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S ! lo
ELEMENTS
OF
QUATERNIONS.
BY THE LATE
SIR WILLIAM ROWAN HAMILTON, LL. D., M. R. I. A.,
D. C. L. CANTAB. ;
FKLr>OW Olf THE AMERICAN SOCIETr OF ARTS AND SCIKNCKS;
OF THE SOCIETY OF AHTS FOR SCOTLAND ; OF THE ROYAL ASTRONOMrCAL SOCIKTT OF LONDON; AND OF THB
ROYAL NORTHKRN SOCtKTY OF ANTIQUAKIES AT COPKNHAGEM :
CORRESPONDING MICMBER OF THE INSTITUTK OF FRANCE ;
HONOHART OR CORRESPONDING MEMBER OF THE IMPERIAL OR ROYAL ACADEMIES OF ST. PETERSBDRGH,
BERLIN, AND TURIN ; OF THE ROYAL SOCIETIES OF EDINBURGH AND DUBLIN;
OFTHK NATIONAL ACADEMY OF THE UNITED STATES;
OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY ; THE NEW YORK HISTORICAL SOCIETY ;
THE SOCIETY OF NATURAL SCIENCES AT LAUSANNE ; THE PHILOSOPHICAL SOCIETY OF VENICB ; .
AND OF OTHER SCIENTIFIC SOCIETIES IN BRITISH AND FORKIGN COUNTRIES ;
ANDREWS' PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF DUBLIN;
AND ROYAL ASTRONOMER OF IRELAND.
EDITED BY HIS SON,
WILLIAM EDWIN HAMILTON, A.B.T.C.D., C.E.
LONDON :
LONGMANS, GREEN, & CO,
1866.
ASTRONOMY UBRARY
DUBLIN:
^rlntetJ at tl)t ©ntijersitp ^regg,
BY M. H. GILL.
ASTRONOMY
UBRARV
^0 THE
EIGHT HONOEABLE WILLIAM EAEL OF EOSSE,
CHANCELLOR OF THE UNIVERSITY OF DUBLIN,
^\im Mximz
IS, BY PERMISSION, DEDICATED,
BY
THE EDITOR.
m^772ao
In my late father's Will no instructions were left as
to the publication of his Writings, nor specially as to
that of the " Elements of Quaternions," which, but
for his late fatal illness, would have been before now,
in all their completeness, in the hands of the Public.
My brother, the Rev. A. H. Hamilton, who was
named Executor, being too much engaged in his cle-
rical duties to undertake the publication, deputed this
task to me.
It was then for me to consider how I could best
fulfil my triple duty in this matter — First, and chiefly,
to the dead ; secondly, to the present public ; and,
thirdly, to succeeding generations. I came to the con-
clusion that my duty was to publish the work as I found
it, adding merely proof sheets, partially corrected by
my late father and from which I removed a few typo-
graphical errors, and editing only in the literal sense
of giving forth.
Shortly before my father's death, I had several con-
versations with him on the subject of the " Elements."
In these he spoke of anticipated applications of Qua-
ternions to Electricity, and to all questions in which
the idea of Polarity is involved — applications which
he never in his own lifetime expected to be able fully
to develope, bows to be reserved for the hands of
another Ulysses. He also discussed a good deal the
nature of his own forthcoming Preface ; and I may
intimate, that after dealing with its more important
topics, he intended to advert to the great labour which
( vi )
the writing of the " Elements" had cost him — labour
both mental and mechanical; as, besides a mass of
subsidiary and unprinted calculations, he wrote out
all the manuscript, and corrected the proof sheets,
without assistance.
And here I must gratefully acknowledge the ge-
nerous act of the Board of Trinity College, Dublin, in
relieving us of the remaining pecuniary liability, and
thus incurring the main expense, of the publication of
this volume. The announcement of their intention to
do so, gratifying as it was, surprised me the less, when
I remembered that they had, after the publication of
my father's former book, " Lectures on Quaternions,"
defrayed its entire cost ; an extension of their liberality
beyond what was recorded by him at the end of his
Preface to the " Lectures," which doubtless he would
have acknowledged, had he lived to complete the Pre-
face of the " Elements."
He intended also, I know, to express his sense of
the care bestowed upon the typographical correctness
of this volume by Mr. M. H. Gill of the University
Press, and upon the delineation of the figures by the
Engraver, Mr. Oldham.
I annex the commencement of a Preface, left in ma-
nuscript by my father, and which he might possibly
have modified or rewritten. Believing that I have
thus best fulfilled my part as trustee of the unpub-
lished " Elements," I now place them in the hands of
the scientific public.
William Edwin Hamilton.
January \st^ 1866.
PREFACE.*
[1.] The volume now submitted to the public is founded on
the same principles as the " LECTURES, "^^^ which were pub-
lished on the same subject about ten years ago : but the plan
adopted is entirely new, and the present work can in no sense
be considered as a second edition of that former one. The
Table of Contents^ by collecting into one view the headings of
the various Chapters and Sections, may suffice to give, to
readers already acquainted with the subject, a notion of the
course pursued : but it seems proper to offer here a few intro-
ductory remarks, especially as regards the method of expo-
sition, which it has been thought convenient on this occasion
to adopt.
[2.] The present treatise is divided into Three Books, each
designed to develope one guiding conception or^view, and to
illustrate it by a sufficient but not excessive number of exam-
ples or applications. The First Book relates to the Concep-
tion of a Vector^ considered as a directed right line^ in space of
three dimensions. The Second Book introduces a First Con-
ception of a Quaternion^ considered as the Quotient of two such
Vectors. And the Third Book treats of Products and Powers
of Vectors^ regarded as constituting a Second Principal Form
of the Conception of Quaternions in Geometry.
* This fragment, by the Author, was found in one of his manuscript books
by the Editor.
. TABLE OF CONTENTS.
BOOK I.
Pages.
ON VECTORS, CONSIDERED WITHOUT REFERENCE TO
ANGLES, OR TO ROTATIONS, . • . . 1-102
CHAPTER* I.
FUNDAMENTAL PBINCIPLES EESPECTING TECTOES, . 1-11
SECTiONf 1. — On the Conception of a Yector ; and on Equa-
lity of Vectors, . 1-3
Section 2. — On Differences and Sums of Yectors, taken two
by two, 3-5
Section 3. — On Sums of Three or more Yectors, .... 5-7
Section 4. — On Coefficients of Yectors, 8-11
This short First Chapter should be read with care by a beginner ;
any misconception of the meaning of the word "Vector" being fatal
to progress in the Quaternions. The Chapter contains explana-
tions also of the connected, but not all equally important, words
or phrases, " revector," " pro vector," " transvector," "actual and
null vectors," "opposite and successive vectors," " origin and term of
a vector," " equal and unequal vectors," "addition and subtraction
of vectors," "multiples and fractions of vectors," &c. ; with the nota-
tion B - A, for the Vector (or directed right line) ab : and a deduction
of the result, essential but not peculiarX to quaternions, that (what
is here called) the vector-sum^ of two co-initial sides of a parallelo-
gram, is the intermediate and co-initial diagonal. The term " Scalar"
is also introduced, in connexion with coefficients of vectors.
* This Chapter may be referred to, as I. i. ; the next as I. ii. ; the first Chap-
ter of the Second Book, as II. i. ; and similarly for the rest.
t This Section may be referred to, as I. i. 1 ; the next, as I. i. 2 ; the sixth
Section of the second Chapter of the Third Book, as III. ii. 6 ; and so on.
X Compare the second Note to page 203.
b
11 CONTENTS.
Pages,
CHAPTER II.
APPLICATIONS TO POINTS AND LINES IN A GIVEN PLANE, 11-49
Sectfon 1. — On Linear Equations connecting two Co-initial
Vectors, 11-12
Section 2. — On Linear Equations between three Co-initial
Vectors, 12-20
After reading these two first Sections of the second Chapter, and
perhaps the three first Articles (31-33, pages 20-23) of the following
Section, a student to whom the subject is new may find it convenient
to pass at once, in his first perusal, to the third Chapter of the present
Book; and to read only the two first Articles (62, 63, pages 49-51)
of the first Section of that Chapter, respecting Vectors in Space, before
proceeding to the Second Book (pages 103, &c.), which treats of Qua-
ternions as Quotients of Vectors.
Section 3. — On Plane Geometrical Nets, ...*.. 20-24
Section 4. — On Anharmonic Co-ordinates and Equations
of Points and Lines in one Plane, 24-32
Section 5. — On Plane Geometrical !N'ets, resumed, . . . 32-35
Section 6. — On Anharmonic Equations and Vector Ex-
pressions, for Curves in a given Plane, 35-49
Among other results of this Chapter, a theorem is given in page 43,
which seems to offer a new geometrical generation of (plane or spheri-
cal) curves of the third order. The anharmonic co-ordinates and equa-
tions employed, for the plane and for space, were suggested to the
writer by some of his own vector forms ; but their geometrical inter-
pretations are assigned. The geometrical nets were first discussed by
Professor Mobius, in his Barycentric Calculus (Note B), but they are
treated in the present work by an entirely new analysis : and, at least
for space, their theory has been thereby much extended in the Chapter
to which we next proceed.
CHAPTER III.
APPLICATIONS OF VECTOKS TO SPACE, . . . 49-102
Section 1. — On Linear Equations between Vectors not Com-
planar, 49-56
It has already been recommended to the student to read the first
two Articles of this Section, even in his first perusal of the Volume ;
and then to pass to the Second Book.
Section 2 — On Quinary Symbols for Points and Planes in
Space, 57-62
CONTENTS. iii
Pages.
Section 3, — On Anharmonic Co-ordinates in Space, . . 62-67
Section- 4. — On Greometrical ]S"ets in Space, 67-85
Section 5. — On Earycentres of Systems of Points ; and on
Simple and Complex Means of Vectors, 85-89
Section 6. — On Anharmonic Equations, and Yector Ex-
pressions, of Surfaces and Curves in Space, .... 90-97
Section 7. — On Differentials of Yectors, 98-102
An application oi finite differences^ to a question connected with ha-
ry centres, occurs in p. 87. The anharmonic generation of a ruled hy-
perboloid (or paraboloid) is employed to illustrate anharmonic equa-
tions ; and (among other examples) certain cones, of the second and third
orders, have their vector equations assigned. In the last Section, a defi-
nition of differentials (of vectors and scalars) is proposed, which is
afterwards extended to differentials of quaternions, and which is in-
dependent of developments and of infinitesimals, but involves the
conception of limits. Vectors of Velocity and Acceleration are men-
tioned ; and a hint of Hodographs is given.
BOOK II.
ON QUATERNIONS, CONSIDERED AS QUOTIENTS OF
VECTORS, AND AS INVOLVING ANGULAR RELA-
TIONS, • 103-300
CHAPTER I,
fundamental peinciples respecting quotients op vectors, 103-239
Very little, if any, of this Chapter II. i., should be omitted, even
in a first perusal ; since it contains the most essential conceptions
and notations of the Calculus of Quaternions, at least so far as quo-
tients of vectors are concerned, with numerous geometrical illustra-
tions. Still there are a few investigations respecting circumscribed
cones, imaginary intersections, and ellipsoids, in the thirteenth Sec-
tion, which a student may pass over, and which will be indicated in
the proper place in this Table.
Section 1 Introductory Remarks ; First Principles
adopted from Algebra, 103-106
Section 2. — First Motive for naming the Quotient of two
Vectors a Quaternion, 106-110
Sections. — Additional Illustrations, .110-112
It is shown, by consideration of an angle on a desk, or inclined
plane, that the complex relation of one vector to another, in length and
IV CONTENTS.
Pages,
in direction, involves generally a system oifour nvmerical elements.
Many other motives, leading to the adoption of the name, " Quater-
nion," for the suhject of the present Calculus, from its fundamental
connexion with the number " Four," are found to present themselves
in the course of the work.
Section 4 On Equality of Quaternions ; and on the Plane
of a Quaternion, 112-117
Section 5. — On the Axis and Angle of a Quaternion j and
on the Index of a Eight Quotient, or Quaternion, . . 117-120
Section 6. — On the Reciprocal, Conjugate, Opposite, and
iN'orm of a Quaternion; and on Null Quaternions, , . 120-129
Section 7. — On Radial Quotients ; and on the Square of a
Quaternion, 129-133
Section 8. — On the Yersor of a Quaternion, or of a Vec-
tor ; and on some General Formulae of Transformation, 133-142
In the five foregoing Sections it is shown, among other things,
that the plane of a quaternion is generally an essential element of its
constitution, so that diplanar quaternions are unequal; but that the
tquare of every right radial (or right versor) is equal to negative unity^
whatever its plane may be. The Symbol V — 1 admits then of a real in-
terpretation, in this as in several other systems ; but when thus treated
as real, it is in the present Calculus too vague to be useful : on which
account it is found convenient to retain the old signification of that
symbol, as denoting the (uninterpreted) Imaginary of Algebra, or
what may here be called the scalar imaginary, in investigations re-
specting non-real intersections, or non-real contacts, in geometry.
Section 9. — On Yector-Arcs, and Vector- Angles, consi-
dered as Representatives of Versors of Quaternions ;
and on the Multiplication and Division of any one such
Versor by another, 142-157
This Section is important, on account of its constructions of mul-
tiplication and division ; which show that the product of two diplanar
versors, and therefore of two such quaternions, is not independent of
the order of the factors.
Section 10. — On a System of Three Right Versors, in
Three Rectangular Planes ; and on the Laws of the
Symbols, ijl, 157-162
The student ought to make himself /awjt7/«r with these laws,
which are all included in the Fundamental Formula,
CONTENTS. V
In fact, a Quaternion may be symbolically defined to be a Quadrino-
mial Expression of the form,
q = w-\-ix+jy + kZj (B)
in which w, x, y, z are four scalars, or ordinary algebraic quantities,
while i,j, k are three new symbols, obeying the laws contained in the
formula (A), and therefore not subject to all the usual rules of alge-
bra : since we have, for instance,
ij= + k, but ji=^-k; and i'^pk^ =^- ^jk)-i.
Section 1 1 . — On the Tensor of a Vector, or of a Quater-
nion ; and on the Product or Quotient of any two Qua-
ternions, 162-174
Section 12 On the Sum or Difference of any two Qua-
ternions ; and on the Scalar (or Scalar Part) of a Qua-
ternion, 175-190
Section 13. — On the Right Part (or Yector Part) of a
Quaternion ; and on the Distrihutive Property of the
Multiplication of Quaternions, 190-238
Section 14. — On the Reduction of the General Quaternion
to a Standard Quadrinomial Porm ; with a Pirst Proof
of the Associative Principle of Multiplication of Qua-
ternions, . . . 233-239
Articles 213-220 (with their sub-articles), in pp. 214-233, maybe
omitted at first reading.
CHAPTER II.
ON COMPLANAE QITATEENIONS, OE QUOTIENTS OF VECTOES IN
ONE PLANE ; AND ON POWEES, EOOTS, AND LOGAEITHMS OF
QUATEENIONS, 240-285
The first six Sections of this Chapter (II. ii.) may be passed over
in a first perusal.
Section 1. — On Complanar Proportion of Vectors ; Fourth
Proportional to Three, Third Proportional to Two,
Mean Proportional, Square Root ; General Reduction
of a Quaternion in a given Plane, to a Standard Bino-
mial Porm, 240-246
Section 2. — On Continued Proportion of Four or more Vec-
tors ; whole Powers and Roots of Quaternions ; and
Roots of Unity, 246-251
vi CONTENTS.
Pages.
Section 3. —On the Amplitudes of Quaternions in a given
Plane; and on Trigonometrical Expressions for such
Quaternions, and for their Powers, 251-257
Section 4. — On the Ponential and Logarithm of a Quater-
nion ; and on Powers of Quaternions, with Quaternions
for their Exponents, 257-264
Section 5. — On Finite (or Polynomial) Equations of Alge-
braic Form, involving Complanar Quaternions ; and on
the Existence of n Eeal Quaternion Boots, of any such
Equation of the n'^ Degree, 265-275
Section 6. — On the n^ - n Imaginary (or Symbolical)
Roots of a Quaternion Equation of the n*'' Degree, with
Coefficients of the kind considered in the foregoing
Section, 275-279
Section 7. — On the Reciprocal of a Vector, and on Har-
monic Means of Vectors ; with Remarks on the Anhar-
monic Quaternion of a Group of Four Points, and on
Conditions of Concircularity, 279-285
In this last Section (II. ii. 7) the short first Article 258, and the
following Art. 259, as far as the formula VIII. in p. 280, should be
read, as a preparation for the Third Book, to which the Student may
next proceed.
CHAPTER III.
ON DIPLA.NAR QUATERNIONS, OR QUOTIENTS OF VECTORS IN
^^ SPACE : AND ESPECIALLY ON THE ASSOCIATIVE PRINCIPLE
OF MULTIPLICATION OF SUCH QUATERNIONS, 286-300
This Chapter may be omitted, in a first perusal.
Section 1. — On some Enunciations of the Associative Pro-
perty, or Principle, of Multiplication of Diplanar Qua-
ternions, 286-293
Section 2. — On some Geometrical Proofs of the Associative
Property of Multiplication of Quaternions, which are
independent of the Distributive Principle, .... 293-297
Section 3. — On some Additional Formulae, .... 297-300
CONTENTS. vii
BOOK III.
Pages.
ON QUATERNIONS, CONSIDERED AS PRODUCTS OR
POWERS OF VECTORS; AND ON SOME APPLICA-
TIONS OF QUATERNIONS, 301 to the end.
CHAPTER I.
ON THE INTEEPEETATION OF A PRODUCT OF VECTORS, OR POWER
OF A VECTOR, AS A QUATERNION, . . . 301-390
The first six Sections of this Chapter ought to be read, even in a
first perusal of the -work.
Section 1 . — On a First Method of Interpreting a Product
of Two Vectors as a Quaternion, 301-303
Section 2. — On some Consequences of the foregoing Inter-
pretation, 303-308
This^r*^ interpretation treats th.e product 13. a, as equal to the
quotient /3 : a-i ; where a"i (or Ra) is the previously defined Eeeiprocal
(II, ii. 7) of the vector a, namely a second vector.^ -which has an in-
verse length, and an opposite direction. Multiplication of Vectors is
thus proved to be (like that of Quaternions) a Bistributive, but not
generally a Commutative Operation. The Square of a Vector is shown
to be always a Negative Scalar, namely the negative of the square of
the tensor of that vector, or of the number which expresses its length ;
and some geometrical applications of this fertile principle, to spheres,
&c., are given. The Index of the JRight Fart of a Product of Two Co-
initial Vectors, OA, ob, is proved to be a right line, perpendicular to
the Flane of the Triangle oab, and representing by its length the
Double Area of that triangle ; while the Eolation round this Index,
from the Multiplier to the Multiplicand, is positive. This right part,
or vector part, Va/3, of the product vanisJies, when the factors are
parallel (to one common line) ; and the scalar part, Sa/3, when they
are rectangular.
Section 3. — On a Second Method of arriving at the same
Interpretation, of a Binary Product of Vectors, . . . 308-310
Section 4. — On the Symbolical Identification of a Eight
Quaternion with its own Index : and on the Construc-
tion of a Product of Two Rectangular Lines, by a Third
Line, rectangular to both, 310-313
Section 5. — On some Simplifications of N'otation, or of
Expression, resulting from this Identification ; and on
the Conception of an Unit-Line as a Right Versor, . 313-316
vni CONTENTS.
Pages.
In this second interpretation^ which is found to agree in all its re-
sults with the first, but is better adapted to an extension of the theory,
as in the following Sections, to ternary products of vectors, a product
of two vectors is treated as the product of the two right quaternions, of
which those vectors are the indices (II. i. 5). It is shown that, on
the same plan, the Sum of a Scalar and a Vector is a Quaternion.
SECTioif 6. — On the Interpretation of a Product of Three
or more Vectors as a Quaternion, 316-330
This interpretation is effected by the substitution, as in recent
Sections, of Eight Quaternions for Vectors, without change oiorder of
the factors. Multiplication of Vectors, like that of Quaternions, is
thus proved to be an Associative Operation. A vector, generally, is
reduced to the Standard Trinomial Form,
p = ix-Vjy-\-Jcz; (C)
in which i,j, h are the peculiar symbols already considered (II. i.
10), but are regarded now as denoting Three Rectangular Vector- Units,
while the three scalars x, y, z are simply rectangular co-ordinates ; from
the known theory of which last, illustrations of results are derived.
The Scalar of the Product of Three coinitial Vectors, oa, ob, oc, is found
to represent, with a sign depending on the direction of a rotation, the
Volume of the Parallelepiped under those three lines ; so that it va-
nishes when they are complanar. Constructions are given also for ^ro-
ducts of successive sides of triangles, and other closed polygons, inscribed
in circles, or in spheres ; for example, a characteristic property of the
circle is contained in the theorem, that the product of the four suc-
cessive sides of an inscribed quadrilateral is a scalar : and an equally
characteristic (but less obvious) property of the sphere is included in
this other theorem, that the product of the ^t?^ successive sides of an
inscribed gauche pentagon is equal to a tangential vector, drawn from
the point at which the pentagon begins (or ends). Some general For-
mula of Transformation of Vector Expressions are given, with which
a student ought to render himself very familiar, as they are of con-
tinual occurrence in the practice of this Calculus ; especially the four
formulae (pp. 316, 317) :
V.yV/3a=aS/3y-)3Sya; (D)
Vy/3a = aS|3y-/3S7a + ySa/3; (E)
pSajSy = aS/3yp + /3Syap + ySa^Sp ; (F)
|0Sa/3y = VjSySap + VyaS^p + Va/3Syp ; (G)
in which a, (3, y, p are any four vectors, while S and V are signs of
the operations of taking separately the scalar and vector parts of a qua-
ternion. On the whole, this Section (III. i. 6) must be considered
to be (as regards the present exposition) an important one ; and if
it have been read with care, after a perusal of the portions previously
indicated, no difficulty will be experienced in passing to any subse-
quent applications of Quaternions, in the present or any other work.
CONTENTS. ix
Pages.
Section 7. — On the Fourth Proportional to Three Diplanar
Vectors, 331-349
Section 8.. — On an Equivalent Interpretation of the Fourth
Proportional to Three Diplanar Vectors, deduced from
the Principles of the Second Book, 349-361
Section 9. — On a Third Method of interpreting a Product
or Function of Vectors as a Quaternion; and on the
Consistency of the Eesults of the Interpretation so ob-
tained, with those which have been deduced from the
two preceding Methods of the present Book, . . .361-364
These three Sections may be passed over, in a first reading. They
contain, however, theorems respecting composition of successive rota-
tions (pp. 334, 335, see also p. 340); expressions for the sem^are« of a
spherical polygon, or for half the opening of an arbitrary pyramid^ as
the angle of a quaternion product, with an extension, by limits, to the
semiarea of a spherical figure bounded by a closed curve, or to half the
opening of an arbitrary cone (pp. 340, 341) ; a construction (pp. 358-
360), for a series of spherical parallelograms, so called from a partial
analogy to parallelograms in o. plane ; a theorem (p. 361), connecting
a certain system of such (spherical) parallelograms with ih^foci of a
spherical conic, inscribed in a certain quadrilateral ; and the concep-
tion (pp. 353, 361) of a Fourth Unit in Space (?^, or + I), which is of
a scalar rather than a vector character, as admitting merely of change
of sign, through reversal of an order of rotation, although it presents
itself in this theory as the Fourth Troportional {if'^h;) to Three Beet-
angular Vector Units.
Section 10. — On the Interpretation of a Power of a Vector
as a Quaternion, 364-384
It may be well to read this Section (III. i. 10), especially for
the Exjjonential Connexions which it establishes, between Quaternions
and Splierical Trigonometry, or rather Folygonometry, by a species of
extetision of Moivr^s theorem, from the plane to space, or to the spliere.
For example, there is given (in p. 381) an equation of six terms^
which holds good for every spherical j^entagon, and is deduced in this
way from an exfetided exponential formula. The calculations in the
sub-articles to Art. 312 (pp. 375-379) may however be passed over;
and perhaps Art. 315, with its sub-articles (pp. 383, 384). But Art
314, and its sub-articles, pp. 381-383, should be read, on account of
the exponential forms which they contain, of equations of the circle,
ellipse, logarithmic spirals (circular and elliptic), h^liz, a.nd screw sur-
face.
Section 11 — On Powers and Logarithms of Diplanar Qua-
ternions ; with some Additional Formulae, .... 384-390
X CONTENTS.
It may suffice to read Art. 316, and its first eleven sub-articles,
pp. 384—386. In this Section, tlie adopted Logarithm, \q, of a Qua-
ternion q, is the simplest root, q\ of the transcendental equation,
and its expression is found to be,
l^ = lT^ + Z?.UVj, (H)
in which T and U are the signs of tensor and versor, while Z. $■ is the
angle of q, supposed usually to be between and tt. Such logarithms
are found to be often useful in this Calculus, although they do not gene-
rally possess the elementary property, that the sum of the logarithms
of two quaternions is equal to the logarithm of their ^ro^wc^ ; this ap-
parent paradox, or at least deviation from ordinary algebraic rules,
arising necessarily from the corresponding property of quaternion
multiplication, which has been already seen to be not generally a com-
mutative operation {q'q" not = q'q\ unless (f and j" be complanar^.
And here, perhaps, a student might consider his first perusal of this
work as closed.*
Pages.
CHAPTER II.
ON DIFFERENTIALS AND DEVELOPMENTS OF FUNCTIONS OF QUA-
TERNIONS ; AND ON SOME APPLICATIONS OF QUATERNIONS
TO GEOMETRICAL AND PHYSICAL QUESTIONS, 391-495
It has been already said, that this Chapter may be omitted in a
first perusal of the work.
Section 1. — On the Definition of Simultaneous Differen-
tials, 391-393
* If he should choose to proceed to the Differential Calculus of Quaternions in
the next Chapter (III. ii.), and to the Geometrical and other Applications in the
third Chapter (III. iii.) of the present Book, it might be useful to read at this
stage the last Section (I. iii. 7) of the First Book, which treats of Differentials of
Vectors (pp. 98-102); and perhaps the omitted parts of the Section II. i. 13,
namely Articles 213-220, with their subarticles (pp. 214-233), which relate,
among other things, to a Oonstruction of the Ellipsoid, suggested by the present
Calculus. But the writer will now abstain from making any further suggestions
of this kind, after having indicated as above what appeared to him a minimum
course of study, amounting to rather less than 200 pages (or parts of pages)
of this Volume, which will be recapitulated for the convenience of the student
at the end of the present Table.
CONTENTS. XI
Pages.
Section 2. — Elementary Illustrations of the Definition,
from Algebra and Geometry, 394-398
In the view here adopted (comp. I. iii. 7), differentials are not ne-
cessarily, nor even generally, small. But it is shown at a later stage
(Art. 401, pp. 626-630), that the principles of this Calculus a^/ot^ us,
whenever any advantage may be thereby gained, to treat differentials
as infinitesimals ; and so to abridge calculation, at least in many ap-
plications.
Section 3 — On some general Consequences of the Defini-
tion, 398-409
Partial differentials and derivatives are introduced ; and differen-
tials of functions of functions.
Section 4 — Examples of Quaternion Differentiation, . . 409-419
One of the most important rules is, to differentiate the /ac^or* of a
c^dXemion. product, in situ ; thus (by p. 405),
6..qq' = diq.q'-VqAq'. (I)
The formula (p. 399), d. ^-» = - q-^^q.q-\ (J)
for the differential of the reciprocal of a quaternion (or vector), is also
very often useful ; and so are the equations (p. 413),
dT^ d^ dU^ d^
Tq q Vq q
and (p. 411), ^ • "' = Y "'^^^^ ' ^^)
g being any quaternion, and a any constant vector-unit, while tisa
variable scalar. It is important to remember (comp. III. i. 11), that
we have not in quaternions the usual equation,
Q
unless q and d^ be complanar ; and therefore that we have not generally,
dlp = ^,
P
if p be a variable vector ; although we have, in this Calculus, the
scarcely less simple equation, which is useful in questions respecting
orbital motion,
dlP-=^, (M)
a p
if a be any constant vector, and if the plane of a and p be given (or
constant).
Section 5. — On Successive Differentials and Developments,
of Functions of Quaternions, 420-435
xii CONTENTS.
Pages.
In this Section principles are established (pp. 423-426), respect-
ing qnatermon functions which vanish together ; and a form of deve-
lopment (pp. 427, 428) is assigned, analogous* to Taylor's Seriesy
and like it capable of being concisely expressed by the symbolical
equation^ 1 + A = £<i (p. 432). As an example of partial and succes-
Bive differentiation, the expression (pp. 432, 433),
p = r¥j^kj-^k~\
■which may represent any vector ^ is operated on ; and an application
is made, by means of definite integration (pp. 434, 435), to deduce the
known area and volume of a sphere, or of portions thereof ; together
with the theorem, that the vector sum of the directed elements of a
spheric segment is zero : each element of surface being represented by an
inward normal, proportional to the elementary area, and correspond-
ing in hydrostatics to \he pressure of a fluid on that element.
Section 6. — On the Differentiation of ImpKcit runctions
of Quaternions ; and on the General Inversion of a Li-
near Function, of a Yector or a Quaternion : with
some connected Investigations, . . . ' 435-495
In this Section it is shown, among other things, that a Linear
and Vector Symbol, 0, of Operation on a Vector, p, satisfies (p. 443) a
Symbolic and Cubic Equation, of the form,
= w - m> + m"(p^ - ^3 ; (N)
whence m(}>~^ — m'— m"<p -\- <p^=->p, (N')
= anotJier symbol of linear operation, which it is shown how to de-
duce otherwise from 0, as well as the three scalar constants, m, m, m'.
The connected algebraical cubic (pp. 460, 461),
Jlf = w + m'c + m"c2 + c3 = 0, (0)
is found to have important applications ; and it is provedf (pp. 460,
462) that if SX^p = Sp^X, independently of X and p, in which case
the function is said to be self-conjugate, then this last cubic has three
real roots, ci, cz, cz ; while, in the same case, the vector equation,
\p^p = 0, (P)
is satisfied by a system of Three Heal and Rectangular Directions :
namely (compare pp. 468, 469, and the Section III. iii. 7), those of
the axes of a (biconcyclic) system of surfaces of the second order, re-
presented by the scalar equation,
* At a later stage (Art. 375, pp. 509, 510), a neiv Enunciation of Taylor's
Tlieorem is given, with a new proof , but stiU in a form adapted to quaternions.
t A simplified proof, of some of the chief results for this important case of
self-conjugation, is given at a later stage, in the few first subarticles to Art. 415
(pp. 698, 699).
CONTENTS. Xlll
Pages.
Sp(f>p = <7p2 -f C", in which C and C are constants. (Q,)
Cases are discussed; and general forms {coX^Qdi cyclic, rectangular,
focal, bifocal, &c., from their chief geometrical uses) are assigned,
for the vector and scalar functions ^p and Sp^/o : one useful pair of
such (cyclic) forms being, with real and constant values of ^, X, j«,
(l>p=ffp + YXpfi, Bp^p=ffp'^ + S\pnp. (R)
And finally it is shown (pp. 491, 492) that if fg be a linear and qua-
ternion function of a quaternion, q, then the Symbol of Operation, f
satisfies a certain Symbolic and Biquadratic Equation, analogous to the
cubic equation in ^, and capable of similar applications.
CHAPTER III.
ON SOME ADDITIONAL APPLICATIONS OF QUATERNIONS, WITH
SOME CONCLUDING REMARKS, . . 495 to the end.
This Chapter, like the one preceding it, may be omitted in a first
perusal of the Volume, as has indeed been already remarked.
Section 1. — Remarks Introductory to this Concluding
Chapter, 495-496
Section 2 On Tangents and Kormal Planes to Curves in
Space, 496-501
Section 3. — On J^ormals and Tangent Planes to Surfaces, 501-510
Section 4. — On Osculating Planes, and Absolute ]N"ornials,
to Curves of Double Curvature, ........ 511-515
Section 5. — On Geodetic Lines, and Families of Surfaces, 515-531
In these Sections, dp usually denotes a tangent to a curve, and v
a normal to a surface. Some of the theorems or constructions may
perhaps be new ; for instance, those connected with the cone of paral-
lels (pp. 498, 513, &c.) to the tangents to a curve of double curvature ;
and possibly the theorem (p. 525), respecting reciprocal curves in
space : at least, the deductions here given of these results may serve
as exemplifications of the Calculus employed. In treating of Families
of Surfaces by quaternions, a sort of analogue (pp. 629, 530) to the for-
mation and integration of Partial Differential Equations presents
itself; as indeed it had done, on a similar occasion, in the Lectures
(p. 674).
Section 6. — On Osculating Circles and Spheres, to Curves
in Space; with some connected Constructions, . . . 531-630
The analysis, however condensed, of this long Section (III. iii. 6),
cannot conveniently be performed otherwise than under the heads of
the respective Articles (389-401) which compose it: each Article
XIV CONTENTS.
Pages,
being followed by several subarticles, which form with it a sort of
Series*
Article 389. — Osculating Circle defined, as the limit of a circle,
which touches a given curve (plane or of double curvature) at a given
point p, and cuts the curve at a near point q (see Fig. 77, p. 511).
Deduction and interpretation of general expressions for the vector k
of the centre k of the circle so defined. The reciprocal of the radius
KP being called the vector of curvature, we have generally,
Vector of Curvature = (p - k)-i = -=~ = — Y ~ = &c. ; (S)
•^ vr- y rp^i^ dp dp
and if the arc (s) of the curve be made the independent variable, then
d2p
Vector of Curvature = p" = Ds^p = ~j. (S')
Examples : curvatures of helix, ellipse, hyperbola, logarithmic spiral ;
locus of centres of curvature of helix, plane e volute of plane ellipse, 531-535
A.RTICLE 390 — Abridged general calculations; return from (S')
to (S), 535, 536
Article 391 Centre determined by three scalar equations ;
Folar Axis, Polar Developable, 537
Article 392. — Vector Equation of o^cvloXm^ civc\e, 538,539
Article 393. — Intersection (or intersections) of a circle with a
plane curve to which it osculates ; example, hyperbola, 539-541
Article 394. — Intersection (or intersections) of a spherical curve
with a small circle osculating thereto ; example, spherical conic ; con-
structions for the spherical centre (or pole) of the circle osculating to
such a curve, and for the point of m^ersec^ww above mentioned, . . 541-549
Article 395. — Osculating Sphere, to a curve of double curvature,
defined as the limit of a sphere, which contains the osculating circle to
the curve at a given point p, and cuts the same curve at a near point
Q (comp. Art. 389). The centre s, of the sphere so found, is (as usual)
the point in which th.Q polar axis (Art. 391) touches the cusp-edge of
tlie polar developable. Other general construction for the same centre
(p. 551, comp. p. 573). General expressions for the vector, a = os,
and for the radius, R = Wp', -K'' is the spherical curvature (comp. Art.
897). Condition of Sphericity {8=1), and Coefficient of Non- sphericity
(^S — 1), for a curve in space. When this last coefficient is positive
(as it is for the helix), the curve lies outside the sphere, at least in the
neighbourhood of the point of osculation, 549-553
Article 396. — Notations r, r, . . for D«p, Bs^p, &c. ; properties
of a curve depending on the square (s^) of its arc, measured from a
given point p ; r = unit-tangent, t' = vector of curvature, r~^ = Tr' = cur-
vature (oT first curvature, comp. Art. 397), v = tt' = binormal ; the
* A Table of initial Pages of all the Articles will be elsewhere given, which will
much facilitate reference.
CONTENTS. XV
Pages.
three planes, respectively perpendicular to r, r', v, are the normal
plane, the rectifying plane, and the osculating plane ; general theory
of emanant lines and planes, vector of rotation, axis of displacement, oscit-
lating screw surface ; condition of developahility of surface of emanants, 554-559
Article 397. — Properties depending on the cube (s^) of the are ;
Radius r (denoted here, for distinction, by a roman letter), and Vector
ir^T, oi Second Curvature ; this radius r may be either positive or ne-
gative (whereas the radius r of first curvature is always treated as
positive), and its reciprocal r^ may be thus expressed (pp. 663, 669),
d^o r"
Second Curvature* = r-i = S ^,, \^ , (T), or, r-i = S — , CT')
the independent variable being the arc in (T'), while it is arbitrary in
(T) : but quaternions supply a vast variety of other expressions for this
important scalar (see, for instance, the Table in pp. 574, 675). "We
have also (by p. 560, comp. Arts. 389, 395, 396),
Vector of Spherical Curvature = sp~i = (p— <y)"^ = &c., (U)
= projection of vector (r') of (simple or first) curvature, on radius (J2)
of osculating sphere : and if p and P denote the linear and angular
elevations, of the centre (s) of this sphere above the osculating plane,
then (by same page 560),
p = r tan F- R&m P = r'r = rD^r. (XT')
Again (pp. 660, 561), if we write (comp. Art. 396),
\ = V — =r-ir + Tr' = Vector of Second Curvature plus Binormal, (V)
T
this line \ may be called the Rectifying Vector ; and if TL denote the
inclination (considered first by Lancret), of this rectifying line (\) to
the tangent (r) to the curve, then
tan JT=r'-i tan P = y-ir. (V')
Known right cone with rectifying line for its axis, and with H. for its
seniiangle, which osculates at p to the developable locus of tangents to
the curve (or by p. 568 to the cone of parallels already mentioned) ;
new right cone, with a new scmiangle, C, connected with H by the
relation (p. 562),
tanC=^tanjEr, (V")
which osculates to the cone of chords, drawn from the given point p
* In this Article, or Series, 397, and indeed also in 396 and 398, several re-
ferences are given to a very interesting Memoir by M. de Saint- Venant, " Sur
les lignes courbes non planes :" in which, however, that able writer objects to such
known phrases as second cirvature, torsion, &c., and proposes in their stead a new
name " cambrure," which it has not been thought necessary here to adopt.
{Journal de V E'cole Poly technique, Cahier xxx )
XYl CONTENTS.
Pages,
to other points q of the'given curve. Other osculating cones, cylinders,
helix, &nd parabola ; this last being (pp. 662, 566) the parabola which
osculates to the projection of the curve, on its own osculating plane. De-
viation of curve, at any near point q, from the osculating circle at p,
decomposed (p. 666) into two rectangular deviations, from osculating
helix and parabola. Additional formulae (p. 676), for the general
theory of emanants (Art. 396) ; case o£ normally emanant lines, or of
tangentially emanant planes. General auxiliary spherical curve (pp.
576-578, comp. p. 515) ; new proof of the second expression (V) for
tan H, and of the theorem that if this ratio of curvatures be constant,
the proposed curve is a geodetic on a cylinder : new proof that if each
curvature (r'l, r~i) be constant, the cylinder is right, and therefore
the curve a helix, . , ' 659-578
Article 398. — Properties of a curve in space, depending on the
fourth and Jlfth powers (si, s^) oiita arc (s), 578-612
This Series 398 is so much longer than any other in the Volume,
and is supposed to contain so much original matter, that it seems
necessary here to subdivide the analysis under several separate heads,
lettered as (a), (b), (c), &c.
(«). Neglecting s^, we may write (p. 578, comp. Art. 396),
OP, = ps = p + 57 + -|«2 r' + XsH" + JjS^r'" ; (W)
or (comp. p. 587), ps = p + XsT + y,rr' + z^rv, (W)
with expressions (p. 588) for the coefficients (or co-ordinates) Xs, ys, Zg,
in terms of r', r, r", r, r', and s. If ^ be taken into account, it be-
comes necessary to add to the expression (W) the term, i^s^t^^ ;
with corresponding additions to the scalar coefficients in (W), intro-
ducing r'" and r" : the laws for forming which additional terms, and
for extending them to higher powers of the arc, are assigned in a
subsequent Series (399, pp. 612, 617).
(4). Analogous expressions for t", v", k", X', cr', and p', B', F, K',
to serve in questions in which s^ is neglected, are assigned (in p. 579) ;
r" v', K, X, <T, and p, R, P, H, having been previously expressed (in
Series 397) ; while r", v", k", \", a", &c. enter into investigations
which take account of s^ : the arc » being treated as the independent
variable in all these derivations.
(c). One of the chief results of the present Series (398), is the
introduction (p. 681, &c.) of a new auxiliary angle, J, analogous in
several respects to the known angle H (397), but belonging to a
higher order of theorems, respecting curves in space : because the new
angle / depends on the fourth (and lower) powers of the arc s, while
Lancret's angle H depends only on s^ (including s^ and s"^). In fact,
while tan jffis represented by the expressions (V), whereof one is
»•'-» tan P, tan /admits (with many transformations) of the following
analogous expression (p. 681),
tan/=:i2'-itanP; (X)
CONTENTS. xvii
Pages.
where JR' depends* by (A) on s^, while r' and F depend (397) on no
higher power than s^.
(d). To give a more distinct geometrical meaning to this new angle
J", than can he easily gathered from such a formula as (X), respecting
which it may he observed, in passing, that /is in general more simply
defined by expressions for its cotangent (pp. 581, 588), than for its
tangent, we are to conceive that, at each point p of any proposed
curve of double curvature, there is drawn a tangent plane to the sphere ^
which osculates (395) to the curve at that point ; and that then the
envelope of all these planes is determined, which envelope (for reasons
afterwards more fully explained) is called here (p. 581) the " Cir-
cumscribed Developable :" being a surface analogous to the ^^ Rectifying
Developable'^ of Lancret, but belonging (c) to a higher order of ques-
tions. And then, as the A'woww angle -ff denotes (Z^l) the inclina-
tion^ suitably measured, of the rectifying line (\), which is a genera-
trix of the rectifying developable, to the tangent (r) to the curve ; so
the new angle / represents the inclination of a generating line (^), of
what has just been called the circumscribed developable, to the same
tangent (r), measured likewise in a defined direction (p. 581), but
in the tangent plane to the sphere. It may be noted as another ana-
logy (p. 582), that while JS'is a right angle for deplane curve, so J
is right when the curve is spherical. For the helix (p. 585), the an-
gles H and / are equal ; and the rectifying and circumscribed deve-
lopables coincide, with each other and with the right cylinder, on
which the helix is a geodetic line.
(e). If the recent line be measured from the given point p, in
a suitable direction (as contrasted with the opposite), and with a suit-
able length, it becomes what may be called (comp. 396) the Vector of
Eolation of the Tangent Plane (d) to the Osculating Sphere ; and then
it satisfies, among others, the equations (pp. 579, 581, comp. (V)),
^ = V^, T0=i2-icosec/; (X')
V
this last being an expression for the velocity of rotation of the plane
just mentioned, or of its normal, namely the spherical radius R, if the
given curve be conceived to be described by a point moving with a con-
* In other words, the calculation of r' and P introduces no differentials
higher than the third order ; but that of R' requires 'Cine fourth order of differen-
tials. In the language of modern geometry, the/on?2^r can be determined by
the consideration oifour consecutive points of the curve, or by that of two consecu-
tive osculating circles ; but the latter requires the consideration of two consecu-
tive osculating spheres, and therefore oifive con^QCMiive points of the curve (sup-
posed to be one of double curvature). Other investigations, in the present and
immediately following Series (398, 399), especially those connected with what
we shall shortly call the Osculating Twisted Cubic, will be found to involve the
consideration of six consecutive points of a curve.
d
xviii CONTENTS.
stant velocity^ assumed = 1. And if we denote by v the point in which
the given radius R or PS is nearest to a consecutive radius of the same
kind, or to the radius of a consecutive osculating sphere^ then this point v
divides the line ps internally, into segments which may (ultimately) be
thus expressed (pp. 580, 581),
PV = -B sin2 /, vs = i2 coss /. (X")
But these and other connected results, depending on s*, have their
known analogues (with H for /, and r for R), in that earlier theory
(c) which introduces only s^ (besides s^ and s2) : and they are all m-
cluded in the general theory oiemanant lines and planes (396, 397), of
which some new geometrical illustrations (pp. 582-584) are here
given.
(/). New auxiliary scalar n {=p-^RR' = cot J'secP= &c.), = ve-
locity of centre s of osculating sphere, if the velocity of the point p of
the given curve be taken as unity (e) ; n vanishes with Rf, cot J", and
(comp. 395) the coefficient S-1 (=wn'"i) of non- sphericity, for the
case of a spherical curve (p. 584). Arcs, first and second curvatures,
and rectifying planes and lines, of the cusp-edges of the polar and
rectifying* developables ; these can all be expressed without going
beyond s\ and some without using any higher power than s\ or diffe-
rentials of the orders corresponding ; r\ = wr, and ri = nr, are the
scalar radii of first and second curvatvire oiihe former cusp-edge, r\
being positive when that curve turns its concavity at s towards the
given curve at p : determination of the point b, in which the latter
cusp-edge is touched by the rectifying line X to the original curve
(pp. 584-587).
