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CONTENTS.

• Chapter I.—Vectors and their Composition . pp. 1—28

  • Sketch of the attempts made to represent geometrically the imaginary of algebra. §§1—13.

  • De Moivre’s Theorem interpreted in plane rotation. §§7, 8.

  • Curious speculation of Servois. §11.

  • Elementary geometrical ideas connected with relative position. §14.

  • Definition of a Vector. It may be employed to denote translation. Definition of currency. §16.

  • Expression of a vector by one symbol, containing implicitly three distinct numbers. Extension of the signification of the symbol = . §18.

  • The sign + defined in accordance with the interpretation of a vector as representing translation. §19.

  • Definition of − . It simply reverses the currency of a vector. §20.

  • Triangles and polygons of vectors, analogous to those of forces and of simultaneous velocities. §21.

  • When two vectors are parallel we have

    α = xβ. §22.

  • Any vector whatever may be expressed in terms of three distinct vectors, which are not coplanar, by the formula

    ρ = xα + yβ + zγ ,
    which exhibits the three numbers on which the vector depends. §23.

  • Any vector in the same plane with α and may be written

    ρ = xα + yβ. §24.

  • The equation

    ϖ = ρ,

    between two vectors, is equivalent to three distinct equations among numbers. §25.

  • The Commutative and Associative Laws hold in the combination of vectors by the signs + and − . §27.

  • XVIII

    The equation

    ρ = xβ,

    where ρ is a variable, and β a fixed, vector, represents a line drawn through the origin parallel to β.
    ρ = α + xβ,
    is the equation of a line drawn through the extremity of α and parallel to β. §28.
  • ρ = yα + xβ,
    represents the plane through the origin parallel to α and β, while
    ρ = γ + yα + xβ,
    denotes a parallel plane through the point γ. §29.

  • The condition that ρ, α, β may terminate in the same line is

    pρ + qα + rβ = 0,
    subject to the identical relation
    p + q + r = 0,

    Similarly

    pρ + qα + rβ + sγ = 0,

    with

    p + q + r + s = 0,

    is the condition that the extremities of four vectors lie in one plane. §30.

  • Examples with solutions. Conditions that a vector equation may represent a line, or a surface.

    The equation

    ρ = φt

    represents a curve in space ; while
    ρ = uφt

    is a cone, and

    ρ = φt + uα

    is a cylinder, both passing through the curve. §31.

  • Differentiation of a vector, when given as a function of one number. §§32—37.

  • If the equation of a curve be

    ρ = φ(s)
    where s is the length of the arc, dρ is a vector-tangent to the curve, and its length is ds. §§38, 39.

  • Examples with solutions. §§40–43.

    Examples to Chapter I. . . . . . . 28—30

• Chapter II.—Products and Quotients of Vectors . pp. 31—57

  • Here we begin to see what a quaternion is. When two vectors are parallel their quotient is a number. §§45, 46.

  • When they are perpendicular to one another, their quotient is a vector perpendicular to their plane. §§47, 64, 72.

  • When they are neither parallel nor perpendicular the quotient in general involves four distinct numbers—and is thus a Quaternion. §47.

  • A quaternion q regarded as the operator which turns one vector into another. It is thus decomposable into two factors, whose order is indifferent, the stretching factor or Tensor, and the turning factor or Versor. These are denoted by Tq, and Uq. §48.

  • XIX

    The equation

    β = qα

    gives β / α = q or βα⁻¹ = q, but not in general
    α⁻¹β = q . §49.
    q or βα⁻¹ depends only on the relative lengths, and directions, of β and α. §50.

  • Reciprocal of a quaternion defined,

    q = β / α gives 1 / q or q⁻¹ = α / β ,
    T.q⁻¹ = 1 / Tq, U.q⁻¹ = (U.q)⁻¹, §51.

  • Definition of the Conjugate of a quaternion,

    Kq = (Tq)²q⁻¹i

    and

    qKq = Kq.q = (Tq). §52.

