Elements of Quaternions

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

and tho laws of the centre of gravity of areas and of living force
18
tion of the deviation from the given sphere of that other near point
33
so that this new surface is cut hy
53
On Anharmonic Coordinates in Space 6267
62
in which i and k are real and constant vectors in the directions
63
tegrability of the equation Yj is expressed by the very simple for
70
Geometrical connexions p 599 between these various results i
83
On Barycentres of Systems of Points and
85
If
89
a given curve in space may be represented rigorously by the vector
92
the last cited Section with the known Modular and Umbilicar Gene
94
On Differentials of Vectors 98102
98
in which a 3 y are any three vector constants represents a central
100
First Motive for naming the Quotient of
106
38
129
5
177
42
185
ptip for the propagation of a rectilinear vibration p 737 we
188
43
193
turfacc d is easily found p 738 to be represented by this other
200
Section of the second Chapter of the Third Book as III ii 6 and so
203
V + v 0 D
207
may be mentioned
227
scribed cone Qi or Qii may be represented p 655 by the very
246
48
269
whereof the conjugate obtained by changing 1 to + 1 in the last
271
16
275
19
281
23
287
313
313
II
321
31
329
62
335
are rigorously proportional to the numbers 1 and 3 tho three forces
354
and 2 i2mDaP+J3 D4
385
F may be called the Principal Function and V the Characteristic
411
311
413
mDa + DP 0 G4 iD0a + D0P 0 H4
419
11
433
300
439
303
445
805
453
306
459
ii
463
and the cone of normals to the last men
489
in which vtTr and w w the vector of an arbitrary point
585
comp the Note to p 600
600
dence o one of these can be at once translated into Mongcs equa
604
higher than the third oritur bat that of It requires the fourth order of differen
609
given point o have the directions p 712 of the three rectangular
610
is thus completely and generally determined without any such difficulty
621
ii
627
302
635
central quadric Spp fp I 636638
636
with some quaternion formulas
643
and y any constant values consistent with Ni the equation Ni
649
other known results are easy consequences of the present analysis
659
65
663
141
665
surface may be the tangents to the lines of curvature bisect the angles
668
comp pp 800 459 662 671 672 and conversely that when this last symbol
669
45
671
inverse function 0 + where e is any scalar and thus by chang
676
91
677
section of an arbitrary surface which touches one of the two lines
679
squares of the two last scmiaxes ore tho reofr may be written p 683
683
consistent with the equations of the surface of centres and its recipro
684
Salmon namely that the centres of curvature of a given quadric at
685
Umbilics of a central quadric 6536G3
686
surface and n Bi Rs the three corresponding points near to each other
690
thus the very simple form p 692
692
47
693
face p 694 s is also the centre of the sphere which osculate
694
the section of the surface made by the normal plane to the given
699
equation fp 2Stp const with fp Sppp is generally npe
705
and VQ is the rector y of a point c upon the central axis which does
709
whereof the system GO contains what may bo called tho Interme
715
so that t and r are unit tangents to tho lines of curvature it is easily
719
through or tends towards a common centre of force be cut perpendicu
726
whence pp 684 689
727
TdDa
729
only to principal terms pp 698 599 we have the expressions
733
ii
735
made respecting any smallness of excentricilies or inclinations p 786
736
connecting the two new rectors with each other they are con
738
19
741
comp the formula W3 in p xlvi by the symbolic and cubic equa
742
77
753
direction of the projection of the ray p on the tangent plane to
757

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