(^). Equation with one arbitrary constant (p. 587), of a cone of
the second order, which has its vertex at the given point p, and has
contact of the third order (or four-side contacf) with the cone of chords
(397) from that point; equation (p. 590) of a cylinder of the second
order, which has an arbitrary line pe from p as one side, and has
contact of \h.e fourth order (or fwe-point contact^ with the curve at p ;
the constant above mentioned can be so determined, that the right line
PE shall be a side of the cone also, and therefore apart of the intersect
Hon of cone and cylinder; and then the remaining or curvilinear
part, of the complete intersection of those two siirfaces of the second
* The rectifying plane, of the cusp-edge of the rectifying developable, is the
plane of \ and t', of which the formula LIV. in p. 587 is the equation ; and the
rectifying line rh, of the same cusp-edge, intersects the absolute normal pk to the
given curve, or the radius (r) of first curvature, in the point h in which that
radius is nearest (e) to a consecutive radius of the same kind. But this last theo-
rem, which is here deduced by quaternions, had been previously arrived at by
M. de Saint- Venant (comp, the Note to p. xv.), through an entirely different
analysis, confirmed by geometrical considerations.
CONTENTS. xix
Pages,
order, is (by known principles) a gauche curve of the third order,
or what is briefly called* a Twisted Cubic : and this last curve, in
virtue of its construction above described, and whatever the as-
sumed direction of the auxiliary line pe may be, has contact of the
fourth order {pv Jive-point contact) with the given curve of double cur-
vature at p (pp. 687-590, comp. pp. 663, 672).
(Ji). Determination (p. 690) of the cow«^«w# in the equation of the
cone {g), so that this cone may have contact of the fourth order (or
Jwe-side contact) with the cone of chords from p ; the cone thus found
may be called the Osculating Oblique Gone (comp. 397), of the second
order, to that cone of chords ; and the coefficients of its equation in-
volve only r, r, /, r', r', x", but not r"\ although this last derivative
is of no higher order than r", since each depends only on s^ (and lower
powers), or introduces only fifth differentials. Again, the cylinder
(g) will have contact of the ffth order (or six-point contact) with the
given curve at p, if the line pe, which is by construction a side of that
cylinder, and has hitherto had an arbitrary direction, be now obliged
to be a side of a certain cubic cone, of which the equation (p. 690) in-
volves as constants not only rrr'rVr", like that of the osculating cone
just determined, but also r"'. The two cones last mentioned have the
tangent (r) to the given curve for a common side,f but they have also
three other common sides, whereof one at least is real^ since they are
assigned by a cubic equation (same p. 690) ; and by taking this side
for the line pe in (g), there results a new cylinder of the second order,
which cuts the osculating oblique cone, partly in that right line pe itself,
and partly in a gauche curve of the third order, which it is proposed to
call an Osculating Twisted Cubic (comp. again (y)), because it has con-
tact of the fifth order (or six-point contact) with the given curve at p
(pp. 690, 691).
(i). In general, and independently of any question of osculation,
a Twisted Cubic (jf), if passing through the origin o, may be repre-
sented by any one of the vector equations (pp. 692, 693),
* By Dr. Salmon, in his excellent Treatise on Analytic Geometry of Three
Dimensions (Dublin, 1862), which is several times cited in the Notes to this final
Chapter (III. iii.) of these Elements. The gauche curves, above mentioned, have
been studied with much success, of late years, by M. Chasles, Sig. Cremona, and
other geometers : but their existence, and some of their leading properties, ap-
pear to have been first perceived and published by Prof. Mobius (see his Bary^
centric Calculus, Leipzig, 1827, pp. 114-122, especially p. 117).
t This side, however, counts as three (p. 614), in the system of the six lines of
intersection (real or imaginary) of these two cones, which have a common vertex p,
and are respectively of the second oxiA. third orders (or degrees). Additional light
will be thrown on this whole subject, in the following Series (399) ; in which also
it will be shown that there is only one osculating twisted cubic, at a given point,
to a given curve of double curvature ; and that this cubic curve can be determined^
without resolving any cubic or other equation.
XX CONTENTS.
Yap + Yp<pp = 0, (Y); or (^ + e)p = a, (¥')
or p = (^ + c)-^a, (Y"); or Yap + pYyp + YpY\pfi = 0, (Y'")
in wliicli a, y, \, fi are real and constant vectors, but <j is a variable sca-
lar ; while 0p denotes (comp. the Section III. ii. 6, or pp. xii., xiii.) a
linear and vector function, which is A^r^ generally woiJ self -conjuff ate,
of the variable vector p of the cubic curve. The number of the scalar
constants, in the form (Y'"), or in any other form of the equation, is
found to be ten (p. 593), with the foregoing supposition that the curve
passes through the origin, a restriction which it is easy to remove.
The curve (Y) is cut, as it ought to be, in three points (real or imagi-
nary), by an arbitrary secant plane ; and its three asymptotes (real or
imaginary) have the directions of the three vector roots /3 (see again
the last cited Section) of the equation (same p. 693),
V/3^/3 = 0: (Z)
so that by (P), p. xii., these three asymptotes compose a real and rect-
angular system, for the case of self -conjugation of the function
in (Y).
(/). Deviation of a near point Ps of the given curve, from the sphere
(395) which osculates at the given point p ; this deviation (by p. 593,
comp. pp. 653, 584) is
r\^ R's^ n&^
it is ultimately equal (p. 696) to the quarter of the deviation (397)
of the same near point Pj from the osculating circle at p, multiplied by
the sine of the small angle spSs, which the small arc sss of the locus of
the spheric centre s (or of the cusp-edge of the polar developable) stib-
tends at the same point p ; and it has an outward or an inward direc-
tion, according as this last arc is concave or convex (/) at s, towards the
given curve at p (pp. 585, 695). It is also ultimately equal (p. 696)
to the deviation pss - TsSs, of the given point p from the near sphere,
which osculates at the near point p^; and likewise (p. 597) to the com-
ponent, in the direction of sp, of the deviation of that near point from
the osculating circle at p, measured in a direction parallel to the nor-
mal plane at that point, if this last deviation be now expressed to the
accuracy of the fourth order : whereas it has hitherto been considered
sufficient to develope this deviation from the osculating circle (397) as
far as the third order (or third dimension of s) ; and therefore to treat
it as having a direction, tangential to the osculating sphere (comp.
pp. 666, 694).
(k). The deviation (Ai) is also equal to the third part (p. 698) of
the deviation of the near point Vg from the given circle (which osculates
at p), if measured in the near normal plane (at p^), and decomposed in
the direction of the radius Rs of the near sphere; or to the third part
(with direction preserved) of the deviation of the new near point in
which the given circle is cut by the near plane, /rom the near sphere : or
finally to the third part (as before, and still with an unchanged direc-
CONTENTS. xxi
Pages,
tion) of the deviation from the given sphere, of that other near point
c, in which the near circle (osculating at Ps) is cut by the given normal
plane (at p), and which is found to satisfy the equation,
sc = 3sps - 2sp. (Bi)
Geometrical connexions (p, 599) between these various results (/) (^),
illustrated by a diagram (Fig. 83).
(J). The Surface, which is the Locus of the Osculating Circle to
a given curve in space, may be represented rigorously by the vector
expression (p. 600),
Ws, u^ps-^- rsTs sin u + n^r/ vers u ; (Ci)
in which s and u are two independent scalar variables, whereof * is
(as before) the arc pp^ of the given curve, but is not now treated as
small : and u is the (small or large) angle subtended at the centre k* of
the circle, by the arc of that circle, measured from its point of oscula-
tion Ps. But the same superficial locus (comp. 392) may be repre-
sented also by the vector equation (p. 611), inyolvmg a2)parentlg only
one scalar variable (s),
Y-^ + Vs = 0, (Di)
<t>-ps
in which Vs—Tst/, and u)= (vs,u = the vector of an arbitrary point
of the surface. The general method (p. 501), of the Section III. iii.
3, shows that the normal to this surface (Ci), at any proposed point
thereof, has the direction of w*, « - o-j ; that is (p. 600), the direction
of the radius of the sphere, which contains the circle through that
point, and has the same point of osculation p* to the given curve. The
locus of the osculating circle is therefore found, by this little calculation
with quaternions, to be at the same time the Envelope of the Osculat-
ing Sphere, as was to be expected from geometrical considerations
(comp. the Note to p. 600).
(m). The curvilinear locus of the point c in (Jc) is one branch of
the section of the surface (I), made by the normal plane to the given
curve at p ; and if d be the projection of c on the tangent at p to this
new curve, which tangent pd has a direction perpendicular to the ra-
dius PS or H of the osculating sphere at p (see again Fig. 83, in p.
599), while the ordinate dc ia parallel to that radius, then (attending
only to principal terms, pp. 598, 599) wc have the expressions,
and therefore ultimately (p. 600),
DC3 81 w3^5r((T-p) ^ ,_.
from which it follows that p is a singular 2>oint of the section here
considered, but not a cusp of that section, although the curvature
at p is infinite : the ordinate dc varying ultimately as the power
with exponent ^ of the abscissa pd. Contrast (pp. 600, 601), of this
xxii CONTENTS.
section, with that of the developable Locus of Tangents, made by the
same normal plane at p to the given curve ; the vectors analogous to
PD and DC are in this case nearly equal to - fs^/ and — ^s^v^v ; so
that the latter varies Tiltimately as the power f of the former, and the
point p is (as it is known to be) a cusp of this last section.
(n). A given Curve of double curvature is therefore generally a
Singular Line (p. 601), although not a cusp-edge, upon that Surface {T)j
which is at once the Locus of its osculating Circle, and the Envelope
of its osculating Sphere : and the new developable surface {d), as being
circumscribed to this superficial locus (or envelope), so as to touch it
along this singular line (p. 612), may naturally be called, as above,
ihe Circumscribed Developable (;;^. h^i).
(o). Additional light may be thrown on this whole theory of the
singular line (n), by considering (pp. 601-611) a problem which was
discussed by Monge, in two distinct Sections (xxii. xxvi.) of his well-
known Analyse (comp. the Notes to pp. 602, 603, 609, 610 of these
Elements') ; namely, to determine the envelope of a sphere with varying
radius R, whereof the centre s traverses a given curve in space ; or
briefly, to find the Envelope of a Sphere with One varying Parameter
(comp. p. 624) : especially for the Case of Coincidence (p. 603, &c.), of
what are usually two distinct branches (p. 602) of a certain Charac-
teristic Curve (or arete de rebroussement), namely the curvilinear enve-
lope (real or imaginary) of all the circles, along which the superficial
envelope of the spheres is touched by those spheres themselves.
{^p)' Quaternion forms (pp. 603, 604) of the condition of coinci-
dence (o) ; one of these can be at once translated into Monge' s equa-
tion of condition (p. 603), or into an equation slightly more general,
as leaving the independent variable arbitrary ; but a simpler and
more easily interpretable form is the following (p. 604),
ridr = ±MB, (Gi)
in which r is the radius of the circle of contact, of a sphere with its
envelope (o), while ri is the radius of (first) curvature of the curve (s),
which is the locus of the centj-e s of the sphere.
(^). The singular line into which the two branches of the curvi-
linear envelope ^refused, when this condition is satisfied, is in general
an orthogonal trajectory (p. 607) to the osculating planes of the curve
(s) ; that curve, which is noiv the given one, is therefore (comp. 391,
395) the cusp-edge (p. 607) of the^o^ar developable, corresponding to
the singular line just mentioned, or to what may be called the curve
(p), which was formerly the given curve. In this way there arise
many verifications of formulae (pp. 607, 608) ; for example, the
equation (Gi) is easily shown to be consistent with the results of (/).
(r). With the geometrical hints thus gained from interpretation
of quaternion results, there is now no difficulty in assigning the Com-
plete and General Integral of the Equation of Condition {p), which was
presented by Monge under the form (comp. p. 603) of a non-linear
differential equation of the second order, involving three variables
Pages.
CONTENTS. xxiii
Pages.
(0, \jj, tt) considered as functions of a fourth (a), namely the co-or-
dinates of tlie centre of the sphere, regarded as varying with the ra-
dius, but which does not appear to have been either integrated or
interpreted by that illustrious analyst. The general integral here
found presents itself at first in a ^^wa^^rw/ow/orm (p. 609), but is easily
translated {^. 610) into the usual language of analysis. A less ge-
neral integral is also assigned, and its geometrical signification exhi-
bited, as answering to a case for which the singular line lately consi-
dered reduces itself to a singular point (pp. 610, 611).
(s). Among the verifications (jf) of this whole theory, it is shown
(pp. 608, 609) that although, when the two branches (o) of the general
curvilinear envelope of the circles of the system are real and distinct,
each branch is a cusp-edge (or arete de rebroussement, as Monge per-
ceived it to be), upon the superficial envelope of the spheres, yet in the
case of fusion (p) this cuspidal character is lost (as was likewise
seen by Monge*) : and that then a section of the surface, made by
a normal plane to the singular line, has precisely the form (on), ex-
pressed by the equation (Fi). In short, the result is in many ways
confirmed, by calciilation and by geometry, that when the condition of
coincidence (j») is satisfied, the Surface is, as in (n), at once the JEnve-
lope of the osculating Sphere and the Locus of the osculating Circle, to
that Singular Line on itself, into which by ((?) the two branches (o)
of its general cusp- edge are fused.
({). Other applications of preceding formulae might be given ;
for instance, the formula for k" enables us to assign general ex-
pressions (p. 611) for the centre and radius of the circle, which oscu-
lates at K to the locus of the centre of the osculating circle, to a given
curve in space : with an elementary verification, for the case of the
plane evolute of the plane evolute of a plane curve. But it is time to con-
clude this long analysis, which however could scarcely have been
much abridged, of the results of Series 398, and to pass to a more
brief account of the investigations in the following Series.
Akticle 399. — Additional general investigations, respecting that
gauche curve of the third order (or degree), which has been above
called an Osculating Twisted Cubic (398, (A))) to any proposed curve
of double curvature ; with applications to the case, where the given
curve is a Me:r, 612-621
(a). In general (p. 614), the tangent pt to the given curve is a
nodal side of the cubic cone 398, (A) ; one tangent plane to that cone
(C3), along that side, being the osculating plane (P) to the curve, and
therefore touching also, along the same side, the osculating oblique cone
(C2) of the second order, to the cone of chords (397) from p ; while the
other tangent plane to the cubic cone (Ca) crosses ihsit first plane (P),
or the quadric cone (C2), at an angle of which the trigonometric cotan-
* Compare the first Note to p. 609 of these Elements.
XXIV CONTENTS.
Pages.
gent (^r') is equal to half the differential of the radius (r) of second
curvature, divided hy the differential of the are (s). And the three
common sides, pe, pb', pe", of these two cones, which remain when the
tangent pt is excluded, and of which one at least must be real, are the
parallels through the given point p to the three asymptotes (398, (t))
to the gauche curve sought ; being also sides of three quadric cylin-
ders, say (Z2), (X'2), (-^"2), which contain those asymptotes as other
sides (or generating lines) : and of which each contains the twisted
cubic sought, and is cut in it by the quadric cone ( G2).
(b). On applying this First Method to the case of a given h,elix, it
is found (p. 614) that the general cubic cone (^C^ breaks up into the
system of a new quadric cone, (jO-i), and a new plane (P') ; which lat-
ter is the rectifying plane (396) of the helix, or the tangent plane at p
to the right cylinder, whereon that given curve is traced. The two
quadric cones, (Co) and (C2), touch each o^Aer andthe plane (P) along
the tangent pt, and have no other real common side : whence tivo of
the sought asymptotes, and tivo of the corresponding cylinders (a), are
in this case imaginary, although they can still be used in calculation
(pp. 614, 615, 617). But the plane (P') cuts the cone (C2), not only
in the tangent pt, but also in a second real side pe, to which the real
asymptote is parallel (a) ; and which is at the same time a side of a
real quadric cylinder (Z2), which has that asymptote for another side
(p. 617), and contains the twisted cubic : this gauche curve being thus
the curvilinear part (p. 615) of the intersection of the real cone (C2),
with the real cylinder (Zo)-
(c). Transformations and verifications of this result ; fractional ex-
pressions (p. 616), for the co-ordinates of the twisted cubic ; expres-
sion (p. 615) for the deviation of the helix irom that osculating curve,
which deviation is directed inwards, and is of the sixth order : the
least distance, between the tangent pt and the real asymptote, is a right
line PB, which is cut internally (p. 617) by the axis of the right cylin-
der (h), in a point a such that pa is to ab as three to seven.
{cT). The First Method (a), which had been established in the pre-
ceding Series (398), succeeds then for the case of the Jielix, with a faci-
lity which arises chiefly from the circumstance (J)), that for this case
the general cubic cone (Cz) breaks up into two separate loci, whereof
one is a. plane (P'). But usually the foregoing method requires, as in
398, (Ji)), the solution of a cubic equation : an inconvenience which is
completely avoided, by the employment of a Second General Method,
as follows.
(e). This Second Method consists in taking, for a second locus of the
gauche osculatrix sought, a certain Cuhic Surface (63), of which
every point is the vertex* of a quadric cone, having six-point con-
* It is known that the locics of the vertex of a quadric cone, which passes
through six given points of space, a, b, c, d, e, f, whereof no four are in one
CONTENTS. XXV
Pages.
tact with the given curve at p : so that this new surface is cut by the
plane at infinity^ in the same cubic curve as the cubic cone ipz). It is
found (p. 620) to be a Ruled Surface^ with the tangent pt for a Sin-
gular Line ; and when this right line is set aside, the remaining (that
is, the curvilinear') part of the intersection of the two loci, (C2) and
(aSs), is the Osculating Twisted Cubic sought : which gauche osculatrix
is thus completely and generally determined, without any such difficulty
or apparent variety, as might be supposed to attend the solution of a
cubic equation (d), and with new verifications for the case of the helix
(p. 621).
Article 400. — On Involutes and Evolutes in Space, .... 621-626
{a). The usual points of Monge's theory are deduced from the two
fundamental quaternion equations (p. 621),
S((r-p)p'=0, V(or-p)(T'=0, (Hi)
in which p and a are corresponding vectors of involute and evolute ;
together with a theorem of Prof. De Morgan (p. 622), respecting the
case when the involute is a spherical curve.
(b). An involute in space is generally the only real part (p. 624) of
the envelope of a certain variable sphere (comp. 398), which has its
centre on the evolute, while its radius R is the variable intercept be-
tween the two curves : but because we have here the relation (p. 622,
comp. p. 602),
i2'2 H- <t'2 = 0, (Hi')
the circles of contact (398, (0)) reduce themselves each to a point (or
rather to a pair of imaginary right lines, intersecting in a real point),
and the preceding theory (398), of envelopes of spheres with one
varying parameter, undergoes important modifications in its results,
the conditions of the application being different. In particular, the
involute is indeed, as the equations (Hi) express, an orthogonal tra-
jectory to the tangents of the evolute; but not to the osculating planes
plane, is generally a Surface, say {S^), of the Fourth Degree : in fact, it is cut by
the plane of the triangle abc in a system of four right lines, whereof three are
the sides of that triangle, and the fourth is the intersection of the two planes,
ABC and DEF. If then we investigate the intersection of this surface (^S\) with
the quadric cone, (a.bcdef), or say ((72), which has a for vertex, and passes
through the five other given points, we might expect to find (in some sense) a
curve of the eighth degree. But when we set aside ^efive right lines, ab, ac, ad,
AE, AF, which are common to the two surfaces here considered, we find that the
(remaining or) curvilinear part of the complete intersection is reduced to a curve
of the third degree, which is precisely the twisted cubic through the six given points.
In applying this general (and perhaps new) method, to the problem of the oscu-
lating twisted cubic to a curve, the osculating ^?(m« to that curve may be excluded,
as foreign to the question : and then the quartic surface {Si) is reduced to the
cubic surface {S3), above described.
e
XXvi CONTENTS.
Pages,
of that curve, as tlie singular line (398, {q)) of the former envelope
was, to those of the curve which was the locus of the centres of the
spheres hQiovQ considered, when a certain condition of coincidence (or
oi fusion, 398, {p)") was satisfied.
(c). Curvature of hodograph of evolute (p. 625) ; if p, Pi, P2, • • and
s, Si, S2, . . he corresponding points of involute and evolute, and if we
draw right lines sti, st2, . . in the directions of SiPi, S2P2, • • and with
a common length = sp, the spherical curve PT1T2 . . will have contact
of the second order at p, with the involute PP1P2 • • (pp. 625, 626).
Article 401. — Calculations abridged, by the treatment of quater-
nion differentials (which have hitherto been finite, comp. p. xi.) as
infinitesimals ;* new deductions of osculating plane, circle, and sphere,
with the vector equation (392) of the circle ; and of the first and se-
cond curvatures of a curve in space, 626—630
Section 7. — On Surfaces of the Second Order; and on
Curvatures of Surfaces, 630-706
Article 402. — References to some equations of Surfaces, in earlier
parts of the Volume, 630, 631
Article 403. — Quaternion equations of the Sphere (p2 = - 1, &c.), 631-633
In some of these equations, the notation N for norm is employed
(comp. the Section II. i. 6).
Article 404. — Quaternion equations of the Ellipsoid, .... 633-635
One of the simplest of these forms is (pp. 307, 635) the equation,
T(tp + pfc) = »e2_t2^ (Ii)
* Although, for the sake of brevity, and even of clearness, some phrases have
been used in the foregoing analysis of the Series 398 and 399, such di^ four-side
or five-side contact between cones, and five-point or six-point contact between
curves, or between a curve and a surface, which are borrowed from the doctrine
of consecutive points and lines, and therefore from that of infinitesimals ; with a
few other expressions of modern geometry, such as the plane at infinity, &c. ;
yet the reasonings in the text of these Elements have all been rigorously reduced,
so far, or are all obviously reducible, to the fundamental conception of Limits :
compare the definitions of the osculating circle and sphere, assigned in Articles
389, 395. The object of Art. 401 is to make it visible how, without abandoning
such ultimate reference to limits, it is possible to abridge calculation, in several
cases, by treating (at this stage) the differential symbols, dp, d^p, &c., as if
they represented infinitely small differences, Ap, A'^p, &c. ; without taking the
trouble to write these latter symbols first, as denoting finite differences, in the
rigorous statement of a problem, of which statement it is not always easy to assign
the proper form, for the case of points, &c., at finite distances : and then having
the additional trouble of reducing the complex expressions so found to simpler forms,
in which differentials shall finally appear. In short, it is shown that in Quater-
nions, as in other parts of Analysis, the rigour of limits can be combined with
the facility of infinitesimals.
CONTENTS, xxvii
Pages,
in which i and k are real and constant vectors, in the directions of
the ci/ch'c normals. This form (Ii) is intimately connected with, and
indeed served to suggest, that Construction of the Ellipsoid (II. i. 13),
by means of a Diacentric Sphere and a Point (p. 227, comp. Fig. 53,
p. 226), which was among the earliest geometrical results of the Qua-
ternions. The three semiaxes, a, b, c, are expressed (comp. p. 230) in
terms of i, k as follows :
a = THT/c; ^=r^^-:^'-y o = Tc-Tk; (!,')
whence «*-»<; = T (t - /c). (Ii")
Article 405. — General Central Surface of the Second Order (or
central quadric), Sp^p -fp = 1, 636-638
Article 406. — General Cone of the Second Order (or quadric cone),
Sp(pp^fp=0, • . . 638-643
Article 407. — Bifocal Form of the equation of a central but non-
conical surface of the second order : with some quaternion formulae,
relaiing to Confocal Surf aces, 643-663
(a). The bifocal form here adopted (comp. the Section III. ii. 6)
is the equation,
Cfp = (Sap)2 - 2^SapSa'p + (Sa'p)2 + (1 - e^) p2 = C, (Ji)
in which, C= (e« - 1) (^ + Saa') l^. (Ji')
a, a' are two (real) focal unit-lines, common to the whole system of
confocals ; the (real and positive) scalar I is also constant for that sys-
tem : but the scalar e varies, in passing from surface to surface, and
may be regarded as a parameter, of which the value serves to distin-
guish one confocal, say (<?), from another (pp. 643, 644).
(i). The squares (p. 644) of the three scalar semiaxes (real or ima-
ginary), arranged in algebraically descending order, are,
a2 = (e+l)^, i2=(g+Saa')/2, c^ = (e-l)P; (Ki)
whence ''=-Y~' ^"'^^^^' ^^'^
and the three vector semiaxes corresponding are,
aU(a + a'), iUVatt', cU(a-a'). (Mi)
(c). Rectangular, unifocal, and cyclic forms (pp. 644, 648, 650),
of the scalar function fp, to each of which corresponds a form of the
vector function 0p ; deduction, by a new analysis, of several known
theorems* (pp. 644, 645, 648, 652, 653) respecting confocal surfaces,
* For example, it is proved by quaternions (pp. 652, 653), that the focal
lines of the focal cone, which has any proposed point p for vertex, and rests on
the focal hyperbola, are generating lines of the single- sheeted hyperboloid (of the
given confocal system), which passes through that point : and an extension of
this result, to the focal lines of any cone circumscribed to a confocal, is deduced
by a similar analysis, in a subsequent Series (408, p. 656). But such known
theorems respecting confocals can only be alluded to, in these Contents.
XXVIU CONTENTS.
Pages,
and their focal conies ; the lines a, a' are asymptotes to the focal hy-
perbola (p. 647), whatever the species of the svirface may be : refe-
rences (in Notes to pp. 648, 649) to the Lectures* for the/om^ ellipse
of the Ellipsoid, and for several different generations of this last sur-
face.
(<?). General Exponential Transformation (p. 651), of the equation
of any central quadric ;
p = xa + yYa% (Ni), with x-^fa + y^fVYaa = 1, (Ni')
^ (a - ea) JJY aa .„ ,,.
this auxiliary vector /3 is constant, for any one confocal (e) ; the expo-
nent, t, in (Ni), is an arbitrary or variable scalar ; and the coefficientSj
X and y, are two other scalar variables, which are however connected
with each other by the relation (Ni').
(i). If onj fixed value be assigned to t, the equation (Ni) then re-
presents the section made by a plane through a (p. 651), which sec-
tion is an ellipse if the surface be an ellipsoid, but an hyperbola for
either hyperboloid ; and the cutting plane makes with the focal plane
of a, a', or with the plane of the focal hyperbola, an angle = J^tt.
(/). If, on the other hand, we allow t to vary, but assign to
X and y any constant values consistent with (Ni'), the equation (Ni)
then represents an ellipse (p. 651), whatever the species of the surface
may be ; x represents the distance of its centre from the centre o of the
surface, measured along the focal line a; y is the radius of a right
cylinder, with a for its axis, of which the ellipse is a section, or the
radius of a circle in a plane perpendicular to a, into which that ellipse
can be oxthogonallj projected : and the angle J^tt is now the excentric
anomaly. Such elliptic sections of a central quadric may be otherwise
obtained from the unifocal form (c) of the equation of the surface ;
they are, in some points of view, almost as interesting as the known
circular sections : and it is proposed (p. 649) to call them Centro-
Focal Ellipses.
(g). And it is obvious that, by interchanging the two focal lines
a, a' in ((?), a Second Exponential Transformation is obtained, with a
Second System of centro-focal ellipses, whereof the proposed surface is
the locus, as well as of the first system (/), but which have their
centres on the line a', and are projected into circles, on a plane per-
pendicular to this latter line (p. 649).
(A). Equation of Confocals (p. 652),
Vv,0v, = Yvf,v. (Oi)
Article 408. — On Circumscribed Quadric Cones; and on the
Umbilics of a central quadric, 653-663
* Lectures on Quaternions (by the present author), Dublin, Hodges and
Smith, 1853.
COMTENTS. XXIX
Pages.
{a). Equations (p. 653) of Conjugate Points^ and of Conjugate Di-
rections, with respect to the surface /p = 1,
fdp, p') = 1, (Pi), and/(p, p') = ; (Pi')
Condition of Contact, of the same surface with the right line pp',
(/(p,p')-i)^ = (/p-i)(/f>'-i); (QO
this latter is also a form of the equation of the Cone, with vertex at
p', which is circumscribed to the same quadric (/p = 1).
{b). The condition (Qi) may also he thus transformed (p. 654),
FYpp' = aH^c^f<ip-p), (QO
F being a scalar function, connected with / by certain relations of
reciprocity (comp. p. 483) ; and a simple geometrical interpretation
may be assigned, for this last equation.
(c). The Reciprocal Cone, or Cone of Normals a at p',to the circum-
scribed cone (Q,i) or (Qi'), may be represented (p. 655) by the very
simple equation,
i?'((r:Sp'(T)=l; (Qi")
which likewise admits of an extremely simple interpretation.
(<?). A given right line (p. 656) is touched by two confocals, and
other known results are easy consequences of the present analysis ;
for example (pp. 658, 659), the cone circumscribed to any surface of
the system, from any point of either of the two real focal curves^ is a
cone of revolution (real or imaginary) : but a similar conclusion holds
good, when the vertex is on the third (or imaginary) focal, and even
more generally (p. 663), when that vertex is any point of the (known
and imaginary) developable envelope of the confocal system.
(e). A central quadric has in general Twelve Umbilics (p. 659),
whereof only /owr (at most) can be real, and which are its intersections
with the three focal curves : and these twelve points are ranged, three by
three, on eight imaginary right lines (p. 662), which intersect the circle
at infinity, and which it is proposd to call the Eight Umbilicar Ge-
neratrices of the surface.
(jT). These (imaginary) umbilicar generatrices of a quadric are
found to possess several interesting properties, especially in relation
to the lines of curvature : and their locus, for a confocal system, is a
developable surface (p. 663), namely the known envelope (d) of that
system.
Article 409 — Geodetic Lines on Central Surfaces of the Second
Order, 664-667
(a). One form of the general differential equation of geodetics on
an arbitrary surface being, by III. iii. 5 (p. 515),
VvdV = 0, (Ri), if Tdp=: const., (R/)
this is shown (p. 664) to conduct, for central quadrics, to the first
integral,
p-2^-2 = Ti.2/Udp = /i = const; (Si)
vfhexe F is the perpendicular from the centre o on the tangent plane,
XXX CONTENTS.
Pages,
and D is tlie (real or imaginary) semidiameter of the surface, which
is parallel to the tangent (dp) to the curve. The known equation
of Joachimstal, F.B = const., is therefore proved anew ; this last
cotistant, however, heing hy no means necessarily real, if the surface
be not an ellipsoid.
(b). Deduction (p. 665) of a theorem of M. Chasles, that the tan-
gents to a geodetic, on any one central quadric {e), touch also a common
eonfocal (e,) ; and of an integral (p. 666) of the form,
e\ sin^ vi + e^ cos^ v\ = e, = const. , (Si')
which agrees with one of M. Liouville.
(c). Without the restriction (Ri'), the differential of the scalar h
in (Si) may be thus decomposed into factors (p. 666),
dA = d. P-22)-2 = 2Svdjvdp-i. Sj/dp-id2p ; (Si")
but, by the lately cited Section (III. iii. 5, p. 515), the differential
equation of the second order ^
Sj^dpd2p=0, (Ri")
with an arbitrary scalar variable, represents the geodetic lines on any
surface : the theorem («) is therefore in this way reproduced.
(d). But we see, at the same time, by (Si"), that the quantity h,
ox P.D = h-\ is constant, not only for the geodetics on a central quadric,
but also for a certain other set of curves, determined by the differen-
tial equation of the Jirst order, Svdvdp = 0, which will be seen, in the
next Series, to represent the lines of curvature.
Article 41 0. — On Lines of Curvature generally ; and in particu-
lar on such lines, for the case of a Central Quadric, 667-674
(a). The differential equation (comp. 409, («?)),
Svdvdp = 0, (Ti)
represents (p. 667) the Zincs of Curvature, upon a,n arbitrary surface ;
because it is a limiting form of this other equation,
SrAi/Ap = 0, (Ti')
which is the condition of intersection (or of parallelism), of the normals
drawn at the extremities of the two vectors p and p + Ap.
(b). The normal vector v, in the equation (Ti), may be multiplied
(pp. .673, 700) by any constant or variable scalar n, without any real
change in that equation ; but in this whole theory, of the treatment
of Curvatures of Surfaces by Quaternions, it is advantageous to con-
sider the expression Srdp as denoting the exact differential of some
scalar function of p ; for then (by pp. 486, 487) we shall have an equa-
tion of the form,
dj/ = 0dp = a self -conjugate function of dp, (Ui)
which iwually involves p also. For instance, we may write generally
(p. 669, comp. (R), p. xiii),
di/ = ^dp+V\dp/^; (Ui')
. CONTENTS. xxxi
Pages,
the scalar g, and the vectors X, fi being real, and being gemrally* func-
tions of p, but not involving dp.
(c). This being understood, the tivo^ directions of the tangent dp,
•vrhich satisfy at once the general equation (Ti) of the lines of curva-
ture, and the differential equation S^-dp = of the surface, are easily-
found to be represented by the two vector expressions (p. 669),
XJVj/X + UVj/At; (Ti")
they are therefore generally rectangular to each other, as they have
long been known to be.
(^). The surface itself remaining still quite arbitrary, it is found
useful to introduce the conception of an Auxiliary Surface of the Se-
cond Order (p. 670), of which the variable vector is p + p', and the
equation is,
Sp>p' = gp'^ -f SXpVp' = 1, (Ui")
or more generally = const. ; and it is proposed to call this surface, of
which the ce^itre is at the given point p, the Index Surface, partly
because its diametral section, made by the tangent plane to the given
surface at p, is a certain Index Curve (p. 668), which may be consi- ,
dered to coincide with the known " itidicatrice" of Dupin.
(e). The expressions (Ti") show (p. 670), that whatever the given
surface may be, the tangents to the lines of curvature bisect the angles
formed by the traces of the two cyclic planes of the Index Surface (^d),
on the tangent plane to the given surface ; these two tangents have
also (as was seen by Dupin) the directions of the axes of the Index
Curve (p. 668) ; and they are distinguished (as he likewise saw) from
all otJier tangents to the given surface, at the given point p, by the
condition that each is perpendicular to its own conjugate, with respect ^
to that indicating curve : the equation of such conjugation, of two
tangents r and r', being in the present notation (see again p. 668),
Sr0r' = 0, or Sr^r = 0. (Ui'")
(/). New proof (p. 669) of another theorem of Dupin, namely
that if a developable be circumscribed to any surface, along any curve
thereon, its generating lilies are everywhere conjugate, as tangents to
the surface, to the corresponding tangents to the curve.
{g). Case of a central quadric ; new proof (p. 671) of still another
theorem of Dupin, namely that the curve of orthogonal intersection
(p. 645), of two confocal surfaces, is a line of curvature on each.
Qi). The system of the eight umbilicar generatrices (j^(i%, (^)), of a
central quadi'ic, is the imaginary envelope of the lines of curvature on
that surface (p. 671) ; and each such generatrix is itself &.n imaginary
* For the case of a central quadric, g, X, /i are constants.
t Generally two ; but in some cases more. It will soon be seen, that three
lines of curvature pass through an wnbilic of a quadric.
XXXll CONTENTS.
Pages.
line of curvature thereon : so that through each of the twelve umUlics
(see again 408, (e)) there pass three lines of curvature (comp. p. 677),
whereof however only one, at most, can be real : namely two genera-
trices, and a principal section of the surface. These last results, which
are perhaps new, will be illustrated, and otherwise proved, in the
following Series (411).
Article 411. — Additional illustrations and confirmations of the
foregoing theory, for the case of a Central* Quadric ; and especially
of the theorem respecting the Three Lines of Curvature through an
Umbilic, whereof two are always imaginary and rectilinear, .... 674-679
(a). The general equation of condition (Ti'), or Si/AvAp = 0, for
the intersection of two finitely distant normals, may be easily trans-
formed for the case of a quadric, so as to express (p. 675), that when
the normals at p and p' intersect (or are parallel), the chord pp' is per-
pendicular to its own polar.
(b). Under the same conditions, if the point p be given, the locus
of the chord pp' is usually (p. 676) a quadric cone, say (C) ; and there-
fore the locus of the point p' is usually a quartic curve, with p for a
double poinf, whereat two branches of the curve cut each other at right
angles, and touch the two lines of curvature.
(c). If the point p be one of aprincipal section of the given surface,
but not an umbilie, the cone (C) breaks up into a. pair of planes, whereof
one, say (P), is the plane of the section, and the other, {F'), is perpen-
dicular thereto, and is not tangential to the surface ; and thus the
quartic (J) breaks up into a pair of conies through p, whereof one is
the principal section itself, and the other is perpendicular to it.
(<?). But if the given point p be an umbilie, the second plane (P')
becom^es a tangent plane to the surface ; and the second conic (/) breaks
up, at the same time, into a pair of imaginary f right lines, namely
the two umbilicar generatrices through p (pp. 676, 678, 679).
(e). It follows that the normal pn at a real umbilie p (of an ellip-
soid, or a double-sheeted hyperboloid) is not -intersected by any other
real normal, except those which are in the same principal section ; but
that this real normal pn is intersected, in an imaginary sense, by all
the no7'mals p'n', which are drawn at points p' oi either of the two ima-
ginary generatrices through the real umbilie p ; so that each of these
* Many, indeed most, of the results apply, without modification, to the case of
the Paraboloids ; and the rest can easily be adapted to this latter case, by the con-
sideration of infinitely distant points. We shall therefore often, for conciseness,
omit the term central, and simply speak of quadrics, or surfaces of the second
order.
t It is well known that the single-sheeted hyperboloid, which (alone of
central quadrics) has real generating lines, has at the same time no real umbilies
(comp. pp. 661, 662).
CONTENTS. xxxiii
Pages.
imaginary right lines is seen anew to be a Urn* of curvature^ on the sur-
face (comp. 410, (7i)), because all the normals p'n', at points of this
line, are situated in one common {imaginary) normal plane (p. 676) :
and as before, there are thus three lines of curvature through an um-
bilie.
(/). These geometrical results are in various ways deducible from
calculation with quaternions ; for example, a form of the equation of
the lines of ctrrvature on a quadric is seen (p. 677) to become an
identity at an umbilic (y || \) : while the differential of that equation
breaks up into two factors, whereof one represents the tangent to the
principal section, while the other (SXd^p = 0) assigns the directions of
the two generatrices.
(g). The equation of the cone (C), which has already presented
itself as a certain locus of chords (i), admits of many quaternion
transformations ; for instance (see p. 675), it may be written thus,
SapAp SaVAp
SaAp ^ Sa'Ap"""' ^ '^
p being the vector of the vertex p, and p + Ap that of any other point
p' of the cone ; while a, a' are still, as in 407, (a), two redl focal lines,
of which the lengths are here arbitrary, but of which the directions
are constant, as before, for a whole confocal system.
(A). This cone (C), or (Vi), is also the locus (p. 678) of a system
* It might be natural to suppose, from the known general theory (410, (c))
of the ttvo rectangular directions, that each such generatrix pp' is crossed perpendi-
cularly, at every one of its non-umbilicar points p', by a second (and distinct,
although imaginary') line of curvature. But it is an almost equally well known
and received result of modem geometry, paradoxical as it must at first appear, that
when a right line is directed to the circle at infinity, as (by 408, (e)) the gene-
ratrices in question are, then this imaginary line is everyivhere perpendicular to
itself. Compare the Notes to pages 459, 672. Quaternions are not at all re-
sponsible for the introduction of this principle into geometry, but they recognise
and employ it, under the following very simple form : that if a non-evanescent
vector be directed to the circle at infinity, it is an imaginary value of the symbol Oi
(comp. pp. 300, 459, 662, 671, 672) ; and conversely, that ivhen this last symbol
represents a vector which is not null, the vector thus denoted is an imaginary line,
which cuts that circle. It may be noted here, that such is the case with the reci-
procal polar of every chord of a quadric, connecting any two mnbilics which are not
in one principal plane' ; and that thus the quadratic equation (XXI., in p. 669)
from which the two directions (410, (cj) can usually be derived, becomes an iden-
tity for every umbilic, real or imaginary : as it ought to do, for consistency with
the foregoing theory of the three lines through that umbUic. And as an addi-
tional illustration of the coincidence of directions of the lines of curvature at any
non-umbilicar point p' of an umbilicar generatrix, it may be added that the cone
of chords (C), in 411, (b), is found to touch the quadric along that generatrix.
when its vertex is at any such point p'.
f
XXXIV CONTENTS.
Pages.
of three rectangular lines ; and if it be cut by any plane perpendicular
to a side, and not passing through the vertex, the section is an equila-
teral hyperbola.
(i). The same cone (C) has, for three of its sides pp', the normals
(p. 677) to the three eonfocals (p. 644) of a given system which pass
through its vertex p ; and therefore also, by 410, (^), the tangents
to the three lines of curvature through that point, which are the inter-
sections of those three eonfocals.
(/). And because its equation (Vi) does not involve the constant
/, of 407, (a), (3), we arrive at the following theorem (p. 678) : — If
indefinitely many quadrics. with a common centre o, have their asymp'
totic cones biconfocal, and pass through a common point p, their normals
at that point have a quadric cone (G) for their locus.
Article 412 — On Centres of Curvature of Surfaces, .... 679-689
(a). If a be the vector of the centre s of curvature of a normal
section of an arbitrary surface^ which touches one of the two lines of
curvature thereon, at any given point p, we have the two fundamental
equations (p. 679),
a = p-{RVv, ("Wi), and i2->dp -1- dUi/ = ; (Wi')
whence
VdpdUa/ = 0, (Wi"), and ^+S^ = 0; (Wi'")
M up
the equation (Wi") being a new form of the general differential equa-
tion of the lines of curvature.
(J). Deduction (pp. 680, 681, &c.) of some known theorems from
these equations ; and of some which introduce the new and general
conception of the Index Surface (410, (<?)), as well as that of the
known Index Curve.
(c). Introducing the auxiliary scalar (p. 682),
in which r (|| dp) is a tangent to a line of curvature, while dv = ^dp,
as in (Ui), the two values of r, which answer to the two rectangular
directions (Ti") in 410, (c), are given (p. 680) by the expression,
r = ~ ^r - TX/i . cos {I ~+ L -^), (X'l)
^ /*.
in which ^, X, [x, are, for any given point p, the constants in the equa-
tion (Ui") of the index surface; the difference of the tioo curvatures
jK"> therefore vanishes at an umbilic of the given surface, whatever the
form of that surface may be : that is, at a point, where v || X or || ^,
and where consequently the index curve is a circle.
(d). At any other point p of the given surface, which is as yet en-
tirely arbitrary, the values of r may be thus expressed (p. 681),
n = ar2,r2=ao-2, (Xi")
ai, &2 being the scalar semiaxes (real or imaginary) of the index curve
(defined, comp. 410, {d), by the equations Sp'^p' = 1, S»'p' = 0),
CONTENTS. XXXV
Pages.
(<?). The quadratic equation, of wMch. ri and rg, or the inverse
squares of the two last semiaxes, are the roots, maybe ■written (p. 683)
under the symhoUeal form,
Sv-i (^ + r) -ij/ = ; (Yi)
■which may be developed (same page) into this other form,
y2 + rSv-ixi' + Sv-J 1//J/-0, (Y'l)
the linear and vector functions, i// and x> being derived from the func-
tion <p, on the plan of the Section III. ii. 6 (pp. 440, 443).
(/). Hence, generally, the product of the two curvatures of a sur-
face is expressed (same p. 683) by the formula,
JSi-ii^a-i = n ^-z Ti/ -» = - S — ii/ — ; (Zi)
V V
•which -will be found useful in the foUo-wing series (413), in connexion
■with the theory of the Measure of Curvature.
{g). The given surface being still quite general, if ■we ■write
(p. 686),
r = TJdjO, r' = U (I'dp), (A2), and therefore tt = Uv, ' (A'2)
so that T and r' are unit tangents to the lines of curvature, it is easily
proved that
dr' = rSr'dr, (B2), or that Yrdr' = 0; (B'2)
this general parallelism of dr to r being geometrically explained, by
observing that a line of curvature oti any surface is, at the same time,
a line of curvature on the developable normal surface, -which rests upon
that line, and to -which r' or vt is normal, if r be tangential to the
line.