  • Representation of versors by arcs on the unit-sphere. §53.

  • Versor multiplication illustrated by the composition of arcs. The process proved to be not generally commutative. §54.

  • Proof that

    K(qr) = Kr.Kq. §55.

  • Proof of the Associative Law of Multiplication

    p.qr = pq.r . §§57—60.

  • [Digression on Spherical Conics. §59*.]

  • Quaternion addition and subtraction are commutative. §61.

  • Quaternion multiplication and division are distributive. §62.

  • Integral powers of a versor are subject to the Index Law. §63.

  • Composition of quadrantal versors in planes at right angles to each other. Calling them i, j, k, we have

    i² + j² + k² = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j,
    ijk = −1. §§64—71.

  • A unit-vector, when employed as a factor, may be considered as a quadrantal versor whose plane is perpendicular to the vector. Hence the equations just written are true of any set of rectangular unit-vectors i, j, k. §72.

  • The product, and also the quotient, of two vectors at right angles to each other is a third perpendicular to both. Hence

    Kα = −α,

    and

    (Tα)2 = αKα = −α2. §73.

  • Every versor may be expressed as a power of some unit-vector. §74.

  • Every quaternion may be expressed as a power of a vector. §75.

  • The Index Law is true of quaternion multiplication and division. §76.

  • Quaternion considered as the sum of a Scalar and a Vector.

    q = β / α = x + γ = Sq + Vq §§77, 78.

  • Proof that

    SKq = Sq, VKq = −Vq, ΣKq = KΣq. §79.

  • Quadrinomial expression for a quaternion

    q = w + ix + jy + kz.

  • An equation between quaternions is equivalent to four equations between numbers (or scalars). §80.

  • XX Second proof of the distributive law of multiplication. §§81–83.

  • Algebraic determination of the constituents of the product and quotient of two vectors. §83, 84.

  • Second proof of the associative law of multiplication. §85.

  • Proof of the formulae

    Sαβ = Sβα,

    Vαβ = −Vβα,
    αβ = Kβα,
    2Sαβ = αβ + βα,
    2Vαβ = αβ − βα,
    S.qr = S.rq,
    S.qrs = S.rsq = S.sqr,
    S.αβγ = S.βγα = S.γαβ = −S.αγβ = &c.,

    2S.αβ...φχ 2V.αβ...φχ

    }

    = αβ...φχ ± χφ...βα,

    the upper sign belonging to the scalar if the number of factors is even. §§86—89.

  • Proof of the formulae

    V.αVβγ = γSαβ − βSγα,
    V.αβγ = αSβγ − βSγα + γSαβ,
    V.αβγ = V.γβα,
    V.VαβVγδ = αS.βγδ − βS.γαδ,
    = δS.αβγ − γS.αβδ,
    δS.αβγ = αS.βγδ + βS.γαδ + γS.αβδ,
    = VαβSγδ + VβγSαδ + VγαSβδ, §§90—92.

  • Hamilton’s proof that the product of two parallel vectors must be a scalar, and that of perpendicular vectors, a vector; if quaternions are to deal with space indifferently in all directions. §93.

  • Examples to Chapter II. . . . . . . 57—58

• Chapter III.—Interpretations and Transformations of Quaternion Expressions . . . 59—88

  • If θ be the angle between two vectors, α and β, we have

    S(β/α) = (Tβ/Tα) cos θ , Sαβ = −TαTβ cos θ ,
    TV(β/α) = (Tβ/Tα) sin θ , TVαβ = TαTβ sin θ. §§94—96.

  • Applications to plane trigonometry. §97.

  • The condition

    Sαβ = 0

    shews that α is perpendicular to β, while
    Vαβ = 0,
    shews that α and β are parallel.

    The expression

    S.αβγ

    is the volume of the parallelepiped three of whose conterminous edges are α, β, γ. Hence
    S.αβγ = 0
    shews that α, β, γ are coplanar.
    XXI Expression of S.αβγ as a determinant. §§98—102.

