Elements of Quaternions

Longmans, Green, & Company, 1866 - Quaternions - 762 pages

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Y	xxxix

and the latter	xlv

sad if A be the radius of the hodograph then p 719	lii

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

SOME ADDITIONAL APPLICATIONS OF QUATERNIONS WITH	3

7 The equation Sv w p 0 or more simply	7

R�R�� r₁ r₂ Tv �� 8	8

then represents an ellipse p 651 whatever the species of the surface	9

19	507

On Osculating Planes and Absolute Normals	511

10	527

and if the arc s of the curve be made the independent variable then	533

22	547

r	553

projection of vector 7 of simple or first curvature on radius	560

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

somewhat more briefly and perhaps more clearly than in the Lectures	12

several less general but interesting results arise whereof many	14

from the common centre o of the two surfaces but v terminating	17

noted by P geometrical constructions for this quantity P are assigned	21

12341796	37

21	55

measure of the force	60

30	61

40	67

Geometrical connexions p 599 between these various results j k	83

all which	86

cited Fig	89

a given curve in space may be represented rigorously by the vector	92

36	95

164	143

in which s and u are two independent scalar variables whereof s	206

the times dt de of hodographically describing the small circular	226

may be mentioned	227

Pages	263

118	269

whereof the conjugate obtained by changing 1 to + 1 in the last	271

265	289

128	307

129	315

63	321

Art Page	345

131	347

while the separate intensities of the first forces in these three first	354

Pages	371

312	379

153	387

the independent variable being the are in T while it is arbitrary	389

292	391

see again 408 e there pass three lines of curvature comp p 677	411

one plane or in space for example in the motion of the centre of	418

342	427

346	435

of position of the point the time dt which corresponds to any vector	453

in which y is a vector function of p not generally linear and deduced	483

SOME CONCLUDING REMARKS	495

On Normals and Tangent Planes to Surfaces 501510	501

ing been those of the principal normal sections of a surface the present	504

130	569

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

Krown right cone with rectifying line for its axis and with H for	575

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

in which v TT and wws the vector of an arbitrary point	585

Pages	591

or p p+c�a Y or Vap+ pVyp+VpVλp�0	593

the section of the surface 1 made by the normal plane to the given	595

ferences are given to a very interesting Memoir by M de SaintVenant Sur	598

10	601

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

is thus completely and generally determined without any such difficulty	621

with a small circle osculating thereto example spherical conic con	629

it is ultimately equal p 595 to the quarter of the deviation 397	634

in which 7 dp is a tangent to a line of curvature while dv pdp	641

in fact it is cut	642

a a in d a Second Exponential Transformation is obtained with	649

in S₁ may be thus decomposed into factors p 666	666

comp pp 300 459 662 671 672 and conversely that when this last symbol	669

14	671

foregoing theory for the case of a Central Quadric and especially	674

inverse function p+e1 where e is any scalar and thus by chang	676

tirely arbitrary the values of r may be thus expressed p 681	681

Umbilics of a central quadric	686

136	689

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

being 1D�p its normal and tangential components are found	694

700	700

The vector of the centre of the quadric represented by the equation fp 28ep const with fp Sppp is generally x�ε	705

Arch with illustration by a diagram Fig 85 p 706	706

15	709

law of the Inverse Square 713717	713

so that	719

to become a tangent this Theorem of Hodographie Isochronism which	726

introducing the two new integrals p 729	729

884	735

made respecting any smallness of excentricities or inclinations p 736	736

connecting the two new vectors ƒ with each other they are con	738

comp the formula W3 in p xlvi by the symbolic and cubic equa	742

direction of the projection of the ray p on the tangent plane to	757

17	761

Smith 1853

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Elements of Quaternions
Sir William Rowan Hamilton
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Common terms and phrases

ABCD algebra angle anharmonic axis centre CHAP circle coefficients collinear comp Compare the Note complanar cone conic conjugate considered construction curvature curve cyclic deduced denote derived differential diplanar direction ellipsoid equal equation expression formula four fourth proportional function geometrical given plane given points harmonic conjugate imaginary interpreted intersection length line oa linear locus multiplication negative notation osculating osculating circle osculating plane P₁ parallelogram perpendicular positive quadratic quadrilateral quinary radius reciprocal relative direction represented right line right quaternions right quotient right versors roots rotation round scalar SECTION sides sphere spherical spherical angle sub-articles supposed surface symbol tangent tensor ternion theorem tion triangle ABC twisted cubic variable vector VIII whence whereof write

Bibliographic information

Title	Elements of Quaternions
Author	Sir William Rowan Hamilton
Editor	William Edwin Hamilton
Publisher	Longmans, Green, & Company, 1866
Original from	the University of California
Digitized	25 Sep 2007
Length	762 pages

Export Citation	BiBTeX EndNote RefMan

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