(A). If the vector of curvature (389) of a line of curvature be
projected on the normal v to the given surface, the projection
(p. 686) is the vector of curvature of the normal section of that sur-
face, -which has the same tangent r ; but this result, and an analo-
gous one (same page) for the developable normal surface {g), are
virtually included in Meusnier's theorem, -which -will be proved by
quaternions in Series 414.
if). The vector a of a centre s of curvature of the given surface,
ans-wering to a given point p thereon, may (by (Wi) and (Xi)) be ex-
pressed by the equation,
(T = p + r-iv; (C2)
•w^hich may be regarded also as a general form of the Vector Equation
of the Surface of Centres, or of the locus of the centre s : the vari-
able vector p of the point p of the given surface being supposed (p. 501)
to be expressed as a vector function of two independent and scalar
variables, whereof therefore v, r, and <t become also functions,
although the two last involve an ambiguous sign, on account of the
Two Sheets of the surface of centres.
{j ). The normal at s, to -what may be called the First Sheet, has
the direction of the tangent r to -what may (on the same plan) be
called the First Line of Curvature at r ; and the vector v of the point
XXXvi CONTENTS.
corresponding to 8, on tho corresponding sA^^^ of tho Heeiproeal (comp.
pp. 607, 508) of the Surface of Centres^ has (by p. 684) the expres-
sion,
i; = r(Spr)-i; (Dg)
which may also be considered (comp. (ff) to be a form of the Vector
Equation of that Reciprocal Surface.
(Jc). The vector v satisfies generally (by same page) the equations
of reciprocity,
Su(r = Say = l, Su5<r = 0, Sc^u = 0, (Dg')
^(T, 5v denoting any infinitesimal variations of the vectors o and v,
consistent with the equations of the surface of centres audits recipro-
cal, or any linear and vector elements of those two surfaces, at two
corresponding points ; we have also the relations (pp. 684, 685),
Spv=l, Sj/v = 0, Si/v0u = O. (D2")
{t). The equation Sv (w - p) = 0, or more simply,
Svw = 1, (E2)
in which w is a variable vector, represents (p. 684) the normal plane
to \h.Q first line (/) of curvature at p ; or the tangent plane at s to the
first sheet of the surface of centres : or finally, the tangent plane to
that developable normal surface (y), which rests upon the second line of
curvature, and touches the first sheet oion^ a certain mrw, whereof we
shall shortly meet with an example. And if v be regarded, comp. («),
as a vector function of two scalar variables, the envelope of the variable
2)lane (E2) is a sheet of the surface of centres ; or rather, on account of
tho ambiguous sign {i\ it is that surface of centres itself : while, in
like manner, the reciprocal surface (j) is the envelope of this other
Pages.
S(Ta> = 1. (E2')
(m). The equations (Wi), (Wi) give (comp. the Note to p. 684),
d(T=di2.Uv; (F2)
combining which with (C2), we see that the equations (Hi) of p. xxv.
are satisfied, when the derived vectors p' and tr' are changed to the cor-
responding differentials, dp and d<r. The known theorem (of Monge),
that each Line of Curvature is generally an involute, with the corre-
sponding Curve of Centres for one of its evolutes (400), is therefore in
this way reproduced : and the connected theorem (also of Monge),
that tJiis evolute is a geodetic on its oicn sheet of the surface of centres,
follows easily from what precedes.
(n). In the foregoing paragraphs of this analysis, the given sur-
face has throughout been arbitrary, or general, as stated in {d) and
(^). But if we now consider specially the case of a central quadric,
several less general but interesting results arise, whereof many, but
perhaps not all, are known ; and of which some may be mentioned
here-
CONTENTS. xxxvii
Pages,
(o). Supposing, then, that not only dj/ = ^d/o, but also v — 0p, and
Spv =fp = 1, the Index Surface (410, {d)) becomes simply (p. 670) the
given surface, with its centre transported from o to p ; whence many
simplifications foUow.
(p). For example, the semiaxes ai, 0.2 of the index curve arc now
equal (p. 681) to the semiaxes of the diametral section of the given
surface, made by a plane parallel to the tangent plane ; and Tv is, as
in 409, the reciprocal P-i of the perpendicular^ from the centre on this
latter plane ; whence (by (Xi) and Xi")) these known expressions
for the two* curvatures result :
iJi-i = Par* ; /?2-> = i^3-'. (G2)
(§'). Hence, by (e), if a neio surface be derived from a given cen-
tral quadric (of any species'), as the locus of the extremities of normals,
erected at the centre, to the planes of diametral sections of the given
surface, each such normal (when real) having the letigth of one of the
semiaxes of that section, the equation of this new surf ace f admits
(p. 683) of being written thus :
Sp(0-p-2)-ip = o. (H2)
(r). Under the conditions (o),the expression (C2) for a gives (p. 684)
the two converse forms,
(r = r-i(^ + r)p, (I2), p = r(0 + y)-i<T; (I2')
whence (pp. 684, 689),
v = r(^ + r)-i^(r, (J2), cr = (^-1 +^"0 1^; (J2')
and therefore (p. 689), by (<?), (p), and by the theory (407) of con-
focal surfaces,
<Tl = (p2'^V = 02" ' 0p, (K3)
if 02 be formed from (p by changing the semiaxes abc to ^252^2 ; it
being understood that the given quadric (abc) is cut by the two confo-
cals (^aibiCi) and {aib^e^), in the first and second lines of curvature
through the given point p : and that <ri is here the vector of that^rs^
centre s of curvature, which answers to the first line (comp. (y ). Of
course, on the same plan, we have the analogous expression.
* Throughout the present Series 412, we attend only (comp. («)) to the curva-
tures of the two normal sections of a surface, which have the directions of the two
lines of curvature : these being in fact what are always regarded as the tivo princi-
pal curvatures (or simply as the two curvatures) of the surface. But, in a shortly
subsequent Series (414), the more general case will be considered, of the curva-
ture of any section, normal or oblique.
t When the given surface is an ellipsoid, the derived SMriQ.CQ is the celebrated
Wave Surface of Fresnel : which thus has (H2) for a symbolical form of its equa-
tion. When the given surface is an hyperboloid, and a semiaxis of a section is
imaginary, the (scalar and now positive) square, of the (imaginary) normal erected,
is still to be made equal to the square of that semiaxis.
Pages.
XXXVUl CONTENTS.
(T2= ^1-1^ = ^1-1^/3, (K2')
for the vector of the second centre.
(s). These expressions for «ri, 02 include (p. C89) a theorem of Dr.
Salmon, namely that the centres of curvature of a given quadric at a
given point are ihe poles of the tangent plane, with respect to the two
confocals through that point ; and either of them may he regarded,
by admission of an ambiguous sign (comp. (?)), as a new Vector Form*
of the Equation of the Surface of Centres, for the case (0) of a given
central quadric.
(t). In connexion with the same expressions for ci, (T2, it may be
observed that if ri, r^ be the corresponding values of the auxiliary
scalar r in (c), and if r, r' stiU denote the unit tangents (g) to the
first and second lines of curvature, while abc, aibiCi, and a2hc2 retain
their recent significations (r), then (comp. pp. 686, 687, see also p.
652),
n =fT =fTJdp = (a-i - fl22)-» = &c., (L2)
and rz ^fr' =f'Uvdip = (a^ — «i2)-i = &c. ; (L2')
this association of ri and ci with 02, &c., and of r^ and 02 with ai,
Sec, arising from the circumstance that the tangents t andr' have re-
spectively the directions of the normals vz and vi, to the two confocal
surfaces, (ozhcz) and (aihiCi').
(ti). By the properties of such surfaces, the scalar here called rz is
therefore constant, in the whole extent of o. first line of curvature ;
and the same constancy of r^, or the equation,
d/Ui/dp = 0, (M2)
may in various ways be proved by quaternions (p. 687).
(w). "Writing simply r and r' for ri and r^, so that r' is constant,
but r variable, for afrst line of curvature, while conversely r is con-
stant and r' variable for a second line, it is found (pp. 684, 685, 586),
that the scalar equation of the surface of centres (i) may be regarded
as the result of the elimination of r-i between the two equations,
I = S.er(l + r-V)-2^(T, (N2), and = S.(r (l + y-i^)-^^^; (N2O
whereof the latter is the derivative of the former, with respect to the
scalar r'K It follows (comp. p. 688), that the First Sheet of the Sur-
face of Centres is touched hy an Auxiliary Quadric (N2), along a Quartic
Curve (N2) (N'2')» which curve is the Locus of the Centres of First Cur-
vature, for all the points of a Line of Second Curvature ; the same
sheet being also touched (see again p. 688), along the same curve, by
the developable normal surface (I), which rests on the same second line :
with permission to interchange the words, frst and second, through-
out the whole of this enunciation.
(tv). The given surface being still a central quadric (0), the vec-
tors p, (T, V can be expressed as functions of v (comp. {j) (/t) (t)),
* Dr. Salmon's result, that this surface of centres is of the twelfth degree, may
be easily deduced from this form.
CONTENTS. XXXix
Pages,
and conversely the latter can be expressed as a function of any one of
the former ; we have, for example, the reciprocal equations (p. 685),
ff={l+r'Hy^-% (O2), and t; = (l + y-i^)-2 ^(t ; (O2')
from which last the formula (N2) may be obtained anew, by observ-
ing (A-) that Scru = 1. Hence also, by (r), we can infer the expres-
sions,*
p = (^ -I + r-i) u = 02 "^ V, (P2), and v = (p^p = v%; (P2')
and in fact it is easy to see otherwise (comp. p. 645), that vg |1 r jj v,
and Spj/2 = 1 = Spy, whence V2=^vqs, before.
{x). More fully, the two sheets of the reciprocal (/) of the surface
of centres may have their separate vector equations written thus,
vi = 02 10 = v%, V2 = 0ip = vi ; (Pa")
and the scalar equationf of this reciprocal surface itself, considered
as including both sheets, may (by page 685) be thus written, the func-
tions/and i?' being related as in 408, (i),
t;4 = (i^y-l)/v, (Q2)
with several equivalent forms ; one way of obtaining this equation
being the elimination of r between the two following (same p. 685) :
Fu + r-^v^ = l, (Q2') ; fv + rv^ = 0. (Q2")
(y). The two last equations may also be written thus, for the^rst
sheet of the reciprocal surface,
F2 VI = 1, (Hz), and/Uvi = r, (RgO
in which (comp. pp. 685, 689),
Fzv = S V02 -iv = Su (0-1 +r-i) v ; (R2")
and accordingly (comp. pp. 483, 645), we have F2V2=Fv=\, oxi^
/Uj/2=/r = n
(z). For a line of second curvature on the given surface, the scalar
r is constant, as before ; and then the two equations (0,2') » (Q-s'Oj or
(R2), (Il'2), represent jointly (comp. the slightly different enunciation
in p. 688) a certain quartic curve, in which the quadric reciprocal (^-2),
of the second confocal {0,2 h ^2)* intersects the first sheet (j/) of the Re-
ciprocal Surface (Q2) ; this quartic curve, being at the same time the
intersection of the quadric surface (Q2') or (R2), with the quadric cone
(Qa") or (R2')) which is biconcyclic with the given quadric, fp= 1.
* The equation v = V2,= the normal to the confocal (^2 h C2) at r, is not ac-
tually given in the text of Series 412 ; but it is easily deduced, as above, from
the formulge and methods of that Series.
t The equation (Q2) is one oi^Q fourth degree; and, when expanded by co-
ordinates, it agrees perfectly with that which was first assigned by Dr. Booth
(see a Note to p. 685), for the Tangential Equation of the Surface of Centres of a
quadric, or for the Cartesian equation of the Reciprocal Surface,
xl CONTENTS.
Pages.
Article 413. — On the Measure of Curvature of a Surface, . . 689-693.
The object of this short Series 413 is the deduction by quaternions,
somewhat more briefly and perhaps more clearly than in the Lectures,
of the principal results of Gauss (comp. Note to p. G90), respecting
the Measure of Curvature of a Surface, and questions therewith con-
nected.
(a). Let p, Pi, P2be any three near points on a given but arbitrary
surface, and n, Ri, R3 the three corresponding points (near to each other)
on the U/nit spliere, which are determined by the parallelism of the radii
OR, ORi, 0R2 to the normals pn, piNi, P2 N2 ; then the areas of the two
small triangles thus formed will bear to each other the ultimate ratio
p. 690),
.. ARR1R2 V.dUi/^Ui/ a 1 , 1 /QN
lim. = =rr-r = _ S — ^Z — ; (S2)
APP1P2 Yapcp V V
whence, with Gauss's definition of the measure of ctirvature, as the
ultimate ratio of corresponding areas on surface and sphere, we have, by
the formula (Zi) in 412, (/), hia fundamental theorem,
Measure of Curvature — Hi "' R<i "', (Sg')
= Product of the two Principal Curvatures of Sections.
(b). If the vector p of the surface be considered as a function of
two scalar variables, t and u, and if derivations with respect to these
be denoted by upper and lower accents, this general transformation
results (p. 691),
Measure of Curvature =S^S^'-(S^^, (T2)
V V \ V j
in which v = Npp, ; (T2')
with a verification for the notation pqrst of Monge.
(c). The square of a linear element d«, of the given but arbitrary
surface, may be expressed (p. 691) as follows :
ds2 = (Tdp2 =) edif^ + 2/dMw + ^dw^ ; (U2)
and with the recent use (J) of accents, the measure (T2) is proved
(same page) to be an explicit function of the ten scalars,
^yfy9\ e\f\9'\ ^.J.^g.\ and e,-1f:^g"; (U2')
the form of this function (p. 692) agreeing, in all its details, with the
corresponding expression assigned by Gauss. *
(<?). Hence follow at once (p. 692) two of the most important
results of that great mathematician on this subject; namely, that
every Reformation of a Surface, consistent with the conception of it as
an infinitely thin and flexible but inextensible solid, leaves unaltered,
* References are given, in Notes to pp. 690, «&;c. of the present Series 413,
to the pages of Gauss's beautiful Memoir, " Bisquisitiones generales circa Superfi-
cies Curvas,^^ as reprinted in the Additions to Liouvillo's Monge.
CONTENTS. xK
Pages.
1st, the Measure of Curvature at any Point, and Ilnd, the Total
Curvature of any Area : this last being the area of the corresponding
portion (a) of the unit-sphere.
(e). By a suitable choice of t and ti, as ccTtsdn yeodetic co-ordinates,
the expression (Uo) naay be reduced (p. 692) to the following,
ds2 = d^2 ^ y,2(1^^2 . (-XJ2")
where ^ is the length of a geodetic arc ap, from a fixed point a to a
variable point p of the surface, and u is the angle bap which this
variable arc makes with a fixed geodetic ab : so that in the immediate *
neighbourhood of a, we have n=t, and n' — Dtn = 1.
(/). The general expression (c) for the measure of curvature takes
thus the very simple form (p. 692),
i?i-i J?2-i =r - n-^n" = - n-^Dt^n ; (V2)
and we have (comp. (^)) the equation (p. 693),
Total Curvature of Area apq, = Aw - J w'dw ; (Vg')
this area being bounded by two geodetics, ap and aq, which make with
each other an angle = Am, and by an arc pq, of an arbitrary curve on
the given surface, for which t, and therefore n, may be conceived to
be a given function of u.
(jl'). If this arc pq be itself a geodetic, and if we denote by v the
variable angle which it makes at p with ap prolonged, so that tan v
= ndu:dt, it is found that df = - ^/dw ; and thus the equation (V2')
conducts (p. 693) to another very remarkable and general theorem of
Gauss, for an arbitrary surface, which may be thus expressed,
Total Curvature of a Geodetic Triangle abc = a+b + c — tt, (V2")
= what may be called the Spheroidal Excess of that triangle, the total
area (47r) of the unit-sphere being represented by eight right angles :
with extensions to Geodetic Polygons, and modifications for the case of
what may on the same plan be called the Spheroidal Defect, when the
two curvatures of the surface are oppositely directed.
Article 414. — On Curvatures of Sections (Normal and Oblique)
of Surfaces ; and on Geodetic Curvatures, 694-698
(a). The curvatures considered in the two preceding Series hav-
ing been those of the principal normal sections of a surface, the present
Scries 414 treats briefly the more general case, where the section is
made by an arbitrary plane, such as the osculating plane at p to an
arbitrary curve upon the surface.
(J>). The vector of curvature (389) of any such curve or section
being (p - k)-i = T>s'^p, its normal and tangential components are found
to be (p. 694),
(p - (t)-i = y-^S -^ = (p - (Ti)-i cos2 V + {p - (To)-! sin2 v, (Wa)
and (p - !)-!= j/-'dp-iSj/dp-id2p ; (W2')
the former component being the Vector of Normal Curvature of the
g
xlii CONTENTS.
Pages.
Surface, for the direction of the tangent to the curve : and the latter
being the Vector of Geodetic Curvature of the same Curve (or section).
(c). In the foregoing expressions, <r and ^ are the vectors of the
points s and x, in which the axis of the osculating circle to the curve
intersects respectively the normal and the tangent plane to the sur-
face (p. 694) ; s is also the centre of the sphere, which osculates to
the surface in the direction dp of the tangent ; ci, (Xz are the vectors
of the two centres Si, S2, of curvature of the surface, considered in Se-
ries ^2, which are at the same time the centres of the two osculating
spheres, of which the curvatures are (algebraically) the greatest and
least : and v is the angle at which the curve here considered crosses
the^rs^ line of curvature.
(d). The equation (W2) contains a theorem of Euler, under the
form (p. 695),
E- 1 = i?i- 1 cos2 v + i22-5 sin2 v ; (W2")
it contains also Meusnier's theorem (same page), under the form
(comp. 412, (7i)) that the vector of normal ctcrvature (J) of a surface,
for any given direction, is the projection on the normal v, of the vector
of oblique curvature, whatever the inclination of the plane of the sec-
tion to the tangent plane may be.
(e). The expression (W2'), for the vector of geodetic curvature, ad-
mits (p. 697) of various transformations, with corresponding expres-
sions for the radius T(p — ^) of geodetic curvature, which is also the
radius of plane curvature of the developed curve, when the developable
circumscribed to the given surface along the given curve is unfolded
into a plane : and when this radius is constant, so that the developed
curve is a circle, or part of one, it is proposed (p. 698) to caU the given
curve &Didonia (as in the Lectures), from its possession of a certain iso-
perimetrical property, which was first considered by M. Delaunay,
and is represented in quaternions by the formula (p. 697),
^JS(U»/.dp^jo) + c^JTdp = 0; (X2)
or c-1 dp = V(U J/ . dUdp), (X'2)
by the rules of what may be called the Calculus of Variations in Qua-
ternions : c being a constant, which represents generally (p. 698)
the radius of the developed circle, and becomes infinite for geodetic
lines, which are thus included as a case of JDidonias.
Article 41 5. — Supplementary Remarks, 698-706
(a). Simplified proof (referred to in a Note to p. xii), of the gene-
ral existence of a system oi three real and rectangular directions, which
satisfy the vector equation Yp<pp = 0, (P), when ^ is a linear, vector,
and self-conjugate function ; and of a system of three real roots of the
cubic equation M=Q (p. xii), under the same condition (pp. 698-
700).
(h). It may happen (p. 701) that the differential equation,
S»'dp = 0, (Y2)
CONTENTS. xliii
Pages,
is inteffrable, or represents a system of surfaces, without the expression
Svd/o being an exact differential, as it was in 410, (b). In this case,
there exists some scalar /ac^or, n, such that S^^vdp is the exact diffe-
rential of a scalar function of p, without the assumption that this vec-
tor p is itself 0. function of a scalar variable, t; and then if we write
(pp. 701, 702, comp. p. xxx),
div = ^dp, d . wv = (idp, (Y2')
this new vector function d) will be self-conjugate, although the function
is not such now, as it was in the equation (Ui).
(tf). In this manner it is found (p. 702), that the Condition* ofln-
tegrability of the equation (Y3) is expressed by the very simple for-
mula,
Syv=0; (Y2")
in which y is a vector function of p, not generally linear, and deduced
from ^ on the plan of the Section Ill.^ii. 6 (p. 442), by the relation,
0dp-fdp = 2Vydp; (Ya"')
0' being the conjugate of <j), but not here equal to it.
(d). Connexions (pp. 702, 703) of the Mixed Transformations in
the last cited Section, with the known Modular and Umbilicar Gene-
rations of a surface of the second order.
(/). The equation (p. 704),
T(p-V.^Vya) = T(a-V.yV/3p), (Z,)
in which a, (3, y are ant/ three vector constants, represents a central
quadric, and appears to offer a new mode of generation\ of such a sur-
face, on which there is not room to enter, at this late stage of the
work.
(/). The vector of the centre of the quadric, represented by the
equation /p - 2S£p = const., with /p = Sp^p, is generally k = ^-'f
= m"it//f (p. 704) ; case oi paraboloids, and of cylinders.
(g). The equation (p. 705),
^qpq'pq'p + Sp^p + Syp + C = 0, (Z2')
represents the general surface of the third degree, or briefly the General
Cubic Surface ; C being a constant scalar, y a constant vector, and q,
q', q" three constant quaternions, while ^p is here again a linear,
vector, and self-conjugate function of p.
(Ji). The General Cubic Cone, with its vertex at the origin, is thus
represented in quaternions by the monomial equation (same page).
* It is shown, in a Note to p. 702, that this monomial equation (^"-i) be-
comes, when expanded, the known equation of six terms, which expresses the con-
dition of integrability of the differential equation ^;daJ4- g'd?/ + rdz = 0.
t In a Note to p. 649 (akeady mentioned in p. xxviii), the reader will find
references to the Lectures, for several different generations of the ellipsoid, derived
from quaternion forms of its equation.
xliv CONTENTS.
Pages.
Sqpq'pq'p = 0. (Z-i")
(t). Scretv Surface, Screw Sections (p. 705) ; Skew Centre ofS/cew
Arch, with illustration by a diagram (Fig. 85, p. 708).
Section 8. — On a few Specimens of Physical Applications
of Quaternions, with some Concluding Remarks, 707 to the end.
Article 416.— On the Statics of a Rigid Body, 707-709
(«). Equation of Equilibrium,
Vr2/3 = SVa^; (Ag)
each a is a vector of application ; (3 the corresponding vector of applied
force ; y an arbitrary/ vector : and this one quaternion formula (A3)
is equivalent to the system of the six usual scalar equations
(X = 0, r= 0, ^= 0, 2 = 0, M= 0, ]sr= o).
(A.) When S (2/3. SVaiS) = 0, (B3), but not ^(5 = 0, (C3)
the applied forces have an unique resultant = 2^3, which acts along
the Hne whereof (A3) is then the equation, with y for its variable
vector.
(c). When the condition (C3) is satisfied, the forces compound
themselves generally into one couple, of which the nxis='S,Ya(3, what-
ever may be the position of the assumed origin o of vectors.
(d). When 2 V«/3 = 0, (D3), with or without (C3),
the forces have no tendency to turn the body round that point o ; and
when the equation (A3) holds good, as in (a), for an arbitrary/ vector
y, the forces do not tend to produce a rotation* round anf/ point c,
so that they completely balance each other, as before, and both the
conditions (C3) and (D3) are satisfied.
(e). In the general case, when neither (C3) nor (D3) is satisfied, if g
be an auxiliary quaternion, such that
j2/3 = 2Va/3, (E3)
then \g is the vector perpendicular from the origin, oa the central
axis of the system ; and if c = S-7, then c2/3 represents, both in quan-
tity and in direction, the axis of the central couple.
(/). If Q be another auxiliary quaternion, such that
Q2/3 = 2fl/3, (F3)
with T2/3 > 0, then SQ = c = central moment divided by total force ;
* It is easy to prove that the moment of \!ne force (3, acting at the end of the
vector a from o, and estimated with respect to any unit-line i from the same ori-
gin, or the energy with which the force so acting tends to cause the body to turn
round that line t, regarded as a. fixed axis, is represented by the scalar, - Sfa/3, or
St"ia^; so that when the condition (D3) is satisfied, the applied forces have no
tendency to produce rotation round any axis through the origin : which origin
becomes an arbitrary ^jo in t c, when the equation of equilibrium (A3) holds good.
CONTENTS.
and V^ is the vector y of a point c ujjon the central axis which does
not vary Math the origin o, and which there are reasons for considering
as the Central Foint of the system, or as the general centre of applied
forces : in fact, for the case of parallelism, this point c coincides with
what is usually called the centre of parallel forces.
(^g). Conceptions of the Total Ifoment Iia(3, regarded as being ge-
nerally a quaternion ; and of the Total Tension, — Sa/3, considered as
a scalar to which that quaternion with its sign changed reducesitself
for the case of equilibrium (a), and of which the value is in that
case independent of the origin of vectors.
(A). Frinciple of Virtual Velocities,
^S(3Sa = 0, (G3)
Article 417. — On the Dynamics of a Eigid Body,
{a). General Eqication of Bgnamics,
2wS(Di2a-4)^a = 0; (H3)
the vector ^ representing the accelerating force, or m% the moving
force, acting on a particle m of which the vector at the time z! is a ;
and ha being any infinitesimal variation of this last vector, geometri-
cally compatible with the connexions between the parts of the
system, which need not here be a rigid one.
(5). For the case oi^free system, we may change each ^a to e + Vta,
£ and t being any two infinitesimal vectors, which do not change in
passing from one particle m to another ; and thus the general equa-
tion (H3) furnishes two general vector equations, namely,
2w (Di2a - ^) = 0, (I3), and 2mVa (D^^a - ?) = ; (J3)
which contain respectively the law of the motion of the centre of
gravity, and the law of description of areas,
{c). If a body be supposed to be rigid, and to have o, fixed point
o, then only the equation (J3) need be retained ; and we may write,
D<a=Vta, (K3)
t being here o. finite vector, namely the Vector Axis of Instantaneous
Rotation : its versor TJt denoting the direction of that axis, and its
tensor Tt representing the angular velocity of the body about it, at the
time t.
{d). "When the forces vanish, or balance each other, or compound
themselves into a single force acting at the fixed point, as for the case
of a heavy body turning freely about its centre of gravity, then
SwVa4 = 0, (L3) ; and if we write, ^i='2maYai, (M3)
so that (p again denotes a linear, vector, and self- conjugate function,
we shall have the equations,
0Dii + V*0i=O, (N3); 0t+r = O, (O3); St0t=A2; (P3)
whence Siy + h^ = 0, (Q3), and 0D<t = Vty; (E3)
the vector y being what we may call the Constant of Areas, and the
scalar h^ being the Constant of Living Force.
xlv
Pages.
709-713
xlvi CONTENTS.
(e). One of Poinsot's representations of the motion of a body, under
tlie circumstances last supposed, is thus reproduced under the form,
that the Ellipsoid of Living Force (P3), with its centre at ilnQ fixed
point o, rolls witJiout gliding on the f zed plane (Q3), which is parallel
to the Plane of Areas (Sty = 0) ; the variable semidiameter of contact^
I, being the vector-axis (c) of instantaneous rotation of the body.
(/). The Moment of Inertia, with respect to ang axis i through 0,
is equal to the living force (Ji^) divided by the square (Tt^) of the
semidiameter of the ellipsoid (P3), which has the direction of that axis ;
and hence may be derived, with the help of the first general construc-
tion of an ellipsoid, suggested by quaternions, a simple geometrical
representation (p. 711) of the square-root of the moment of inertia
of a body, with respect to any axis ad passing through a given point
A, as a certain right line bd, if cd = ca, with the help of two other
points B and c, which are likewise fixed in the body, but may be
chosen in more ways than one.
(y). A cone of the second degree,
Stj/=0, (S3), with V = -y^^t _ ^202t^ (T3)
ia fixed in the body, but rolls in space on that other cone, which is the
locKs of the instantaneous axis i ; and thus a second representation,
proposed by Poinsot, is found for the motion of the body, as the rolling
of one cone on another.
(A). Some of Mac Cullagh's results, respecting the motion here
considered, are obtained with equal ease by the same quaternion
analysis ; for example, the line y, although fxed in space, describes
in the body an easily assigned cone of the second degree (p. 712), which
cuts the reciprocal ellipsoid,
Sy0-iy = A2, (U3)
in a certain sphero-conic : and the cone of normals to the last men-
tioned cone (or the locus of the line t + h^y-^) rolls on the plane of areas
(Sty = 0).
(0- The Three {Frincipat) Axes of Inertia of the body, for the
given point o, have the directions (p. 712) of the three rectangular and
vector roots (comp. (P), p. xii., and the paragraph 415, (a), p. xlii.)
of the equation
Vt^i= 0, (V3), because, for each, D<t = ; (V3')
and if ^, B, C denote the three Principal Moments of inertia corre-
sponding, then the Symbolical Cubic in (comp. the formula (N) in
page xii.) may be thus written,
(0 \A) (^ + ^) (0 + C) = 0. (W3)
(». Passage (p. 713), from moments referred to axes passing
through a given point o, to those which correspond to respectively
parallel axes, through any other point Q of the body.
Pages
. CONTENTS. xlvii
Pages.
Article 418. — On the motions of a System of Bodies, considered
as free particles m, m, . . whicli attract each other according to the
law of the Inverse Square 713-717
(a). Equation of motion of the system,
SmSD^Sa^a + ^P= 0, (X3), if P= 2mm'T (a - a')"' ; (Y3)
a is the vector, at the time t, of the mass or particle m ; P is the po-
tential (jav force-function) ; and the infinitesimal variations ^a are ar-
bitrary.
(i). Extension of the notation of derivatives,
dP= 2S (DaP. Sa). (Z3)
(<?), The differential equations of motion of the separate masses
m, . . become thus, •
mDt2a+DaP=0, . . ; (A4)
and the laws of the centre of gravity, of areas, and of living force,
are obtained under the forms,
2mD<a = /3, (B4); 2MVaD<a = y; (C4)
and r=-i5:w(D<a)2=P+^; (d/)
(3, y being two vector constants, and S a scalar constant
(d). Writing,
P= r (P+ T) df, (E4), and r= r 2 Pdi{ = P + tR, (Ft)
F may be called the Principal* Function, and V the Characteristic
Function, of the motion of the system ; each depending on the final
vectors of position, a, a', . . and on the initial vectors, uq, a'o, . . ; but
F depending also (explicitly) on the time, t, while V (= the Action^
depends instead on the constant JBTof living force, in addition to those
final and initial vectors : the masses m, m', . . being supposed to be
known, or constant.
(e). We are led thus to equations of the forms,
mBta + DaP= 0, . . (G4) ; -mB^a + Da^F= 0, . . (H4) ;
(BtF) = -Sr, (I4)
whereof the system (G4) contains what may be called the Interme-
diate Integrals, while the system (H4) contains the Final Integrals,
of the differential Equations of Motion (A4),
(/). In like manner we find equations of the forms,
Dar=-mD<a, .. (J4); D„^r=wDoa, . . (K4); DjF=*; (L4)
the intermediate integrals (e) being here the result of the elimination
* References are given to two Essays by the present writer, " On a General
Method in Dynamics," in the Philosophical Transactions for 1834 and 1835, in
which the Action (V), and a certain other function (S), which is here denoted by P,
were called, as above, the Characteristic and Principal Fimctions. But the ana-
lysis here used, as being founded on the Calculus of Quaternions, is altogether
unlike the analysis which was employed in those former Essays.
xlviii CONTENTS.
Pages.
of H, between the system (J4) and the equation (L4) ; and the final
integrals, of the same system of differential equations (A4), being now
(theoretically) obtained, by eliminating the same constant R between
(K4) and (L4).
{g). The functions F and V are obliged to satisfy certain Partial
Differential Equations in Quaternions, of which those relative to the
final vectors a, a\ . . are the following,
(D,i^)-i2m-i(D„jF)2=P, (M4); |2m-i(D^r)2 + P+Jf = 0; (N4)
and they are subject to certain geometrical conditions, from which
can be deduced, in a new way, and as new verifications, the law of mo-
tion of the centre of gravity, and the law of description of areas.
(A). General appro:^mate expressions (p. 717) for the functions
i^and V, and for their derivatives jH" and t, for the case oi o. short mo-
tion of the system.
Article 419. — On the Relative Motion of a Binary System ; and
on the Law of the Circular Hodograph, 717-72
(«). The vector of one body from the other being a, and the dis-
tance being r (= Ta), while the sum of the masses is M, the differen-
tial equation of the relative motion is, with the law of the inverse
square,
D^a = jlfa-»r-i ; (O4;
D being here used as a characteristic of derivation, with respect to the
time t.
(J)). As a first integral, which holds good also for any other law
of central force, we have
VaDa = /8 = a constant vector ; (P4)
which includes the two usual laws, of the constant plane {-^ j3), and
of the constant areal velocity ( - = |T/3
(c). Writing r = Da = vector of relative velocity, and conceiving this
new vector r to be drawn from that one of the two bodies which is
here selected for the origin o, the locus of the extremities of the vector
T is (by earlier definitions) the Hodograph of the Relative Motion ;
and this hodograph is proved to be, for the Law of the Inverse Square,
a Circle.
(d). In fact, it is>hown (p. 720), that for any /««<? of central force,
the radius of curvature of the hodograph is equal to the force, multi-
plied into the square of the distance, and divided by the doubled areal
velocity ; or by the constant parallelogram c, under the vectors (a
and r) ot position and velocity, or of the orbit and the hodograph.
(e). It follows then, conversely, that the law of the inverse square
is the only law which renders the hodograph generally a circle ; so
that the law of nature may be characterized, as the Law of the Circular
Hodograph : from which latter law, however, it is easy to deduce
the form of the Orbit, as a conic section with di focus at o.
CONTENTS. Xlix
Pages.
(/). If the semiparameter of this orbit be denoted, as usual, by
Pf and if h be the radius of the hodograph^ then (p. 719),
h = Mc-^ = cp-^ = {Mp-^yi. (Qi)
(jg). The orbital excentricity e is also the hodographic excentri-
city, in the sense that eh is the distance of the centre h of the hodo-
graph, from the point o which is here treated as the centre of force.
(A). The orbit is an ellipse^ when the point o is interior to the
hodographic circle (^ < 1) ; it is a parabola, when o is on the circum-
ference of that circle (e= 1) ; and it is an hyperbola, when o is an an-
terior point (e> 1). And in all these cases, if we write
a=p(l-e^y^ = ch-^(l-e^y\ (R^)
the constant a will have its usual signification, relatively to the
orbit.
(0- The quantity Mr-^ being here called the Potential, and de-
noted by P, geometrical constructions for this quantity P are assigned,
with the help of the hodograph (p. 723) ; and for the harmonic mean,
2M(r + y')-», between the two potentials, P and P', which answer to
the extremities t, t' of any proposed chord of that circle : all which
constructions are illustrated by a new diagram (Fig. 86).
ij). If u be the pole of the chord tt' ; m, m' the points in which
the line ou cuts the circle ; l the middle point, and n the pole, of the
new chord mm', one secant from which last pole is thus the line ntt' ;
u' the intersection of this secant with the chord mm', or the harmonic
conjugate of the point u, with respect to the same chord ; and nt,t/
any near secant from n, while u, (on the line ou) is the pole of the
near chord Tjs I : then the two small arcs, Tjr and t't/, of the hodo-
graph, intercepted between these two secants, are proved to be xHiii-
maielj proportional to the ttvo potentials, P andP'; or to the two
ordinates tv, t'v', namely the perpendiculars let fall from t and t', on
what may here be called the hodographic axis ln. Also, the harmonic
mean between these two ordinates is obviously (by the construction)
the line u'l; while ux, ut', and u,t, u,t/ oxe four tangents to the
hodograph, so that this circle is cut orthogonally, in the two pairs of
points, T, t' and t,, t/, by two other circles, which have the two near
points TJ, u^ for their centres (pp. 724, 725).
(k). In general, for any motion of a point (absolute or relative, in
one plane or in space, for example, in the motion of the centre of the
moon about that of the earth, under the perturbations produced by the
attractions of the sun and planets), with a for the variable vector (418)
oi position of the point, the time dit which corresponds to any vector-
element dDa of the hodograph, or what may be called the time of ho-
dographically describing that element, is the quotient obtained by
dividing the same element of the hodograph, by the vector of accelera-
tion D«a in the orbit ; because we may write generally (p. 724),
J, dDa , TdDa ., ,
d. = __, or d.= .jj^, .f d*>0. (S.)
h
1 CONTENTS.
Q). For tlie law of the inverse square (comp. («) and (O)? the
measure oi the force is,
TD2a = Mr-^ = M-^P^ ; (T4)
the times d^, d^, of hodographically describing the small circular
arcs T,T and t't/ of the hodograph, being found by multiplying the
lengths (y) of those two arcs by the mass, and dividing each product
by the square of the potential corresponding, are therefore inversely
as those two potentials, P, P', or directly as the distances, r, r', in the
orbit : so that we have the proportion,
d^:df :di + d<'=r:/:r + r'. (U4)
(m). If we suppose that the mass, M, and ihe Jive points 0, l, m,
"U, u^ upon the chord mm' are given, or constant, but that the ra-
dius, h, of the hodograph, or the position of the centre h on the hodo-
graphic axis ln, is altered, it is found in this way (p. 725) that
although the two elements of time, d^, dd', separately vary, yet their
sum remains unchanged : from which it follows, that even if the two
circular arcs, tt, t't/, be not small, but still intercepted (/) between
two secants from the pole n of ihe fixed chord mm', the sum (say, M +
A^') of the two times is independent of the radius, h.
(n). And hence may be deduced (p. 726), by supposing one secant
to become a tangent, this Theorem of Sodographic Isochronism, which
was communicated without demonstration, several years ago, to the
Royal Irish Academy,* and has since been treated as a subject of
investigation by several able writers :
If two circular hodographs, having a common chord, which passes
through, or tends towards, a common centre of force, he cut perpendicu-
larly by a third circle, the times of hodographically describing the inter-
cepted arcs will be equal.
(0). This common time can easily be expressed (p. 726), under the
form of the definite integral,
, 2MC^ dw
Time of TMT = -^ ; ; (V4)
9^ Jo (l-e'cosw)2' ^ '^
2g being the length oi the fixed chord mm'; e' the quotient lo : lm,
which reduces itself to - 1 when is at m', that is for the case of a pa-
rabolic orbit ; e lying between ± 1 for an ellipse, and outside those limits
for an hyperbola, but being, in all these cases, constant ; while w is a
certain auxiliary angle, of which the sine = ut : ul (p. 727), or
= 5 (r + r')"i, if s denote the length pp' of the chord of the orbit, cor-
responding to the chord tt' of the hodograph ; and w varies from to 7r,
when the yjhiAe periodic time 2'7rn~^ for a closed orbit is to be computed :
with the verification, that the integral (V4) gives, in this last case,
M=ahi^, as usual. (Wi)
* See the Proceedings of the 16th of March, 1847. It is understood that the
common centre o oi force is occupied by a common mass, M.
CONTENTS.
(p). By examining the general composition of the definite inte-
gral (V4), or by more purely geometrical considerations, which are
illustrated by Fig. 87, it is found that, with the law of the inverse
square, the time t of describing an are pp' of the orbit (closed or un-
closed) is Q. function (p. 729) of the three ratios^
a3 ,.+ / s
M' "^' ^^" ^^^
and therefore simply a function of the chord (s, or fp') of the orbit,
and of the sum of the distances (r + r*, or op + op') when M and a are
given : which is a form of the Theorem of Lambert.
(q). The same important theorem may be otherwise deduced,
through a quite different analysis, by an employment of partial deri-
vatives, and of partial differential equations in quaternions, which is
analogous to that used in a recent investigation (418), respecting the
motions of an attracting system of any number of bodies, m, m', &c.
(r). "Writing now (comp. p. xlvii) the following expression for the
relative living force, or for the mass {M= m + m'), multiplied into the
square of the relative velocity (TDa),
2T=-ifDa2= 2(P+ J?) = if (2r-i - «-i) ; (Y4)
introducing the two new integrals (p. 729),
J5'=r(P+T)d^, (Z4), and r=[*^2TdLt = F+tH, (A5)
which have thus (comp. (E4) and (r4)) the same forms as before, but
with different (although analogous) significations, and may stiU be
called the Principal and Characteristic Functions of the motion ; and
denoting by a, a' (instead of ao, a) the initial and final vectors of po-
sition, or of the orbit, while r, r' are the two distances, and r, r' the
two corresponding vectors of velocity, or of the hodograph : it is found
that when M is given, F may be treated as a function of a, a', t, or
of r, r, s, t, and Fas a function of a, a, a, oxofr, r, s, andJS"; and
that their partial derivatives, in the first view of these two functions,
are (p. 729),
BaF^DaV^T, (Bo); Ba'F=J)a'V=-T'; (Cs)
(J)t)F=-H, (Ds); and D^r = — Dar=<; (E5)
while, in the second view of the same functions, they satisfy the two
partial differential equations (p. 730),
DrF=^Dr'F, (F5), and D,.F=D/r; (G5)
along with two other equations of the same kind, but of the second
degree, for each of the functions here considered, which are analogous
to those mentioned in p. xlviii.
(5). The equations (Fa) (G5) express, that the two distances, r
and /, enter into each of the two functions only by their sum ; so that,
if M be still treated as given, F may be regarded as a function of the
lii CONTENTS.
Pages.
three quantities, r + Z, s, and t\ while F, and therefore also t by
(Es), is found in like manner to be a function of the three scalars,
r + r', s, and a : which last result respecting the time agrees with
(p), and furnishes a new proof of Lambert' s Theorem.
(0- The three partial differential equations (r) in F conduct, by
merely algebraical combinations, to expressions for the three partial
derivatives, DrF, D,' V {=J)rV), and D^F; and thus, with the help
of (E5), to twoneiv definite integrals* (p. 731), which express respec-
tively the Action and the Time, in the relative motion of a binary
system here considered, namely, the two following :
]-s\r^r'-^s a j
whereof the latter is not to be extended, without modification, be-
yond the limits within which the radical is finite.
Article 420. — On the determination of the Distance of a Comet,
or new Planet, from the Earth, 733, 734
(a). The masses of earth and comet being neglected, and the mass
of the sun being denoted by M, let r and w denote the distances of
earth and comet from sun, and z their distance from each other, while
a is the heliocentric vector of the earth (Ta = r), known by the theory
of the sun, and p is the unit- vector, determined by observation, which
is directed from the earth to the comet. Then it is easily proved by
quaternions, that we have the equation (p. 734),
SpDpDV r[M M\
CJ5)
SpDpUa
with t<;2 = r2 + 2;2 _ 2zSa|0 ; (K5)
eliminating w between these two formulae, clearing of fractions, and
dividing by a, we are therefore conducted in this way to an algebrai-
cal equation of the seventh degree^ whereof owe root is the sought dis-
tance, z.
(J}). The final equation, thus obtained, differs only by its notation,
and by the facility of its deduction, from that assigned for the same
purpose in the Mecanique Celeste; and the rw/^ofLaplace there given,
for determining, by inspection of a celestial globe, which of the two
* References are given to the First Essay, &c., by the present writer (comp.
the Note to p. xlvii.), in which wore assigned integrals, substantially equivalent
to (H5) and (I5), but deduced by a quite different analysis. It has recently been
remarked to him, by his friend Professor Tait of Edinburgh, that while the area
described, with Newton's Law, about the full focus of an orbit, has long been
known to be proportional to the time corresponding, so the area about the empty
foam represents (or is proportional to) the action.
CONTENTS. liii
Pages,
bodies (earth and comet) is the nearer to the sun, results at sight from
the formula (Js)-
Article 421. — On the Development of the Disturbing Force of
the Sun on the Moon ; or of one Planet on another, which is nearer
than itself to the Sun, 734-736
(«). Let a, <T be the geocentric vectors of moon and sun ; r (= Ta),
and s(=T(t), their geocentric distances ; JLTthe sum of the masses of
earth and moon ; S the mass of the sim ; and D (as in recent Series)
the mark of derivation with respect to the time : then the differential
equation of the disturbed motion of the moon about the earth is,
D2a = Jf^a4-»7, (Lg), if 0a = 0(a) = a-iTa-', (M5)
and rj — Vector of Disturbing Force = S {(pa - (tr — a)) ; . (N5)
denoting here a vector function, but not a linear one.
(Z»). If we neglect rj, the equation (L5) reduces itseK to the form
T>-a = M<pa ; which contains (comp. (O4)) the laws of undisturbed
elliptic motion.
(c). If we develope the disturbing vector rj, according to ascend-
ing powers of the quotient r : s, of the distances of moon and sun from
the earth, we obtain an infinite series of terms, each representing a
finite group oi partial disturbing forces, which may be thus denoted,
»?=»?i+»?2+»;3 + &c. ; (O5)
n\ = nh\^*lh2l »72=»72,l+»?2j2+ J?2,3, &C. ; (P5)
these partial forces increasing in number, but diminishing in intensity,
in the passage from any one group to the following ; and being con-
nected with each other, within any such group, by simple numerical
ratios and angular relations.