  • Proof that

    (Tq)2 = (Sq)2 + (TVq)2,

    and

    T(qr) = TqTr. §103.

  • Simple propositions in plane trigonometry. §104.

  • Proof that −αβα⁻¹ is the vector reflected ray, when β is the incident ray and α normal to the reflecting surface. §105.

  • Interpretation of αβγ when it is a vector. §106.

  • Examples of variety in simple transformations. §107.

  • The relation among the distances, two and two, of five points in space. §108.

  • De Moivre’s Theorem, and Plane Trigonometry. §§109—111.

  • Introduction to spherical trigonometry. §§112—116.

  • Representation, graphic, and by quaternions, of the spherical excess. §§117, 118.

  • Interpretation of the Operator

    q( )q⁻¹
    in connection with rotation. Astronomical examples. §§119—122.

  • Loci represented by different equations—points, lines, surfaces, and volumes. §§123—126.

  • Proof that

    r−1(r2q2)1/2q−1 = U(rq + KrKq). §127.

  • Proof of the transformation

    (Sαρ)² + (Sβρ)² + (Sγρ)² = [ T(ιρ + ρκ) / (κ² − ι²) ]²
    where 2ι/κ = [ (Tα + Tγ) / (TαTγ) ] {√[(β² − α²) / (γ² − α²) ] Uα ± √[(γ² − β²) / (γ² − α²) ] Uγ}. §§128—129.

  • Biquaternions. §§130—132.

  • Convenient abbreviations of notation. §§133, 134.

    Examples to Chapter III....... 89—93

• Chapter IV.—Differentiation of Quaternions . 94—105

  • Definition of a differential,

    dr = dFq = L_∞n {F(q + dq/n) − F(q)},
    where dq is any quaternion whatever.
    We may write dFq = f (q, dq), where f is linear and homogeneous in dq ; but we cannot generally write dFq = f (q)dq. §§135—138.

  • Definition of the differential of a function of more quaternions than one.

    d(qr) = qdr + dq.r, but not generally d(qr) = qdr + rdq. §139.

  • Proofs of fundamental differential expressions :—

    dTρ/Tρ = Sdρ/ρ
    dUρ/Uρ = Vdρ/ρ, &c. §140.

  • Successive differentiation; Taylor’s theorem. §§141—142.

  • XXII If the equation of a surface be

    F(ρ) = C,
    the differential may be written
    Sνdρ = 0,
    where ν is a vector normal to the surface. §144.

  • Definition of Hamilton’s Operator

    ∇ = id/dx + jd/dy + kd/dz

    Its effects on simple scalar and vector functions of position. Its square the negative of Laplace’s Operator. Expressions for the condensation and rotation due to a displacement. Application to fluxes, and to normals to surfaces. Precautions necessary in its use. §§145—149.

    Examples to Chapter IV. 106, 107

• Chapter V.—The Solution of Equations of the First Degree ....... 108—141

  • The most general equation of the first degree in an unknown quaternion q, may be written

    ΣV.aqb + S.cq = d,
    where a, b, c, d are given quaternions. Elimination of Sq, and reduction to the vector equation
    φρ = Σ.αSβρ = γ. §§150,151.

  • General proof that φ³ρ is expressible as a linear function of ρ, φρ, and φ²ρ. §152.

  • Value of for an ellipsoid, employed to illustrate the general theory. §§153—155.

  • Hamilton’s solution of

    φρ = γ.

    If we write

    Sσφρ = Sρφ′σ,

    the functions φ and φ′ are said to be conjugate, and
    mφ⁻¹Vλμ = Vφ′σ.
    Proof that m, whose value may be written as
    (S.φ′λφ′ν) / (S.λμν) ,
    is the same for all values of λ, μ, ν. §§156—158.

