gyroscopic couple is set up about the \p sin 9 axis and the resulting velocity about this axis counterbalances the gravitational effort. A hoop, if mchning a little to the left, does not fall but simply turns to the left. If inclining to the right, it turns to the right and continues rolling. If the rotational velocity is high, the inclinations and the corresponding turns become very small, and the hoop runs practically upright in a straight course. This principle, as we shall see, has been applied by Howell to the torpedo. It is the same with a bicycle. The front wheel turns easily and automatically. It is a rolling hoop — a gyro- scope. With a high speed, the bicyclist instinctively feels himself in equilibrium. If there is the slightest tendency to tip to one side or the other, he learns that a touch, the merest trace of a movement about a vertical axis — through the handle bar — sets up a powerful gyroscopic couple 67 68 THE GYROSCOPE about a horizontal axis which immediately rights him. Of course, if he executes an appreciable curve, centrifugal forces, tending to throw his body to the outside of the curve, act in the same direction, but in direct running the front wheel does the righting. If anybody doubts the force of this gyroscopic couple, let him spin a bicycle wheel on a bar through the axle held in his hands, and give the bar the slightest turn. He will experience a very violent jerk, but not in the direction he expected it. If he gives it too sudden a turn, it will take him off his guard and twist out of his hands. A top, if it has a sharp point and the supporting surface is rough, will precess about the vertical at a practically constant height. It actually does fall rapidly back and forth through its nutations, but these cannot be seen; only heard as a humming. If, however, the point is rounded, so that it has some surface, and the surface on which it spins is rough, the toe will roll on the surface, thus causing, at the expense of the initial rotational energy, a new rotation about a vertical axis which, as we have seen, will set up a gyroscopic couple tending to bring the axis of the top into coincidence with this vertical axis. And, if the rotational energy is sufficient, it will, in fact, raise the top to an up- right position. The top now seems motionless, and it is soundless. No more precession, no more nutations, no more humming. The boys say it is asleep. Nutation means nodding, so that after so much nodding it is eminently proper that it should go to sleep. It would remain asleep forever, if it were not for the friction of the air and that on the point of support. Whenever vehicles running on wheels turn, gyroscopic couples are set up. When a train rounds a curve, gyro- scopic couples are set up tending to overturn the cars outward. It is well known that the simple centrifugal APPLICATIONS 69 moment tends to overturn the cars outward and this is provided against by raising the outer rail. Engineers, however, overlook the fact that besides this simple centrif- ugal moment, there is an added gyroscopic couple, due to the rotation of the wheels, which always acts with the cen- trifugal moment, and which helps the centrifugal couple to overturn the car, in case such an accident occurs. It is true that, owing to the mass of the car being much greater than the mass of the wheels, and its center of gravity much higher above the rails, the simple centrifugal moment is much greater than the gyroscopic couple. The gyroscopic couple, however, is very appreciable, and where the limit has been calculated solely upon the basis of the centrifugal couple, as is usually the case, and this limit is approached, the unsuspected gyroscopic couple is the agent which gives the "coup de grace." Such accidents have occurred, but neither before nor afterwards does there seem to have been any clear comprehension of the principles involved. The cloudy ideas enshrouding gyroscopics in the popular mind, and even in that of engineers, is perhaps due on the one side to the fear of certain pure mathematicians * that their work and demonstrations may by any possible means be put to a practical use, and on the other side to the dread and distrust of the practical man of what he styles the higher mathematics. When an aviator spirals downward in his machine, the aeroplane as a whole becomes a gyroscope, even though the motor be stopped. In general, the axis about which it is turning at any instant is not a principal axis, and when such a rotating body is turned sharply it will be subjected to gyroscopic couples which are especially dangerous, since they act in directions unapprehended and unforeseen by * Was it not Dirichlet who boasted that nothing of his work could be put to practical use ? 7 99 NOTE. ON THE MOTION OF CYCLONES. The earth being a rotating spheroid, every particle on its surface is sub- jected to a tangential gravitational component towards the pole, and to a tangential centrifugal component towards the equator. The amount of the latter is R sin cos 0^ 2 , where R is the radius of the earth, and ^ the horizontal angular velocity of the particle at any instant. Hence a movable particle, at rest relatively to the earth, must have a gravitational component towards the pole equal to R sin cos 0« 2 , where a is the angular velocity of the earth. When the horizontal angular velocity of a particle is greater, or less, than that of the earth, the particle will experience an acceleration along a meridian, towards the equator or towards the pole, as the case may be, and the amount of this acceleration is R sin cos 6 (\P — a; 2 ). In the case of a cyclone, every particle in it will experience this meridianal acceleration according as the value of if/, which it possesses at any instant, is greater, or less, than u>. The sum of these accelerations will be the total meridianal force acting upon the cyclone. Considering, at first, only a ring of rotating matter, let \j/ c be the horizontal angular velocity of its center, r the radius of the ring, the angle which a particle makes at any instant with a parallel of latitude through the center, and 4> the angular velocity of the ring relatively to this parallel. Then the horizontal angular velocity of a particle is • • T(t> sin 4 > if cos i :, ; „ 2^ c r<£sin0 rVsin 2 ^ I ft i rsinA where 6 C is the latitude of the center, 2 f c rj> sin (j> R0 = ^ sin 2 (e c + ^p) (co* - W) + 4c T4> sin ^sin ($„ + f -^\ rVanV / , rsjnjA B— taa { e ' + -R-)' Since — = — is, in a cyclone, usually a small quantity, in developing a Hence MOTION OF CYCLONES IOI function of this angle by Taylor's theorem, we can neglect its squares and higher powers without appreciable error. Hence R6 = ^Tsin 20 c + 4cos 2 c ^j^ , l(a> J - £ c ») + 2 i/ c rtj> sin f sin 8 C H — cos 8 C ) r 2 * 2 sin 2 <£ / The sum, or integral, of all these accelerations, for the complete ring, is irr R sin 20 o (co 2 — ^ c 2 ) + — cos 8 C trfc ir v tan 8 C = MRB C . Putting A for the moment of inertia of the ring about the center of the earth, since 2vr is the mass of the ring, we have ■ . d 2 A sin C cos 8 C (u 2 — iAc 2 ) + C 4> c cos 8 C tan 8 C = A8 C . (i) 2 We, therefore, see that the meridianal forces acting upon a cyclone are ■equivalent to four forces acting on its center, viz., a gravitational force •acting towards the pole, a revolutional centrifugal force acting towards the equator, a gyroscopic force {C$4>c cos B c ) acting towards the pole, and a rotational centrifugal force acting towards the equator. If it should happen that these four forces (two positive and two negative) were balanced, or 8 = o, then the cyclone would remain on the same parallel of latitude. This rarely happens, for, when 4> is large, the gyroscopic force, at first, always urges the cyclone towards the pole. Again, we see from Equation (i) that it cannot move continually towards the pole, but must reach a latitude — its latitude of equilibrium— where 8 C vanishes. Here it performs a quasi Poinsot motion about the axis of the earth, providing there are no frictional forces. For the hypothetical frictionless case, our equations of motion are, A sin0cos0 (w 2 - cos 2 8 = C