{d). For example, the two forces r\\,\, »;i,2 of the /rs^ group
are, rigorously, proportional to the numbers 1 and 3 ; the three forces
»72,i» »72,2, >72,3 of the second %xo\y^ are as the numbers 1, 2, 5; and
the /02<r forces of the ^Aw-<f group are proportional to 5, 9, 15, 35 :
while the separate intensities of i\ie first forces, in these three first
groups, have the expressions,
'Sr _, 3<Sr« ^ 5Sr3
, (J). All ih.QS>Q partial forces are conceived to act at the moon ; but
their directions may be represented by the respectively jj^mW^/ unit-
lines \J r]\, i, &c., drawn /rom the earth, and terminating on a great
circle of the celestial sphere (supposed here to have its radius equal to
unity), which passes through the geocentric (or apparent) places,
and ]), of the sun and moon in the heavens.
(/). Denoting then the geocentric elongation D oimoon from sun
(in the plane of the three bodies) by 4 ; and by 0i, 03, and ])i, 1)2,
Da, what may be called tivo fictitious suns, and three fictitious moons,
of which the corresponding elongations from 0, in the same great
liv CONTENTS.
Pages.
circle, are +29,- 29, and -0, +B9,-39, as illustrated by Fig. 88
(p. 735) ; it is found that tte directions of the two forces of the Jirst
group are represented by the two radii of this unit-circle, which termi-
nate in D and ])i ; those of the three forces of the secowc? group, by the
three radii to 0i, 0, and 03 ; and those ot the four forces of the
third group, by the radii to h, D, Dij and %', with facilities for ex-
tending all these results (with the requisite modifications), to the
fourth and subsequent groups, by the same quaternion analysis.
(g). And it is important to observe, that no supposition is here
made respecting any smallness of excentricities or inclinations (p. 736) ;
so that all the formulce apply, with the necessary changes oi geocen-
tric to heliocentric vectoT^, &c., to the perturbations of the motion of a
coinet aboict the sun, produced by the attraction of a planet, which is
(at the time) more distant than the comet from the sun.
Article 422— On Fresnel's Wave, 736-756
(a). If p and fi be two corresponding vectors, of ray-velocity and
wave-slowness, or briefly Hay and Index, in a biaxal crystal, the velo-
city of light in a vacuum being unity ; and if dp and Sfx, be any infi-
nitesimal variations of these two vectors, consistent with the equa-
tions (supposed to be as yet unknown), of the Wave (or wave- surface),
and its reciprocal, the Index-Surface {or surface of ivave-sloivness) : we
have then first the fundamental Equations of ^Reciprocity (comp. p.
417),
S/ip=-l, (Ra); S/ti5p = 0, (Ss); Sp^/i = 0, (T5)
which are independent of any hypothesis respecting the vibrations of
the ether.
(b). If dp he next regarded as a displacement (or vibration), tan-
gential to the wave, and if de denote the elastic force resulting, there
exists then, on Fresnel's principles, a relation between these two small
vectors ; which relation may (with our notations) be expressed by
either of the two following equations,
de = r'^p, (U5), or dp = ^ds; (Vg)
the function ^ being of that linear, vector, and self- conjugate kind,
which has been frequently employed in these Elements.
{c). The fundamental connexion, between the functional symbol
<p, and the optical constants abc of the crystal, is expressed (p. 741,
comp. the formula (W3) in p. xlvi) by the symbolic and cubic equa-
tion,
i<p + «-2) (^ + i-2) (0 + c-2) = ; ( W5)
of which an extensive use is made in the present Series.
(d). The normal component, /x-iS/x^c, of the elastic force de, is in-
effective in Fresnel's theory, on account of the supposed incompressi-
bility of the ether; and the tangential component, ^-^dp~ fi-^S/xds, is
(in the same theory, and with present notations) to be equated to
CONTENTS, Iv
Pages.
fi-^Sp, for the propagation of a rectilinear vibration (p. 737) ; we ob-
tain then thus, for such a vibration or tangential displacement, dp, the
expression,
^p = (r^-/i-2)-V-»S;/5€; CX5)
and therefore by (S5) the equation,
O = S/i-K0-»-/x-2)-V-S (Y5)
which is a Symbolical Form of the scalar Equation of the Index-Sur-
face, and may be thus transformed,
l = S;u(/*2-^)-V. (Z5)
(e). The Wave- Surface, as being the reciprocal (a) of the index-
surface {d), is easily found (p. 738) to be represented by this other
Symbolical Equation,
O=Sp-i(0-p-2)-'p-i; (Ae)
or l = Sp(p2-^-i)-ip. (Be)
(/). In such transitions, from one of these reciprocal surfaces to
the other, it is found convenient to introduce two auxiliary vectors,
V and w(= ^v), namely the lines ou and ow of Fig. 89 ; both drawn
from the common centre o of the two surfaces ; but v terminating (p.
738) on the tangent plane to the wave, and "being parallel to the direc-
tion of the elastic force de ; whereas w terminates (p. 739) on the tan-
gent plane to the index- surface, and is parallel to the displacement dp.
{g). Besides the relation,
b) = <i>v, or V = ^"'w, (Ce)
connecting the two new vectors (/) with each other, they are con-
nected with p and ft by the equations (pp. 738, 739),
S^t; = -1, (De); Spi; = 0; (Ee)
Spw=-1, (Fe); S/^a; = 0; (Ge)
and generally (p. 739), the following Rule of the Interchanges holds
good: In any formula involving p, fi, v, w, and 0, or some of them,
it is permitted to exchange p with /a, v -with a>, and with 0'' ; pro-
vided that we at the same time interchange dp with Se, but not gene-
rally* Sfi with dp, when these variations, or any of them occur.
(A). We have also the relations (pp. 739, 740),
_ p-i = v-iVv/i = fi + v-i^; (He)
— /*-J = (o'^Ywp = p + 0)-' ; (le)
* This apparent exception arises (pp. 739, 740) from the circumstance, that
dp and ^6 have their directions generally fixed, in this whole investigation
(although subject to a common reversal by +), when p and p. are given ; whereas
dfi continues to be used, as in (a), to denote any infinitesimal vector, tangential to
the index- surface at the end of /u.
Ivi CONTENTS.
with others easily deduced, whichmay all be illustrated by the above-
cited Fig. 89.
(i). Among such deductions, the following equations (p. 740)
may be mentioned,
(Yv<pvy + Sv<pv = 0, (Je); (Vw0-iw)2 + Sw^-iw = ; (Ke)
which show that the Zocus of each of the itvo Auxiliary Points, v and
w, wherein the two vectors v and w terminate (/), is a Surface of
the Fourth Degree, or briefly, a Quartic Surf ace ; of which two loathe
constructions xii9.Y\>e connected (as stated in p. 741) with those of the
two reciprocal ellipsoids,
Sp<pp=l, (Lg), and Sp^-ip = l; (Me)
p denoting, for each, an arbitrary semidiameter.
(y). It is, however, a much more interesting use of these two
ellipsoids, of which (by (W5), &c.) the scalar semiaxes are a, b, c for
the first, and <?"i, b~'^, c-^ for the second, to observe that they may be
employed (pp. 738, 739) for the Constructions of the Wave and the
Index- Surface, respectively, by a very simple rule, which (at least for
t\Q first of these two reciprocal surfaces (a)) was assigned by Fres-
nel himself.
(ky In fact, on comparing the symbolical form (Ae) of the equa-
tion of the Wave, with the form (H2) in p. xxxvii, or with the equa-
tion 412, XLI., in p. 683, we derive at once FresneVs Construction :
namely, that if the ellipsoid (abc) be cut, by an arbitrary plane
through its centre, and \i perpendiculars to that plane be erected at
that central point, which shall have the lengths of the semiaxes of
the section, then the locus of the extremities, of the perpendiculars so
erected, will be the sought Wave-Surface.
(J). A precisely similar construction applies, to the derivation of
the Index- Surface from the ellipsoid (a"'Z>"'c-i) : and thus the two
auxiliary surfaces, (Lg) and (Me), may be briefly called the Generat-
ing Ellipsoid, and the Reciprocal Ellipsoid.
(jn). The cubic (W5) in (j) enables us easily to express (p. 741) the
inverse function (^ + e)-J, where e is any scalar ; and thus, by chang-
ing 6 to — p-3, &c., new forms of the equation (Ac) of the wave are
obtained, whereof one is,
= (0-ip)2 + (p2 + «2 + j2 + c2) Sp^-'p - ame^ ; (Ne)
with an analogous equation in fx (comp. the rule in (y)), to represent
the index-surface : so that each of these two surfaces is of the fourth
degree, as indeed is otherwise known.
(n). If either Sp(p-^p or p2 be treated as constant in (Ne), the
degree of that equation is depressed from the fourth to the second;
and therefore the Wave is cut, by each of the two concentric quadrics,
Sp^-ip = AS (Oe), p2 + r2 = 0, (Po)
in a (real or imaginary) curve of the fourth degree : of which two quar-
Pages.
CONTENTS. Ivii
Pages.
tic curves, answering to all scalar values of the constants h and r, the
wave is the common locus.
(o). The new ellipsoid (Oe) is similar to the ellipsoid (Me), and
similarly placed, while the sphere (Pe) has r for radius ; and every
quartic of the second system (n) is a sphero-conic, because it is, by the
equation (A^) of the wave, the intersection of that sphere (Pe) with
the concentric and quadrie coney
O = Sp(0 + r2)-ip; (Qe)
or, by (Be), with this other concentric quadrie,*
-l = Sp(0-i + y2)-ip^ (Re)
whereof the conjugate (obtained by changing - 1 to + 1 in the last
equation) has
fl;2_y2^ ^2_y2j c2_y2, (Se)
for the squares of its scalar semiaxes, and is therefore confocal with
the generating ellipsoid (Le).
(^). For any point p of the wave, or at the end of any ray p, the
tangents to the two curves (w) have the directions of a> and /iw ; so
that these two quartics cross each other at right angles, and each is a . •
common orthogonal in all the curves of the other system.
((?). But the vibration dp is easily proved to be parallel to (o ;
hence the curves of the^rs^ system (n) are Zincs of Vibration of the
Wave : and the curves of the second system are the Orthogonal Trajec-
toriesf to those Zines.
(r). In general, the vibration dp has (on Fresnel's principles) the
direction of the projection of the ray p on the tangent plane to the
wave ; and the elastic force de has in like manner the direction of the
projection of the index-vector fi on the tangent plane to the index-
surface : so that the ray is ^ms, perpendicular to the elastic force
Article423.— Mac Cullagh's Theorem of the Polar Plane, . . 757-762
********
********
* For real curves of the second system (n), this new quadrie (Ee) is an hy-
perboloid, with one sheet or with two, according as the constant r lies between a
and b, or between b and c ; and, of course, the conjugate hyperboloid (o) has two
sheets or one, in the same two cases respectively.
t In a different theory of light (comp. the next Series, 423), these sphero-
conics on the wave are themselves the lines of vibration.
Iviii
CONTENTS.
Table* of Initial Pages of Aeticles.
Art.
Page.
Art.
Page.
Art.
I
Page, t
Art.
1
Page.
Art.
Page.
Art.
Page.
1
1
49
37
97
1
88
145
i
126
193
173
241
260
2
2
50
38
98
90
146
129
194
174
242
262
3
8
51
39
99
95
147
130
195
175
243
264
4
52
))
100
98
148
11
196
176
244
265
5
4
53
40
101
103
149
131
197
183
245
n
6
6
54
41
102
104
150
132
198
184
246
266
7
55
42
103
t)
151
133
199
185
247
11
8
5
66
43
104
105
152
11
200
187 ;
248
11
9
6
57
44
105
^^
153
134
201
190 1
249
267
10
))
58
46
106
106
154
11
202
11
250
»»
11
7
59
j»
107
„ ;
155
135
203
192 :
251
12
8
60
47
108
»j
156
11
204
193 i
252
268
13
61
48
109
107
157
136
205
200
253
269
14
9
62
49
110
108
158
137
206
202 I
254
272
15
63
50
111
t)
159
138
207
203
255
274
16
10
64
51
112
109
160
139
208
204 ;
256
275
17
65
53
113
110
161
140
209
207 !
257
277
18
11
66
);
114
111
162
142
210
208
268
279
19
))
67
54
115
)>
163
143
211
213
269
11
20
12
68
55
116
;;
164
144
212
214
260
281
21
))
69
n
117
112
165
11
213
11
261
283
22
13
70
57
118
It
166
145
214
217
262
286
23
14
71
;;
119
113
167
146
215
219
263
287
24
)9
72
58
120
11
168
147
216
223
264
11
25
15
73
))
121
114
169
148
217
225
265
289
26
16
74
59
122
11
170
149
218
227
266
290
27
17
75
>)
123
115
171
11
219
229
267
291
28
18
76
60
124
116
172
150
220
232
268
292
29
19
77
61
125
11
173
11
221
233
269
293
30
))
78
J)
126
11
174
151
222
234
270
11
81
20
79
62
127
117
175
^j
223
236
271
295
82
22
80
)j
128
11
176
152
224
239
272
11
33
))
81
J)
129
V
177
153
225
240
273
297
34
23
82
63
130
118
178
^,
226
11
274
298
35
24
83
64
131
It
179
154
227
241
275
301
36
26
84
?>
132
119
180
155
228
244
276
^j
37
28
85
65
133
120
181
157
229
246
277
302
38
29
86
j»
134
11
182
158
230
11
278
11
39
30
87
66
135
121
183
159
231
247
279
303
40
ii
88
67
136
11
184
161
232
11
280
11
41
31
89
68
137
11
185
162
233
248
281
42
32
90
»
138
122
186
163
234
250
282
305
43
83
91
69
139
11
187
166
235
251
283
308
44
5)
92
»5
140
123
188
167
236
253
284
jj
45
34
93
77
141
11
189
168
237
255
285
310
46
35
94
80
j 142
124
190
169
238
257
286
11
47
36
95
83
143
1
191
170
239
11
287
311
48
37
96
85
144
125
192
171
240
259
288
312
* This Table was mentioned in the Note to p. xiv. of the Contents, as one
likely to facilitate reference. In fact, the references in the text of the Elements
are almost entirely to Articles (with their sub -articles), and not to pages.
. CONTENTS.
Table of Initial Taq-es— continued.
lix
Art.
Page.
Art.
Page.
Art.
337
Page.
Art.
Page.
Art.
Page.
Art. Page.
289
312
313
379
417
361
482
385
524
\ 409 664
290
313
314
381
338
420
362
484
386
625
410 667
291
5)
315
383
339
421
363
485
387
527
i 411 674
292
314
316
384
340
422
364
487
388
529
! 412 679
293
315
317
391
341
423 '
365
491
389
631
413 (
389
294
316
318
))
342
427
366
495
390
535
414 694
295
321
319
343
429
367
496
391
637
415 698
296
324
320
292
! 344
431
368
392
538
416 '
ro7
297
331
321
393
345
432
369
)>
393
639
1 417 709 1
298
343
322
394
346
435
370
498
394
641
418 '
ri3
299
347
323
399
347
436
371
500
395
649
i 419 '
'17
300
349
324
400
348
439
372
501
396
664
420 '
r33
301
351
325
401
349
441
373
602
397
659
421 '
r34
302
))
326
403
350
443
374
508
398
578
422 '
r36
303
352
327
404
351
445
375
509
399
612
423 '
757
304
354
328
405
352
447
376
611
400
621
424
305
356
329
406
353
453
377
512
401
626
425
306
358
330
407
354
459
378
613
402
630
426
307
361
331
408
355
464
379
))
403
631
427
308
364
332
409
356
466
380
515
404
633
428
309
366
333
411
357
468
381
519
405
636
429
310
370
334
412
358
470
382
620
406
638
430
311
373
335
414
359
474
383
622
1 407
649
. ,
312
374
336
416
360
481
384
624
1 408
1
663
• •
Table of Pages foe the Figuees.
Figure.
Page.
Figure.
1
Page.
Figure.
Page.
Figure.
Page.
Figure.
Page.
1
1
21
21 i
38
119
64
247
72
348
2
2
i 22
25 1
39
129
65
269
73
369
3
1 23
27
40
130
66 bis
74
397
4
! 24
33
41
56
75
426
6
3
25
36
41 bis
)i
67
274
76
499
6
26
37
42
132
68
280
77
611
7
4
27
42
42 bis
141
69
288
78
517
8
5
28
60
43
144
60
290
79
620
9
6
29
64
44
151
61
80
543
10
>»
30
82
46
152
62
295
81
669
11
7
! 31
91
46 bis
63
324
82
673
12
8
1 32
98
46
164
63 bis
325
83
599
13
10
i 33
108
47
167
64
84
660
14
11
33bis
120
47 bis
158
65
326
85
706
15
13
34
110
48
168
66
327
86
724
16
14
35
112
49
172
67
332
87
727
17
16
i 85bis
143
60
190
68
334
88
736
18
17
; 36
112
61
216
69
^^
89
740
19
20
1 B6bis
126
62
220
70
343
90
. .
20
»
i "
116
63
226
71
344
91
• •
Note. — It appears by these Tables tbat tbe Author intended to have com-
pleted the work by the addition of Seven Articles, and Two Figures.— Ed.
ELEMENTS OF QUATERNIONS.
BOOK I.
ON VECTORS, CONSIDERED WITHOUT REFERENCE TO ANGLES,
OR TO ROTATIONS.
CHAPTER I.
FUNDAMENTAL PRINCIPLES RESPECTING VECTORS.
Section l,— 0?i the Conception of a Vector; and on Equality
of Vectors.
Art, 1 . — A right line ab, considered as having not only length,
but also direction, is said to be a Vector. Its initial point a
is said to be its origin; and its final point b is said to be its
term, A vector ab is conceived to be (or to construct) the
differerice of its two extreme points ; or, more fully, to be the
result of the subtraction of its own origin from its own term ;
and, in conformity with this conception, it is also denoted by
the symbol b - a : a notation which will be found to be exten-
sively useful, on account of the analogies which it serves
to express between geometrical and algebraical operations.
When the extreme points a and b are distinct, the vector ab
or B - A is said to be an actual (or an effective) vector ; but
when (as a limit) those two points are conceived to coincide,
the vector aa or a - a, which then results, is said to be null.
Opposite vectors, such as ab and ba,
or B - a and a - b, are sometimes
called vector and revector. Succes-
sive vectors, such as ab and bc, or Kevector.
B - a and c - b, are occasionally said ^'S- ^•
to be vector and provector: the line ac, or c - a, which is
A
Vector,
b-'a
A
^7^
ELEMENTS OF QUATERNIONS.
[book I,
Fig. 2.
drawn from the origin a of the first to the term c of the second,
being then said to be the trans-
vector. At a later stage, we shall
have to consider vector-arcs and
vector-angles ; but at present, our
only vectors are (as above) right
lines.
2. Two vectors are said to be equal to each other, or the
equation ab = CD, or b - a = d - c, is said to hold good, when
(and only when) the origin and term of the one can be brought
to coincide respectively with the corresponding points of the
other, by transports (or by translations) without rotation. It
follows that all null vectors are equal, and may therefore be
denoted by a common symbol, such as that used for zero ; so that
wemaywrite, ^_ ^ = b _b =&«. = O;
but that two actual vectors, ab and cd, are not (in the present
full sense) equal to each other, unless they have not merely
equal lengths, but also similar directions. If then they do not
happen to be parts of one common line, they must be opposite
sides of a parallelogram, /^ c. ^^ ,d
abdc ; the two lines ad, bc
becoming thus the two dia-
gonals of such a figure, and
consequently bisecting each
other, in some point e.
Conversely, if the two equa-
tions,
D - E = E - A, and
are satisfied, so that the two lines
AD and BC are commedial, or have
a common middle point e, then even
if they be parts of one right line,
the equation D-c=B-Ais satis-
fied. Two radii, ab, ac, of any
one circle (or sphere), can never be equal vectors ; because their
directions differ.
Pig. 4.
CHAP. I.J FUNDAMENTAL PRINCIPLES VECTORS. 3
3. An equation between vectors^ considered as an equidif-
ference of points, admits of inversion and ^ ^
alternation ; or in symbols, if
D - C = B - A,
then
c - D =A-B,
and
D - B = C - A.
Fig. 5.
Two vectors, cd and ef, which are
equal to the same third vector, ab, ^(
are also equal to each other ; and
these three equal vectors are, in
general, the three parallel edges of '^p. g
a prism.
Section 2. — On Differences and Sums of Vectors taken two
by two,
4. In order to be able to write, as in algebra,
(c' - a') - (b - a) = c - B, if c' - a' = c - a,
we next define, that when a first vector ab is subtracted from
a second vector ac which is co-initial with it, or from a third
vector a'c' which is equal to that second vector, the remainder
is that fourth vector bc, which is drawn from the term b of the
first to the term c of the second vector : so that if a vector be
subtracted from a transvector (Art. 1), the remainder is the
provector corresponding. It is evident that this geometrical
subtraction of vectors answers to a decomposition of vections (or
of motions) ; and that, by such a decomposition of a null vec-
tion into two opposite vections, we have the formula,
- (b - a) = (a - a) - (b - a) = A - b ;
so that, if an actual vector ab be subtracted from a null vector
A A, the remainder is the revector ba. If then we agree to
abridge, generally, an expression of the form - « to the
shorter form, - «, we may write briefly, - ab = ba; a and - a
being thus symbols of opposite vectors, while a and - (- a) are,
4 ELEMENTS OF QUATERNIONS. [bOOK I.
for the same reason, symbols of one common vector : so that
we may write, as in algebra, the identity^
5. Aiming still at agreement with algebra, and adopting
on that account the formula of relation between the two signs^
+ and -,
(b -a) + a = b,
in which we shall say as usual that b- ais added to «, and that
their sum is b, while relatively to it they may be jointly called
summands, we shall have the two following consequences :
I. If a vector, ab or b - a, be added to its own origin a,
the sum is its term b (Art. 1) ; and
II. If a provector bc be added to a vector ab, the sum is
the transvector ac ; or in symbols,
I. . (b - a) + A = B ; and II. . (c - b) + (b - a) = c - a.
In fact, the first equation is an immediate consequence of the
general formula which, as above, connects the signs + and -,
when combined with the conception (Art. 1 ) of a vector as a dif-
ference of two points ; and the second is a result of the same
formula, combined with the definition of the geometrical sub-
traction of one such vector from another, which was assigned
in Art. 4, and according to which we have (as in algebra) for
any three points^ a, b, c, the identity,
(c - a) - (b - a) = c - B.
It is clear that this geometrical addition of successive vectors
corresponds (comp. Art. 4) to a composition of successive vec-
tions, or motions ; and that the sum of
two opposite vectors (or of vector and
revector) is a null line ; so that
ba + ab = 0, or (a - b) + (b - a) = 0.
It follows also that the sums of equal
pairs of successive vectors are equal; ^,.
or more fully that
if b' - a' = b - a, and c' - b' = c - b, then c' - a' = c
CHAP. I.] FUNDAMENTAL PRINCIPLES VECTORS. 5
the two triangles, abc and a'b'c', being in general the two oppo-
site faces of ^ prism (comp. Art. 3).
6. Again, in order to have, as in algebra,
(c' - b') + (b - a) = c - A, if c' - b' = c - B,
we shall define that if there be two successive vectors, ab, bc,
and if a third vector b'c' be equal to the second, but not suc-
cessive to the first, the sum obtained by adding the third to the
first is that fourth vector, ac, which is drawn from the origin
A of the first to the term c of the se-
cond. It follows that the sum of any
two co-initial sides, ab, ac, of 2iny paral-
lelogram abdc, is the intermediate and
co-initial diagonal ad ; or, in symbols,
(C - a) + (b - a) = D - A, if D - C = B - A ; Fig. 8.
because we have then (by 3) c-a = d-b.
7. The sum of any two given vectors has thus a value which
is independent of their order ; or, in symbols, a -f j3 = j3 + a.
If equal vectors be added to equal vectors, the sums are equal
vectors, even if the summands be not given as successive
(comp. 5) ; and if a null vector be added to an actual vector,
the sum is that actual vector ; or, in symbols, + a = a. If
then we agree to abridge generally (comp. 4) the expression
+ « to + fl, and if a still denote a vector, then + a, and + (+ a),
&c., are other symbols for the same vector; and we have, as
in algebra, the identities,
- (- a) = + a, + (- a) = - (+ a) = - a, (+ a) + (- a) = 0, &c.
Section 3. — On Sums of three or more Vectors.
8. The sum of three given vectors, a, j3, y, is next defined
to be that fourth vector,
^ = 7 + (/3 + a), or briefly, S=7 + /3 + a,
which is obtained by adding the third to the sum of the first
and second ; and in like manner the sum of any number of
vectors is formed by adding the last to the sum of all that
6 ELEMENTS OF QUATERNIONS. [bOOK I.
precede it: also, for any four vectors, a, /3, 7, S, the sum
S + (7 + j3 + a) is denoted simply by 8 + 7 + /3 + a, without pa-
rentheses, and so on for any number of summands.
9. The sum of any number of successive vectors, ab, bc,
CD, is thus the line ad, which is
drawn from the origin a of the first,
to the term d of the last ; and be-
cause, when there are three such vec-
tors, we can draw (as in Fig. 9) the
two diagonals ac, bd of the (plane "^ p. 9
or gauche) quadrilateral abcd, and
may then at pleasure regard ad, either as the sum of ab, bd,
or as the sum of ac, cdj we are allowed to establish the follow-
ing general formula of association ^ for the case oi' any three
summand lines, a, f5, y '•
(7 + /3) + a = 7 + (j3 + a)=7 + j3 + a;
by combining which with the formula of commutation (Art. 7),
namely, with the equation,
a + j3 = |3 + a,
which had been previously established for the case of any two
such summands, it is easy to conclude that the Addition of
Vectors is always both an Associative and a Commutative Ope-
ration. In other words, the sum oYany number of given vectors
has a value which is independent of their order, and of the
mode of grouping them ; so that if the lengths and directions of
the summands be preserved, the length and direction of the
sum will also remain unchanged : except that this last direction
may be regarded as indeterminate, when the Zew^^A of the sum-
line happens to vanish, as in the case
which we are about to consider.
1 0. When any n summand-lines,
AB, bc, CA, or AB, bc, CD, DA, &C.,
arranged in any one order, are the n
successive sides of a triangle ab c, or of f" 10
a quadrilateral abcd, or of any other
closed polygon, their sum is a 7iull line, aa ; and conversely.
CHAP. I.] FUNDAMENTAL PRINCIPLES VECTORS.
when the sum of any given system of n vectors is thus equal
to zero, they may be made {in any order ^ hy transports without
rotatioTi) the n successive sides of a closed polygon (plane or
gauche). Hence, if there be given any such polygon (p), sup-
pose a pentagon abcde, it is possible to construct another
closed polygon (p'), such as a'b'c'd'e', with an arbitrary initial
point a', but with the same number of sides, a'b', . . e'a', which
new sides shall be equal (as vectors) to the old sides ab, . . ea,
taken in any arbitrary order. For example, if we draw^wr
successive vectors, as follows,
A B = CD, B C
AB.
CD = EA,
D E = BC,
and then complete the new pentagon by drawing the line e'a',
this closing side of the second figure (p') will be equal to the
remaining side de of the^rs^ figure (p).
11. Since a closed figure abc . . is still a closed one, when
all its points ^vq projected on any assumed joZawe, by any system
of parallel ordinates (although the
area of the projected figure a'b'c' . . .
may happen to vanish), \t follows that
if the sum of any number of given
vectors a, j3, y, . . be zero, and if we
project them all 07i any one plane by
parallel lines drawn from their extre-
mities, the sum of the projected vec-
tors a, /3') y'i . . will likeivise be null; ^'
so that these latter vectors, like the
former, can be so placed as to become the successive sides of a
closed polygon, even if they be not already such. (In Fig. 1 1 ,
a"b"c" is considered as such a polygon, namely, as a triangle
loith evanescent area ; and we have the equation,
Fig. 11.
as well as
a"b" + b"c" + c"a" = 0,
a'b' + b'c' -f cV = 0, and ab + bc + ca = 0.)
8 ELEMENTS OF QUATERNIONS. [bOOK I.
Section 4. — On Coefficients of Vectors,
12. The simple or single vector, a, is also denoted by la,
or by 1 . a, or by (+ 1 ) a ; and in like manner, the double vector,
a-\^a, is denoted by 2a, or 2 . a, or (+ 2) a, &c. ; the rule being,
that for any algebraical integer, m^ regarded as a coefficient by
which the vector a is multiplied, we have always,
\a + ma = {\ -^r m) a I
the symbol 1 + m being here interpreted as in algebra. Thus,
Oa = 0, the zero on the one side denoting a null coefficient, and
the zero on the other side denoting a null vector ; because by
the rule,
la -f Oa = (l + 0)a = la = a, and .'.Oa = a-a = 0.
Again, because (I) a + (- 1) a = (1 - 1 ) a = Oa = 0, we have
(- l)a = - a = -a = -(la); in like manner, since(l)a+ (-2)a
= (l-2)a = (- l)a = -a, we infer that (- 2)a = -a - a = - (2a) ;
and generally, (^-m) a = - (ma), whatever whole number m
may be : so that we may, without danger of confusion, omit
the parentheses in these last symbols, and write simply, - la,
- 2a, -ma.
13. It follows that whatever two whole numbers (positive or
negative, or null) may be represented by m and n, and what-
Fig. 12.
ever two vectors may be denoted by a and j3, we have always,
as in algebra, the formulae,
na±ma = {n± m) a, n (ma) = (nm) a =« nma,
and (compare Fig. 12),
m (/3 ± a) = /w/3 ±ma;
CHAP. I.J FUNDAMENTAL PRINCIPLES VECTORS. 9
SO that the multiplication of vectors by coefficients is a doubly
distributive operation^ at least if the multipliers be whole
numbers; a restriction which, however, will soon be re-
moved.
14. If ma = j3, the coefficient m being still whole, the vector
|3 is said to be a multiple ol' a ; and conversely (at least if the
integer m be different from zero), the vector a is said to be a
sub-multiple of /3. A multiple of a sub-multiple of a vector is
said to be infraction of that vector ; thus, if /3 = ma, and y = na,
n
then y is a fraction of j3j which is denoted as follows, 7 = — jS ;
m
n
also j3 is said to be multiplied by the fractional coefficient — ,
and y is said to be the product of this multiplication. It fol-
lows that if a; and y be any two fractions (positive or negative
or null, whole numbers being included), and if a and (3 be any
two vectors, then
ya±xa==(y±x)a, ' y{xa) = {yx)a = yxa, x(P ± a) = xj3 ±Xa ;
results which include those of Art. 1 3, and may be extended
to the case where x and y are incommensurable coefficients, con-
sidered as limits oi' fractional ones.
15. For any actual vector a, and for any coefficient x, of
any of the foregoing kinds, ihaproduct xa, interpreted as above,
represents always a vector j3, which has the same direction as
the multiplicand-line a, if x> 0, but has the opposite direction
if aj < 0, becoming null if x= 0. Conversely, if a and /3 be any
two actual vectors, with directions either similar or opposite, in
each of which two cases we shall say that they are parallel
vectors, and shall write j3 H a (because both are then parallel,
in the usual sense of the word, to one common line), we can
always find, or conceive as found, a coefficient x^O, which shall
satisfy the equation j3 = xa; or, as we shall also write it,
f3 = ax; and the positive or negative number x, so found, will
bear to ± 1 the same ratio, as that which the lenyth of the line
3 bears to the lengtli of a.
10 ELEMENTS OF QUATERNIONS. [bOOK I.
16. Hence it is natural to say that this coefficient x is the
quotient which results, from the division of the vector j3, hy the
parallel vector a ; and to write, accordingly,
x = Q-7-a, orx=Q:a, or^ = ^;
a
SO that we shall have, identically, as in algebra, at least if the
divisor-line a be an actual vector, and if the dividend-line ^hQ
parallel thereto, the equations,
(j3 : a) .a = — a = j3, and Xa\a=- — = x',
which will afterwards be extended, by definition, to the case of
non-parallel vectors. We may write also, under the same
conditions, d = — , and may say that the vector a is the quotient
X
of the division of the other vector j3 hy the numher x ; so that
we shall have these other identities,
— .a3 = (aa;=)j3, and — = a.
17. The positive or negative quotient, x-=^, which is thus
obtained by the division of one of two parallel vectors by ano-
ther, including zero as a limit, may also be called a Scalar ;
because it can always be found, and in a certain sense con-
structed, by the comparison of positions upon one common scale
(or axis) ; or can be put under the form,
c - A AC
b-a~ab'
where the three points, a, b, c, are collinear (as in the figure
annexed). Such scalar s are, there- ^
fore, simply the Re a ls (or real quan- ' ^, '
tities) oi Algebra; but, in combina-
tion with the not less real Vectors above considered, they
form one of the main elements of the System, or Calculus, to
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 11
which the j)resent work relates. In fact it will be shown, at
a later stage, that there is an important sense in which we can
conceive a scalar to be added to a vector ; and that the sum
so obtained, or the combination,
Scalar plus Vector^'*
is a Quaternion.
CHAPTER II.
APPLICATIONS TO POINTS AND LINES IN A GIVEN PLANE.
Section 1. — On Linear Equations connecting two Co-uiitial
Vectoj's.
18. When several vectors, oa, ob, . . are all drawn from
one common point o, that point is said to be the Origin of the
System ; and each particular vector, such as oa, is said to be
the vector of its own term, a. In the present and future sec-
tions we shall always suppose, if the contrary be not expressed,
that all the vectors a, j3, . . which we may have occasion to
consider, are thus drawn from one common origin. But if it
be desired to change that origin o, without changing the term-
points a, . . we shall only have to subtract, from each of their
old vectors a, . . one common vector w, namely, the old vector
oo' of the new origin d ; since the remainders, a - w, j3 - w, • •
will be the new vectors a, /3', . . of the old points a, b, . . . For
example, we shall have
a = o'a = a - o' = (a - o) - (o' - o) = oa - oo' = a - w.
19. If tivo vectors a, /3, or oa, ob, be thus drawn from a
given origin o, and if their o a b
directions be either similar or ' "■; ~ '
opposite, so that the three
points, o, A, B, are situated on one right line (as in the figure
12 ELEMENTS OF QUATERNIONS. [bOOK I.
annexed), then (by 16, 17) their quotient — is some positive or
negative scalar, such as x ; and conversely, the equation
j3 = xa, interpreted with this reference to an origin, expresses
the condition of collinearity , of the points o, a, b ; the particu-
lar values, 03 = 0, x=\, corresponding to the particular /)052-
tions, o and a, of the variable point b^ whereof the indefinite
right line OA is the locus.
20. The linear equation, connecting the two vectors a and
j3, acquires a more symmetric ^or/w, wlien we write it thus :
aa + ^/3 = ;
where a and b are two scalars, of which however only the ratio
is important. The condition of coincidence, of the two points
A and B, answering above to a? = 1, is now -j- = 1 ; or, more
symmetrically,
« + 5 = 0.
Accordingly, when a=-b, the linear equation becomes
b{(5-a)-^0, or i3-a = 0,
since we do not suppose that both the coefficients vanish ; and
the equation j3 = a, or ob = oa, requires that ihepointB should
coincide with the point a : a case w^hich may also be conve-
niently expressed by the formula,
B = a;
coincident points being thus treated (in notation at least) as
eqy^L In general, the linear equation gives,
a . OA + 6 . OB = 0, and therefore « : 6 = bo : oa.
Section 2. — On Linear Equations between three co-initial
Vectors.
21. If two (actual and co-initial) vectors, a, /3, be not con-
nected by any equation of the form aa 4 Z>/3 = 0, with any two
scalar coefficients a and b whatever, their directions c^n neither
be similar nor opposite to each other ; they therefore determine
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE.
13
a plane aob, in which the (now actual) vector, represented by
the sum aa + Z>/3, is situated. For if, for the sake of symmetry,
we denote this sum by the
symbol - cy, where c is some
third scalar, and 7=00 is
some third vector, so that the
three co-initial vectors, a, )3,
7, are connected by the linear
equation,
«a -f ^>/3 + C7 = ;
and if we make
, - aa
oa = ,
c
then the two auxiliary points, a' and b', will be situated (by
19) on the two indefinite right lines, oa, ob, respectively:
and we shall have the equation,
oc = oa'+ ob',
so that the figure a'ob'c is (by 6) a parallelogram, and conse-
quently plane.
22. Conversely, if c be any point in the plane aob, we can
draw from it the ordinates, ca' and cb', to the lines oa and ob,
and can determine the ratios of the three scalars, a, b, c, so as
to satisfy the two equations.
OA
oa'
OB
OB
after which we shall have the recent expressions for oa', ob',
with the relation oc = oa' + ob' as before ; and shall thus be
brought back to the linear equation aa + b^ + cy = 0, which
equation may therefore be said to express the condition ofcom-
plariarity of the^wr points, o, a, b, c. And if we write it under
the form,
Xa + 7/f5 + zy = 0,
and consider the vectors a and j3 as ^iven, but 7 as a variable
vector, while x, y, z are variable scalars, the locus of the va-
riable poirit will then be the given plane, oab.
14
ELEMENTS OF QUATERNIONS.
[book
23. It may happen that the point c is situated on the right
line ABj which is here considered as a given one. In that
AC
case (comp. Art. 17, Fig. 13), the quotient — must be equal
AB
to some scalar, suppose t ; so that we shall have an equation of
the form,
= t, or y = a + t(f5-a), or (1 - #) a + ^/3 - 7 = ;
jS-a
by comparing which last form
with the linear equation of Art.
21, we see that the condition
of collinearity of the three
points A, B, c, in the given
plane oab, is expressed by the
formula,
« + i + c = 0.
This condition may also be thus written,
Fig. 10.
-a -b
c c '
OA OB
or — + — = 1 ;
OA OB
and under this last form it expresses a geometrical relation,
which is otherwise known to exist.
24. When we have thus the two equations,
«a + 6/3 + c-y = 0, and « + 6 + c = 0,
so that the three co-initial vectors a, /3, 7 terminate on one
right line, and may on that account be said to be ternwio-col-
linear, if we eliminate, successively and separately, each of
the three scalars a, b, c, we are conducted to these three other
equations, expressing certain ratios of segments :
b(j5-a) + c{y-a) = 0, dy - (5) + a(a - (^) = 0,
a(a-7) + i(/3-7) = 0;
or
= 6.AB 4 C.AC = C.BC + «.BA = a.CA + 6.CB.
Hence follows this proportion, between coefficients and seg-
ments,
« :6:c = Bc : CA : ab.
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 15
We might also have
! observed that the
proposed equations
give,
bf3 + Cy
P-
cy + aa
aa + bf3
a + b '
whence
AC_
y -a
= ^=-*<S
;c.
AB j3 - a a +b c
25. If we still treat a and j3 as given, but regard y and
- as variable, the equation
xa-\- yfi
^~ ^ + y
will express that the variable point c is situated someivhere
on the indefinite right line ab, or that it has this line for its
locus : while it divides the Jinite line ab into segments, of which
the variable quotient is,
CB x'
Let c' be another point on the same line, and let its vector be,
>
1
then, in
like
manner,
we
shall have this
other
ratio
ofseg^
ments,
AC _ 2/'
c'b ~ a?'*
If, then, we agree to employ, generally, ^o?- any group offo
collinear points, the notation.
^ ab CD AB AD
(abcd) = — = — : —
^ bc da bc dc
SO that this symbol,
(abcd),
may be said to denote the anharmonic function, or anharmonic
quotient, or simply the anharmonic of the group, a, b, c, d : we
shall have, in the present case, the equation,
„ AC Ac' yx
(acbc ) = — :-T- = ^.
^ ^ CB CB xy
16 ELEMENTS OF QUATERNIONS. [bOOK I.
26. When the anharmonic quotient h^QomQ^ equal to nega-
tive unity, the group becomes (as is well known) harmonic.
If then we have the two equations,
xa + y(^ , xa- yj5
' x + y X -y
the two points c and c' are harmonically conjugate to each other,
with respect to the two given points^ a and b ; and when they
vary together, in consequence of the variation of the value of
-, they form (in a well-known sense), on the indefinite right
line AB, divisions in involution; the double points (ov foci) of
this involution, namely, the points of which each is its oion
conjugate, being the points a and b themselves. As a verifi-
cation, if we denote by p. the vector of the middle point m of
the given interval ab, so that ^
A M C B C'
/3-/i=/x-a, or/i = J(a+/3), Fig. 17.
we easily find that
y - f-i _y - X P -luL MCMB^
/3-jU y ^ X~ y' - fx MB MC'*
so that the rectangle under the distances mc, mc', of the two
variable but conjugate points^ c, c', from the centre m of the
involution, is equal to the constant square of half the interval
between the two double points, a, b. More generally, if we
write
xa+ y(5 , _ Ixa + my (5
' X +y ^ lx + my '
where the anharmonic quotient — = — ,- is any constant scalar,
then in another known and modern* phraseology, the points
c and c' will form, on the indefinite line ab, tivo homographic
divisions, of which a and b are still the double points. More
generally still, if we establish the two equations,
* See the Gtometrie Supe'rieure of M. Chasle?, p. 107. (Paris, 1852.)
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 17
xa + vQ , , lxa + my 3'
y= ^, and 7'=— ^^,
x^y lx-\- my
I , , y .
— beinof still constant, but - variable, while a = oa', 3' = ob',
and y' = oc', the two given lines, ab and a'b', are then homo-
graphically divided, by the two variable points, c and c', not
now supposed to move along one common line.
27. When the linear equation aa + bf3 + cy = subsists,
without the relation « -i- ^ + c = between its coefBcients, then
the three co-initial vectors a, /3, y are still complanar, but they
no longer terminate on one right line ; their term-points a, b, c
being now the corners of a triangle.
In this more general case, we may propose to find the vec-
tors a', j3', y' of th€ three points,
a' = oabc, b'=obca,
C'= OCAB ;
that is to say, of the points in
which the lines drawn from the
origin o to the three corners of
the triangle intersect the three
respectively opposite sides. The three collineations oaa', &c.,
give (by 19) three expressions of the forms,
a = Xa, (5' = yj3, y' = Z.y,
where x, y, z are three scalars, which it is required to deter-
mine by means of the three other collineations, a'bc, &c., with
the help of relations derived from the principle of Art. 23.
Substituting therefore for a its value re 'a', in \)i\^ given linear
equation, and equating to zero the sum of the coefficients of
the new linear equation which results, namely,
and eliminating similarly j3, 7, each in its turn, from the ori-
ginal equation ; we find the values,
-a -h -c
X = , y = , z = 7 ;
ft + c ^c + a a^ b
18 ELEMENTS OF QUATERNIONS. [boOK I.
whence the sought vectors are expressed in either of the two
following ways :
or
J , -aa
1. , . a =7 ,
b + c
^ c + a
'^~a + b'
II.
, bfi + Cy
C + a
, aa + b[5
^ a + b
In fact we see, by one of these expressions for a, that a' is on
the line oa ; and by the other expression for the same vector
a', that the same point a' is on the line bc. As another veri-
fication, we may observe that the last expressions for a, j5', y\
coincide with those which Avere found in Art. 24, for a, /3, y
themselves, on the particular supposition that the three points
a, B, c were collinear.
28. We may next propose to determine the ratios of the
segments of the sides of the triangle abc, made by the points
a', b', c'. For this purpose, we may write the last equations
for a', j3', y under the form,
0=^b{a'-(5)-c{y-a') = c((5'-y)-a{a-(5') = a{y'-a)
and we see that they then give the required ratios, as follows :
ba'_ c cb' a Ac'_ b
a'c b' b'a c' c'b a'
whence we obtain at once the known equation of six segments,
ba' cb' ac'
a'c b'a c'b '
as the condition of concurrence of the three right lines a a', bb',
cc', in a common point, such as o. It is easy also to infer, from
the same ratios of segments, the following proportion of coeffi-
cients and areas,
a:b:c= OBC : oca : gab,
in which we must, in general, attend to algebraic signs ; a tri-
angle being conceived to pass {through zero) from positive to
negative, or vice versa, as compared with any give?i triangle in
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 19
its own plane, when (in the course of any continuous change)
its vertex crosses its base. It may be observed that with this
conveiition (which is, in fact, a necessary one, for the establish-
ment o{ general for mulce) we have, for any three points^ the
equation
ABC + BAC = 0,
exactly as we had (in Art. 5) for any two points, the equa-
tion
AB+ BA= 0.