  • Proof that if

    mg = m − m1g + m2g2 − g3,

    where

    m1 = S(λφ′μφ′ν + φ′λμφ′ν + φ′λφ′μν) / S.λμν ,

    and

    m1 = S(λμφ′ν + φ′λμν + λφ′μν) / S.λμν ,

    (which, like m, are Invariants,)

    then

    m2(φ − g)−1Vλμ = (mφ−1 − gχ + g2)Vλμ ,

    Also that

    χ = m2 − φ ,

    whence the final form of solution
    mφ⁻¹ = m₁ − m₂φ + φ². §§159, 160.

  • XXIII Examples. §§161—173.

  • The fundamental cubic

    φ³ − m₂φ² + m₁φ − m = (φ − g₁)(φ − g₂)(φ − g₃) = 0.
    When φ is its own conjugate, the roots of the cubic are real; and the equation
    Vρφρ = 0,

    or

    (φ − g)ρ = 0,

    is satisfied by a set of three real and mutually perpendicular vectors. Geometrical interpretation of these results. §§174—178.

  • Proof of the transformation of the self-conjugate linear and vector function

    φρ = fρ + hV.(i + ek)ρ(i − ek)

    where

    (φ − g1)i = 0,

    (φ − g₂)k = 0,
    e₂ = (g₃ − g₂) / (g₁ − g₂),
    f = (g₁ + g₂) / 2,
    h = −(g₁ − g₂) / 2.
    Another transformation is
    φρ = aαVαρ + bβSβρ. §§179—181.

  • Other properties of φ. Proof that

    Sρ(φ − g)⁻¹ρ = 0, and Sρ(φ − h)⁻¹ρ = 0
    represent the same surface if
    mSρφ⁻¹ρ = ghρ².
    Proof that when φ is not self-conjugate
    φρ = φ′ρ + Vερ. §§182—184.

  • Proof that, if

    q = αφα + βφβ + γφγ,

    where α, β, γ are any rectangular unit-vectors whatever, we have
    Sq = −m² , Vq = ε,

    where

    Vερ = (φ − φ′) ρ / 2.

    This quaternion can be expressed in the important form
    q = ∇φρ. §185.

  • A non-conjugate linear and vector function of a vector differs from a self-conjugate one solely by a term of the form

    Vερ. §186.

  • Graphic determination of the conditions that there may be three real vector solutions of

    Vρφρ = 0. §187.

  • Degrees of indeterminateness of the solution of a quaternion equation—Examples. §§188—191.

  • The linear function of a quaternion is given by a symbolical biquadratic. §192.

  • Particular forms of linear equations. Differential of the nth root of a quaternion. §§193—196.

  • A quaternion equation of the nth degree in general involves a scalar equation of degree m⁴. §197.

  • Solution of the equation

    q2 = qa + b. §198.

    Examples to Chapter V....... 142—145

• XXIV

  Chapter VI.—Sketch of the Analytical Theory of Quaternions 146—159

• Chapter VII.—Geometry of the Straight Line and Plane . 160—174

  • Examples to Chapter VI....... 175—177

• Chapter VIII.—The Sphere and Cyclic Cone . 178—198

  • Examples to Chapter VIII....... 199—201

• Chapter IX.—Surfaces of the Second Degree . 202—224

  • Examples to Chapter IX....... 224—229

• Chapter X.—Geometry of Curves and Surfaces 230—269

  • Examples to Chapter X....... 270—278

• Chapter XI.—Kinematics ..... 279—304

  • A. Kinematics of a Point. §§354—366.

    If ρ = φt be the vector of a moving point in terms of the time, ρ˙ is the vector velocity, and ρ˙˙ the vector acceleration.
    σ = ρ˙ = φ′(t) is the equation of the Hodograph.
    ρ˙˙ = v˙ρ′ + v²ρ′′ gives the normal and tangential accelerations.
    Vρρ˙ = 0 if acceleration directed to a point, whence Vρρ˙˙ = γ.
    Examples.—Planetary acceleration. Here the equation is

    ρ˙˙ = μUρ / Tρ²,
    giving Vρρ˙ = γ ; whence the hodograph is
    ρ˙ = εγ⁻¹ − μUp.γ⁻¹,
    and the orbit is the section of
    μTρ = Sε(γ²ε⁻¹ − ρ).
    by the plane Sγρ = 0.
    Cotes’ Spirals, Epitrochoids, &c. §§354—366.

