Elements of Quaternions

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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 Spp 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

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