Elements of Quaternions

Sir William Rowan Hamilton

Longmans, Green, & Company, 1866 - 762 páginas

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and their focal conics the lines a a are asymptotes to the focal	xxxiii

so that through each of the twelve umbilics	xxxiii

Τβφβ 0	lii

somewhat more briefly and perhaps more clearly than in the Lectures	liii

of reciprocity	liv

section of an arbitrary surface which touches one of the two lines	lvi

Singular Line p 601 although not a cuspedge upon that Surface 1	1

On Differences and Sums of Vectors taken	3

whence	443

a given curve in space may be represented rigorously by the vector	458

duce otherwise from ø as well as the three scalar constants m m	460

ON SOME ADDITIONAL APPLICATIONS OF QUATERNIONS WITH	495

On Normals and Tangent Planes to Surfaces 501510	501

and	507

On Osculating Planes and Absolute Normals	511

an arbitrary surface being by III iii 5 p 515	515

CHAPTER II	11

whence	15

which likewise admits of an extremely simple interpretation	33

and is supposed to contain so much original matter that it seems	42

in which and x are real and constant vectors in the directions	53

any misconception of the meaning of the word Vector being fatal	66

tegrability of the equation Y2 is expressed by the very simple for	70

whereof only four at most can be real and which are its intersections	94

Radius r denoted here for distinction by a roman letter and Vector	102

any misconception of the meaning of the word Vector being fatal	106

in direction involves generally a system of four numerical elements	110

On Equality of Quaternions and on the Plane	112

On the Axis and Angle of a Quaternion	117

On the Reciprocal Conjugate Opposite	120

On Radial Quotients and on the Square of	129

On VectorArcs and VectorAngles consi	142

On a System of Three Right Versors	157

On the Tensor of a Vector or of a Quater	162

On the Sum or Difference of any two Qua	175

a suitable direction as contrasted with the opposite and with a suit	177

On the Right Part or Vector Part of	190

p 226 which was among the earliest geometrical results of the Qua	196

tion of the deviation from the given sphere of that other near point	198

higher than the third order but that of R requires the fourth order of differen	200

Section of the second Chapter of the Third Book as III ii 6 and so	203

so that by P p xii these three asymptotes compose a real and rect	222

relative living force or for the mass M m+m multiplied into	232

On the Reduction of the General Quaternion	233

which are all included in the Fundamental Formula	234

CHAPTER II	240

On Continued Proportion of Four or more Vec	246

On the Amplitudes of Quaternions in a given	251

On the Ponential and Logarithm of a Quater	257

On Finite or Polynomial Equations of Alge	265

On the nn Imaginary or Symbolical	275

CHAPTER III	286

On some Geometrical Proofs of the Associative	293

BOOK III	301

On a Second Method of arriving at the same	308

On the Fourth Proportional to Three Diplanar	331

On an Equivalent Interpretation of the Fourth	349

On a Third Method of interpreting a Product	361

a is the vector at the time t of the mass or particle m P is the	367

the Exponential Connexions which it establishes between Quaternions	381

6 y being two vector constants and H a scalar constant	385

Elementary Illustrations of the Definition	394

it is ultimately equal p 595 to the quarter of the deviation 397	397

tials of functions of functions	408

2 and if h be the radius of the hodograph then p 719	409

which may represent any vector is operated on and an application	435

p being the vector of the vertex P and p + Ap that of any other point	439

2T MDa² 2 P+ H M 2r¹ a¹	521

the independent variable being the arc in T while it is arbitrary	522

Polar Axis Polar Developable	537

projection of vector r of simple or first curvature on radius	560

equal p 681 to the semiaxes of the diametral section of the given	563

of the same near point P from the osculating circle at P multiplied	566

this line A may be called the Rectifying Vector and if I denote	568

a normal to a surface Some of the theorems or constructions	574

to serve in questions in which so is neglected are assigned in p 579	579

which is at once the Locus of its osculating Circle and the Envelope	581

in which v₁ T Ts and wws the vector of an arbitrary point	585

the function is said to be selfconjugate then this last cubic has three	592

Journal de lEcole Polytechnique Cahier xxx	598

dence o one of these can be at once translated into Monges equa	604

so that this new surface is cut by	621

and therefore ultimately p 600	622

the constant a will have its usual signification relatively to	625

On Osculating Circles and Spheres to Curves	630

analogous expression p 581	633

which have thus comp E and F₁ the same forms as before	639

a a are two real focal unitlines common to the whole system	643

in which w is a variable vector represents p 684 the normal plane	647

and y any constant values consistentwith N₁ the equation N₁	649

in S1 may be thus decomposed into factors p 666	666

surface may be the tangents to the lines of curvature bisect the angles	668

of the two rectangular directions that each such generatrix PP is crossed perpendi	669

F may be called the Principal Function and V the Characteristic	671

the equation W1 being a new form of the general differential equa	679

tral quadric of any species as the locus of the extremities of normals	683

m The equations W₁ W1 give comp the Note to p 684	684

the envelope of a certain variable sphere comp 398 which has	685

Umbilics of a central quadric	686

surface and R R1 R2 the three corresponding points near to each other	690

jectory to the tangents of the evolute but not to the osculating planes	692

points s and x in which the axis of the osculating circle to the curve	694

selfconjugation is given at a later stage in the few first subarticles to Art 415	698

700	700

in which y is a vector function of p not generally linear and deduced	702

and p1 v¹dp1Svdo¹d²p W₂	706

and VQ is the vector y of a point c upon the central axis which does	709

is equal to the living force h2 divided by the square Ti² of	711

integrals of the same system of differential equations A4 being	716

the line ou cuts the circle L the middle point and N the pole of	724

measure of the force	725

whence pp 684 689	727

and if A B C denote the three Principal Moments of inertia corre	729

while in the second view of the same functions they satisfy the	730

results of that great mathematician on this subject namely that	731

DaVmDia J₁ DaVmDa K DÁVt L4	740

i² j² k² ijk 1	756

Otras ediciones - Ver todas

Elements of Quaternions
Sir William Rowan Hamilton
Vista completa - 1866

Elements of Quaternions, Volumen2
Sir William Rowan Hamilton
Vista completa - 1901

Ver todas »

Términos y frases comunes

ABCD algebra angle anharmonic arbitrary axis centre CHAP chord coefficients collinear comp Compare the Note complanar cone conic conjugate considered constant construction cubic cubic equation cyclic cylinder deduced denote derived differential diplanar direction ellipsoid equal expression formula four function geometrical given curve given plane given point harmonic conjugate hodograph imaginary intersection length line oa linear lines of curvature locus multiplication normal notation osculating circle osculating plane P₁ perpendicular positive quinary quotient radius reciprocal rectangular represented right line right quaternions right quotient roots scalar second order sides sphere spherical sub-articles supposed surface symbol tangent plane tensor theorem tion transformation twisted cubic variable vector versor VIII whence whereof write

Información bibliográfica

Título	Elements of Quaternions
Autor	Sir William Rowan Hamilton
Editor	William Edwin Hamilton
Editor	Longmans, Green, & Company, 1866
Procedencia del original	Universidad de Harvard
Digitalizado	24 Sep. 2008
Largo	762 páginas

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