  • B. Kinematics of a Rigid System. §§367—375.

    Rotation of a rigid system. Composition of rotations. If the position of a system at time t is derived from the initial position by q( )q⁻¹, the instantaneous axis is

    ε = 2Vq˙q⁻¹.
    Rodrigues’ cöordinates. §§367—375.
  • XXV

    C. Kinematics of a Deformable System. §§376—386.

    Homogeneous strain. Criterion of pure strain. Separation of the rotational from the pure part. Extraction of the square root of a strain. A strain φ is equivalent to a pure strain √φ′φ followed by the rotation φ / √φ′φ. Simple Shear. §§376—383.

    Displacements of systems of points. Consequent condensation and rotation. Preliminary about the use of ∇ in physical questions. For displacement σ, the strain function is

    φτ = τ − Sτ∇.σ. §§384—386.

  • D. Axes of Inertia. §387.

    Moment of inertia. Binet’s Theorem. 387.

    Examples to Chapter XI....... 304—308

• Chapter XII.—Physical Applications . . 309—409

  • A. Statics of a Rigid System. §§389—403.

    Condition of equilibrium of a rigid system is ΣS.βδα = 0, where β is a vector force, a its point of application. Hence the usual six equations in the form Σβ = 0, ΣVαβ= 0. Central axis. Minding’s Theorem, &c. §§389—403.

  • B. Kinetics of a Rigid System. §§404—425.

    For the motion of a rigid system

    ΣS(mα − β)δα = 0,
    whence the usual forms. Theorems of Poinsot and Sylvester. The equation
    2q˙ = qφ⁻¹(q⁻¹γq),
    where γ is given in terms of t and q if forces act, but is otherwise constant, contains the whole theory of the motion of a rigid body with one point fixed. Reduction to the ordinary form
    dt/2 = dw/W = dx/X = dy/Y = dz/Z.
    Here, if no forces act, W, X, Y, Z are homogeneous functions of the third degree in w, x, y, z. §§404—425.

  • C. Special Kinetic Problems. §§426—430.

    Precession and Nutation. General equation of motion of simple pendulum. Foucault’s pendulum. §§426—430.

  • D. Geometrical and Physical Optics. §§431—452.

    Problem on reflecting surfaces. §431.

    Fresnel’s Theory of Double Refraction. Various forms of the equation of Fresnel’s Wave-surface;
    S.ρ(φ − ρ²)⁻¹ρ = −1, T(ρ⁻² − φ⁻¹)^−1/2ρ = 0, 1 = −pρ² ∓ (T ± S)VλρVμρ.
    The conical cusps and circles of contact. Lines of vibration, &c. §§432—452.

  • XXVI

    E. Electrodynamics. §§453—472.

    Electrodynamics. The vector action of a closed circuit on an element of current α₁ is proportional to Vα₁β where

    β = ∫ (Vα / Tα³)da = ∫ dUα / α
    the integration extending round the circuit. It can also be expressed as −∇Ω, where Ω is the spherical angle subtended by the circuit. This is a many-valued function. Special case of a circular current. Mutual action of two closed circuits, and of solenoids. Mutual action of magnets. Potential of a closed circuit. Magnetic curves. §§453—472.

  • F. General Expressions for the Action between Linear Elements. §473.

    Assuming Ampère’s data I, II, III, what is the most general expression for the mutual action between two elements? Particular cases, determined by a fourth assumption. Solution of the problem when I and II, alone, are assumed. Special cases, including v. Helmholtz’ form. §473 (a)...(l).

