# Elements of Quaternions

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

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### Contents

 Su u o 25 25 m become thus 33 Art Page Art Page Art Page 1 2 1 2 3 37 45 and their focal conics the lines a a are asymptotes to the focal 53 surface d is easily found p 738 to be represented by this other 60 Deted by P geometrical constructions for this quantity P are assigned 86 with others easily deduced which may all be illustrated by the above 89 the last cited Section with the known Modular and Umbilicar Gene 94 130 95 265 96 40 99 102 104 134 120 60 121 7 125 8 128 21 142 47 157 48 168 136 171 are rigorously proportional to the numbers 1 and 3 the three forces 172 138 175 140 176 142 177 144 182 245 184 246 266 184 268 185 145 188 Pages 191 Vector of Curvature p 192 147 193 206 203 in which the near circle osculating at P is cut by the given normal 206 The Surface 212 To Ricosec 220 transformations for instance see p 675 it may be written thus 222 may be mentioned 227 154 244 and if the arc 8 of the curve be made the independent variable then 247 158 260 159 267 161 279 167 288 175 289 177 290 183 321 185 322 the other it is found convenient to introduce two auxiliary vectors 329 99 333 366 333 411 310 370 334 412 311 373 335 414 312 374 336 416 337 417 338 420 339 59 340 427 342 103 351 105 352 the four forces of the third group are proportional to 5 9 15 35 354 470 358 109 360 361 362 112 363 120 365 Empression p 600 379 128 384 131 385 this last being an expression for the velocity of rotation of the plane 387 the independent variable being the arc in T while it is arbitrary 389 B y being two vector constants and I a scalar constant 408 513 409 140 410 namely two genera 411 setted also by the rector equation p 611 involving apparently only 428 713 434 141 464 143 481 W 485 SOME ADDITIONAL APPLICATIONS OF QUATERNIONS WITH 495 On Normals and Tangent Planes to Surfaces 501510 501 99 502 16 506 149 507 On Osculating Planes and Absolute Normals 511 23 513 60 534 24 548 hence the curves of the first system n are Lines of Vibration of 557
 41 590 in which a y i pe are real and constant vectors but c is a variable sca 593 117 597 SP 599 this line A may be called the Rectifying Vector and if H denote 600 79 601 83 602 84 603 dence one of these can be at once translated into Monges equa 604 bis 42 42 617 Knoun right cone with rectifying line for its axis and with H for 618 it is ultimately equal p 595 to the quarter of the deviation 397 621 8 622 12 623 14 624 18 625 295 627 in fact it is cut 628 with a small circle osculating thereto example spherical conic con 629 On Osculating Circles and Spheres to Curves 630 tan C tan 631 gral Vd or by more purely geometrical considerations which 637 bis 56 57 58 59 60 61 640 const 642 bis 64 65 66 67 68 69 646 71 647 which show that the Locus of each of the two Auxiliary Points u 649 4 650 in which w is a variable vector represents p 684 the normal plane 652 31 653 32 654 599 660 33 662 36 663 in S1 may be thus decomposed into factors p 666 666 Function of the motion of the system each depending on the final 671 foregoing theory for the case of a Central Quadric and especially 674 ing e to p2 c new forms of the equation A6 of the wave 676 tirely arbitrary the values of r may be thus expressed p 681 681 squares of the two last semiaxes are the roots may be written p 683 683 Umbilics of a central quadric 686 37 688 surface and R R1 R2 the three corresponding points near to each other 690 the the very simple form p 692 692 11 693 points s and x in which the axis of the osculating circle to the curve 694 represents p 667 the Lines of Curvature upon an arbitrary surface 699 700 700 63 702 Equation fp 28p const with fp Spøp is generally p1 705 Arch with illustration by a diagram Fig 85 p 706 706 a1 az being the scalar semiaxes real or imaginary of the index curve 715 so that t and r are unit tangents to the lines of curvature it is easily 719 26 722 30 723 1 725 SA 729 87 732 280 733 283 735 made respecting any smallness of excentricities or inclinations p 736 736 connecting the two new vectors f with each other they are con 738 comp the formula W3 in p xlvi by the symbolic and cubic equa 742 33 744 41 746 44 747 171 748 325 749 80 753 87 754 90 755 94 756 direction of the projection of the ray p on the tangent plane to 757 96 758 projected on the normal v to the given surface the projection 761 255 Smith 1853 172 112 281 310 286 287 311 288 312 99 124 34 35 36 37 17 19