More fully, we have, on this plan, the formula3,
ABC = - BAC = BCA = - CBA = CAB = - ACB ;
and any two complanar triangles, abc, a'b'c', bear to each other
a positive or a negative ratio, according as the two rotations,
which may be conceived to be denoted by the same symbols
ABC, a'b'c', are similarly or oppositely directed.
29. If a' and b' bisect respectively the sides bc and ca,
then
a = b = c,
and c' bisects ab ; whence the known theorem follows, that
the three bisectors of the sides of a triangle concur, in a point
which is often called the centre of gravity, but which we pre-
fer to call the mean point of the triangle, and which is here the
origiji o. At the same time, the first expressions in Art. 27
for a, ft', y' become,
"~~2' ^^"2' ^^"2'
whence this other known theorem results, that the three bisec-
tors trisect each other,
30. The linear equation between a, ft, y reduces itself, in
the case last considered, to the form,
a + /3 4 7 = 0, or oa + ob + oc = ;
the three vectors a, ft, y, or oa, ob, oc, are therefore, in this
ca^e, adapted (by Art. 10) to become the successive sides of a.
20
ELEMENTS OF QUATERNIONS.
[book I.
triangle, by transports without rotation ; and ticcordingly, if
we complete (as in Fig. 19) the /^c
parallelogram aobd, the triangle
GAD will have the property in
question. • It follows (by 11)
that if we project the four points
o, A, B, c, by any system of pa-
rallel ordinates, into four other A^
points, o^, A^, B^, c , on any as-
sumed pZ«we, the sum of the three j^
projected vectors^ a^, j3^, y^, or Fig. 19.
o A , &c., will be null; so that we shall have the new linear
equation,
or.
o A^ + o B^ + o^c^ = ;
and in fact it is evident (see
Fig. 20) that the projected
mean point o^ will be the mean
point of the projected triangle, ^'^" ^^•
A^, B^, c^. We shall have also the equation,
(a,-o) + (/3,-^) + (y,--y) = 0;
where
hence
a^- a = O^A - OA = (O^A + AA ) - (OO^ + O^a) = AA^ - 00^ ;
OO^ = ^ (aA^ -\ BB^ + CC ).
or the ordinate of the mean point of a triangle is the mean of
the ordinates of the three corners.
Section 3. — On Plane Geometrical Nets,
31. Resuming the more general case of Art. 27, in which
the coefficients «, b, c are supposed to be unequal, we may next
inquire, in what points a", b", c" do the lines b'c', c'a', a'b'
meet respectively the sides bc, ca, ab, of the triangle ; or may
seek to assign the vectors a\ /3", y" of the points of intersec-
tion (comp. 27),
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE.
21
A =BC*BC, B =CA*CA, C -ABAB.
The first expressions in Art. 27 for |3', 7', give the equa-
tions,
B"
Fig. 21.
(c -f «) j3' + ^>i3 = 0, (a + &)y + C7 = ;
whence
b[5-cy _ (a + b)y-(c+a)j5\
b- c {a + b) ~ {c -\- a)
but (by 25) one member is the vector of a point on bc, and
the other of a point on b'c' ; each therefore is a value for the
vector a" of a", and similarly for j3" and 7". We may there-
fore write,
„_bfi- Cy ^„ Cy - aa „ tta- b[5
a = -7 , ~
o - c
^,.^cy-aa^
c - a
7 =
and by comparing these expressions with the second set of
values of a', /3', 7' in Art. 27, we see (by 26) that the points
a", b", c" are, respectively, the harmonic conjugates (as they
are indeed known to be) of the points a', b', c', with respect
to the three pairs of points, b, c ; c, a ; a, b ; so that, in the
notation of Art. 25, we have the equations,
(baca") = (cb'ab") = (ac'bc") =- I.
And because the expressions for a", /3", 7" conduct to the fol-
lowing linear equation between those three vectors,
22 ELEMENTS OF QUATERNIONS. [boOK I.
{b-c)a'+ (c-«)j3"+ {a - b)y"=0,
with the relation
(b-c)+ {c-a) + (a-b) =
between its coefficients, we arrive (by 23) at this other known
theorem, that the three points a", b", c" are collifiear, as indi-
cated by one of the dotted lines in the recent Fig. 2 1 .
32. The line a"b'c' may represent any rectili?iear transver-
sal, cutting the sides of a triangle abc ; and because we have
ba"_ «"-/3 ^ c
a"c 7 - a" b
while -7- = -, and —r- = -, as before, we arrive at this other
ba c cb a
equation of six segments, for any triangle cut by a right line
(comp. 28),
ba" cb' ac' _
a"c b'a c'b
which again agrees with known results.
33. Eliminating j3 and 7 between either set of expressions
(27) for j3' and y', with the help of the given linear equation,
we arrive at this other equation, connecting the three vectors
a, /3', 7' :
O = - «a + (c + «) j3' + (a + ^) 7'.
Treating this on the same plan as the given equation between
a, j3, 7> we find that if (as in Fig. 21) we make,
a'" = OA • Bc', b"' = OB • c'a', C ' = DC ' a'b',
the vectors of these three new points of intersection may be ex-
pressed in either of the two following ways, whereof the first
is shorter, but the second is, for some purposes (comp. 34, 36)
more convenient :
'" ^ «« n.n^ bP ,„^ Cy ^
2a + b + c ^ 2b^c + a ^ 2c + a + b'
or
„, _ 2aa + bj5 + Cy ^,„ _ 2^/3 + cy + aa
^ 2a + b^c ' ^ ~ 26 + c + « '
,„ _ 2cy -{ aa^bf5
^ 2c + « + ft
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 23
And the three equations, of which the following is one,
{h-c)a:'- (26+ c + «)/3'"+ (2c+ « + 6)7'" = 0,
with the relations between their coefficients w^hich are evident
on inspection, show (by 23) that we have the three additional
collineations, a"b'"c'", b"c"'a'", c"a"'b'", as indicated by three of
the dotted lines in the figure. Also, because we have the two
expressions,
„, (a-\-b)y+(c + a)(5' „ _(a +b)y - (c -¥ a)^'
." ~ {a-\-b) + (€ + a) ' {a + b)-(c + a) ^
we see (by 26) that the two points a", a'" are harmonically con-
jugate with respect to b' and c' ; and similarly for the two
other pairs of points, b", b'", and c", c'", compared with c', a',
and with a', b': so that, in a notation already employed (25,
31), we may write,
(b a'"c a") = (c b'Vb") = (a'c'"b'c") = - 1 .
34. If we beyin^ as above, with any four complanar points,
o, A, B, c, of which no three are collinear, we can (as in Fig.
18), by what may be called a First Construction, derive from
them six lines, connecting them two by two, and intersecting
each other in three new points, a', b', c' ; and then by a Second
Construction (represented in Fig. 21), we may connect these
by three new lines, which will give, by their intersections with
the former lines, six new points, a", . . c"\ We might pro-
ceed to connect these with each other, and with the given
points, by sixteen new lines, or lines of a Third Construction,
namely, the four dotted lines of Fig. 21, and twelve other
lines, whereof three should be drawn from each of the four
given points : and these would be found to determine eighty-
four new points of intersection, of which some may be seen,
although they are not marked, in the figure.
But however far these processes oi linear construction may
be continued, so as to form what has been called* a plane
* By Prof. A. F. INIobujs, in page 274 of his Bary centric Calculus (dcr baryrcu-
trische Calcul, Leipzig, 1827).
24 ELEMENTS OF QUATERNIONS. [bOOK I.
geometrical net, the vectors of the points thus determined have
all one common property : namely, that each can be represented
by an expression of the form,
xaa H- yh^ -1- zcy
xa + yh + zc
where the coefficients x, y, z are some whole numbers. In fact
we see (by 27, 31, 33) that such expressions can be assigned
for the nine derived vectors, a', . . . y", which alone have been
hitherto considered ; and it is not 'difficult to perceive, from
the nature of the calculations employed, that a similar result
must hold good, for every vector subsequently deduced. But
this and other connected results will become more completely
evident, and their geometrical signification will be better un-
derstood, after a somewhat closer consideration of anharmonic
quotients, and the introduction of a certain system o^ anhar-
monic co-ordinates, for points and lines in one plane, to which
we shall next proceed : reserving, for a subsequent Chapter,
any applications of the same theory to space.
Section 4. — On Anharmonic Co-ordinates and Equations of
Points and Lines in one Plane.
35. If we compare the last equations of Art. 33 with the
corresponding equations of Art. 31, we see that the harmowc
group ba'ca", on the side bc of the triangle abc in Fig. 21,
has been simply reflected into another such group, b V'c'a", on
the line b'c', by a harmonic pencil of four rays, all passing
through the point o ; and similarly for the other groups.
More generally, let oa, ob, oc, od, or briefly o.abcd, be
any pencil, with the point o for vertex ; and let the new ray
OD be cut, as in Wig. 22, by the three sides of the triangle
ABC, in the three points Ai, Bi, Ci ; let also
yh^ + zcy
OAi = ai = — ^ ^,
yb 4- zc
so that (by 25) we shall have the anharmonic quotients,
y , ^
(ba'cai) = -, (ca'b.\i) = -;
^ ^ 2 y
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE.
25
and let us seek to express the two other vectors of intersec-
tion, j3i and 71, with a view to
determining the anharmonic ra-
tios of the groups on the two
other sides. The given equation
(27),
«a + 6/3 + cy = 0,
shows us at once that these two
vectors are.
OB
1 - Pl ;
Fig. 22.
001 = ^1 =
{z-y)h^-^zaa
{z-y)b + za *
whence we derive (bj 25) these two other anharmonics,
(cb'aBi) =
(bCACi)
y -2
so that we have the relations,
(CB'aBi) + (ca'bAi) = (bc'aCi) + (ba'cAi) = 1.
Bat in general, for any four collinear points a, b, c, d, it is
not difficult to prove that
AB AC
CD+ BD= DA
BC CB
whence by the definition (25) of the signification of the sym-
bol (abcd), the following identity is derived,
(abcd) + (acbd)= 1.
Comparing this, then, with the recently found relations, we
have, for Fig. 22, the following anharmonic equations ;
(cab'Bi) = (ca'bAi) = - ;
y
(bac'Ci) = (ba'cAi) =-;
and we see that (as was to be expected from known princi-
26 ELEMENTS OF QUATERNIONS. [bOOK I.
pies) the anharmonic of the group does not change, when we
pass from one side of the triangle, considered as a transversal
of the pencil, to another such side, or transversal. We may
therefore speak (as usual) of such an anharmonic of a group^
as being at the same time the Anharmonic of a Pencil ; and,
with attention to the order of the rays, and to the definition
(25), may denote the two last anharmonics by the two following
reciprocal expressions:
z y
(o.cabd) = -; (o.bacd) = -;
y ^
with other resulting values, when the order of the rays is
changed ; it being understood that
(o . cabd) = (c'aVd'),
if the rays oc, oa, ob, od be cut, in the points c', a\ b\ d\
by any one right line.
36. The expression (34),
xaa + yh^ + zcy
p- J
xa +yo + zc
may represent the vector o^ any point p in the given plane ^ by a
suitable choice of the coefficients x, y, x, or simply of their ra-
tios. For since (by 22) the three complanar vectors pa, pb,
PC must be connected by some linear equation, of the form
«' . PA + i' . PB -r c' . PC = 0,
or
aXa-p) + b'(f5-p) + c(y-p) = 0,
which gives
a a + b'Q + cy
P~
a' + b' + c
we have only to write
a' b' d
a b " c
and the proposed expression for p will be obtained. Hence
it is easy to infer, on principles already explained, that if we
write (compare- the annexed Fig. 23),
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 27
Pi=PABC, P2 = PB'CA, P3 = PCAB,
we shall have, with the same coefficients xyz, the following
expressions for the vectors opj, 0P2,
0P3, or |0i, /02, /03, of these three points
of intersection, Pi,*P25 P3 :
yh^ + zcy
^^~ yb + zc
^2=-
zcy + xaa
J
zc\ xa '
xaa + yhfi ^
^ xa^yh Fig. 23
which give at once the following anharmonics of pencils, or of
groups,
(a . BOCP) = (ba CPi) = - ;
z
z
(B . COAP) = (cb'aPz) = - ;
X
X
(C . AOBP) = (ac'bPs) = - ;
y
whereof we see that the product is unity. Any two of these
three pencils suffice to determine the position of the point P,
when the triangle abc, and the origin o are given ; and there-
fore it appears that the three coefficients x, y, z, or any scalars
proportional to them, of which the ^'z^o^zVw^a- thus represent the
anhai^monics of those pencils, may be conveniently called the
Anharmonic Co-ordinates of that point, p, with respect to
the given triangle and origin : while the point p itself may be
denoted by the Symbol,
p = (07, y, z).
With this notation, the thirteen points of Fig. 21 come to be
thus symbolized ;
a =(1,0,0), b =(0,1,0), c =(0,0,1), = (1,1,1);
a' =(0,1,1), B' =(1,0,1), €'=(1,1,0);
a" = (0,1,-1), B" = (-1,0, 1), €"=(1,-1,0);
A'"=(2, 1, 1), B'"= (1,2,1), €'"=(1,1,2).
28 ELEMENTS OF QUATERNIONS. [bOOK I.
37. If Pi and Pa be any two points in the given plane,
Pi = (^H yi, zi), P2 = (^2> y2, Z2),
and if t and u be any two scalar coefficients, then the following
third pointy
p = (toi + UX2, tyi + uy^i tzx + uz^,
is collinear with the two former points, or (in other words) is
situated on the right line PiPg. For, if we make
a; = ^a!i + 11X2, y=ty\^ wyz) z = #Zi + uz^r
and
a^ifla + . . x^aa + . . xaa + . .
p\ = J Pa"" > /> = J
aJia + . . ^2« + • • a:a + . .
these vectors of the three points P1P2P are connected by the
linear equation,
t (xia -h . .)pi + u (x^a + . 0/02 - {xa + . .) /o = ;
in which (comp. 23), the s?im of the coefficients is zero. Con-
versely, the point p cannot be collinear with Pi, Pg, unless its
co-ordinates admit of being thus expressed in terms of theirs.
It follows that if a variable point p be obliged to move along a
given right line PiPg, or if it have such a line (in the given
plane) for its locusy its co-ordinates xyz must satisfy a homo-
geneous equation of the first degree, with constant coefficients ;
which, in the known notation of determinants, may be thus
written,
X, y, z
= Xu yi, z^
«^2> y^i Z2
or, more fully,
= x {yxZ^ - z{y^ + y {zix^ ~ ofiZz) + z {x^y^ - y^x^) ;
or briefly,
= l.v + my + nz,
where /, m, n are three constant scalars, whereof the quotients
determine the position of the right line A, which is thus the
locus of the point p. It is natural to call the equation, which
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 29
thus connects the co-ordinates of the point p, the Anharmonic
Equation of the Line A ; and we shall find it convenient also
to speak of the coefficients /, w, n, in that equation, as being
the Anharmonic Co-ordinates of that Line: which line may
also be denoted by the Symbol^
A = [Z, m, w] .
38. For example, the three sides bc, ca, ab of the given
triangle have thus for their equations,
a; = 0, y = 0, 2=0,
and for their symholsy
[1,0,0], [0,1,0], [0,0,1].
The three additional lines oa, ob, oc, of Fig. 18, have, in hke
manner, for their equations and symbols,
3/-0 = O, 2-37 = 0, x-y=0,
[0,1,-1], [-1,0,1], [1,-1,0].
The lines b'c'a", c'a'b", a'b'c", of Fig. 21, are
y + z -x = 0, z-rx-i/ = 0) x + y -z = 0,
or
[-1,1,1], [1,-1,1], [1,1,-1];
the lines aV'c'", b"c'V", cV'b'', of the same figure, are in like
manner represented by the equations and symbols,
y + z-Sx = 0, z + x-3y=0, x-\^y-3z = 0,
[-3,1,1], [1,-3,1], [1,1,-3];
and the line a"b "c" is
X -^ y + z=0, or [1, 1, 1].
Finally, we may remark that on the same plan, the equation
and the symbol of what is often called the line at infinity, or
of the locus of all the irifinitely distant points in the given plane,
are respectively,
ax -v by ^ cz = 0, and [a, b, c] ;
30 ELEMENTS OF QUATERNIONS. [bOOK I.
because the linear function, ax + hy + cz, of the co-ordinates
z, y, 2r of a point p in the plane, is the denominator of the ex-
pression (34, 36) for the vector p of that point : so that the
point p is at an infinite distance from the origin o, when, and
only when, this linear function vanishes.
39. These anharmonic co-ordinates of a line, although
above interpreted (37) with reference to the equation of that
line, considered as connecting the co-ordinates of a variable
point thereof, are capable of receiving an independent geome-
trical interpretation. For the three points l, m, n, in which
the line A, or [/, m, w], or lx\my \nz = 0, intersects the three
sides BC, CA, ab of the given triangle abc, or the three given
lines a? = 0, 7/=0, 2:=0 (38), may evidently (on the plan of
36) be thus denoted :
L = (0, 7i, - m) ; M = (- w, 0, /) ; n = (m, - I, 0).
But we had also (by 36),
a" = (0,1,-1); b"=(- 1,0,1); c"= (1,-1,0);
whence it is easy to infer, on the principles of recent articles,
that
— = (ba"cl) ; - = (cb"am) ; — = (ac'bn) ;
m ^ n ^ ' I ^
with the resulting relation,
(ba"cl) . (cb"am) . (ac"bn) = 1.
40. Conversely, this last equation is easily proved, with
the help of the known and general relation between segments
(32), applied to any two transversals, a"b"c" and lmn, of any
triangle abc. In fact, we have thus the two equations,
ba" cb" ac"_ bl cm an
a"c b"a c"b ' LC MA NB '
on dividing the former of which by the latter, the last formula
of the last article results. We might therefore in this way
have been led, without any consideration of a variable point p,
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 31
to introduce three auxiliary scalar s^ /, ?w, n^ defined as having
71 I Tfl
their quotients — , -, — equal respectively, as in 39, to the
three anharmonics of groups,
(ba"cl), (cb"am), (ac"bn);
and then it would have been evident that these three scalars,
/, m, n (or any others proportional thereto), are sufficient to
determine the position of the right line A, or lmn, considered
as a transversal of the given triangle abc : so that they might
naturally have been called, on this account, as above, the an--
harmonic co-ordinates of that line. But although the anhar-
monic co-ordinates of a point and of a line may thus be inde-
pendently defined^ yet the geometrical utility of such definitions
will be found to depend mainly on their combination : or on the
formula Ix ^-my a- nz=0 of 37, which may at pleasure be con-
sidered as expressing, either that the variable point (re, y, z) is
situated somewhere upon the given right line [/, m, ri\ ; or else
that the variable line [/, tw, n\ passes, in some direction, through
the given point {x, y, z).
41. If Ai and As be any two right lines in the given plane,
Ai = [/i, mi, ni], Aa = [h, m^, Wo],
then any third right line A in the same plane, which passes
through the intersection ArA25 or (in other words) which cow-
curs with them (at a finite or infinite distance), may be repre-
sented (comp. 37) by a symbol of the form,
'A = [til + uli, tmi + um2, tn^ + uji.^,
where t and u are scalar coefficients. Or, what comes to the
same thing, if I, m, n be the anharmonic co-ordinates of the
line A, then (comp. again 37), the equation
/, m, n
= 1 (min-i- nimz) + &c. = h, mi, Ui
hi 'mi, ni
must be satisfied ; because, if {X, Y, Z) be the supposed point
common to the three lines, the three equations
32 ELEMENTS OF QUATERNIONS. [boOK I.
lX+mY+nZ=0, hX + m,Y+n,Z =0, kX + m^Y+n^Z^(S,
must co-exist. Conversely, this coexistence will be possible,
and the three lines will have a common point (which may be
infinitely distant), if the recent condition of concurrence be sa-
tisfied. For example, because [a, J, c] has been seen (in 38)
to be the symbol of the line at infinity (at least if we still re-
tain the same significations of the scalars a, 6, c as in articles
27, &c.), it follows that
A = [Z, m, ri] , and A' = [/+ ua, m + ub, n + uc] ,
are symbols of two parallel lines ; because they concur at infi-
nity. In general, all problems respecting intersections of right
lines, coUineations of points, &c., in the given plane, when
treated by this anharmonic method, conduct to easy elimina-
tions between linear equations (of the scalar kind), on which
we need not here delay : the mechanism of such calculations
being for the most part the same as in the known method of
trilinear co-ordinates : although (as we have seen) the geome-
trical interpretations are altogether different.
Section 5. — On Plane Geometrical Nets, resumed.
42. If we now resume, for a moment, the consideration of
those plane geometrical nets, which were mentioned in Art. 34 ;
and agree to call those points and lines, in the given plane, ra-
tional points and rational lines, respectively, which have their
anharmonic co-ordinates equal (or proportional) to whole num-
bers ; because then the anharmonic quotients, which were dis-
cussed in the last Section, are rational ; but to say that a point
or line is irrational, or that it is irrationally related to the
given system o^four initial points o, a, b, c, when its anhar-
monic co-ordinates are not thus all equal (or proportional) to
integers ; it is clear that ivhatever four points we may assume
as initial, and however far the construction of the net may be
carried, the net-points and net-lines which result will all be ra-
tional, in the sense just now defined. In fact, we begin with
such; and the subsequent eZz/w ma ^20W5 (41) oan never after-
CHAP. II.] PLANE GEOMETRICAL NETS. 33
wards conduct to any, that are of the contrary kind : the right
line which connects two rational points being always a rational
line ; and the point of intersection of two rational lines being
necessarily a rational point. The assertion made in Art. 34
is therefore fully justified.
43. Conversely, every rational point of the given plane,
with respect to the four assumed initial points oabc, is a point
of the net which those four points determine. To prove this,
it is evidently sufficient to show that every rational point
Ai = (0, y, z), on any one side bc of the given triangle abc, can
be so constructed. Making, as in Fig. 22,
Bi = oAi • CA, and Ci = oAi • ab,
we have (by 35, 36) the expressions,
Bi = (2/, 0,2/-2r), Ci=(z, ;2-y, 0);
from which it is easy to infer (by 36, 37), that
c' Bi • BC = (0, y,z- y), b'Ci • bc = (0, 2/ - z, z) ;
and thus we can reduce the linear construction of the rational
point (0, 2/j 2;), in which the two whole numbers y and z may
be supposed to be prime to each other, to depend on that of
the point (0, 1, 1), which has already been constructed as a'.
It follows that although no irrational point Q of the plane can
he a net-point, jet every such point can be indefinitely approached
to, by continuing the linear construction;
so that it can be included within a quadrila-
teral interstice P1P2P3P4, or even within a tri-
angular interstice P1P2P3, which interstice of p^^ -T^^*
the net can be made as small as we may de-
sire. Analogous remarks apply to irrational
lines in the plane, which can never coincide
with net-lines, but may always be indefinitely approximated to
by such.
44. If p, Pi, P2 be any three collinear points of the net, so
that the formulae of 37 apply, and if p'be any ^wr^^ net-point
{x, y, z) upon the same line, then writing
Xxa + yj) + z^c ~ Vx, x^a + y-h + z.c = v^.
34 ELEMENTS OF QUATERNIONS. [bOOK I.
we shall have two expressions of the forms,
_ tVipi + UV2P2 , t'Vipi + UV2P2
tVi + UV2 ' t'Vi + UV2 '
in which the coefficients tut'u are rational, because the co-or-
dinates xyz, &c., are such, whatever the constants abc may be.
We have therefore (by 25) the following rational expression
for the anharmonic of this net-group :
"" ^ tu' {X1/2 - yXi) {x'y, - y'x,) '
and similarly for every other group of the same kind. Hence
every group of four coUinear net-points, and consequently also
every pencil of four concurrent net-lines, has a rational value for
its anharmonic function ; which value depends only on the pro-
cesses of linear construction employed, in arriving at that group
or pencil, and is quite independent of the configuration or ar-
rangement oiihefour initial points : because the three initial
constants, «, b, c, disappear ^vom the expression which results.
It was thus that, in Fig. 21, the niiie pencils, which had the
nine derived points a' . . c"' for their vertices, were all harmo-
nic pencils, in whatever manner the four points o, a, b, c
might be arranged. In general, it may be said that plane
geometrical nets are all homo graphic figures ;* and conversely,
in any two such ^2,wq figures, corresponding points may be con-
sidered as either coinciding, or at least (by 43) as indefinitely
approaching to coincidence, with similarly constructed points
of two plane nets : that is, with points of which (in their re-
spective systems) the anharmonic co-ordinates (36) are equal
integers.
45. Without entering heref on any general theory of trans-
fi)rmation of anharmonic co-ordinates, we may already see that
if we select any fjur net-points Oi, Ai, Bi, Ci, of which no three
are collinear, every other point p of the same net is rationally
related (42) to these ; because (by 44) the three new anhar-
* Compare the Geometrie Svpe'rieure of M. Chasles, p. 362.,
t See Note A, on Anharmonic Co-ordinates.
CHAP. II.] CURVES IN A GIVEN PLANE. 35
monies of pencils, (Aj . BiOiCip) = — , &c., are rational : and
therefore (comp. 36) the new co-ordinates Xi, r/i, Zi of the point
p, as well its old co-ordinates xi/z, are equal or proportional to
whole numbers. It follows (by 43) that everi/ point p of the
net can be linearly constructed, if ani/ four such points be
ffiven (no three being collinear, as above) ; or, in other words,
that the whole net can be reconstructed,* \^ any one of its qua-
drilaterals (such as the interstice in Fig. 24) be known. As
an example, we may suppose that the four points oa'b'c' in
Fig. 21 are given, and that it is required to r^c^juer from them
the three points abc, which had previously been among the
data of the construction. For this purpose, it is only neces-
sary to determine first the three auxiliary points a'", b'", c"', as
the intersections oa' • b'c', &c. ; and next the three other auxi-
liary points a", b", c", as b'c' • b'"c'", &c. : after which the for-
mulae, A = b'b" • c'c"j &c., will enable us to return, as required,
to the points a, b, c, as intersections of known right lines.
Section 6. — On Anharmonic Equations, and Vector Expres-
sions, for Curves in a given Plane.
46. When, in the expressions 34 or 36 for a variable vec-
tor p = OP, the three variable scalars (or anharmonic co-ordi-
nates) X, y, z are connected by any given algebraic equation,
such as
fp{x,y, 2) = 0,
supposed to be rational and integral, and homogeneous of the
p^^ degree, then the locus of the term v (Art. 1) of that vector
is biplane curve of the jo^^ order; because (comp. 37) it is cut
* This theorem (45) of the possible reconstruction of a plane net, from any one
of its quadrilaterals^ and the theorem (43") respecting the possibility of indefi-
nitely approaching by net-lines to the points above called irrational (ii), without
ever reaching such points by any processes of linear constrtiction of the kind here
considered, have been taken, as regards their substance (although investigated by a
totally different analysis), from that highly original treatise of Mobius, which was
referred to in a former note (p. 23). Compare Note B, upon the Bai-ycentric Calcu-
lus ; and the remarks in the following Chapter, upon nets in space.
I
36 ELEMENTS OF QUATERNIONS. [bOOK I.
in p points (distinct or coincident, and real or imaginary), by
any given right line, Ix -^^ my ■\- nz = 0, in the given plane.
For example, if we write
f^aa + u^b^ + v'^cy
where t, u, v are three new variable scalars, of which we shall
suppose that the sum is zero, then, by eliminating these be-
tween the four equations,
a; = t^, y = u\ z=v\ t + u+v = 0,
we are conducted to the following equation of the second
degree, q =^ = ^2 ^ ^2 + ^^ - 22/z - 2zx - 2xy ;
so that here p-% and the locus of p is a conic section. In fact,
it is the conic which touches the sides of the given triangle abc,
at the points above called a', b', c' ; for if we seek its inter sec-
tions with the side bc, by making a; = (38), we obtain a
quadratic with equal roots, namely, {y-zy = 0\ which shows
that there is contact with this side at the point (0, 1, 1), or a'
(36) : and similarly for the two other sides.
47. If the point o, in which the three right lines aa', bb',
cc' concur, be (as in Fig. 18, &c.) interior to the triangle abc,
the sides of that triangle are then all cut internally, by the
points a', b', c' of contact with the conic ; so that in this case
(by 28) the ratios of the constants «, h, c are all positive, and
the denominator of the recent expression (46) for p cannot va-
nish, for any real values of the va-
riable scalars t, u^ v, and conse-
quently no such values can render
infinite that vector p. The conic is
therefore generally in this case, as in
Fig. 25, an inscribed ellipse ; which
becomes however the inscribed cir-
cle, when
«-M &-^ : c"^ = s - a : s - b : s - c ;
a, b, c denoting here the lengths of ^'^' ^^*
the sides of the triangle, and s being their semi-sum.
CHAP. II.] ANHARMONIC EQUATIONS OF PLANE CURVES. 37
48. But if the point of concourse o be exterior to the tri-
angle of tangents abc, so that two of its sides are cut externally^
then two of the three ratios o^ segments (28) are negative; and
therefore one of the three constants a, h, c may be treated as
< 0, but each of the two others as > 0. Thus if we suppose
that
i>0, oO, «<0, « + J>0, a+oO,
a' will be a point on the side b itself, but the points b', c', o
will be on the lines Ac, ab, ka! prolonged, as in Fig. 26 ; and
then the conic a'b'c' will be an
ellipse (including the case of a
circle), or a parabola, or an hy-
perbola^ according as the roots of ^
the quadratic.
Fig. 26.
{a + c) t^ + 2ctu +{b + c)u^ = 0,
obtained by equating the deno- b'
minator (46) of the vector p to
zero, are either, 1st, imaginary ; or Ilnd, real and equal; or
Ilird, real and unequal : that is, according as we have
bc + ca + ab>0, or = 0, or < ;
or (because the product abc is here negative), according as
a'^ + b-^ + c-^ < 0, or =0, or > 0.
For example, if the conic be what is often called the exscribed
circle, the known ratios of segments give the proportion,
a'^ : 6"^ : c'^ = - s : s - c : s - b ;
and
-s + s-c + s-b<0.
49. More generally, if c^ be (as in Fig. 26) a point upon
the side ab, or on that side prolonged, such that cc^ is parallel
to the chord b'c', then
c^c' : Ac' = cb' : ab' = - rt : c, and ab : ac' = « + i : 6 ;
writing then the condition (48) of ellipticity (or circularity)
38 ELEMENTS OF QUATERNIONS. [bOOK I.
under the form, ^— < —7—, we see that the conic is an ellipse,
c
parabola, or hyperbola, according as c^c' < or = or > ab ; the
arrangement being stilU in other respects, that which is repre-
sented in Fig. 26. Or, to express the same thing more sym-
metrically, if we complete the parallelogram cabd, then ac-
cording as the point d falls, 1st, beyond the chord b'c', with
respect to the point a; or llnd, on that chord; or Ilird,
ivithin the triangle ab'c', the general arrangement of the same
Figure being retained, the curve is elliptic^ or parabolic, or
hyperbolic. In that other arrangement or configuration, which
answers to the system of inequalities, Z>>0, c>0, « + 5 + c<0,
the point a' is still upon the side bc itself, but o is on the line
a'a prolonged through a ; and then the inequality,
a (^ + c) + 6c < - (^>2 + 6c + c2) < 0,
shows that the conic is necessarily an hyperbola ; whereof it is
easily seen that one branch is touched by the side bc at a',
while the other branch is touched in b' and c', by the sides
CA and ba prolonged through a. The curve is also hyperbo-
lic, if either a + 6 or a + c be negative, while b and c are posi-
tive as before.
50. When the quadratic (48) has its roots real and un-
equal, so that the conic is an hyperbola, then the directions of
the asymptotes may be found, by substituting those roots,
or the values of t, u, v which correspond to them (or any
scalars proportional thereto), in the numerator of the expres-
sion (46) for p ; and similarly we can find the direction of the
axis of the parabola, for the case when the roots are real but
equal : for we shall thus obtain the directions, or direction, in
which a right line op must be drawn from o, so as to meet the
conic at infinity. And the same conditions as before, for dis-
tinguishing the species of the conic, maybe otherwise obtained
by combining the anharmonic equation, /= (46), of that
conic, with the corresponding equation ax + by ^■cz={) (38) of
the line at infinity ; so as to inquire (on known principles of
modern geometry) whether that line meets that curve in tivo
CHAP. II.] DIFFERENTIALS — TANGENTS POLARS. 39
imaginary points^ or touches it, or cuts it, in points which (al-
though infinitely distant) are here to be considered as real,
51. In general, if /(a?, y, z) = be the anharmonic equa-
tion (46) oi any plane curve, considered as the locus of a varia-
ble point p ; and if the differential* of this equation be thus
denoted,
= d/(a?,3/, ^) = Xdar+ Ydy+^ds';
then because, by the supposed homogeneity (46) of the func-
tion/, we have the relation
Xx^Yy + Zz=^fd,
we shall have also this other but analogous relation,
if ,
x' - x'.y' -y \z' - z = diX',diy\<\.z\
that is (by the principles of Art. 37), if p'=-(a;'j y\ z!) be any
point upon the tangent to the curve, drawn at the point
p = (re, y, z), and regarded as the limit of a secant. The sym-
hoi (37) of this tangent at p may therefore be thus written,
[X,y, ZJ, or [D,/ D,/, D,/];
where d^, d^, d^ are known characteristics of partial deriva-
tion.
52. For example, whenyhas the form assigned in 46, as an-
swering to the conic lately considered, we have d.t/= 2{x-y-z),
&c. ; whence the tangent at any point (x, y, z) of this curve
may be denoted by the symbol,
\_x-y-z, y-z-x, z-x-y];
in which, as usual, the co-ordinates of the line may be replaced
by any others proportional to them. Thus at the point a', or
(by 36) at (0, 1, 1), which is evidently (by the form of/) a
point upon the curve, the tangent is the line [- 2, 0, 0], or
[1, 0, 0] ; that is (by 38), the side bc of the given triangle, as
* In the theory of qziaternions, as distinguished from (although including) that
of vectors, it will he found necessary to introduce a new definition of differentials, on
account of the non- commutative property o{ quaternion-multiplication : hut, for the
present, the usual significations of the signs d and d are sufficient.
40 ELEMENTS OF QUATERNIONS. [bOOK I.
was Otherwise found before (46). And in general it is easy to
see that the recent symbol denotes the right line, which is (in
a well known sense) the polar of the point {x, y, z), with re-
spect to the same given conic ; or that the line [X', F', Z''\ is
the polar of the point (x', y, z) : because the equation
Xx'+Yy' + Zz^O,
which for a conic may be written as X'x + Y'y + Z'z = 0,
expresses (by 51) the condition requisite, in order that a point
(x, y, z) of the curve* should belong to a tangent which passes
through the point {x\ y\ z). Conversely, the point {x, y, z)
is (in the same well-known sense) the po/^ of the line [X, Y, Z"] ;
so that the centre of the conic, which is (by known principles)
i\\Q pole of the line at infinity (38), is the point which satisfies
the conditions a-^X=^h-^Y=c-^Z \ it is therefore, for the pre-
sent conic, the point k = (6 + c, c + «, a + S), of which the
vector OK is easily reduced, by the help of the linear equation,
«a + Z>j3 + cy = (27), to the form,
2 {he + c« + ah) '
with the verification that the denominator vanishes^ by 48,
when the conic is a parabola. In the more general case, when
this denominator is different from zero, it can be shown that
every chord of the curve, which is drawn through the extremity
K of the vector k, is bisected at that point k : which point
would therefore in this way be seen again to be the centre.
53. Instead of the inscribed conic (46), which has been the
subject of recent articles, we may, as another example, consi-
der that exscribed (or circumscribed) conic, which passes
through the three corners a, b, c of the given triangle, and
touches there the lines aa", bb", cc" of Fig. 21. The anhar-
monic equation of this new conic is easily seen to be,
yz -v zx -\^ xy = ;
* If the curve /= were of a degree higher than the second, then the two equa-
tions above written would represent what are called the first polar, and the last or
the line-polar, of the point (x', y\ z'), with respect to the given curve.
CHAP. 11 ] VECTOR OF A CUBIC CURVE. 41
the vector of a variable point p of the curve may therefore be
expressed as follows,
with the condition ^ + m + v = 0, as before. The vector of its
centre k' is found to be,
^2 _^ 52 4. c2 - 2bc - 2ca - lab '
and it is an ellipse, a parabola, or an hyperbola, according as
the denominator of this last expression is negative, or null, or
positive. And because these two recent vectors^ jc, k, bear a
scalar ratio to each other, it follows (by 19) that the three
points o, K, k' are collinear ; or in other words, that the line
of centres kk', of the two conies here considered, passes through
the point of concourse o of the three lines aa', bb', cc'. More
generally, if l be the pole of any given right line A = [/, w, n]
(37), with respect to the inscribed conic (46), and if l' be the
pole of the same line A with respect to the exscribed conic of
the present article, it can be shown that the vectors ol, ol', or
A, X', of these two poles are of the forms,
\ = k (laa + mb^ + ncy)^ A' = h! {laa + mb^ + ncy),
where k and k' are scalar s ; the three points o, l, l' are there-
fore ranged on one right line.
54. As an example of a vector-expression for a curve of an
order higher than the second, the following may be taken :
t^aa + U^bQ + v^Cy
^ t^a + v?b + v^c
with ^ 4- M + r = 0, as before. Making x = t^, y^u^, z = v^, we
find here by elimination of t, u, v the anharmonic equation^
{x-\-y+ zy - 27 xgz--^0;
the locus of the point p is therefore, in this example, a curve of
the third order, or briefly a cubic curve. The mechanism (41)
G
42
ELEMENTS OF QUATERNIONS.
[book I.
Fig. 27.
of calculations with anharmonic co-ordinates is so much the
same as that of the known trilinear method, that it may suffice
to remark briefly here that the sides of the given triangle abc
are the three (real) tangents of inflexion; the points of inflexion
being those which are marked as a", b", c" in Fig. 2 1 ; and the
origin of vectors o being a conjugate point* lia=b = c,in which
case (by 29) this origin o becomes (as in Fig. 19) the mean
point of the trian-
gle, the chord of
inflexion a"b"c" is
then the line at
infinity, and the
curve takes the
form represented
in Fig. 27; hav-
ing three infinite
branches, inscribed within the angles vertically opposite to
those of the given triangle abc, of which the sides are the
three asymptotes.
55. It would be improper to enter here into any details of
discussion of such cubic curves, for which the reader will na-
turally turn to other works.f But it may be remarked, in
passing, that because the general cubic may be represented, on
the present plan, by combining the general expression of Art.
34 or 36 for the vector p, with the scalar equation
s^ = 27kxgz, where s = a; + y-\- z;
k denoting an arbitrary constant, which becomes equal to
unity, when the origin is (as in 54) a conjugate point; it fol-
lows that if p = (x, y, z) and p' = (a?', y', z) be any two points
of the curve, and if we make s' = x' + y' + z, we shall have the
relation,
x^ ys' zs
sx sy sz'
xyzs ^ = xyz s^, or — ;
* Answering to the values ^=1, m = 0, v=Q\ where is one of the imaginary
cube-roots of unity ; which values of t^ u, v give x — y = z, and p = 0.
t Especially the excellent Treatise on Higher Plane Curves^ by the Rev. George
Salmon, F. T. C D., &c. Dublin, 1852.
CHAP. II.] ANHARMONIC PROPERTY OF CUBIC CURVES. 43
in which it is not difficult to prove that
•^'=(a".pbp'b"); ^,= (b".pcp'c"); — , = (c". papV);
sx ^ sy ^ sz
the notation (35) of anharmonics of pencils being retained.
We obtain therefore thus the following Theorem : — " If the
sides of any given plane* triangle abc he cut (as in Fig. 2\)hy
any given rectilinear transversal a"b"c'', and if any two points
p and p' in its plane be such as to satisfy the anharmonic rela-
tion
(a". pbpV) . (b". pcp'c") . (c". papa") = 1,
then these two points p, p' are on one common cubic curve, which
has the three collinear points a", b", c" for its three real points
of inflexion^ and has the sides bc, ca, ab of the triangle for its
three tangents at those points ;" a result which seems to offer
a new geometrical generation for curves of the third order,
5Q. Whatever the order of a plane curve may be, or what-
ever may be the degree p of ihQfunctionf'm. 46, we saw in 51
that the tangent to the curve at any point p = (a:, y, z) is the
right line
A = [/, m, w], if 1= Hxf, rn = Hyfi n = n^f-,
expressions which, by the supposed homogeneity off, give the
relation, Ix -\-my+nz^ 0, and therefore enable us to establish
the system of the two following differential equations,
Idx + mdy + ndz = 0, xdl + ydm + zdn = 0.
If then, by elimination of the ratios of x, y, z, we arrive at a neio
homogeneous equation of the form,
as one that is true for all values of x, y, z which render the
function /= (although it may require to be cleared of factors,
introduced by this elimination), we shall have the equation
F(l,m, n) = 0,
* This Theorem may be exteaded, with scarcely any modification, from plane to
spherical curves., of the third order.
44 ELEMENTS OF QUATERNIONS. [bOOK I.
as a condition that must be satisfied by the tangent A to the
curve, in all the positions which can be assumed by that right line.
And, by comparing the two differential equations,
dr(/, 772, W) = 0, red/ + 7/d77Z + 2:d77 = 0,
we see that we may write the proportion,
x\y\z= D/F : D,rtF : d„f, and the symbol v = (d^f, d„iF, d^f),
if {x, 7/, z) be, as above, the point of contact p of the variable
line [/, 772, n\ in any one of its positions, with the curve which
is its envelope. Hence we can pass (or return) from the tan-
gential equation f = 0, of a curve considered as the envelope of
a right line A, to the local equation f= 0, of the same curve
considered (as in 46) as the locus of a point p : since, if we ob-
tain, by elimination of the ratios of /, m, n, an equation of the
form
0=/(dzF, d,„f, d„f),
(cleared, if it be necessary, of foreign factors) as a conse-
quence of the homogeneous equation f = 0, we have only to
substitute for these partial derivatives, D/F, &c., the anhar-
monic co-ordinates x, 7/, z, to which they are proportional.
And when the functions /"and f are not only homogeneous (as
we shall always suppose them to be), but also rational and
integral (which it is sometimes convenient not to assume them
as being), then, while the degree of the function^ or of the
local equation, marks (as before) the order of the curve, the
degree of the other homogeneous function f, or of the tangential
equation F = 0, is easily seen to denote, in this anharmonic
method (as, from the analogy of other and older methods, it
might have been expected to do), the class of the curve to
which that equation belongs : or the number of tangents (dis-
tinct or coincident, and real or imaginary), which can be drawn
to that curve, from an arbitrary point in its plane.
57. As an example (comp. 52), if we eliminate x, y, z be-
tween the equations,
l = x-y-z, m = y-z - X, n = z-x-y, Ix + my + 7iz== 0,
where /, in, n are the co-ordinates of the tangent to the inscribed
CHAP. II.] LOCAL AND TANGENTIAL EQUATIONS. 45
conic of Art. 46, we are conducted to the following tangen-
tial equation of that conic, or curve of the second class,
f(1, m,n) = mn + nl+ lm = ;
with the verification that the sides [1, 0, 0], &g. (38), of the
triangle abc are among the lines which satisfy this equation.
Conversely, if this tangential equation were given, we might
(by 5Q) derive from it expressions for the co-ordinates of con-
tact X, 2/, z, as follows :
a;=D/F = 772+72, 2/ = n -^ I, Z = I -^ m ',
with the verification that the side [1, 0, 0] touches the conic,
considered now as an envelope, in the point (0, 1, 1), or a', as
before : and then, by eliminating /, m, n, we should be brought
back to the local equation, f= 0, of 46. In like manner, from
the local equation /= yz + zx-\- xy = of the exscribed conic (53),
we can derive by differentiation the tangential co-ordinates,*
I = T>jf^= y -^ z, rn = z-\- X, n = X + y,
and so obtain by elimination the tangential equation, namely,
f(/, 7w, n) = l^ + m^+n''- 2mn - 2nl -2lm = 0;
from which we could in turn deduce the local equation. And
(comp. 40), the very simple formula
Ix + my+nz = 0,
which we have so often had occasion to employ, as connecting
two sets of anharmonic co-ordinates, may not only be consi-
dered (as in 37) as the local equation of a given right line A,
along which a point p moves, but also as the tangential equa-
tion of a given point, round which a right line turns : according
as we suppose the set I, 7n, n, or the set x, y, z, to be given.