  • G. Application of ∇ to certain Physical Analogies. §§474—478.

    The effect of a current-element on a magnetic particle is analogous to displacement produced by external forces in an elastic solid, while that of a small closed circuit (or magnet) is analogous to the corresponding vector rotation.

  • H. Elementary Properties of ∇. §§479—481.

    Sdp∇ = −d = Sdσ∇_σ
    where σ = iξ + jη + kσ and ∇_σ = id/dξ + jd/dη + kd/dσ ;
    so that, if dσ = σdρ,
    ∇_σ = φ′⁻¹∇.

  • J. Applications of ∇ to Line, Surface and Volume Integrals. §§482—501.

    Proof of the fundamental theorem for comparing an integral over a closed surface with one through its content

    ∫∫∫ S∇σds = ∫∫ Sσ∇Uνds.
    Hence Green’s Theorem. Examples in potentials and in conduction of heat. Limitations and ambiguities. Determination of the displacement in a fluid when the consequent rotation is given. §§482—494.

    Similar theorem for double and single integrals

    ∫ Sσdρ = ∫∫ S.Uν∇σds. §§495—497.
    The fundamental forms, of which all the others are simple consequences, are
    ∫ dρμ = ∫∫ dsV(Uν∇σds.
    ∫∫∫ ∇uds = ∫∫ Uνuds. §§498, 499.

    The first is a particular case of the second. §500.

    Expression for a surface in terms of an integral through the enclosed volume. §499.

  • XXVII

    K. Application of the ∇ Integrals to Magnetic &c. Problems. §§502—506.

    Volume and surface distributions due to a given magnetic field. Solenoidal and Lamellar distributions. §502. Magnetic Induction and Vector potential. §503. Ampère’s >Directrice. §504. Gravitation potential of homogeneous solid in terms of a surface-integral. §505.

  • L. Application of ∇ to the Stress-Function. §§507—511.

    When there are no molecular couples the stress-function is self-conjugate. §507. Properties of this function which depend upon the equilibrium of any definite portion of the solid, as a whole. §508. Expression for the stress-function, in terms of displacement, when the solid is isotropic:—

    φω = −n(Sω∇.σ + ∇Sωσ) − (c − 2n/3)ωS∇σ.

    Examples. §509. Work due to displacement in any elastic solid. Green’s 21 elastic coefficients. §511.

  • M. The Hydrokinetic Equations. §§512, 513.

    Equation of continuity, for displacement σ:—

    ∂e / ∂t = de / dt − Sσ∇.e = eS∇σ.
    Equation for rate of change of momentum in unit volume:—
    ∂σ / ∂t = −∇(P + ∫ dp/e) = −∇Q. §512.
    Term introduced by Viscosity:—
    ∝∇²σ + ∇S∇σ/3. (Miscellaneous Examples, 36.)
    Helmholtz’s Transformation for Vortex-motion:—
    (∂/∂t)V∇σ = V.∇V.σV∇₁σ₁.
    Thomson’s Transformation for Circulation:
    −∂f/∂t = [Q − v²/2]_α^ρ, where f = −∫_α^ρ Sσdρ.

  • N. Use of ∇ in connection with Taylor’s Theorem. §§514—517.

    Proof that

    e^−Sσ∇f(ρ) = f(ρ + σ),

    Applications and consequences. Separation of symbols of operation and their treatment as quantities.

  • O. Applications of ∇ in connection with the Calculus of Variations. §§518—527.

    If

    A = ∫QTdρ

    we have

    δA = −[QSUdρδρ] + ∫{δQTdρ + S.δρ(QUδρ)},

    whence, if A is a maximum or minimum,
    (d/ds)(Qρ′) − ∇Q = 0.

    Applications to Varying Action, Brachistochrones, Catenaries, &c. §§518—526.

    Thomson’s Theorem that there is one, and but one, solution of the equation

    S.∇(e²∇u) = 4πr. §527.

    Miscellaneous Examples...... 409—421.