Thus, while the right line a"b"c", or [1, 1, 1], of Fig. 21, was
* This name of " tangential co-ordinates'^ appears to have been first introduced
by Dr. Booth in a Tract published in 1840, to which the author of the present Ele-
ments cannot now more particularly refer : but the system of Dr. Booth was entirely
dilFerent from his own. See the reference in Salmon's Higher Plane Curves, note to
page 16.
46 ELEMENTS OF QUATERNIONS. [bOOK I.
represented in 38 by the equation a; + z/ + 2: = 0, the point o of
the same figure, or the point (1, 1, 1), may be represented by
the analogous equation^
l + m + n = 0;
because the co-ordinates I, ni, n of every line, which passes
through this point o, must satisfy this equation of the first de-
gree, as may be seen exemphfied, in the same Art. 38, by the
lines OA, ob, oc.
58. To give an instance or two of the use of forms, which,
although homogeneous, are yet not rational and integral {pQ),
we may write the local equation of the inscribed conic (46) as
follows :
ai + ?/4 + 22 = ;
and then (suppressing the common numerical factor J), the
partial derivatives are
I = x% m = 2/"2, n = z'h;
so that a form of the tangential equation for this conic is,
/-I + ni-i ^ ^-1 = Q .
Avhich evidently, when cleared of fractions, agrees with the first
form of the last Article : with the verification (48), that
^-1 4. ^-1 4. c-i = when the curve is a parabola ; that is, when
it is touched (50) by the line at infinity (38). For the ex-
scribed conic (53), we may write the local equation thus,
x-'^ + y^ + 2-^ = 0;
whence it is allowed to write also,
Z=a;-2, m = y-'^, n-=z-\
and
lh + mh + n^=0 ;
a form of the tangential equation which, when cleared of radi-
cals, agrees again Avith 57. And it is evident that we could
return, with equal ease, from these tangential to these local
equations.
59. For the cubic curve with a conjugate point (54), the
local equation may be thus written,*
* Compare Salmon's Higher Plane Curves, page 172.
CHAP. II.] LOCAL AND TANGENTIAL EQUATIONS. 47
we may therefore assume for its tangential co-ordinates the
expressions,
/ = x'i, m = ?/-!, n = ^i ;
and a form of its tangential equation is thus found to be,
Conversely, if this tangential form were given, we might re-
turn to the local equation, by making
X = Zf , y = m"f , z = w"2,
which would give x^-vy^-^ zi= 0, as before. The tangential
equation just now found becomes, when it is cleared of radi-
cals,
= 7-2 + ^-2 ^ ^-2 _ 2m-i n' - 2n-' l' - 21' m' ;
or, when it is also cleared of fractions,
= F = m^n^ + ^2/2 4. /2^2 _ 2nl^m - 2Im^n - 2mnH ;
of which the biquadratic form shows (by 5Q) that this cubic
is a curve of the fourth class, as indeed it is known to be.
The inflexional character (54) of the points a", b", c" upon
this curve is here recognised by the circumstance, that when
we make m -n = 0, in order to find the four tangents from
a" =(0, 1,- 1) (36), the resulting biquadratic, = m*- Alm^, has
three equal roots ; so that the line [1, 0, 0], or the side Bc,
counts as three, and is therefore a tangent of inflexion : the fourth
tangent from a" being the line [1, 4, 4], which touches the
cubic at the point (- 8, U 1).
60. In general, the two equations {6Q),
nDj.f- lDzf= 0, nTfyf- mBzf^ 0,
may be considered as expressing that the homogeneous equa-
tion, ^
f{nx,ny, -lx-my) = 0,
which is obtained by eliminating z with the help of the rela-
tion Ix + my-^nz^ 0, from f(x, y, z) = 0, and which we may
48 ELEMENTS OF QUATERNIONS. [bOOK I.
denote by {x, y) = 0, has two equal roots x:y,\{ /, wi, n be
still the co-ordinates of a tangent to the curve/*; an equality
which obviously corresponds to the coincidence of two intersec-
tions of that line with that curve. Conversely, if we seek by
the usual methods the condition of equality of two roots xiy of
the homogeneous equation of the p^^ degree,
= ^ (a;, y) =f{nx, ny, -Ix- my),
by eliminating the ratio x : y between the two derived homo-
geneous equations, = Dj.^, = d,^0, we shall in general be
conducted to a result of the dimension 2p{p- 1) in /, m, n,
and of the ^rm,
= wP^P-i) F (/, m, n) ;
and so, by the rejection of the foreign factor nP^P-'^\ introduced
by this elimination,* we shall obtain the tangential equation
F = 0, which will be in general of the degree /?(p - 1 ) ; such being
generally the known class (pQ) of the curve of which the
order (46) is denoted by p : with (of course) a similar mode of
passing, reciprocally, from a tangential to a local equation.
61. As an example, when the function /has the cubic form
assigned in 54, we are thus led to investigate the condition for
the existence of two equal roots in the cubic equation,
= (p(x,y)= [(n-l)x+ (m - l)y]'^ + ''277i^xy(lx+ my),
by eliminating x : y between two derived and quadratic equa-
tions ; and the result presents itself, in the first instance, as of
the twelfth dimension in the tangential co-ordinates /, m, n ;
but it is found to be divisible by n^, and when this division is
effected, it is reduced to the sixth degree, thus appearing to
imply that the curve is of the sixth class, as in fact the general
cubic is well known to be. A. further reduction is however
possible in the present case, on account of the conjugate point
o (54), which introduces (comp. 57) the quadratic factor,
* Compare the method employed in Sahnon's Higher Plane Curves, page 98, to
find the equation of the reciprocal of a given curve, with respect to the imaginary
conic, *2 4.y3-|- j2 = 0. In general, if the function f be deduced from /as above,
then F(a;y?)= 0, and f(xyz) = are equations of two reciprocal curves.
CHAP. III.] VECTORS OF POINTS IN SPACE. 49
(/+ m + w)2 = ;
and when this factor also is set aside, the tangential equation
is found to be reduced to the biquadratic form* already assigned
in 59 ; the algebraic division, last performed, corresponding
to the known geometric depression of a cubic curve with a
double point, from the sixth to ihQ fourth class. But it is time
to close this Section on Plane Curves ; and to proceed, as in
the next Chapter we propose to do, to the consideration and
comparison of vectors of points in space.
CHAPTER III.
APPLICATIONS OF VECTORS TO SPACE.
Section 1. — On Linear Equations between Vectors not Com^
planar.
62. When three given and actual vectors oa, ob, oc, or
«5 i3j 7 J are not contained in any common plane, and w^hen
the three scalars a, b, c do not all vanish, then (by 21, 22)
the expression aa + b[5 + cy cannot become equal to zero ; it
must therefore represent 50/w^ actual vector (1), which we may,
for the sake of symmetry, denote by the symbol - d^ : where
the new (actual) vector B, or od, is not contained in any one
* If we multiply that form f = (59) by z% and then change nz to-lx- my,
we obtain a biquadratic equation in / : w, namely,
= ;//(;, w) = (^ - m)2 (Ix + myy^ + 2lm {I + m) {Jx -f my) z + I'^nfiz^ \
and if we then eliminate I : m between the two derived cubics, = Dii|/, = d,„i//,
we are conducted to the following equation of the twelfth degree, = x^y'^z^fix, y, z),
where /ha3 the same cubic form as in 54. "We are therefore thus brought hack
(comp. 59) from the tangential to the local equation of the cubic curve (54) ; com-
plicated, however, as we see, with the /ac^or x^y'^z^^ which corresponds to the sys-
tem of the three real tangents of inflexion to that curve, each tangent being taken
three times. The reason why we have not here been obliged to reject also the foreign
factor, 2*2, as by the general theory (60) we might have expected to be, is that we
multiplied the biquadratic function f only by z2, and not by z'^.
H
50
ELEMENTS OF QUATERNIONS.
[book I.
of the three given and distinct planes, boc, coa, aob, unless
some one, at least, of the three given coefficients «, 6, c, va-
nishes ; and where the new scalar^ d, is either greater or less
than zero. We shall thus have a linear equation between four
vectors,
aa + b(5 + cy + dd = ;
which will give
g =
aa
bfi
where oa', ob', oc'.
-Cj
or
or od = oa'+ ob'+ oc'
aa
-b(5 ~Cy
Fig. 28.
—7-5 — -T-j — r, are the
a d d
vectors of the three points
a', b', c', into which the
point D is projected^ on the
three given lines oa, ob, oc,
by planes drawn parallel to
the three given planes, boc,
&c. ; so that they are the
three co-initial edges of a
parallelepiped, whereof the sum, od or §, is the internal
and co-initial diagonal (comp. 6). Or we may project d on
the three planes, by lines da", db", dc" parallel to the three
• . bQ + Cy
given lines, and then shall have oa" = ob' + oc'= — — — ^, &c.,
- d
and
g = OD = oa' + oa" = ob' + ob" = oc' + oc".
And it is evident that this construction will apply to any ffth
point D of space, if the j^wr points oabc be still supposed to be
given, and not complanar : but that some at least of the three
ratios of the four scalars a, b, c, d (which last letter is not
here used as a mark of differentiation) will vary with the^o-
sition of the point d, or with the value of its vector 8. For
example, we shall have a = 0, if d be situated in the plane boc ;
and similarly for the two other given planes through o.
63. We may inquire (comp. 23), ichat relation between
these scalar coefficients must exist, in order that the point d
CHAP. III.] VECTORS OF POINTS IN SPACE. 51
may be situated in the fourth given plane abc ; or what is the
condition of complanarity o^ \hQ four points, a, b, c, d. Since
the three vectors da, db, dc are now supposed to be complanar,
they must (by 22) be connected by a linear equation, of the
form
fl(a-g) + 6(j3-g) + c(y-g) = 0;
comparing which with the recent and more general form (62),
we see that the required condition is,
a + 5 + c + c?= 0.
This equation may be written (comp. again 23) as
-a -b -c , oa' ob' oc' ,
d d d OA OB 00
and, under this last form, it expresses a known geometrical
property of a plane abcd, referred to three co-ordinate axes
OA, OB, oc, which are drawn from any common origin o, and
terminate upon the plane. We have also, in this case of com-
planarity (comp. 28), the following proportion of coefficients
and areas :
a :b: c :- d = dbc : dca : dab : abc ;
or, more symmetrically, with attention to signs of areas,
a :b: c : d = bcd : - cda : dab : - abc ;
where Fig. 1 8 may serve for illustration, if we conceive o in
that Figure to be replaced by d.
64. When we have thus at once the two equations,
aa-¥bf^ + cy + d^ = 0, and a + b + c + d=0,
so that the four co-initial vectors a, /3, y, S terminate (as above)
on one common plane, and may therefore be said (comp. 24) to
be termino-complanar, it is evident that the two right lines,
da and bc, which connect two pairs of the four complanar
points, must intersect each other in some point a' of the plane,
at a finite or infinite distance. And there i no difficulty in
perceiving, on the plan of 31, that the vectors of the three
52
ELEMENTS OF QUATERNIONS.
[book I.
points a', b', c' of intersection, which thus result, are the fol-
lowing :
for a' = bc'Da,
for b'= ca'DB,
for c' = ab • DC,
^'=
b^c -
a +
d
cy + aa
&/3 +
dd
cv a
- b +
d
aa + b^
Cy +
d^
a +b
c +
d
expressions which are independent of the position of the arbi-
trary origin o, and which accordingly coincide with the cor-
responding expressions in 27, when we place that origin in the
point D, or make S = 0. Indeed, these last results hold good
(comp. 31), even when the^wr vectors a, ^, y, ^, or the Jive
points o, A, b, c, d, are all complanar. For, although there
then exist two linear equations between those four vectors,
which may in general be written thus,
a a + ft'j3 + Cy + d'^ = 0, a"a f 6"/3 + c'y + d"8 = 0,
without the relations, a' + &c. = 0, a" + &c. = 0, between the
coefficients, yet if we form from these another linear equation,
of the form,
(a" + ta)a + {b" + tb')fi + (c" + tc')y + (d" + td')^ = 0,
and determine t by the condition,
t =
a" + b" + c" + d"
a+b' + c+d'^
we shall only have to make a = a"+ ta, &c., and the two equa-
tions written at the commencement of the present article will
then both be satisfied; and will conduct to the expressions
assigned above, for the three vectors of intersection : which
vectors may thus be found, without its being necessary to em-
ploy those processes of scalar elimination^ which were treated
of in the foregoing Chapter.
As an Example, let the two given equations be (comp. 27, 33),
aa + ij3 + cy = 0, (2a + fc + c)a'"- aa = ;
CHAP. III.] VECTORS OF POINTS IN SPACE. 53
and let it be required to determine the vectors of the intersections of the three pairs
of lines bc, aa'" ; CA, ba'" ; and ab, ca"'. Forming the combination,
(2a + 6 + c)a" - aa-\- t(aa + JjS + cy) = 0,
and determining t by the condition,
(2a + 6 + c) - a + <(a + 6 + c) = 0,
which gives * = — 1, we have for the three sought vectors the expressions,
bfi + cy cy + 2aa 2aa + bjS
b + c ' c+2a ' 2a + 6 '
whereof the first = a, by 27. Accordingly, in Fig. 21, the line aa'" intersects bc in
the point a' ; and although the two other points of intersection here considered,
which belong to what has been called (in 34) a Third Construction, are not marked
in that Figure, yet their anharmonic symbols (36), namely, (2, 0, 1) and (2, 1, 0),
might have been otherwise found by combining the equations y = and x — lz for the
two lines ca, ba'" ; and by combining z = 0, x = 2y for the remaining pair of lines.
Q5. In the more general case, when the four given points
A, B, c, D, are not in sluj common plane, let k be any fifth given
point of space, not situated on any one o^ the fijur faces of the
given pyramid abcd, nor on any such face prolonged ; and let
its vector oe = c. Then the/owr co-initial vectors, ea, eb, ec,
ED, v^hereof (by supposition) no three are complanar, and which
do not terminate upon one plane, must be (by 62) connected
by some equation of the form,
tf .EA + 6.EB + C.E0 + 6?.ED = 0;
where the^wr scalar s, a, b, c, d, and their sum, which we shall
denote by - e, are all different fiom zero. Hence, because
ea = a - £, &c., we may establish the following linear equation
between five co-initial vectors, a, j3, 7, S, e, whereof wo j^tt?- are
termlno-complanar (64),
aa + Jj3 + Cy + c?S + e£ = ;
with the relation, a+^ + c + c?+e = 0, between ih^five scalars
a, b, c, d, e, whereof no one now separately vanishes. Hence
also, £ = (aa + b(5+cy + d^) : (a+b + c+ d), &c.
66. Under these conditions, if we write
Di = DE*ABC, and ODi = ^i,
that is, if we denote by di the vector of the point Di in which
the right line de intersects the plane abc, we shall have
54 ELEMENTS OF QUATERNIONS. [bOOK I.
Oi = r = — = .
a + b+ e d+ e
In fact, these two expressions are equivalent^ or represent one
common vector, in virtue of the given equations; but the first
shows (by 63) that this vector Si terminates onthe/>Z«we abc,
and the second shows (by 25) that it terminates on the line
DE ; its extremity Di must therefore be, as required, the inter-
section of this line with that plane. We have therefore the two
equations,
I. . .a(«-gi) + *(i3-^i) + c(y-S0 = 0;
II.. .d{d~Si) + e(e-Bi)^0;
whence (by 28 and 24) follow the two proportions,
T, . . a:b:c= DjBC : DiCA : DiAB ;
ir. . . d:e= EDiiDiD ;
the arrangement of the points, in the
annexed Fig. 29, answering to the case
where all the four coefficients a, b, c, d
are positive (or have one common sign),
and when therefore the remaining co- '^" '
efficient e is negative (or has the opposite sign).
67. For the three complanar triangles, in the first propor-
tion, we may substitute any three pyramidal volumes, which
rest upon those triangles as their bases, and which have one
common vertex, such as D or e ; and because the collineation
DEDi gives DDiBc - EDiBc ~ DEBc, &c., wc may write this other
proportion,
F. . . a:b:c = debc : deca : deab.
Again, the same collineation gives
EDi : DDi = EABC : DABC ;
we have therefore, by IP., the proportion,
II". . . d: -e = EABC : DABC.
But
DEBC + DECA + DEAB + EABC = DABC,
and
CHAP. III.] VECTORS OF POINTS IN SPACE. 55
a-^ b + c + d= -e;
we may therefore establish the following fuller formula of
proportion, between coefficients and volumes :
III. . . aibicid: -e = debc : deca : deab : eabc : dabc ;
the ratios of all these five pyramids to each other being consi-
dered as positive^ for the particular arrangement of the points
which is represented in the recent figure.
68. The formula III. may however be regarded as per-
fectly general^ if we agree to say that a pyramidal volume changes
sign, or rather that it changes its algebraical character, as po-
sitive or negative, in comparison with a given pyramid, and
with a given arrangement of points, in passing through zero
(comp. 28) ; namely when, in the course of any continuous
change, any one of its vertices crosses the corresponding base.
With this convention* we shall have, generally,
DABC = -ADBC = ABDC = - ABCD, DEBC = BCDE, DECA = CDEA ;
the proportion III. may therefore be expressed in the follow-
ing more symmetric, but equally general form :
Iir. . , a:b:c:d:.e = bcde : cdea : deab : eabc : abcd ;
the sum of these j^ve pyramids being always equal to zero,
when signs (as above) are attended to.
69. We saw (in 24) that the two equations,
aa + bfi + cy = 0, a + b + c = 0,
gave the proportion of segments,
a : b : c = BC : CA : ab,
whatever might be the position of the origin o. In like man-
ner we saw (in 63) that the two other equations,
♦ Among the consequences of this convention respecting signs of volumes, which
has already been adopted by some modern geometers, and which indeed is necessary
(comp. 28) for the establishment of general formulae, one is that any two pyramids,
ABCD, a'b'c'd', bear to each other a positive or a negative ratio, according as the two
rotations, BCD and b'c'd', supposed to be seen respectively from the points A and a',
have similar or opposite directions, as right-handed or left-handed.
56 ELEMENTS OF QUATERNIONS. [bOOK I.
aa + bfi + Cy+d^^O, a + 6 + c + c? = 0,
gave the proportion of areas,
a:b:c: d= bcd : - cda : dab : - abc ;
where again the origin is arbitrary. And we have just deduced
(in 68) a corresponding proportion of volumes, from the two
analogous equations {65),
fla + 6/3 + cy + </S + ee = 0, a + b + c-\^d+e=0,
with an equally arbitrary origin. If then we conceive these
segments, areas, and volumes to be replaced by the scalars to
which they are thus proportional, we may establish the three
general for mulce. :
I. OA.BC + OB.CA+ OC.AB = ;
II. OA.BCD - OB. CD A + 00. DAB -0D.ABC = ;
III. OA.BCDE + OB.CDEA+ OC.DEAB + OD . EABC+ OE . ABCD = ;
where in I., a, b, c are ani/ three collinear points ;
in II., A, B, c, D are any four complanar points ;
and in III., a, b, c, d, e are any five points of space ;
while o is, in each of the three formulas, an entirely arbitrary
point. It must, however, be remembered, that the additions
and subtractions are supposed to be performed according to the
rules of vectors, as stated in the First Chapter of the present
Book ; the segments, or areas, or volumes, which the equations
indicate, being treated as coefficients of those vectors. We
might still further abridge the notations, while retaining the
meaning of these formulae, by omitting the symbol of the arbi-
trary origin o ; and by thus writing,*
r. A.BC + B.CA + CAB = 0,
for any three collinear points ; with corresponding formulae II'.
and III'., for any four complanar points, and for any five points
of space.
* We should thus have some of the notations of the Barycentric Calculus (see
Note B), but employed here with different interpretations.
CHAP. III.] QUINARY SYMBOLS FOR POINTS IN SPACE. 57
Section 2. — On Quinary Symbols far Points and Planes in
Space.
70. The equations of Art. Q6 being still supposed to hold good,
the vector p of any point P of space may, in indefinitely many ways,
be expressed (comp. 36) under the form :
xaa + yhB + zc<^ + wd^ + vee
I. . . op = /> = ^!-^. ^ ;
in which the ratios of the differences of ihe five coefficients^ xyzwv, de-
termine the position of the point. In fact, because the four points
ABCD are not in any common plane, there necessarily exists (comp.
65) a determined linear relation between the four vectors drawn to
them from the point P, which may be written thus,
a/a . PA + y'b . pb + z^c . PC + w'd . pd = 0,
giving the expression,
_ x'aa + y^h^ + z'c^ + w'dh
x'a + y'b + z'c + w'd *
in which the ratios of the four scalars x'y'z'w'^ depend upon, and
conversely determine, the position of p ; writing, then,
ic=te' + v, y = ty'^v^ z-tz'-^v^ w-tw' + Vy
where t and v are two new and arbitrary scalars, and remembering
that aa + . . + ee = 0, and « + . . + e = (65), we are conducted to the
form for /», assigned above.
71. When the vector p is thus expressed, the point p maybe
denoted by the Quinary Symbol {x, ?/, z^ Wy v) ; and we may write
the equation,
p = (x, y, z, w, v).
But we see that the same point p may also be denoted by this other
symbol, oHhe same kind, (a/, y, z\ w\ v'), provided that the follow-
ing /jropor^eoM between differences of coefficients (70) holds good:
x' -v' '. y' -v''.z' -v''.w' -v' = x-v'.y-v\z-v'.w-v,
Undei' this condition, we shall therefore write the following /orww/a
of congruence,
{x\ y', z', w', v') E {x, y, z, w, v),
to express that these two quinary symbols, although not identical in
composition, have yet the same geometrical signification, or denote one
common point. And we shall reserve the symbolic equation,
{x', y, z', w', v') = {x, y, z, w, v),
I
58 ELEMENTS OF QUATERNIONS. [bOOK I.
to express that the Jive coefficients, x' . . . v\ of the one symbol, are
separately equal to the corresponding coefficients of the other,
a;' = flj, . . v' = v.
72. Writing also, generally,
(to, ty^ tZf tw, tv) = t (x, y, z, w, v),
{x' + a;, . . v' + v) = (x\ . . v') + (a;, . . v), &c.,
and abridging the particular symbol* (1, 1, 1, 1, 1) to (Z7), while
(Q)> (Q0» • • "^^y briefly denote the quinary symbols (a;, . . v),
{x', . . v'), . . we may thus establish the congruence (71),
(Q')=(a), if (Q)=«(ao+w(£^);
in which t and u are arbitrary coefficients. For example,
(0,0, 0,0, 1)E (1,1, 1,1,0), and (0, 0, 0, 1, 1)E(1, 1, 1, 0, 0);
each symbol of the first pair denoting (fi5) the given point e; and
each symbol of the second pair denoting ifiQ) the derived point Di.
When the coefficients are so simple as in these last expressions, we
may occasionally omit the commas^ and thus write, still more briefly,
(00001) = (11110); (00011) E (1 1100).
73. If three vectors, />, /?', p"^ expressed each under the first
form (70), be termino-collinear (24) and if we denote their denomi-
tors, a;a + . . , rc'a + . . , x"a + . . , by m-, m\ m!\ they must then (23) be
connected by a linear equation, with a null sum of coefficients, which
may be written thus :
tmp + t'm'p' + i"m"p" = ; tm^ t'm' + t"m" + 0.
We have, therefore, the two equations of condition^
t {xaa + . . + vee) + 1' {x'aa + . . + v'ee) + 1" {x"aa + . . + v"ee) = ;
t{xa + . . + ve) + 1' {x'a + . . + v'e) + f' {x"a + . . + v"e) = ;
where t, f, t" are three new scalars, while the five vectors a . . e, and
the five scalars a..e, are subject only to the two equations (65);
but these equations of condition are satisfied by supposing that
tx + t'x' + t"x" = . . = a' + t'v' + t"v" = -u,
where u is some new scalar, and they cannot be satisfied otherwise.
Hence the condition of collinearity of the three points p, p', p'', in
which the three vectors />, p', p" terminate, and of which the qui-
nary symbols are (Q), (QOi {.01% "^^y briefly be expressed by the
equation,
* This quinary symbol ( U) denotes no determined point, since it corresponds
(by 70, 71) to the indeterminate vector /o = - ; but it admits of useful combinations
with other quinary symbols, as above.
CHAP. III.] QUINARY SYMBOLS OF PLANES. 59
t{Q) + V {Q) + t" {Q")^-u{U);
so that if ant/ four scalars, <, t\ t'\ u, can he found, which satisfy this
last symbolic equation, then, but not in any other case, those three
points pp'p" are ranged on one right line. For example, the three
points D, E, Di, which are denoted (72) by the quinary symbols,
(00010), (00001), (11100), are coUinear ; because the sum of these
three symbols is ( U). And if we have the equation,
where t, f, u are any three scalars, then {Q") is a symbol for a point
v", on the right line pp'. For example, the symbol (0, 0, 0, t, t') may
denote any point on the line de.
74. By reasonings precisely similar it may be proved, that if
(Q) (QO (^'0 (Q'^0 be quinary symbols for &ny four points pp^p'^p'^'
in any common plane, so that the four vectors pp'p^p'^' are termino-
complanar (64), then an equation, of the form
UQ) + i^QO + 1" (Q'O + i'" ( Q''0 = - «^( C^)»
must hold good; and conversely, that \i the fourth symbol can be
expressed as follows,
{Cl"^) = t{a)^t' {Cl')^t"{Q!') ^u{U\
with any scalar values oit, t', t" , u, then the fourth point 2'^' is situ-
ated in the plane pp'p'^ of the other three. For example, the four
points,
(10000), (01000), (00100), (11100),
or A, B, c, Di {^^\ are complanar; and the symbol {t, t' , t", 0, 0)
may represent any point in the plane abc.
75. When a point p is thus complanar with three given points,
Po, Pi, P2, we have therefore expressions of the following forms, for
ih.Q five coefficients x, ..v oi its quinary symbol, in terms of the fif-
teen given coefficients oi their symbols, and of /owr new and arbitrary
scalars :
X = ^o^^o + <i^i + k^i + «^; . . . V = ^0^0 + t,Vi + kv.i + u.
And hence, by elimination of these four scalars, tQ..u, we are con-
ducted to a linear equation of the form
l{x -v) -^^ m{y - v) + n{z - v) ^- r (w -v) = 0,
which may be called the Quinary Equation of the Plane PqPiPo, or of
the supposed locus of the point p: because it expresses a common
property of all the points of that locus; and because the three ratios
of the/owr new coefficients I, m, n, r, determine the position of the plane
60 ELEMENTS OF QUATERNIONS. [bOOK I.
in space. It is, however, more symmetrical, to write the quinary
equation of a plane 11 as follows,
Ix -h my + nz + rw + sv ~ 0,
■where the ffth coefficient, s, is connected with the others by the rela-
tion,
/4-w + n + r+5 = 0;
and then we may say that [/, w, n, r, 5] is (comp. 37) the Quinary
Symbol of the Plane 11, and mtiy write the equation,
n = [I, m, w, r, s].
For example, the coefficients of the symbol for a point p in the plane
ABC may be thus expressed (comp. 74) :
X=^tQ + U, y = ti + U, Z = t^ + U, W=U, V=U'^
between which the only relation, independent of the four arbitrary
scalars to. .u, is w-v=0; this therefore is the equation of the plane
ABC, and the symbol of that plane is [0, 0, 0, 1, - 1]; which may
(comp. 72) be sometimes written more briefly, without commas, as
[00011]. It is evident that, in any such symbol, the coefficients may
all be multiplied by any common factor.
76. The symbol of the plane P0P1P2 having been thus determined,
we may next propose to find a symbol for the^om^, p, in which that
plane is intersected by a given line P3P4: or to determine the coefficients
a; . . «>, or at least the ratios of their differences (70), in the quinary
symbol of that point,
(x, y, z, w, v) = T = PoPiPg • P3P4.
Combining, for this purpose, the expressions,
X = ^30:3 + tiX4, + u',. . v = t^Vs + ^4^4 + u\
(which are included in the symbolical equation (73),
{Q)=^t,{Q,)-\-t,(CL) + u^iU).
and express the collinearity PP3P4,) with the equations (75),
/a?+ .. +5t;=0, Z+.. + 5 = 0,
(which express the complanarity pPqPiP^,) we are conducted to the
formula,
^3 {Ix^ + . . + svg) -I- «4 {Ix^ + . . + 5^4) = 0;
which determines the ratio t^ : ^4, and contains the solution of the
problem. For example, if p be a point on the line de, then (comp.
73),
X=:y = z-u', w^tz+u', V = «4 + ?/;
CHAP. III.] QUINARY TYPES OF POINTS AND PLANES. 61
but if it be also a point in the plane abc, then w-v-0 (75), and
therefore ^3 - ^4 = ; hence
(Q) = ^3(00011) + w^(ll 111), or (Q) = (00011);
which last symbol had accordingly been found (72) to represent the
intersection (fi^), Dj = abc • de.
77- When the five coefficients, xyzwv, of any given quinary
symbol (Q) for a point p, or those of any congruent symbol (71), are
any whole numbers (positive or negative, or zero), we shall say
(comp. 42) that the point p is rationally related to the five given points,
A . . E ; or briefly, that it is a Kational Point of the System, which
those five points determine. And in like manner, when the five
coefficients, Imnrs, of the quinary symbol (75) of a plane 11 are either
equal or proportional to integers, we shall say that the plane is a Ra-
tional Plane of the same System; or that it is rationally related to the
same five points. On the contrary, when the quinary symbol of a
point, or of a plane, has not thus already whole coefficients, and can-
not be transformed (comp. 72) so as to have them, we shall say that
the point or plane is irrationally related to the given points; or
briefly, that it is irrational. A right line which connects two rational
points, or is the intersection of two rational planes, may be called, on
the same plan, a Rational Line ; and lines which cannot in either
of these two ways be constructed, may be said by contrast to be
Irrational Lines. It is evident from the nature of the eliminations
employed (comp. again 42), that a plane, which is determined as con-
taining three rational points, is necessarily a ra^eowaZ^Zawe; and in
like manner, that o. point, which is determined as the common inter-
section of three rational planes, is always a rational jwint : as is also
every point which is obtained by the intersection of a rational line
with a rational plane ; or of two rational lines with each other (when
they happen to be complanar).
78. Finally, when two points^ or two planes, differ only by the ar-
rangement (or order) of the coefficients in their qn'mar j symbols^ those
points or planes may be said to have one common type; or briefly
to be syntypicaL For example, ihefive given points, a, . . e, are thus
syntypical, as being represented by the quinary symbols (10000), . .
(00001); and the ten planes, obtained by taking all the ternary
combinations of those five points, have in like manner one common
type. Thus, the quinary symbol of the plane abc has been seen
(75) to be [OOOll]; and the analogous symbol [11000] represents
the plane cde, &c. Other examples will present themselves, in a
62 ELEMENTS OF QUATERNIONS. [bOOK I.
shortly subsequent Section, on the subject of Nets in Space. But
it seems proper to say here a few words, respecting those Aiihar-
monic Co-ordinates, Equations^ Symbols, and Types, for Space, which
are obtained from the theory and expressions of the present Section,
by reducing (as we are allowed to do) the number of the coefficients^
in each symbol or equation, from Jive to four.
Section 3. — On Anharmonic Co-ordinates in Space.
79. When we adopt the second form (70) for />, or suppose (as
we may) that the fifth coefficient in the yir5^ form vanishes, we get this
other general expression (comp. 34, 36), for the vector of a point in
space:
xaa + yh3 + zc^ + wdb
xa + yb-\-zc + wd
and may then write the symbolic equation (comp. 36, 71),
p=(a7, y, z, w),
and call this last the Quaternary Symbol of the Point P : although
we shall soon see cause for calling it also the Anharmonic Symbol of
that point. Meanwhile we may remark, that the only congruent
symbols (71), of this last form, are those which differ merely by the
introduction oi s. common factor : the three ratios of the /owr coeffi-
cients, X . ,w, being all required, in order to determine the position of
the point; whereof those four coefficients may accordingly be said
(comp. 36) to be the Anharmonic Coordinates in Space.
80. When we thus suppose that v = 0, in the quinary symbol of
t\ie point p, we may suppress the fifth term sv, in the quinary equation
of 2i plane IT, lx-\- ..+sv = (75) ; and therefore may suppress also (as
here unnecessary) th^ fifth coefficient, s, in the quinary symbol of that
plane, which is thus reduced to the quaternary form,
n = [/, m, n, r].
This last may also be said (37, 79), to he the Anharmonic Symbol of
the Plane, of which the Anharmonic Equation is
Ix + my + nz + rw = 0',
the four coefficients, Imnr, which we shall call also (comp. again 37)
the Anharmonic Co-ordinates of that Plane 11, being not connected
among themselves by any general relation (such as Z+ . .+5 = 0): since
their three ratios (comp. 79) are all in general necessary, in order to
determine the position of the plane in space.
81. If we suppose that the fourth coefficient, w, also vanishes, in
CHAP. III.] ANHARMONIC CO-ORDINATES IN SPACE. 63
the recent symbol of a point, thsit point p is in theplane abc ; and may-
then be sufficiently represented (as in 36) by the Ternary Symbol
(a?, y, z). And if we attend only to the points in which an arbitrary
plane n intersects the given plane abc, we may suppress its fourth co-
efficient, r, as being for such points unnecessary. In this manner,
then, we are reconducted to the equation, lx+my + nz= 0, and to the
symbol, A= [Z, m, w], for a right line (37) in the plane abc, considered
here as the trace, on that plane, of an arbitrary plane H in space. If
this plane n be given by its quinary symbol (75), we thus obtain
the ternary symbol for its tf^ace A, by simply suppressing the two last
coefficients, r and s.
82. In the more general case, when the point p is not confined
to the plane abc, if we denote (comp. 72) its quaternary symbol by
(Q), the lately established formulae of collineation and complanarity
(73, 74) will still hold good: provided that we now suppress the
symbol ( U), or suppose its coefficient to be zero. Thus, the formula,
{Q)=t'{Q)^t"{Q^)-Vt"'{Q"),
expresses that the point p is in the plane -j^'^f'-p'" ; and if the coeffi-
cient t"' vanish, the equation which then remains, namely,
signifies that p is thus complanar with the two given points p^, v",
and with an arbitrary third ^wint; or, in other words, that it is on
the right line v'v" ; whence (comp. 76) problems of intersections of
lines with planes can easily be resolved. In like manner, if we de-
note briefly by [i?] the quaternary symbol \l, m, n, r'] for a plane
n, the formula
[i2] = t' [i?'] + 1" IR"^ + 1"' [R"q
expresses that the plane n passes through the intersection of the thr^
planes, 11', II'', W ; and if we suppose t'^' = 0, so that
[ij]=«'[fi']+«"[fi"3,
the formula thus found denotes that the plane 11 passes through
the point of intersection of the two planes, 11', 11", with any third
jilane; or (comp. 41), that this plane n contains the line of intersec-
tion of n', n" ; in which case the three planes, Tl, 11', 11", may be
said to be coUinear. Hence it appears that either of the two expres-
sions,
I. . . t' ( Q') + ^" ( a^O. II- • • i' [-^G + i" \.Rf'\
may be used as a Symbol of a Right Line in Space : according as we
consider that line A either, 1st, as connecting two given points, or
64 ELEMENTS OF QUATERNIONS. [bOOK I.
Ilnd, as being the intersection of two given planes. The remarks (77)
on rational and irrational points, planes, and lines require no modifi-
cation here; and those on types (78) adapt themselves as easily to
quaternary as to quinary symbols.
83. From the foregoing general formulee of collineation and conj-
planarity, it follows that the point p', in which the line ab inter-
sects the plane cdp through CD and any proposed point P = {xyzw)
of space, may be denoted thus :
p' = AB • CDP = {xy{)Q)) ;
for example, e = (U 1 1), and c' = ab • cde = (1100). In general, if
ABCDEF be any six points of space, the four collinear planes (82), abc,
abd, ABE, ABF, are said to form a pencil through ab; and if this be
cut by any rectilinear transversal, in four points, c, D, e, f', then
(comp. 35) the anharmonic function of this group of points (25) is
called also the Anharmonic of the Pencil of Planes: which may be
thus denoted,
(ab . cdef) = (c'dVf').
Hence (comp. again 25, 35), by what has just been shown respect-
ing c' and p', we may establish the important formula:
(cD . AEBp) = (ac'bpO = - ;
so that this ratio of coefficients, in the symbol {xyzw) for a variable
point p (79), represents the anharmonic of a pencil of planes, of which
the variable plane cdp is one; the three other planes of this pencil
being given. In like manner,
• \ y 1 / \ -2^
(ad . BECP) = -, and (bd . ceap) = - ;
^ Z X
so that (comp. 36) the product of these three last anharmonics is
unity. On the same plan we have also,
(bc.aedp)=— , (ca.bedp) = — , (ab.cedp) = -;
w w ^ ^ w
so that the three ratios, of the three first coefficients xyz to the
fourth coefficient w, suffice to determine the three planes, bcp, cap,
ABP, whereof \h.Q point p is the common intersection, by means of the
anharmonics of thxe pencils of planes, to which the three planes re-
spectively belong. And thus we see a motive (besides that of analogy
to expressions already used for points in a given plane), for calling
the/owr coefficients, xyzw, in the quaterna/ry symbol (Jd) for 9, point in
space, the Anharmonic Co-ordinates of that Point.
84. In general, if there be any four collinear points, Vq, . . P3, so
CHAP. III.] ANHARMONIC CO-ORDINATES IN SPACE. 65
that (comp. 82) their symbols are connected by two linear equations,
such as the following,
(Qi) = «(Qo) + u{Cl,), (as) = t'{Q,) + w'(Q2),
then the anharmonic of their group may be expressed (comp. 25, 44)
as follows :
ut'
(PoPiP.P3) = -,;
as appears by considering the pencil (cd . PoPiPgPa), and the transversal
AB (83). And in like manner, if we have (comp. again 82) the two
other symbolic equations, connecting /om?' collinear planes IIq . . n^,
the anharmonic of their pencil (8.3) is expressed by the precisely
similar formula,
ut'
(n„n,n,n,) = _;
as may be proved by supposing the pencil to be cut by the same
transversal line ab.
85. It follows that ii f{xyzw) and /j (a^^^it') be any two homo-
geneous and linear functions of ic, y, z^w\ and if we determine four
collinear planes IIo . . Ila (82), by the four equations,
■/=0, /i=/, /x = 0, j\ = kf,
where h is any scalar ; we shall have the following value of the an-
harmonic function, of the pencil of planes thus determined:
f
Hence we derive this Theorem^ which is important in the application
of the present system of co-ordinates to space : —
" The Quotient of any two given liomogeneous and linear Functions^
of the anharmonic Co-ordinates (79) of a variable Point p in space, may
be expressed as the Anharmonic (noninalls) of a Pencil of Planes;
w^hereof three are given, while the fourth passes through the variable
point p, and through a given right line A which is common to the three
former planes y
86. And in like manner may be proved this other but analogous
Theorem : —
" The Quotient of any two given homogeneous and linear Functions,
of the anharmonic Co-ordinates (80) of a variable Plane n, may be ex-
pressed as the Anharmonic (PoPiP^Pa) of a Group of Points; whereof
three are given and colliriear, and the fourth is the intersection, A ' 11,
of their common and given right line A, with the variable plane H,"
K
66 ELEMENTS OF QUATERNIONS. [bOOK I.
More fully, if the two given functions of Imnr be f and y^^ and
if we determine three points P0P1P2 by the equations (comp. 57)
F = 0, Fi = F, Fi=:0, and denote by P3 the intersection of their com-
mon line A with n, we shall have the quotient,
^=(P0P,P,P3).
For example, if we suppose that
A2=(1001), B2=(010]), C2=(0011),
A'2 = (1001), B'2 = (OIOT), c'2 = (00 iT),
so that
A2 = DA*BCE, &c., and (dA2Aa'2) = - 1, &c.,
we find that the three ratios of Z, m, n to r, in the symbol n = [/mnr],
may be expressed (comp. 39) under the form of anharmonics of
groups, as follows;
- = (da'sAQ) ; - = (db^^br) ; - = (dc'sCs) ;
where q, r, s denote the intersections of the plane n with the three
given right lines, da, db, dc. And thus we have a motive (comp.
83) besides that of analogy to lines in a given plane (37), for calling
(as above) the, four coefficients I, m, n, r, in the quaternary symbol (80)
for a, plane n, the Anharmonic Co-ordinates of that Plane in Space.
87. It may be added, that if we denote by l, m, n the points in
which the same plane IT is cut by the three given lines bc, ca, ab,
and retain the notations a'', b''', c'^ for those other points on the same
three lines which were so marked before (in 31, &c.), so that we may
now write (comp. 36)
A''= (0110), b'' = (1010), c''= (llOO),
we shall have (comp. 39, 83) these three other anharmonics of groups,
with their product equal to unity :
— = (ca'^bl) ; - = (ab^^cm) ; — = (bc'^an) ;
n V 7ft
and the six given points, a.'\ e", c", A'2, B'2, c'2, are all in one given plane
[e], of which the equation and symbol are:
x + y + z + w = 0\ [e] = [11111].
The six groups of points, of which the anharmonic functions thus
represent the six ratios of the four anharmonic co-ordinates, lmm\
of a variable plane n, are therefore situated on the six edges of the
given pyramid^ abcd; two poi7iis in each group being corners of that
CHAP. III.] GEOMETRICAL NETS IN SPACE. 67
pyramid, and the tiuo others being the intersections of the edge with
the two planes^ [e] and n. Finally, the plane [e] is (in a known
modern sense) the plane of homology ^^' and the point e is the centre
of homology^ of the given pyramid abcd, and of an inscribed pyramid
AiBiCiDi, where Ai = ea*bcd, &c.; so that Di retains its recent signi-
fication (QQ, 76), and we may write the anharmonic symbols,
Ai = (0111), Bi = (1011), Ci=(1101), Di = (IllO).
And if we denote by a'ib'iC^d'i the harmonic conjugates to these
last points, with respect to the lines ea, eb, ec, ed, so that
(eaiAA'i) = . . = (eDiDD'i) = - 1,
we have the corresponding symbols,
A'i=(2111), B^ = (1211), c'i = (1121) D^ = (1112).
Many other relations of position exist, between these various points,
lines, and planes, of which some will come naturally to be noticed,
in that theory of nets in space to which in the following Section we
shall proceed.
Section 4. — On Geometrical Nets in Space,
88. When we have (as in Q5) five given points a . . e, whereof no
four are complanar, we can connect any two of them by a right line^
and the three others by a plane, and determine the point in which
these last intersect one another: deriving t\i\\s a system oHen lines Aj,
ten planes Hi, and ten points Pi, from the given system oi five points
Po, by what may be called (comp. 34) a First Construction. We may
next propose to determine all the new and distinct lines, A,, and
planes, Ila, which connect the ten derived points Pj with the five
given points Fq, and with each other ; and may then inquire what
new and distinct points Pa arise (at this stage) as intersections of lines
with planes, or oHines in one plane with each other: all such new lines,
planes, and points being said (comp. again 34) to belong to a Second
Construction. And then we might proceed to a Third Construction
of the same kind, and so on for ever : building up thus what has
been calledf a Geometrical Net in Space. To express this geome-
trical process by quinary symbols (71, 75, 82) o^ points, planes, and
lines, and by quinary types (78), so far at least as to the end of the
second construction, will be found to be an useful exercise in the
* See Poncelet's Traite des Propriete's Projectives (Paris, 1822).
t By Mbbius, in p. 291 of his already cited Barycentric Calculus,
68 ELEMENTS OF QUATERNIONS. [bOOK I.
application of principles lately established : and therefore ulti-
mately in that Method of Vectoks, which is the subject of the
present Book. And the quinary form will here be more convenient
than the quaternary^ because it will exhibit more clearly the geome-
trical dependence of the derived points and planes on ih^five given
points, and will thereby enable us, through a principle of symmetry^
to reduce the number of distinct types.
89. Of the five given points, Pq, the quinary type has been seen
(78) to be (10000); while of the ten derived points p,, o^ first con-
struction, the corresponding type may be taken as (00011); in fact,
considered as symbols, these two represent the points a andDj. The
nine other points Pi are a Vc/AiBjCiAaBaCa ; and we have now (comp.
83, 87, 86) the symbols,
A'= BC • ADE = (01 100), Ai = EA • BCD = (10001),
A2=DA -BCE^ (10010);
also, in any symbol or equation of the present form, it is permitted
to change a, b, c to b, g, a, provided that we at the same time write
the third, first, and second co-efficients, in the places of the first,
second, and third: thus, b' = ca • bde = (10100), &c. The symbol
(a;^000) represents an arbitrary point on the line ab ; and the sym-
bol [OOm'5], with n + r + 5 = 0, represents an arbitrary plane through
that line : each therefore may be regarded (comp. 82) as a symbol also
of the line ab itselfi and at the same time as a type of the ten lines
Ai; while the symbol [000 ll], of the plane abc (75), may betaken
(78) as a type of the ten planes Hi. Finally, the five pyramids,
bcde, cade, abde, abce, abcd,
and the ten triangles, such as abc, whereof each is a common face of
two such pyramids, may be called pyramids i?i, and triangles T^, of
the First Construction.
90. Proceeding to a Second Construction (88), we soon find that
the lines A, may be arranged in two distinct groups; one group con-
sisting oi fifteen lines Aj, i, such as the line* aa''d„ whereof each coti-
nects two points Pi, and passes also through one point Pq, being the inter-
section of two planes IIi through that point, as here of abc, ade;
while the other group consists of thirty lines Ag, 2, such as b'c', each
connecting two points Pi, but not passing through any point ?„, and
being one of the thirty edges of five new pyramids R^, namely,
C'b'AzA,, A'c'B^B], B^A'C^C,, A.B^C^Di, AiBjCiDj :
* AB1C2, ABoCi, da'Ai, ea'Ao, are other lines of this group.
CHAP. 111.] GEOMETRICAL NETS IN SPACE. 69
which pyramids i?2 may be said (comp. 87) to be inscribed homo-
logues of the five former pyramids i?i, the centres of homology for these
Jive pairs of pyramids being the five given points a . . e ; and \)i\Q. planes
of homology being five planes [a] . . [e], whereof the last has been
already mentioned (87), but which belong properly to a third con-
struction (88). IhQ planes lis, oi second construction, form in like
manner two groups; one consisting o^ fifteen planes U^, i, such as the
plane of the five points, AB1B3C1C2, whereof each passes through one
point Po, and t\iVou^\ four points Pi, and contains two lines Ag, 1, as
here the lines AB1C2, AC1B2, besides containing /<?wr lines A2,2, as here
BiB^, &c. ; while the other group is composed of twenty planes H^, 2,
such as AiBiCi, namely, the twenty faces of the five recent pyramids It^t
whereof each contains three points Pj, and three lines Agjg, but does
not pass through any point Pq. It is now required to express these
geometrical conceptions* of the forty-five lines A^ ; the thirty-five planes
112; and the five planes of homology of pyramids, [a] . . . [e], by qui-
nary symbols and types, before proceeding to determine the points P2
of second construction.
91. An arbitrary joom^ on the right line aa'Dj (90) may be re-
presented by the symbol {tuuOO); and an arhiirsiry plane through
that line by this other symbol, [Ommrr], where m and r are written
(to save commas) instead of-m and -r; hence these two symbols
may also (comp. 82) denote the litie aa'Di itself, and may be used as
types (78) to represent the g7-oup of lines Ag, 1. The particular sym-
bol [01111], of the last form, represents that particular plane
through the last-mentioned line, which contains also the line AB1C2
of the same group ; and may serve as a type for the group of planes
rig,!. The line B^c^ and the group A2,2, may be represented by
(stuOO) and [tttus'], if we agreef to write s = t + Uy and s--s; while
the plane b'c'A2, and the group rig, 2, may be denoted by [111 12].
Finally, the plane [e] has for its symbol [11114]; and the four
other planes [a], &c., of homology of pyramids (90), have this last
for their common type.
92. The points -p^, of second construction (88), are more nume-
* Mbbius (in his Bary centric Calculus, p. 284, &c.) has very clearly pointed
out the existence and chief properties of the foregoing lines and planes ; but besides
that his analysis is altogether different from ours, he does not appear to have aimed
at enumerating, or even at classifying, all the points of what has been above called
(88) the second construction, as we propose shortly to do.
f With this convention, the line ab, and the group Ai, may be denoted by
the plane -symbol [OfXvs] their point-syrnbol being (tuOOO).
70 ELEMENTS OF QUATERNIONS. [bOOK I.
rous than the lines Ag Midi planes Ilg of that construction: yet with
the help of types, as above, it is not difficult to classify and to
enumerate them. It will be sufficient here to write down these
types, which are found to be eighty and to oiFer some remarks re-
specting them ; in doing which we shall avail ourselves of the eight
ioWoYimgtypical points^ whereof the two first have already occurred,
and which are all situated in the plane of abc :
A'' = (0lT00); A^^' = (21100); a'^ =(21100); a^ =(02100);
A"' = (02100) ; A"" = ( 1 2 1 00) ; a^'" = (32 1 00) ; A« = (23 100) ;
the second and third of these having (10011) and (30011) for con-
gruent symbols (71). It is easy to see that these eight types repre-
sent, respectively, ten, thirty, thirty, twenty, twenty, sixty, sixty,
and sixty distinct points, belonging to eight groups^ which we shall
mark as Po, i, . . P25 8; so that the total number of the points v., is 290.
If then we consent (88) to close the present inquiry, at the end of
what we have above defined to be the Second Construction^ the total
number of the net points^ Pi, Pj, which are thus derived by lines
Midi planes from the, five given points Pq, is found to be exactly three
hundred: while i\iQ joint number of the net-lines, A^, A2, and of the
net-planes^ IIi, Ila, has been seen to be one hundred^ so far.
(1.) To the type Pq,! belong the ten points^
a"b"c", a'2B'2C'2, a'iB'iC'iD'i,
with the quinary symbols,
A"=(0ir00),.. A'z =(10010),.. A'l = (10001),.. D'i= (00011),
which are the harmonic conjugates of the ten points Pi, namely, of
a'b'c', A2B2C2, AiBiCiDi,
with respect to the ten lines Ai,on which those points are situated ; so that we have
ten harmonic equations, (ba'ca") = — 1, &c., as already seen (31, 86, 87). Each point
P2, 1 is the common intersection of a line Ai with three lines A2,2 ; thus we may esta-
blish the four following /brwiMZcB of concurrence (equivalent, by 89, to ten such for-
mulae) :
a" =BC'B'c' -Bid -8202; A'2 = DA-DiArB'C2*c'B3;
A'i = EA*DiA3'b'Ci-c'Bi; d'i = DE'AiA2-BiB2*CiC2.
Each point P2, i is also situated in three planes Hi ; in three other planes, of the
group 112,1; and in six planes 112,2; for example, a" is a point common to the
twelve planes,
ABC, BCD, BCE ; AB1C2C1B2, Db'BiC'Ci, Eb'B2C'C2 ;
b'c'Ai, BiCiA], B2C2A2, b'c'Ao, BiCjDi, B2C2D1.
Each line, Ai or Aa,?, contains- one point P^, i; but no line Ao, i contains any. Each
plane, Hi or 112,2, contains f /tree such points; and each plane Uo^\ contains two,
CHAP. III.] GEOMETRICAL NETS IN SPACE. 71
which are the intersections of opposite sides of a quadrilateral Q2 in that plane,
whereof the diagonals intersect in a point Po : for example, the diagonals BiC2, B2C1
of the quadrilateral B]B2C2Ci, which is (by 90) in one of the planes Ila,!, intersect*
each other in the point a ; while the opposite sides CiBi, B2C3 intersect in a" ; and
the two other opposite sides, B1B2, C2C1 have the point d'i for their intersection.
The ten points P2, 1 are also ranged, three hy three, on ten lines of third construction
As, namely, on the axes of homology,
A"b'iC'i, . . a"b'2C'2, . . a'iA'2D'i, . . A"b"c",
of ten pairs of triangles Ti, 22, which are situated in the ten planes ITi, and of
which the centres of homologj' are the ten points pi : for example, the dotted line
a"b"c", in Fig. 21, is the axis of homology of the two triangles, abc, a'b'c', whereof
the latter is inscribed in the former, with the point o in that figure (replaced by Di
in Fig. 29), to represent their centre of homology. The same ten points P2,i are
also ranged six hy six, and the ten last lines A3 are ranged four by four, in fve
planes lis, namely in the planes of homology of five pairs of pyramids, i?x, -R2J
already mentioned (90) : for example, the plane [e] contains (87) the six points
a"b"c"a'2b'2c'2, and the four right lines,
A"b'2C'2, b"c'2A'2, c"a'2B'2, A"b"c" ;
which latter are the intersections of the four faces,
DCB, DAC, DBA, ABC,
of the pyramid abcd, with the corresponding faces,
DiCiBi, DiAiOi, DiBiAi, AiBiCi,
of its inscribed homologue AiBiCiDi ; and are contained, besides, in the four other
planes,
A2B'c', B2C'a', C2A'b', A2B2C2 :
the three triangles, abc, AiBiCi, A2B2C2, for instance, being all homologous, although
in different planes, and having the line a"b"c" for their common axis of homology.
We may also say, that this line a"b"c" is the common trace (81) of two planes 112, 2,
namely of AiBiCi and A2B2C2, on the plane abc ; and in like manner, that the point
a" is the common trace, on that plane TIi, of two lines A2,2, namely of BiCi and B2C2 :
being also the common trace of the two lines b'ic'i and b'2c'2, which belong to the
third construction.
(2.) On the whole, these ten points, of second construction, a". . ., may be
considered to be already well known to geometers, in connexion with the theory
of transversal-]; lines and planes in space : but it is important here to observe,
with what simplicity and clearness their geometrical relations are expressed (88),
by the quinary symbols and quinary types employed. For example, the col-
linearity [^i) of the four planes, ABC, AiBiCi, A2B2C2, and [e], becomes evident
from mere inspection of their jTowr symbols,
* Compare the Note to page 68.
t The collinear, complauar, and harmonic relations between the ten points,
which we have above marked as P2, 1, and which have been considered by Mcibiua
also, in connexion with his theory of nets in space, appear to have been first noticed
by Carnot, in a Memoir upon transversals.
72 ELEMENTS OF QUATERNIONS. [bOOK I.
[OOOllJ, [U121], [11112], [11114],
which represent (75) the four quinary equations^
w-»=0, a:+y+z-2M>-u=0, a; + y + z -u)-2y= 0, x -V y + z^-w -Av = 0',
with this additional consequence, that the ternary symbol (81) of the common trace,
of the three latter on the former, is [111]: so that this trace is (by 38) the line
A"B"c"of Fig. 21, as above. And if we briefly denote the quinary symbols of the
four planes, taken in the same form and order as above, by \_Rq\ [iZi] [-Rg] [-^3], we
see that they are connected by the two relations,
[iJi] =- [/2o] + [i?2] ; [.Rz'\ = 2[/?o] + [Ro] ;
whence if we denote the planes themselves by IIi, 112, n'2, lis, we have (comp. 84)
the following value for the anharmonic of their pencil,
(Hinan'sHs) = - 2 ;
a result which can be very simply verified, for the case when abcd is a regular py-
ramid, and E (comp. 29) is its mean point : the plane lis, or [e], becoming in this
case (comp. 38) the plane at infinity, while the three other planes, abc, AiBiCi,
A2B2C2, axe parallel ; the second being intermediate ioei^eQn the other two, but twice
as near to the third as to the first.
(3.) "We must be a little more concise in our remarks on the seven other types of
points P2, which indeed, if not so well known,* are perhaps also, on the whole, not
quite so interesting : although it seems that some circumstances of their arrangement
in space may deserve to be noted here, especially as affording an additional exercise
(88), in the present system of symbols and types. The type P2, 2 represents, then, a. group
oi thirty points, of which a", in Fig. 21, is an example; each being the intersection
of a line A2,i with a line A2,2, as a'" is the point in which aa' intersects b'c' : but
each belonging to no other line, among those which have been hitherto considered.
But without aiming to describe here all ihe lines, planes, and points, of what we have
called the third construction, we may already see that they must be expected to be
numerous : and that the planes lis, and the hnes A3, of that construction, as well as
the pyramids Ro, and the triangles To, of the second construction, above noticed, can
only be regarded as specimens, which in a closer study of the subject, it becomes ne-
cessary to mark more fully, on the present plan, as lis, i, . . Tz,i. Accordingly it is
found that not only is each point P2, 2 one of the corners of a triangle T3, 1 of third
construction (as a'" is of a"'b"'c"' in Fig. 21), the sides of which new triangle are
lines A3, 2, passing each through one point P2,i and through two points P2,2 (hke
the dotted line a"b"'c"' of Fig. 21) ; but also each such point P2, 2 is the intersection
of two new lines of third construction, A3, 3, whereof each connects a point Pq with a
* It does not appear that any of these other types, or groups, of points P2, have
hitherto been noticed, in connexion with the net in space, except the one which we
have ranked as the fifth, po, 5, and which represents two points on each line Ai, as
the type P2, 1 has been seen to represent one point on each of those ten lines of first con-
struction : but thdX fifth group, which maybe exemplified by the intersections of the
line DE with the two planes AiBiCi and A2B2C2, has been indicated by Mobius (in
page 290 of his already cited work), although with a different notation^ and as the re-
sult of a different analysis.
CHAP. III.] GEOMETRICAL NETS IN SPACE. 73
point P2,i. For example, the point a'" is the common trace (ou the plane abc) of the
two new lines, da'i, EA'g: because, if we adopt for this point a'" the second of its two
congruent symbols, we have (comp. 73, 82) the expressions,
A"'= (10011) = (d) - (A'l) = (e) - (A'2).
We may therefore establish the formula of concurrence (comp. the first sub-article) :
a'" = aa' • b'c' • da'i • E A'2 -,
which represents a system of thirty such formulae,
(4.) It has been remarked that the point a'" may be represented, not only by the
quinary symbol (21100), but also by the congruent symbol, (10011); if then we
write,
Ao = (Ii100), Bo = (iriOO), Co = (11100),
these three new points AqBoOo, in the plane of abc, must be considered to be syntypical,
in the quinary sense C78), with the three points a"'b"'c"', or to belong to the same
group P2,2, although they have (comp. 88) a different ternary type. It is easy to
see that, while the triangle a"'b"'c"' is (comp. again Fig. 21) an inscribed homo-
logue Ty,! of the triangle a'b'c', which is itself (com\). sub-article 1) an inscrihed
homologue To, 1 of a triangle Ti, namely of abc, with a"b"c" for their common a is
of homology, the new triangle AqBoCo is on the contrary an exscrihed homologue
Ti,2, with the same axis As,!, of the same given triangle Ti. But from the syuty-
pical relation, existing as above for space between the points a'" and Ao, we may
expect to find that these two points P2, 2 admit of being similarly consirucfed, when
the^ue points Pq are treated as entering symmetrically (or similarly), as geometri-
cal elements, into the constructions. The point Aq must therefore be situated, not
only on a line A2,i, namely, on aa', but also on a line A2,2, which is easily found to
be A1A2, and on two lines A3, 3, each connecting a point Pq with a point P2,i ; which
latter lines are soon seen to be bb" and cc". We may therefore establish the formula
of concurrence (comp. the last sub-article) :
Ao = aa'*AiA3*bb"-cc";
and may consider the three points Aq, Bq, Co as the traces of the three lines AiAo,
B1B2, C1C2 : while the three new lines aa'', bb", cc", which coincide in position
with the sides of the exscribed triangle AqBoCIo, are the traces A3, 3 of three planes
1X2, 1, such as AB1C2B2C1, which pass through the three given points A, B, c, but do
not contain the Unes A2,i whereon the six points P2,2 in their plane ITi are situated.
Every other plane IIi contains, in like manner, six points P2 of the present group ;
every plane 1X2, 1 contains eight of them ; and every plane 112,2 contains three; each
line A2, 1 passing through two such points, but each line A2, 3 only through one.
But besides being (as above) the intersection of two lines Ao, each point of this group
P2,2 is common to two planes Yli, four planes 113,1, and two planes 112,2; while
each of these thirty points is also a common corner of two different triangles of
^/aVrf construction, of the lately mentioned kinds Ts, 1 and 2^,2, situated respectively
in the two planes oi first construction which contain the point itself. It may be
added that each of the two points P2, 2, on a line A2, 1, is the harmonic conjugate of
one of the two points pi, with respect to the point Pq, and to the other point Pi oa
that line ; thus we have here the two harmonic equations,
(aa'dia'") = (adia'ao) = — 1,
by which the positions of the two points a'" and Ao miglit be determined.
L
74 ELEMENTS OF QUATERNIONS. [bOOK I.
(5.) A third group, P2,3, oi second construction, consists (like the preceding group)
of thirty points, ranged two hxj two on the fifteen lines Aa^i, and six hy six on the
ten planes ITi, but so that each is common to two such planes ; each is also situated
in two planes Zlg,!, in two planes Il2,2, and on one line A3, i in which (by sub art. 1)
these two last planes intersect each other, and two of the five planes lis, i ; each
plane 112,1 contains /owr such points, and each plane 112,2 contains three of them ;
but no point of this group is on any line Ai, or A2,2' The six points P2,3, which
are in the plane abc, are represented (like the corresponding points of the last
group) by two ternary types, namely by (211) and (311) ; and may be exemplified
by the two following points, of which these last are the ternary symbols :
A'^ = AA' • a"b"c" = AA' • AiBiCi ' A2B2C2 ;
Ai'^ = AA' •d'iA'2A 1 = AA' •b'CiC2 •c'BiB2.
The three points of the first sub-group a'^ . . are collinear ; but the three points Ai''^ . .
of the second sub-group are the corners of a new triangle, T3, 3, which is homologous
to the triangle abc, and to all the other triangles in its plane which have been hitherto
considered, as well as to the two triangles AiBiCi and A2B2C2 ; the line of the three
former points being their common axis of homology ; and the sides of the new trian-
gle, Ai'^Bi'^Ci'^, being the traces of the three planes (comp, 90) of homology of pyra-
mids, [a], [b], [c] ; as (comp. sub-art. 2) the line a'^b^'^c'"' or a"b"c" is the com-
mon trace of the two other planes of the same group lis, 1, namely of [d] and [e]. We
may also say that the point Ai'"^ is the trace of the line a'ia'2 ; and because the lines
b'co, c'bo are the traces of the two planes 112,2 in which that point is contained, we
may write the formula of concurrence,
Ai" = A a' • a'ia'2 • b'Co • c'Bo.
(G.) It may be also remarked, that each of the two points P2, 3) on any line A2, 1, is
the harmonic conjugate of a point P2, 2, with respect to the point Pq, and to one of
the two points Pi on that line ; being also the harmonic conjugate of this last point,
with respect to the same point Pq, and the other point P2,2 : thus, on the line aa'dj,
we have the /oMr harmonic equations, which are not however all independent, since
two of them can be deduced from the two others, with the help of the two analogous
equations of the fourth sub-article :
(aa"'a'a''^) = (aa'aqA") = (aaqDiAi'^) = (adia"'ai*'^) = - 1.
And the three pairs of derived points Pi, P2,2, P2,3, on any such line A2, 1, will
be found (comp. 26) to compose an involution, with the given point Pq on the line for
07ie of its two double points (ov foci') : the other double point of this involution being
a point P3 of third construction ; namely, the point in which the line A2, 1 meets that
one of the five planes of homology IT3, 1, which corresponds (comp. 90) to the par-
ticular point Pq as centre. Thus, in the present example, if we denote by A'' the
point in which the line aa' meets the plane [a], of which (by 81, 91) the trace on
ABC is the line [411], and therefore is (as has been stated) the side Bi'^ci*^ of the
lately mentioned triangle T3, 3, so that
A^ = (1 22) = aa' • BC'" • Cb'" • Bi'^Ci"^,
we shall have the three harmonic equations,
(aa'a^Di) = (aa"'a^Ao) = (AA'^A^Ai'^) = - 1 ;
which express that this new point A" is the common harmonic conjvgate of the given
CHAP. III.] GEOMETRICAL NETS IN SPACE. 75
point A, with respect to the three pairs of points^ a'di, a"'Ao, a'^Ai'^ ; and therefore
that these three pairs form (as has been said) an involution, with A and A'^ for its two
double points.
(7.) It will be found that we have now exhausted all the types of points of
second construction, which are situated upon lines A2, 1 ; there being only four
sach points on each such line. But there are still to be considered two new groups
of points P2 on lines Ai, and three others on lines A2,2- Attending first to the former
set of lines, we may observe that each of the two new types, P2,4, P2,5, represents
twenty points, situated two by two on the ten lines of first construction, but not on
any line A2 ; and therefore six by six in the ten planes ITi, each point however being
coinmon to three such planes : also each point P2,4 is common to three planes 172,2,
and each point P2, 5 is situated in one such plane ; while each of these last planes
contains three points P2, 4, but only one point P2, 5- If we attend only to points in the
plane abc, we can represent these two new groups by the two ternary types, (021)
and (021), which as symbols denote the two typical points,
A^ = BC • c'AiA2 • DlAiBi • «iA2B2 ; A^' = BC • c'BiBo = BC c'Bq ;
we have also the concurrence,
A^ = BC • o'Aq • DiC" • AB '",
It may be noted that A^ is the harmonic conjugate of c, with respect to Aq and
Bi'^, which last point is on the same trace c'aq, of the plane c'aiA2 ; and that a^' is
harmonically conjugate to Bi^, with respect to c' and Bq, on the trace of the plane
c'biB2, where bi^ denotes (by an analogy which will soon become more evident) the
intersection of that trace with the line ca : so that we have the two equations,
(AqC'Bi'^A^) = (boBi^o'a^'') = - 1.
(8.) Each line Ai, contains thus two points P2, of each of the two last new
groups, besides the point P2, 1, the point Pi, and the two points Pq, which had been
previously considered : it contains therefore eight points in all, if we still abstain (88)
from proceeding beyond the Second Construction. And it is easy to prove that these
eight points can, in two distinct modes, be so arranged as to form (comp. sub-art. 6)
an involution, with two of them for the two double points thereof. Thus, if we attend
only to points on the line bc, and represent them by ternary symbols, we may write,
B = (010), c=(001), A'=(011), a"=(0i1);
a^=(021), a^' = (021), AiV = (012), Ai^' = (012);
and the resulting harmonic equations
I. . . (ba'oa") = (BA^CA^') = (BAf CAi^O = - I,
II. . . (a'ba'c) = (A'AVA"Af') = (aVa"Ai^') = - I,
will then suffice to show : 1st., that the two points Pq, on any line Ai, are the double
points of an involution, in which the points Pi, Po,i form one pair of conjugates,
while the two other pairs are of the common form, P2,4, P2,5; and Ilnd., that the
two points Pi and P2, 1, on any such line Ai, are the double points of a second iiivo-
lution, obtained by pairing the two points of each of the three other groups. Also
each of the two points Pq, on a line Ai, is the harmonic conjugate of one of the
two points P2,5 on that line, with respect to the other point of the same group, and
to the point Pi on the same line ; thus,
76 ELEMENTS OF QUATERNIONS. [bOOK I.
(ba'ai"a^O = (ca'a^Ai^O = - 1.
(9.) It remains to consider briefly three other groups of points P2, each group
containing sixty points , which are situated, two by two, on the thirty lines A2,2, and
six by six in the ten planes 11 1. Confining our attention to those which are in the
plane abc, and denoting them by their ternary symbols, we have thus, on the line
b'c', the three new typical points, of the three remaining groups, P2.6, P2,7, P2,8 :
A^"= (121) ; A^"' = (321) ; a« = (237) ;
with which may be combined these three others, of the same three types, and on the
same line b'c' :
Ai^" = (112); Ai^'" = (312); Ai« = (213).
Considered as intersections of a line A2,2 with lines A3 in the same plane IIi, or with
planes 112 (in which latter character alone they belong to the second construction),
the three points a"', &c., may be thus denoted :
A"^" = b'c' ■ BB" • Cb"' • AA^^ = b'c' ' BCiA2AiC2 ;
jjni _ 3'^' . j,^b" . ^"^v _ b'c' . DiCiAi • D1C2A2 ;
A™= b'c'* a'CoBi'^Ci"^B^i-BA*^Bi'^'Bi'^" = b'c''a'ciC2 ;
with the harmonic equation,
(CqA'Ci^A^^) = - 1,
and with analogous expressions for the three other points, Ai^", &c. The line b'c' thus
intersects one plane 112,1 (or its trace bb" on the plane abc), in the point a^" ; it
intersects two planes 112,2 (or their common trace Dib") in A"^°' ; and one other plane
112,2 (or its trace a'cq) in a'^ : and similarly for the other points, Ai"^", &c., of the same
three groups. Each plane li^, 1 contains twelve points P2,6, eight points P2,7, and eight
points P2,8; while every plane 112,2 contains six points P2,6) twelve points P2,7,
and nine points P2,8. Each point P2,6 is contained in one plane IIi; in three
planes 112,1; and in two planes n2,2. Each point P2,7 is in one plane ITi, in two
planes 112,1, and mfour planes 02,2. And each point P2,8 is situated in one plane ITi,
in two planes 112,1, and in three planes 112,2.
(10.) The points of the three last groups are situated o/j/y on lines A2,2; but, on
each such hne, two points of each of those three groups are situated ; which, along
with one point of each of the two former groups, P2, 1 and P2,2, and with the two
points Pi, whereby the line itself is determined, make up a system oitenpoints upon
that line. For example, the line b'c' contains, besides the six points mentioned in
the last sub -article, the^wr others:
b'=(101); c'=(110); a" = (011); a"'=(211).
Of these ten points, the two last mentioned, namely the points P2,i and P2,2upon the
line A2,2, are the double poitits (comp. sub-art. 8) of a new involution, in which the two
points of each of the four other groups compose a conjugate pair, as is expressed by
the harmonic equations,
(a"b'a"'c') = (A"A^"A"'Ar") = (A"A^"'A"'Ar"') = (a"a'*a"'Ai«) = - I.
And the analogous equations,
(b'a"c'a"') = (b'a^"c'a^'") = (b'ai^"c'ai^'") =- 1»
show that the two points Pi on any line A2,2 are the double points of of another invo-
lution (comp. again sub-art. 8), whereof the two points P2,i, P2,2 on that line form
CHAP. III.] GEOMETIIICAL NETS IN SPACE. 77
one conjugate pair, while each of the two points P2,6 is paired with one of the points
P2,7 as its conjugate. In fact, the eight-rayed pencil (a.c'b'a'"a"a^'"'a^"Ai^"'Ai'")
coincides in position with the pencil ( A . bca Wa"^'Ai^Ai"^'), and maybe said to be
a pencil in double involution ; the third and fourth, the fifth and sixth, and the se-
venth and eighth rays forming one involution, whereof the first and second are the
two double* rays ; while the first and second, the fifth and seventh, and the sixth
and eighth rays compose another involution, whereof the double rays are the third
and fourth of the pencil.
(11.) If we proceeded to connect systematically the points P2 among themselves,
and with the points Pi and Pq, we should find many remarkable lines and planes of
third construction (88), besides those which have been incidentally noticed above ; for
example, we should have a group IIo,2 of twenty new planes^ exemplified by the
two following,
[E„] = [11103], [D^] = [11130],
which have the same common trace A3, 1, namely the line a"b"c", on the plane abc,
as the two planes AiBiCi, A2B2C2, and the two planes [d], [e], of the groups 1X2,2 and
113, 1, which have been considered in former sub- articles ; and each of these new planes
Ha, 2 would be found to contain one point Pq, three points P2,i, six points P2,25 and
three points P2, 3. It might be proved also that these twenty new planes are the
twenty faces of Jive new pyramids R3, which are the exscribed homologues of the five
old pyramids Ki (89), with the five given points Pq for the corresponding centres of
homology. But it would lead us beyond the proposed limits, to pursue this dis-
cussion further : although a few additional remarks may be useful, as serving to
establish the completeness of the enumeration above given, of the lines, planes, and
points oi second construction.
93. In general, if there be any n given points^ whereof no four
are situated in any common plane, the number N of the derived
points, which are immediately obtained from them, as intersections
A • n of line with plane (each line being drawn through two of the
given points, and each plane through three others), or the number of
points of the/orm ab'CDE, is easily seen to be,
_ n(^^-])(7^-2)(7.-3)(n-4) ^
^'•^^^~ 2.2.3
so that N - 10, as before, when 7t = 5. But if we were to apply this
formula to the case n= 15, we should iSnd, for that case, the value,
iVr=y(i5)=i5.i4. 13.11 = 30030;
and ikiVi^ fifteen given and independent points of space would conduct,
by what might (relatively to them) be called a First Construction
(comp. 88), to a system of more than thirty thousand points. Yet it
has been lately stated (92), that from the fifteen points above called
Po> Pi, there can be derived, in this way, onlu two hundred and ninety
* Compart; page i7'2 of the GJc:::. Srvc'rUure of il. Chasies.
78 ELEMENTS OF QUATERNIONS. [bOOK I.
points P2, as intersections of the form* A -11; and therefore /e^^er
than three hundred. That this reduction of the number of derived
points^ at the end of what has been called (88) the Second Construc-
tion for the net in space, arising from the dependence of the ten points
Pi on thQJive points Pq, would be found to be so considerable, might
not perhaps have been anticipated; and although the foregoing ex-
amination proves that all the eight types (92) do really represent
points P2, it may appear possible, at this stage, that some other type
of such points has been omitted. A study of the manner in which
the types of points result, from those of the lines and planes oi which
they are the intersections, would indeed decide this question ; and
it was, in fact, in that way that the eight types, or groups, Po, 1, . .p^is,
of points of second construction for space, were investigated, and
found to be sufficient: yet it may be useful (compare the last sub-
art.) to verify, as below, the completeness of the foregoing enumeration.
(1.) ThQ ff teen points, V(!, Pi, admit of 105 binary^ and of 455 ternary combina-
tions; but these are far from determining so many distinct lines and planes. In fact,
those 15 points are connected by 25 collineations, represented by the 25 lines Ai,
A2,i; which lines therefore count as 75, among the 105 binary combinations of
points : and there remain only 30 combinations of this sort, which are constructed
by the 30 other lines, A2,2- Again, there are 25 ternary combinations of points,
which are represented (as above) by lines, and therefore do not determine any plane.
Also, in each of the ten planes IIi, there are 29 (=35 - 6) triangles Ti, Tg, because
each of those planes contains 7 points Pq, Pi, connected by 6 relations of coUinearity.
In like manner, each oi the fifteen planes 1X2,1 contains 8 (= 10-2) other triangles
T-z, because it contains 5 points po, Pi, connected by two collineations. There re-
main therefore only 20 (= 455 — 25 — 290 - 120) ternary combinations of points to
be accounted for; and these are represented by the 20 planes 112, 2- The complete-
ness of the enumeration of the lines and planes of the second construction is therefore
verified ; and it only remains to verify that the 305 points, Pq, Pi, P2, above consi-
dered, represent all the intersections A -IT, of the 55 lines A 1, A2, with the 45 planes
III, n2.
(2.) Each plane IIi contains three lines of each of the three groups, Ai, A2, 1,
A 2, 2; each plane 1X2,1 contains two lines A 2,1, and four lines A2,2; and each plane
1X2,2 contains three lines A2,2. Hence (or because each line Ai is contained in three
planes 11 1; each line A 2,1 in two planes IXi, and in two planes 1X2,1; and each
line A2, 2 in one plane ITi, in two planes 1X2, 1, and in two planes IX2, 2), it follows that,
without going beyond the second construction, there are 240 (= 30 i- 30 + 30 + 30
* The definition (88) of the points P2 admits, indeed, intersections A'A ofcom-
planar lines, when they are not already points Pq or Pi ; but all such intersections
are also points of the form A- XI ; so that no generality is lost, by confining ourselves
to this last form, as in the present discussion we propose to do.
CHAP.
u..]
GEOMETRICAL NETS IN SPACE.
79
+ 60 + 60) cases of coincidence of line and plane; so that the number of cases of
intersection is reduced, hereby, from 56 . 45 = 2475, to 2235 (= 2475 — 240).
(3.) Each point Pq represents twelve intersections of the form Ai'Hi ; because it
is common to four lines A\, and to six planes IIi, each plane containing two of those
four lines, but being intersected by the two others in that point Pq ; as the plane
ABC, for example, is intersected in A by the two lines, ad and ae. Again, each
point Po is common to three planes IIo, i, no one of which contains any of the four
lines Ai through that point ; it represents therefore a system of twelve other inter-
sections^ of the form Ai • ITa, i. Again, each point Pq is common to three lines Ai, i,
each of which is contained in two of the six planes IIi, but intersects the four others
in that point Pq ; which therefore counts as twelve intersections, of the form A2, rlli.
Finally, each of the points Pq represents three intersections, A2, 1 * ITo, 1 ; and it re-
presents no o^Aer intersection, of the form A -IT, within the limits of the present
inquiiy. Thus, each of the^re given points is to be considered as representing, or
constructing, thirty-nine (= 12 -f 12 + 12 +3) intersections of line with plane; and
there remain only 2040 (= 2235 — 195) other cases of such intersection A •IT, to be
accounted for (in the present verification) by the 300 derived points, Pi, P2.
(4.) For this purpose, the nine columns, headed as I. to IX. in the following
Table, contain the numbers of such intersections which belong respectively to the
nine forjns,
Ai'iii, Ai-n2,i, Arn2,2; A2,i-ni, A2,i-n2,i, A2,i-n2,2;
A2,2*ni, A2,2*n3, 1, A2,2"n2,2,
for each of the nine typical derived points, a' . . . A'^, of the nine groups Pi, P2, 1, . .
P2,8. Column X. contains, for each point, the sum of the nine numbers, thus tabu-
lated in the preceding columns ; and expresses therefore the entire number of inter-
sections, which any one such point represents. Column XI. states the number of the
points for each type ; and column XII. contains the product of the two last numbers, or
the number of intersections A . Tl which are represented (or constructed) by the group.
Finally, the sum of the numbers in each of the two last columns is written at its foot ;
and because the 300 derived points, of first and second constructions, are thus found
to represent the 2040 intersections Avhich were to be accounted for, the verification is
seen to be complete : and no new type, of points P2, remains to be discovered.
(5.)
Table
of Intersections A
n.
Type.
I.
11,
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
XI.
XII.
a'
1
6
6
6
12
18
18
24
24
115
10
1150
a"
3
6
6
3
12
30
10
300
a'"
2
2
1
2
7
30
210
A'^
2
2
30
60
A'
3
3
20
60
A^'
1
1
20
20
A^"
1
1
60
60
^Tin
2
2
60
120
A'*
"
1
1
60
300
60
1
2040
80 ELEMENTS OF QUATERNIONS. [bOOK I.
(6._) It is to be remembered tbat we have not admitted, by our definition (88),
any points which can only he determined hy intersections of three planes TIi, 02,
as belonging to the second construction : nor have we counted, as lines A2 of that
construction, any lines which can only be found as intersections of two such planes.
For example, we do not regard the traces Aa", &c., of certain pZanes A2,i considered
in recent sub-articles, as among the lines of second construction, although they would
present themselves early in an enumeration of the lines A3 of the third. And any
point in the plane abc, which can only be determined (at the present stage) as the
intersection of two such traces, is not regarded as a point P2. A student might find
it however to be not useless, as an exercise, to investigate the expressions for such
intersections ; and for that reason it may be noted here, that the ternary types (comp,
81) of the forty-four traces of planes ITi, IIo, on the plane abc, which are found to
compose a system of only twenty-two distinct lines in that plane, whereof nine are
lines Ai, A2, are the seven following (comp. 38) :
[100], [Oil], [111], [111], [Oil], [211], [211];
which, as ternary symbols, represent the seven lines,
EC, aa', b'c', a"b"c", aa", Dia'', a'co-
(7.) Again, on the same principle, and with reference to the same definition, that
new point, say f, which may be denoted by either of the two congruent quinary
symbols (71),
F= (43210) E (01234),
and which, as a quinary type (78), represents a new group of sixty points of space
(and of no more, on account of this last congruence, whereas a quinary type, with all
its Jive coefiicients unequal, represents generally a group of 120 distinct points), is
not regarded by us as a point P2 ; although this new point f is easily seen to be the
intersection of three planes of second construction, namely, of the three following,
which all belong to the group IIo, 1 :
[OlIIl], [11011], [iilio],
or aa'diCiB3, cc'diBiA2, eb'b2c'c2. It may, however, be remarked in passing, that
each plane II 2, 1 contains twelve points P3 of this new group : every such point being
common (as is evident from what has been shown) to three such planes.
94. From the foregoing discussion it appears that the^ye given
points Po, and the three hundred derived points Pi, P2, are arranged in
space, upon the fifty-Jive lines A^, A^, and in the forty-Jive planes H^
rig, as follows. Each line Aj contains eight of the 305 points, forming
on it what may be called (see the sub-article (8.) to 92) a double in-
volution. Each line A2, 1 contains seven points, whereof one, namely
the given point, Pq, has been seen (in the earlier sub-art. (6.)) to be
a double point of another involution, to which the thj^ee derived pairs
of points, Pi, p.^, on the same line belong. And each line Aj,jj con-
tains ten points, forming on it a 7iew involution; while eight of these
ten points, with a different order of succession, compose still another
CHAP, in.] GKOxMETRICAL NETS IN SPACE. 81
involution* (92, (10.))- Again, each plane n, contains fifty -two
points, namely three given points, four points of first, and 45 points
of 5ecow<i construction. Each plane 11^, i contains /br^y-seven points,
whereof owe is a given point, four are points Pi, and 42 are points
* These theorems respecting the relations of involution, of given and derived
points on lines oi first and second constructions, for a net in space, are perhaps new ;
although some of the harmonic relations, above mentioned, have been noticed under
other forms by Mobius : to wliom, indeed, as has been stated, the conception of such
a net is due. Thus, if we consider (compare the Note to page 72) the two intersec-
tions,
Ei=DE'AiBiCi, E2 = DE • A2B2C2,
we easily find that they may be denoted by the quinary symbols,
El = (00012), E3= (00021);
they are, therefore, by Art. 9'2, the two points P3, 5 on the line de : and consequently,
by the theorem stated at the end of sub-art. 8, the harmonic conjugate of each, taken
with respect to the other and to the point Di, must be one of the two points d, e on
that line. Accordingly, we soon derive, by comparison of the symbols of these ^»e
points, DED1E1E2, the two following harmonic equations, which belong to the same
type as the two last of that sub-art. 8 :
(D1DE2E1) = — 1 ; (diEEiEj) = — 1 ;
but these two equations have been assigned (with notations slightly different) in the
formerly cited page 290 of the Barycentric Calculus. (Comp. again the recent Note
to page 72.) The geometrical meaning of the last equation may be illustrated, by
conceiving that abcd is a regular pyramid, and that e is its mean pohit; for then
(comp. 92, sub-art. (2.) ), vty is the mean point of the base abc ; DiD is the altitude
of the pyramid ; and the three segments DiE, DiEi, D1E2 are, respectively, the quar-
ter, the third part, and the half of that altitude : they compose therefore (as the for-
mula expresses) a Aarmowtc /jro^ressi'ow; or Di and Ei are conjugate points, with
respect to e and E2. But in order to exemplify the double involution of the same
sub-art. (8.), it would be necessary to consider three other points P2, on the same line
DE ; whereof one, above called d'i, belongs to a known group P2, i (92, (2.)); but
the two others are of the group Po, 4, and do not seem to have been previously noticed.
As an example of an involution on a line of third construction, it may be remarked
that on each line of the group A3, 3, or on each of the sides of any one of the ten tri-
angles T3, 2, in addition to one given point pq, and one derived point Pj, 1, there are
two points P2, 2i and two points P2,6; and that the two first points are the double
points of an involution, to which the two last pairs belong : thus, on the side
Aqbco of the exscribed triangle AqBoCo, or on the trace of the plane bciAzAiCj, we
have the two harmonic equations,
(b AoB"Co) = (BA'"B"crn) = - 1 .
Again, on the trace a'co of the plane ACiCa, (which latter trace is a line not passing
through any one of the given points,) Co and ei'^ are the double points of an invo-
lution, wherein a' is conjugate to cf and a'^ to b''*. But it wouid be tedious to
multiply such instances.
M
82
ELEMENTS OF QUATERNIONS.
[book I.
Pa: of which last, 38 are situated on the six lines Aj in the plane, but
four are irdersections of that plane n^, i with/owr other lines of second
construction. Finally, each plane 112,2 passes through no given
point, but coTii2Lms forty-three derived points, whereof 40 are points
of second construction. And because the planes o^ first construc-
tion alone contain specimens of all the ten groups of points^ Po, Pi,
P2»i> • • 1*25 81 given or derived, and of all the three groups of lines, A^,
■^2)1) ^2,2, at the close of that second construction (since the types
P2»4j P2>5j Ai are not represented by any points or lines in any plane
112,1, nor are the types Pq, Ai, Ag,! represented in a plane 112,2), it
has been thought convenient to prepare the annexed diagram (Fig.
30), which may serve to illustrate, by some selected instances, the
arrangement oi th^ fifty -two points Pq, Pi, P2 in a plane 11^, namely, in
the plane abc; as well as the arrangement of the nine lines A„ A,
in that plane, and the ti^aces A3 of other planes upon it.
View of the Arrangement of the Principal Points and Lines in a Plane
of First Construction,
In this Figure, the triangle abc is suppposed, for simplicity, to be the equilateral
base of a regular pyramid abcd (comp. sub-art. (2.) to 92) ; and Di, again replaced
by o, is supposed to be its mean point (29). The first inscribed triangle, a'b'c',
therefore, bisects the three sides ; and the axis of homology a''b"c" is the line at in-
finity (38): the number 1, on the line c'b' prolonged, being designed to suggest that
CHAP. 111.] GEOxMETRlCAL NETS IN SPACE. 83^
the point a", to which that line tends, is of the type ?•.>, i, or belongs to the y/rs<
group of points of second construction. A second inscribed triangle, a"'b"'c"', for
which Fig. 21 may be consulted, is only indicated by the number 2 placed at the
middle of the side b'c', to suggest that this bisecting point a'" belongs to the second
group of points Pg. The same number 2, but with an accent, 2', is placed near the
corner Aq of the exscribed triangle AqBoCo, to remind us that this corner also belongs
(by a syntypical relation in space) to the group P2,2. The point a''', which is now
infinitely distant, is indicated by the number 3, on the dotted line at the top ; while
the same number with an accent, lower down, marks the position of the point Ai".
Finally, the ten other numbers, unaccented or accented, 4, 4', 5, 5', 6, 6', 7, 7',
8, 8', denote the places of the ten points, a^, Ai^, a^', Ai^', a"', Ai^« a'"', a^'"
A'*, Ai"^. And the principal harmonic relations, and relations of involution, above
mentioned, may be verified by inspection of this Diagram.
95. However far the series of construction of the net in space
may be continued, we may now regard it as evident, at least on com-
parison with the analogous property (42) of the plane net, that every
pointf line, or plane, to which such constructions can conduct, must
necessarily be rational (77); or that it must be rationally related to
the system o^ the f^ve given points : hecause ihm anharmojiic co-ordi-
nates (79, 80) of every net-point, and of every net-plane, are equal or
proportional to whole numbers. Conversely (comp. 43) every pointy
line, OT plane, in space, which is thus rationally related to the system of
points ABODE, is a point, line, or plane of the net, which those five points
determine. Hence (comp. again 43), every irrational point, line, or
plane (77), is indeed incapable of being rigorously constructed, by any
processes of the kind above described; but it admits of being inde-
finitely approximated to, by points, lines, or planes of the net. Every
anharmonic ratio, whether of a. group of net-points, or of a pencil of
net-lines, or of net-planes, has a rational value (comp. 44), which de-
pends only on the processes of linear construction employed, in the
generation of that group or pencil, and is entirely independent of the
arrangement, or configuration, of the five given points in space. Also,,
all relations of collineation, and of complanarity, are preserved, in the
passage from one net to another, by a change of the given system of
points: so that it may be briefly said (comp. again 44) that all geo-
metrical nets in space are homographic figures. Finally, any five points*
of such a net, of which no four are in one plane, are sufficient (comp.
* These general properties (95) of the space-net are in substance taken from
Mobius, although (as has been remarked before) the analysis here employed appears
to be new : as do also most of the theorems above given, respecting ihepoints of second
construction (92), at least after we pass beyond the Jirst group V2, \ of ten such points,
which (as already stated) have been known comparatively long.
84 ELEMENTS OF QUATERNIONS. [bOOK I.
45) for the determination of the whole net: or for the linear construc-
tion of all its points, including the five given ones.
(1.) As an Example, let the five points AiBiCiDi and e be now supposed to be
given ; and let it be required to derive the four points abcd, by linear constructions,
from these new data. In other words, we are now required to exscrihe a pyramid
ABCD to a given pyramid AiBiCiDi, so that it may be homologous thereto, with the
point E for their given centre of homology. An obvious process is (comp. 45) to in.
scribe another homologous pyramid, A3B3C3D3,, so as to have A3 = eai*BiCiDi, &c ;
and then to determine the intersections of corresponding faces, such as AiBiCi and
A3B3C3 ; for these/owr lines of intersection will be in the common plane\E^, of homology
of the three pyramids, and will be the traces on that plane of the /owr sought planes,
ABC, &c., drawn through the four given points Di, &c. If it were only required to
construct one corner A of the exscribed pyramid, we might find the point above
called a'' as the common intersection of three planes, as follows,
A'^ = AiBiCi • Aid/e • A3B3C3 ;
and then should have this other formula of intersection,
A =EAi-DiA''.
Or the point A might be determined by the anharmonic equation,
(EAA1A3) = 3,
yrhich for a regular pyramid is easily verified.
(2.) As regards the general passage from one net in space to another, let the
symbols Pi ={xi . . vi), . . P5 = (a^s . . Pg) denote any Jive given points, wliereof no four
are complanar ; and let a'b'c'd'e and «' be six coefiicients, of which the five ratios are
such as to satisfy the symbolical equation (^comp. 71, 72),
a' (Pi) + bXFz) + c' (P3) + d'(Pi) + ^'(yd ==-u'CU):
or the five ordinary equations which it includes, namely,
a'xi + . . + e'x5 = . . = a'vi + . . 4- e'v^ = - u'.
Let p' be any sixth point of space, of which the quinary symbol satisfies the equa-
tion,
(p')=:ica'(Pi) + 2/5(P2)+ zc'(pi) + wd'(Fi) + ve'(V5)+u{ U) ;
then it will be found that this last point p' can be derived from the five points Pi . . P5
by precisely the same constructions, as those by which the point p = (^xyzwv') is de-
rived from the five points abcde. As an example, if w' = aj + y + « + w — 3w, then
the point {xyzwv) is derived from AiBiCiD]E, by the same constructions as (xyzwv)
from ABCDE ; thus a itself may be constructed from Ai . . E, as the point p = (30001)
is from a . . b ; which would conduct anew to the anharmonic equation of the last
sub-article.
(3.) It may be briefly added here, that instead of anharmonic ratios, as con-
nected with a net in space, or indeed generally in relation to spatial problems, we
are permitted (comp. 68) to substitute products (or quotients) of quotients of volumes
of pyramids; as a specimen of which substitution, it may be remarked, that the an-
harmonic relation, just referred to, admits of being replaced by the following equa-
tion, involving one such quotient of pyramids, but introducing no auxiliary point :
CHAP. III.] MEANS OF VECTORS. 85
EA : AiA = 3eBiCiDi : AiBiCiDi.
In general, if xyzw be (as in 79, 83) the anharmonic co-ordinates of a point p in
space, yve may write,
X PBCD EBCD
^ PCDA " ECDa'
with other equations of the same type, on which we cannot here delay.
Section 5. — On Barycentres of Systems of Points ; and on
Simple and Complex Means of Vectors,
96. In general, when the sum 2a of any number of co-initial
vectors,
ai = OAi, .. a^ = OA„„
is divided (16) by their number, m, the resulting vector ,
a = OM = — 2a = - 2oA,
m m
is said to be the Simple Mean of those m vectors; and ihQ point m,
in which this mean vector terminates, and of which the position
(comp. 18) is easily seen to be independent of the position of the
common origin o, is said to be the Mean Point (comp. 29), of the
system of the m points, Aj, . . A«. It is evident that we have the equa-
tion,
= (ai-^) + . .+(a^-/i) = 2(a-/t)-2MA;
or that the sum of the m vectors, drawn/row the mean point m, to the
points A of the system, is equal to zero. And hence (comp. 10, 11, 30),
it follows, 1st., that these m vectors are equal to the m successive
sides of a closed polygon ; Ilnd., that if the system and its mean
point be projected, by any parallel ordinates, on any assumed plane
(or line), the projection m', of the mean point m, is the mean point of
the projected system : and Illrd., that the ordinate mm', of the mean
point, is the mean of all the other ordinates, AiA'i, . . a^a'„. It fol-
lows, also, that if n be the mean point of another system, Bi, . . b„;
and if s be the mean point of the total system, Aj . . b,„ of the m + tj
= s points obtained by combining the two former, considered as par-
tial systems ; while v and a may denote the vectors, on and os, of
these two last mean points : then we shall have the equations,
7W/*-2a, wi^ = 2y3, 5ff = 2a+ 2)3 = w/i + /ii^,
miff- iJi) = n{v~ a), w.MS=n.SN;
so that the general mean point, s, is situated on the right line mn,
which connects the two partial mean points, m and n; and divides
86 ELEMENTS OF QUATERNIONS. [bOOK I.
that line (internally), into tivo segments ms and sn, which are inversely
proportional to the two whole numbers^ m and n.
(1.) As an Example, let abcd be a gauche quadrilateral^ and let E be its mean
point ; or more fully, let
OE = ;i (OA + OB -t- DC -f Od),
or
that is to say, let o = 6 = c = rf, in the equations of Art. 65. Then, with notations
lately used, for certain derived points Di, &c., if we write the vector formuloe,
OAi = ai = i(i3 + y + 5), .. 5i=K« + /3 + r),
OA3=a2 = |(a + 5), . . r2 = Ky + ^).
oA' = a'=|(/3+r),.. y'=K«+/3),
we shall have seven different expressions for the mean vector^ i ; namely, the follow-
ing:
e = K« + 3ai) = .. = i(^+3^0
= K«'+«2) =.. = §(/ + 72).
And these conduct to the seven equations between segments^
AE = 3eai, . . DB = 3edi ;
a'e = ea2, . . c'b = ec2;
which prove (what is otherwise known) that the four right lines, here denoted by
AAi, . . DDi, whereof each connects a corner of the pyramid abcd with the mean
point of the opposite face, intersect and quadrisect each other, in one common
point, e ; and that the three common bisectors a'as, b'b2, c'co, of pairs of opposite
edges, such as BO and da, intersect and bisect each other, in the same mean point :
so that the /our middle points, c', a', C2, A2, of the four successive sides ab, &c., of
the gauche quadrilateral abcd, are situated in one common plane, which bisects also
the common bisector, b'b2, ofthe^wo diagonals, AC and bd.
(2.) In this example, the number s of the points A . . D being j^wr, the number
of the derived lines, which thus cross each other in their general mean point E is seen
to be seven ; and the number of the derived planes through that point is nine :
namely, in the notation lately used for the net in space, four lines Ai, three lines A2, 1,
six planes Hi, and three planes 112, 1. Of these nine planes, the six former may (in
the present connexion) be called triple planes, because each contains three lines (as
the plane abe, for instance, contains the lines aai, bbi, c'c2), all passing through the
mean point e; and the three latter may be said, by contrast, to he non-triple planes,
because each contains only two lines through that point, determined on the foregoing
principles.
(3.) In general, let («) denote the number of the lines, through the general mean
point s of a total system of s given points, which is thus, in all possible ways, decom-
posed into partial systems ; let/(*) denote the number of the triple planes, obtained
by grouping the given points into three such partial systems ; let ;^ (s) denote the
number of non-triple planes, each determined by grouping those s points in two dif-
ferent ways into two partial systems ; and let f(«) =/(*) + »// (s) represent the entire
number of distinct planes through the point s : so that
^(4) = 7, /(4) = 6, 4'(4) = 3, F(4) = &.
CHAP. III.] MEAN POINTS OF SYSTEMS. 87
Then it is easy to perceive that if we introduce a new point c, each old line mn fur-
nishes two new lines, according as we group the new point with one or other of the
two old partial systems, (M) aud (A') ; and that there is, besides, one other new line,
namely cs : we have, therefore, the eqication infinite differences,
which, with the particular value above assigned for 0(4), or even with the simpler
and more obvious value, ^(2)= 1, conducts to the general expression,
0Cs) = 2*-i-l.
(4.) Again, if (Af) (iV) (P) be any three partial systems, which jointly make
up the old or given total system (-S") ; and if, by grouping a new point c with each
of these in turn, we form three new partial systems, {M') (N') (P') ; then each
old triple plane such as mnp, will furnish three new triple planes,
m'np, mn'p, mnp' ;
while each old line, kl, will give one new triple plane, Ckl ; nor can any new triple
plane be obtained in any other way. We have, therefore, this new equation in dif-
fer eiices :
/(*+l) = 3/(O + 0(*).
But we have seen that
0(» + l) = 20(5) + l;
if then we write, for a moment,
/(s) + 0(O=xW,
we have this other equation in finite differences,
X(« + I) = 3x(«)+1.
Also,
/(3)-l, 0(3) = 3, x(3) = 4:
therefore,
2x (s) = 3»-i - 1,
and
2/(«) = 3»-»-2»+l.
(5.) Finally, it is clear that we have the relation,
3/(*) + ^(*) = l0(O-(^(O-l) = (2-'-l) (2-2-1);
because the triple planes, each treated as three, and the non-triple planes, each treated
as one, must jointly represent all the binary combinations of the lines, drawn through
the mean point s of the whole system. Hence,
2»//(«) = 22«-2 + 3 . 2«-» - 3* - 1 ;
and
F(s) = 22»-3+2«-2-3«-i;
so that
P(» + 1) - 4f(») = 3*-» - 2«-i,
and
^(* + l)-4,^(*) = 3/(.);
which last equation in finite differences admits of an independent geometrical inter-
pretation.
(6.) For instance, these general expressions give,
0(5) = 15; /(5) = 25; <//(5) = 30; f(5) = 55;
so that if we assume a gauche pentagon^ or a system of^i-e points in space, A . . e,
88 ELEMENTS OF QUATERNIONS. [bOOK 1.
and determine the jnean point f of this system, there will in general be a set ofjif-
teen lines, of the kind above considered, all passing through this sixth point f : and
these will be arranged generally m fifty- five distinct planes, -whereof twenty-five will
be what we have called triple, the thirty others being of the non-triple kind.
97. More generally, if a^ . . a^ be, as before, a system of m given
and co-initial vectors^ and if osi, . . a^he any system of m given sea-
lars (17), then that new co-initial vector /S, or OB, which is deduced
from these by the formula,
a,a 4- . . + «,„«,» 2aa 2aoA
3 = = , or OB = ,
«i + . • + «« 2a 2a
or by the equation
2a(a -/3) = 0, or Saba = 0,
may be said to be the Complex Mean of those m given vectors a, or
OA, considered as affected (or combined) with that system of given
scalars, a, as coefficients, or as multipliers (12, 14). It may also be said
that the derived point b, of which (comp. 96) the position is inde-
pendent of that of the origin o, is i\\e Barycentre (or centre of gravity)
of the given system of points Ai . . ., considered as loaded with the
given weights ai . . . ; and theorems of intersections of lines and planes
arise, from the comparison of these complex means, or harycentres, of
partial and total systems, which are entirely analogous to those lately
considered (96), for simple means of vectors and oi points.
(1.) As an Example, in the case of Art. 24, the point c is the barj'centre of the
system of the two points, a and b, with the weights a and h ; while, under the con-
ditions of 27, the origin o is the bary centre of the three points A, b, c, with the three
weights a,h,c; and if we use the formula for p, assigned in 34 or 36, the same three
given points A, b, c, when loaded with xa, yh, zc as weights, have the point p in
their plane for their bary centre. Again, with the equations of 65, e is the bary cen-
tre of the system of the ybwr given points. A, b, o, d, with the weights a, b, c, d;
and if the expression of 79 for the vector op be adopted, then xa, yh, zc, wd are
equal (or proportional) to the weights with which the same four points A . . D must
be loaded, in order that the point p of space may be their barycentre. In all these
cases, the weights are thus proportional (by 69) to certain segments, or areas, or
volumes, of kinds which have been already considered ; and what we have called the
anharmonic co-ordinates of a variable point p, in a plane (36), or in space (79),
may be said, on the same plan, to be quotients of quotients of weights.
(2.) The circumstance that the position of a barycentre (97), like that of a sim-
ple mean point (96), is independent of the position of the assumed origin of vectors,
might induce us (comp. 69) to suppress the symbol o of that arbitrary and foreign
point; and therefore to write' simply, under the lately supposed conditions,
* We should thus have some of the principal notations of the Barycentric Calcu-
lus : but used mainly with a reference to vectors. Compare the Note to page 56.
CHAP. III.] BARYCENTRES OF SYSTEMS OF POINTS. 81)
B = — — or 65=20.4, if 6 = a.
2a
It is easy to prove (comp. 96), by principles already established, that the ordi-
nate of the barycentre of any given system of points is the complex mean (in
the sense above defined, and with the same system oi weights)^ of the ordinates of
the points of that system, with reference to any given plane : and that the projection
of the barycentre, on any such plane, is the barycentre of the projected system.
(3. ) Without any reference to ordinates, or to any foreign origin, the barycentrie
notation B = may be interpreted, by means of our fundamental convention
2a
(Art. 1) respecting the geometrical signification of the symbol b— A, considered as
denoting the vector from A to B : together with the rules for midtiplying such vec-
tors by scalars (14, 17), and for taking the sums (6, 7, 8, 9) of those (generally
new) vectors, which are (16) the products of such multiplications. For we have only
to write the formula as follows,
2a(A-B) = 0,
in order to perceive that it may be considered as signifying, that the system of the
vectors from the barycentre B, to the system of the given points Ai, A2, . . when mul-
tiplied respectively by the scalars (or coefficients) of the given system ai, 02, . . be-
comes (generally) a new system of vectors with a null sum : in such a manner that
these last vectors, ai . b Ai, 02 • BA2, . • can be made (10) the successive sides of a closed
polygon, by transports without rotation.
(4.) Thus if we meet the formula,
B = ^(Ai + A2),
we may indeed interpret it as an abridged form of the equation,
OB = |(OAi + OA2);
which implies that if o be any arbitrary point, and if o' be the point which completes
(comp. 6) the parallelogram AiOA20', then B is the point which bisects the diagonal
00', and therefore also the given line AiA2, which is here the other diagonal. But we
may also regard the formula as a mere symbolical transformation of the equation,
(a3-b)+(ai-b) = 0;
which (by the earliest principles of the present Book) expresses that the two vectors,
from B to the two given points Ai and A2, have a null sum; or that they are equal in
length, but opposite in direction : which can only be, by B bisecting A1A2, as before.
(5.) Again, the formula, bi = ^(ai + A2 4- A3), may be interpreted as an a&Hcf^-
ment of the equation,
OBi = J (OAi + OA3 + OA3) ,
which expresses that the point B trisects the diagonal 00' of the parallelepiped
(comp. 62), which has OAi, 0A2, OA3 for three co-initial edges. But the same for-
mula may also be considered to express, in full consistency with the foregoing inter-
pretatiim, that the sum of the three vectors, from b to the three points Ai, A2, A3, va-
nishes : which is the characteristic property (30) of the mean point of the triangle
A1A2A3. And similarly in more complex cases : tlie legitimacy of such transforma-
tions being here regarded as a consequence of the original interpretation (1) of the
symbol n - A, and of the rules for operations on vectors, so far as as they have been
hitherto established.
N
90 KLEMENTS OF QUATERNIONS. [bOOK I.
Section 6 On Anharmonic Equations, and Vector- Expres-
sions, of Surfaces and Curves in Space.
98. When, in the expression 79 for the vector /> of a variable
point P of space, the four variable scalars, or anharmonic co-ordi-
nates, xi/zw, are connected (comp. 46) by a given algebraic equation,
f,{x, y, z, w) = 0, or briefly /= 0,
supposed to be rational and integral, and homogeneous of the p'''
dimension, then the point P has for its locus a surface of the p^^ orde?',
whereof /= may be said (comp. 56) to be the local equation. For
if we substitute instead of the co ordinates x . .w, expressions of the
forms,
X = tXo + UXx^ .. w= tWo + UWi^
to indicate (82) that p is collinear with two given points, Po, Pi, the
resulting algebraic equation int'.u is of the p*^ degree ; so that (ac-
cording to a received modern mode of vspeaking), the surface may be
said to be cut in p points (distinct or coincident, and real or imagi-
nary*), hy any arhitrary right line, PyPi- And in like manner, when
the four anharmonic co-ordinates Imnr of a variable plane 11 (80) are
connected by an algebraical equation, of the form,
F^(/, m, n, r) =0, or briefly F = 0,
where F denotes a rational and integral function, supposed to be ho-
mogeneous of the q^^ dimension, then this plane n has for its enve-
lope (comp. 5%) a surface of the q*'' class, with f= for its tangential
equation: because if we make
l = tlQ+ uli,.. . r = tro-\-uri,
to express (comp. 82) that the variable plane 11 passes through a given
right line ITo'IIi, we are conducted to an algebraical equation of the
q^^ degree^ which gives q (real or imaginary) values for the ratio t:u,
and thereby assigns q (real or imaginary!) tangent planes to the sur-
* It is to be observed, that no interpretation is here proposed, for imaginary in-
tersections of this kind, such as those of a sphere with a right line, which is wholly
external thereto. The language of modern geometry requires that snch imaginary
intersections should be spoken of, and even that they should be cnwrnera/ec? : exactly
as the language of algebra requires that we should count what are called the imagi-
nary roots of an equation. But it would be an error to confound geometrical imagi-
naries, of this sort, with those square roots of negatives, for which it will soon be seen
that the Calculus of Quaternions supplies, from the outset, a di finite and real in-
terpretation.
f As regards the uninterpreted character of such imaginary contacts in geometry,
the preceding Note to the present Article, resptcting imaginary intersections, may be
consulted.
CHAP. III.] ANHARMONIC EQUATIONS OF SURFACES.
91
face^ drawn through any such given but arbitrary right line. We
may add (comp. 51, 56), that if the functions / and f be only ho-
mogeneous (without necessarily being rational and integral)^ then
is the anharmonic symbol (80) of the tangent plane to the surface
/= 0, at the point (xyziv) ; and that
(DjF, d,„f, d„f, d,f)
is in like manner, a symbol for the point of contact of the plane
\_lmnr'], with its enveloped surface^ f= 0; d^, . . d^, . . being charac-
teristics of partial derivation.
(1.) As an Example, the surface of the second order, which passes through the
nine points called lately
A, c', B, a', C, C2, D, A2, E,
has for its local equation,
0=f=xz-yw;
which gives, by differentiation,
I = T)xf— z; m = Dy/= — w ;
n=Dzf=X', r =DM,/=-y:
so that
lz,-w, a!,-2/]
is a symbol for the tangent plane, at the point (x, y, z, w).
(2.) In fact, the swrface here considered is the ruled (or hyper'holic) hyperboloid,
on which the gauche quadrilateral abcd is superscribed, and which passes also
through the point e. And if we write
p = (xyziv), Q = (aryOO), R = (OyzO),
then Qs and rt (see the annexed Figure 31),
namely, the lines drawn through p to intersect the
two pairs, ab, cd, and bc, da, of opposite sides
of that quadrilateral abcd, are the two generating
lines, or generatrices, through that point ; so that
their plane, qrst, is the tangent plane to the sur-
face, at the point p. If, then, we denote that tan-
gent plane by the symbol [Imnr], we have the
equations of condition,
= Zar + my = my + nz = nz + rw = rw+lx;
whence follows the proportion,
l:m:n:r = otr^ : — y~^ : z*' : — w • ;
or, because xz = yw,
I: m: n: r= z : —w: x
as before.
(3.) At the same time we see that
(ac'bq) = - =
= (002u;), T = {xOOw\
Fig. 3
(ncacs) ;
92 * ELEMENTS OF QUATERNIONS. [bOOK I.
so that the variable generatrix QS divides (as is known) the two Jixed generatrices
AB and DC homographically* ; ad, bc, and c'cj being three of its positions. Con-
versely, if it were proposed to find the locus of the right liiie Q3, which thus divides
homographically (comp. 26) two given right lines in space, we might take ab and DC
for those two given lines, and ad, bc, c'c2 (with the recent meanings of the letters)
for three given positions of the variable line ; and then should have, for the two va-
riiible but corresponding (or homologous^ points % s themselves, and for any arbitrary
point p collinear with them, anharmonic symbols of the forms,
Q = (s, M, 0, 0), s = (0, 0, M, s), P = (st, tu, uv, vs) ;
because, by 82, we should have, between these three symbols, a relation of the form,
(p) = ^(q) + »(s)!
if then we write p= (ar, y, z, w), we have the anharmonic equation xz = yw, as before ;
80 that the locus, whether of the line qs, or of the point p, is (as is known) a ruled
surface of the second order.
(4.) As regards the known double generation of that surface, it may suflSce to
observe that if we write, in like manner,
K=(Of«0), T = (<00f), (p)=«(r) + «(t),
we shall have again the expression,
p = {st, tu, uv, vs), giving xz = yw,
as before : so that the same hyperboloid is also the locus of that other line rt, which
divides the other pair of opposite sides bc, ad of the same gauche quadrilateral abcd
homographically ; ba, cd, and A'Ag being three of its positions ; and the lines a'a2,
c'c2 being still supposed to intersect each other in the given point e.
(5.) The symbol of an arbitrary point on the variable line kt is (by sub-art. 2)
of the form, t(0, y, z, 0) +u(x, 0, 0, w), or (ux, ty, tz, uw) ; while the symbol of an
arbitrary point on the given line c'C2 is (t', f, u, u'). And these two symbols repre-
sent one common point (comp. Fig. 31),
p' = RT-c'c2=(y,y,2,2),
when we su[)pose
, , y 2
t =y, u =z, t=\, «=-=-.
X w
Hence the known theorem results, that a variable generatrix, kt, of one system, in-
tersects three fixed lines, BC, AD, c'Cg, which are generatrices of the other system.
Conversely, by the same comparison of symbols, for points on the two lines rt and
c'c2, "we should be conducted to the equation xz =yw, as the condition for their inter-
section ; and thus should obtain this other known theorem, that the locus of a right
line, which intersects three given right lines in space, is generally an hyperboloid
with tliose three lines for generatrices. A similar analysis shows that QS intersects
a'a2, in a point (comp. again Fig. 31) which may be thus denoted :
p" = QS • a'a2 = (xyyx).
(6.) As another example of the treatment of surfaces by their anharmonic and
local equations, we may remark that the recent symbols for p' and p'', combined with
Compare p. 298 of the Geometric Superieure.
CHAP. III.] ANHAllMONIC EQUATIONS OF SURFACES. 93
those of sub-art. 2 for p, q, r, s, t; with the symbols of 83, 86 for c', a', C2, A2, e;
and with the equation xz = y w, give the expressions :
(p)=(q) + (8) = (r) + (t); (P') = y(c') + ^(C2)=(R)+^(T);
(E) = (c') + (C2) = (A-) + (A2) ; (p") = y{A')-^x (a^) = (q) + ^ (s) ;
whence it follows (84) that the two points p', p", and the sides of the quadrilateral
ABCD, divide the four generating lines through p and e in the following anharmonic
ratios :
(c'eCzP') = (qp"sp) = - = (bA'CR) = (AAgDT) ;
/ y
(a'eA2P ') = (rp'tp) = - = (bc'Aq) = (CC2DS) J
so that (as again is known) the variable generatrices, as well as the fixed ones, of the
hyperboloid, are all divided homographically .
(7.) The tangential equation of the present surface is easily found, by the expres-
sions in sub-art. 1 for the co-ordinates Imnr of the tangent plane, to be the follow-
ing:
= F = /n — wir ;
which may be interpreted as expressing, that this hyperboloid is the surface of the
second class, which touches the nine planes,
[1000], [0100], [0010], [0001], [1100], [0110], [0011], [1001], [1111] ;
or with the literal symbols lately employed (comp. 86, 87),
BOD, CDA, DAB, ABC, CDc", DAa", ABc'o, BCA'2, and [e].*
Or we may interpret the same tangential equation f = as expressing (comp. again
86, 87, where q, l, n are now replaced by t, r, q), that the surface is the envelope of
a plane qrst, which satisfies either of the two connected conditions of homography :
(bc'aq) = = = (ccaDs) ;
m n
(CA Br) = = = (dA2 at) ;
n r
a double generation of the hyperboloid thus showing itself in a new way. And as re-
gards the. passage (or return)^ from the tangential to the local equation (comp. 66),
we have in the present example the formulae :
X = DiF = n ; y = d^f = — r; z = d„f = Z ; w = d^-f = — to ;
whence
xz — yw = 0,
as before.
(8.) More generally, when the surface is of the second order, and therefore also
of the second class, so that the two functions / and f, when presented under rational
and integral forms, are both homogeneous of the second dimension, then whether we
derive I . .r from x . .why the formulae.
* In the anharmonic symbol of Art. 87, for the plane of homology [e], the co-
efficient 1 occurred, through inadvertence, five times.
94 ELEMENTS OF QUATERNIONS. [bOOK I.
or a; . . M7 from / . . r by the converse formulae,
X = DiF, y = DmF, Z = D„F, W = D^F,
the /)oin< p = (xyzvi) is, relatively to that surface, what is usually called (corap. 62)
the pole of the plane 11 = [Imnr] ; and conversely, the plane 11 is the polar of the
point p ; wherever in space the point P and plane 11, thus related to each other,
may be situated. And because the centre of a surface of the second order is known
to be (comp. again 52) the pole of (what is called) the plajie at infinity ; while (comp.
38) the equation and the symbol of this last plane are, respectively,
aa; + &y + cz -f rfw = 0, and [a, 6, c, d],
if the four constants aftccZ have still the same significations as in 05, 70, 79, &c.,
with reference to the system of the five given points abode : it follows that we may
denote this centre by the symbol,
K=(DaFo, DfcFo, DcFq, DrfFo) ;
where Fq denotes, for abridgment, the function f (abcd)^ and d is still a scalar con-
stant.
(9.) In the recent example, we have YQ = ac — ld; and the anharmonic symbol
for the centre of the hyperboloid becomes thus,
K = (c, — d, a, — 6),
Accordingly if we assume (comp. sub- arts. 3, 4),
p = (.<si, tu, w», »s), p' = (s't\ — t'u, uv\ = r's'),
where s, ;f, «, v are any four scalars, and p' is a new point, while
&' = 6^-1- cw, <' = CM + ds, u =dv ■\^ at, v =as-\-hu;
if also we write, for abridgment,
e = ac — hd, w' = ast + htu + cuv + dvs ;
we shall then have the symbolic relations,
e' (p) + (P ) = w (k), e' (p) - (p') = (p"),
if p" = {x"y"z"w") be that new point, of which the co-ordinates are,
x" = lest — cw\ y" = 2e'tu -\- dw\ z" = 2e'uv — aw\ w" = 2e'vs + hw\
and therefore,
ax" + by" + cz" + dw" = 0.
That is to say, if pp' be any chord of the hyperboloid, which ]^SiSses through the fixed
point K, and if p" be the harmonic conjugate of that fixed point, with respect to that
variable chord, so that (pkp'p") = - 1, then this conjugate point p" is on the infinitely
distant plane [abed] : or in other words, the fixed point K bisects all the chords pp'
which pass through it, and is therefore (as above asserted) the centre of the surface.
(10.) With the same meanings (65, 79) of the constants a, b, c, d, the mean
point (96) of the quadrilateral abcd, or of the system of its comers, may be denoted
by the svmbol,
M = («-!, 6 1, cS rf-i);
if then this mean point be on the surface, so that
ac-bd=0,
the centre K is on the plane [a, /», r, d] ; or in other words, it is infinitely distant : so
CHAP. III.] VECTORS OF SURFACES AND CURVES IN SPACE. 95
that the surface becomes, in this case, a ruled (or hyperbolic) paraboloid. In gene-
ral (comp. sub-art. 8), if Fo = 0, the surface of the second order is a paraboloid of
some kind, because its centre is then at infinity^ in virtue of the equation
(aD« + bDb + cDc + dUd) Fo = ;
or because (comp. 60, 58) the plane [abed'] at infinity is then one of its tangent
planes, as satisfying its tangential equation, F = 0.
(11.) It is evident that a curve in space may be represented by a system of two
anharmonic and local equations ; because it may be regarded as the intersection nf
two surfaces. And then its order, or the number of points (real or imaginary*"), in
which it is cut by an arbitrary plane, is obviously the product of the orders of those
two surfaces; or iho. product of the degrees of their two local equati(,ns, supposed to
be rational and integral.
(12.) A curve of double curvature may also be considered as the edge of regres-
sion (or arete de rebrovssement) of a developable surface, namely of the locus of the
tangents to the curve ; and this surface may be supposed to be circumscribed at once
to two given surfaces, which are envelopes of variable planes (98), and are repre-
sented, as such, by their tangential equations. In this view, a ciirve of double cur-
vature may itself he represented by a system of two anharmonic and tangential equa-
tions ; and if the class of such a curve be defined to be the number of its osculating
planes, which pass through ah arbitrary point of space, then this class is the product
of the classes of the two curved surfaces just now mentioned: or (what comes to the
same thing) it is the product of the dimensions of the two tangential equations, by
which the curve is (on this plan) symbolized. But we cannot enter further into these
details ; the mechanism of calculation respecting which would indeed be found to be
the same, as that employed in the known method (comp. 41) of quadriplanar co-or-
dinates.
99. Instead of anharmonic co-ordinates, we may consider any
other system of n variable scalars, x^, .. x„, which enter into the ex-
pression of a variable vector, p\ for example, into an expression of
the form (comp. 96, 97),
p-Xa^-ir XSH + • . = Ixa.
And then, if those n scalars x be ^\\ functions of one independent and
variable scalar^ t, we may regard this vector p as being itself a func-
tion of that single scalar; and may write,
!.../>= (2(0.
But if the n scalars x . ,hQ functions of two independent and scalar
variables, t and u, then p becomes a function of those two scalars^
and we may write accordingly,
II. . . /> = <|)(;, v).
In the 1st case, the term p (comp. 1) of the variable vector /> has
• Compare the Notes to page 90.
96 ELEMENTS OF QUATERNIONS. [bOOK I.
generally for its locus a curve in space^ which may be plane or of
double curvature, or may even become a right line^ according to the
form of the vector-function cp ; and p may be said to be the vector of
this line, or curve. In the Ilnd case, p is the vector of a surface, plane
or curved, according to the form of <p (t, u) ; or to the manner in which
this vector p depends on the two independent scalars that enter into
its expression.
(1.) As Examples (comp. 25, 63), the expressions,
signify, 1st, that p is the vector of a variable point p on the right line ab ; or that
it is the vector of that line itself, considered as the locus of a point; and Ilnd, that
p is the vector of the plane abc, considered in like manner as the locus of an arbitrary
point P thereon.
(2.) The equations,
1. .. p = xa^-y(i, II. .. p = jca + y/3 + zy,
>nth
a;2 + y2 = 1 for the 1st, and a:^ + y2 + ^2 = i for the Ilnd,
signify 1st, that p is the vector of an ellipse, and Ilnd, that it is the vector of an
ellipsoid, with the origin o for their common centre, and with OA, OB, or OA, ob,
DC, for conjugate semi-diameters.
(3.) The equation (comp. 46),
p = t''a^ui^^(t^uyy,
expresses that p is the vector of a cone of the second order, with o for its vertex (or
centre), which is touched by the three planes obc, oca, gab ; the section of this cone,
/> made by the plane abc, being an ellipse (comp. Fig. 25), which is inscribed in the
/t"'' triangle ABO ; and the middle points A, b', c', of the sides of that triangle, being tlje
points of contact of those sides with that conic.
(4.) The equation (comp. 53),
p = r'a + «"i/3 + r-iy, with < + u + v = 0,
expresses that p is the vector of another cone of the second order, with o still for
vertex, but with OA, ob, oc for three of its sides (or rays). The section by the
plane abc is a new ellipse, circumscribed to the triangle abc, and having its tangents
at the corners of that triangle respectively parallel to the opposite sides thereof.
(5 J The equation (comp. 54),
p=t^a + m'/3 + v^y, with t +- m + « = 0,
signifies that p is the vector of a cone of the third order, of wliich the vertex is still
the origin ; its section (comp. Fig. 27) by the plane abc being a cubic curve, whereof
the sides of the triangle abc are at once the asymptotes, and the three (real) tangents
of inflexion; while the mean point (say o') of that <na«^Ze is Si conjugate point oi
the curve; and therefore the right line oo', from the vertex o to that mean point,
may be said to be a conjugate ray of the cone.
(6.) The equation (comp. 98, sub-art. (3.) ),
CHAP. III.] VECTORS OF SURFACES AND CURVES IN SPACE. 97
staa + tuhfi + uvcy + vsdS
p =: ■ ,
sta + tub + uvc + vsd
s t
in which - and - are two variable scalars, while o, 6, c, d are still four constant
u V
scalars, and a, /3, y, d are four constant vectors, but p is still a variable vector, ex-
presses that p is the vector of a ruled (or single-sheeted^ hyperholoid^ on which the
gauche quadrilateral abcd is superscribed, and which passes through the given point
E, whereof the vector e is assigned in 65.
(7.) If we make (comp. 98, sub-art (9.)),
, s't'aa - t'u'hfi + u'v'cy — v's'dd
P =; _ __ — . — ^
s't'a — t'u'b + u'v'c — vsd
where
s'=bt + cv, t' = cu + ds, u'=dv-\-at, v' = as + bu,
then p' = op' is the vector of another point p' on the same hyperboloid ; and because
it is found that the sum of these two last vectors is constant,
„+.„'-2« if,. °<« + r)- K^ + ^)
p+-p-2«,,l.: 2(ac-6rf)
it follows that k is the vector o^ a, fixed point k, which bisects evert/ chord pp' that
passes through it : or in other words (comp. 52), that this point k is the centre of
the surface.
(8.) The three vectors,
a + y (3+d
"' 2 ' 2 '
are termino-collinear (24) ; if then a gauche quadrilateral abcd be superscribed on
a ruled hyperboloid, the common bisector of the two diagonals, AC, bd, passes through
the centre K.
(9.) When ac = bd, or when we have the equation,
sta + tu(3 + uvy -f vsS
n =
st + tu + uv -{- vs
or simply,
p = sta + tuj3 + uvy + vsd, with s +u=t + v = l,
p is then the vector of a ruled paraboloid, of which the centre (comp. 52, and 98, sub-
art. (10.) ), is infinitely distant, but upon which the quadrilateral abcd is still super-
scribed. And this surface passes through the mean point M of that quadrilateral, or
of the system of the four given points A . . D ; because, when s = t = u = v = -^, th«
variable vector p takes the value (comp. 96, sub-art. (1.)),
)tt = i(a + /3 + y + ^).
(10.) In general, it is easy to prove, from the last vector-expression for p, that
this paraboloid is the locus of a right line, which divides similarly the two opposite
tides AB and DC of the same gauche quadrilateral abcd; or the other pair of oppo-
site sides, EC and ad.
98 ELEMENTS OF QUATERNIONS. [bOOK I.
Section 7 — On Differentials of Vectors.
100. The equation (99, I.),
in which /> = op is generally the vector of a point p of sl curve in space,
PCI . . ., gives evidently, for the vector oq of another point Q of the
same curve, an expression of the form
p + Ap^<p(ti-At);
so that the chord pq,, regarded as being
itself a vector, comes thus to be repre-
sented (4) by the Jlnite difference,
PQ = A/> = A(p (t) = (p(t + At)-(p (t).
Suppose now that the other finite dif-
ference, A^, is the n*^ part of a new
scalar, u ; and that the chord A/>, or pq, is in like manner (comp.
Fig. 32), the n^^ part of a new vector, ff„, or pr ; so that we may
write,
nAt = u, and ?iA/3 = w . pq = o-,^ = pr.
Then, if we treat the two scalars, t and u, as constant, but the num-
ber n as variable (the, form of the vector function (f), and the origin o,
being given), the vector p and the;?om^ p will he fixed: but the two
points Qt and R, the two differences At and Ap, and the multiple vector
nAp, or <T„, will (in general) vary together. And if this number n
be indefinitely/ increased, or made to tend to infinity, then each of the
two differences At, Ap will in general tend to zero ; such being the
common limit, of n~^u, and of <|> (^ + n~^u) - ^(f)'. so that the variable
•point Q of the curve will tend to coincide with the fixed point p. But
although the chord pq will thus be indefinitely shortened, its n^^ mul-
tiple, PR or a,,, will tend (generally) to Vi finite liinit,* depending on
the supposed continuity oi the function <j>(^); namely, to a certain
definite vector, pt, or «t„, or (say) t, which vector pt will evidently
be (in general) tangential to the curve: or, in other words, the variable
point R will tend to a fixed position t, on thetangent to that curve at p.
We shall thus have a limiting equation, of the form
T = PT = lim. PR = croo = lim. 7iA0(^), if ?iA^ = w;
M = 00
t and u being, as above, two given and (generally) /wiVe scalars. And
* Compare Newton's Privcipia.
CHAP. III.] DIFFERENTIALS OF VECTORS. 99
if we then agree to call the second of these two given scalars the dif-
ferential of the first, and to denote it by the symbol d^, we shall de-
fineih2i,\, the vector-limit^ r or o-», is the (corresponding) differential of
the vector p, and shall denote it by the corresponding symbol^ d/>; so
as to have, under the supposed conditions,
u = dt, and t = dp.
Or, eliminating the two symbols u and t, and not necessarily suppos-
ing that p is SL point of a curve, we may express our Definition"^ of the
Differential of a Vector />, considered as a Function ^ of a Scalar t,
by the following General Formula :
dp = d^{t)=\m-i.n\cl^{t+-\-^(f)\,
n = cc ( n J )
in which t and d^ are two arbitrary and independent scalars, both ge-
nerally finite ; and dp is, in general, a new and finite vector, depending
on those two scalars, according to a law expressed by the formula,
and derived from that given law, whereby the old ov former vector, p
or <p (t), depends upon the single scalar, t.
(1.) As an example, let the given vector-function have the form,
p — ^(f) — ^t^a, "where a is a given vector.
u
Then, making Af = -, where u is any given scalar, and n is a variable whole number,
we have
(Tn = nAp = au{t + —]; a^^ — atu ;
and finally, writing dt and dp for u and (Tx,
dp=d0(O = df^U«<d^
(2.) In general, let <p(t)=af(t), where a is still a given or constant vector, and
f(f) denotes a scalar function of the scalar variable, t. Then because a is a common
factor within the brackets { } of the recent general formula (100) for dp, we may
write,
dp = d0(O=d.a/(O = ad/(O;
provided that we now define that the differential of a scalar function, f{t), is a new
scalar function of two independent scalars, t and dt, determined by the precisely
similar formula :
d/(0 = lhn.n|/(^ + ^']-/(0};
* Compare the Note to page 39.
100 ELEMENTS OF QUATERNIONS. [bOOK 1.
which can easily be proved to agree^ in all its consequences^ with the usual rules for
differentiating functions of one variable.
(3.) For example, if we write dt = nh, where A is a new variable scalar, namely,
the »*'» part of the given and (generally) finite differential, At, we shall thus have
the equation,
4/*(0 ,. /(^ + / 0-/(0 .
— — - = lim. ;
dt 7^=0 h
in which the first member is here considered as the actual quotient of two finite sca-
lars, df(i) : d^, and not merely as a differential coefficient. We may, however, as
usual, consider this quotient, from the expression of which the differential dt disap-
pears, as a derived function of the former variable, t ; and may denote it, as such, by
either of the two usual symbols,
fit) and J)tf{t).
(4.) In like manner we may write, for the derivative of a vector-function,* ^(t),
the formula :
,, dp d0(O
p' = f(<) = D<p = D<5&(0=^= -^;
these two last forms denoting that actual and finite vector, p' or ^' (t), which is
obtained, or deri')ed, by dividing (comp. 16) the not less actual (or finite) vector,
dp or d<p(t), by the finite scalar, dt. And if again we denote the n'^ part of this
last scalar by h, we shall thus have the equally general formula :
Dtp = Dt(}) (t) = hm. ;
A = «
with the equations,
dp = Dtp . d^ = pdt ; d0 (t) = Dt(p (t) . dt = ^'(t) . dt,
exactly as if the vector-function, p or ^, were a scalar function, f.
(5.) The particular value, dt — 1, gives thus dp = p'\ so that the derived vector
p' is (with our definitions) a particular but important case of the differential of a
vector. In applications to mechanics, if t denote the time, and if the term v of
the variable vector p be considered as a moving point, this derived vector p' may be
called the Vector of Velocity : because its length represents the amount, and its di-
rection is the direction of the velocity. And if, by setting off vectors ov = p' (comp.
again Fig. 32) /rom one origin, to represent thus the velocities of a point moving in
space according to any supposed law, expressed by the equation p = (p(t), we con-
struct a new curve vw . . of which the corresponding equation may be written as
p' = <p'(t), then this new curve has been defined to be the HoDOGKAPH,t as the old
curve FQ. . mav be called the orbit of the motion, or of the moving point.
* In the theory of Differentials of Functions of Quaternions, a definition of the
differential d^{q) will be proposed, which is expressed by an equation of precisely
the same form as those above assigned, for df(t), and for d<p {t) ; but it will be found
that, for qyafernions, the quotient d^(«7): d^' is not generally independent of dq ;
and consequently that it cannot properly be called a derived function, such as ^'(9),
of the quaternion q alone. (Compare again the Note to page 39.)
t The subject of the Hodograph will be resumed, at a subsequent stage of this
work. In fact, it almost requires the assistance of Quaternions, to connect it, in
what appears to be the best mode, with Newton's Law of Gravitation.
CHAP. III.] DIFFERENTIALS OF VECTORS. 101
(6.) We may differentiate a vector-function twice (or oftener), and so obtain its
successive differentials. For example, if we diff^erentiate the derived vector p', we
obtain a result of the form,
dp' = p"dt, where p" = D(p' = D<2p,
by an obvious extension of notation ; and if we suppose ihat the second differential,
dd^ or d-^, of the scalar t is zero, then the second differential of the vector p is,
d2(0 = ddp = d. p'd^ = dp'. At^p'Afi ;
-where At^, as usual, denotes (d<)2 ; and where it is important to observe that, with
the definitions adopted, d^p is as finite a vector as dp, or as p itself. In applications
to motion, lit denote the time, p" may be said to be the Vector of Acceleration.
(7.) "We may also say that, in mechanics, i\\Q finite differential dp, of the Vector
of Position p, represents, in length and in direction, the right line (suppose pt in
Fig. 32) which would have been described, by a freely moving point p, in the finite
interval of time At, immediately /oZ/owzw^r the time t, z/at the end of this time t all
foreign forces had ceased to act.*
(8.) In geometry, if p = <p(t) be the equation of a curve of double curvature, re-
garded as the edge of regression (comp. 98, (12.) ) of a developable surface, then the
equation of that surface itself, considered as the locus of the tangents to the curve,
may be thus written (comp. 99, II.) :
p = (p(t) + u(p'(ty, or simply, p = (p(t)+ d(p(t),
if it be remembered that u, or d^, may be any arbitrary scalar.
(9.) If any other curved surface (comp. again 99, 11.) be represented by an equa-
tion of the form, p = (p(x, y), where now denotes a vector -function of two indepen-
dent and scalar variables, x and y, we may then differentiate this equation, or this
expression for p, with respect to either variable separately, and so obtain what may
be called two partial (hwt finite) differentials, d^p, dyp, and two partial derivatives,
X)xp, Dyp, whereof the former are connected with the latter, and with the two arbitrary
(hut finite') scalar s, dx, dy, by the relations,
dxp = D^-p . dx ; dyp = Dyp . dy.
And these two differentials (or derivatives) of the vector p of the surface denote two
tangential vectors, or at least two vectors parallel to two tangents to that surface at
the point P : so that their plane is (or is parallel to) the tangent plane at that point.
(10.) The mechanism of all such differentiations of vector-functions is, at the
present stage, precisely the same as in the usual processes of the Differential Calcu-
lus; because the most general form of such a vector- function, which has been consi-
dered in the present Book, is that of a sum of products (comp. 99) of the form xa,
where a is a constant vector, and a? is a variable scalar : so that we have only to
operate on these scalar coefficients a; . ., by the usual rules of the calculus, the vec-
tors a. . being treated as constant factors (comp. sub-art. 2). But when we shall
come to consider quotients or products oi vectors, or generally those new functions of
vectors which can only be expressed (in our system) by Quaternions, then some few
new rules of differentiation become necessary, although deduced from the same (or
nearly the same) definitions, as those which have been established in the present
Section.
As is well illustrated by Atwood's machine.
102 ELEMENTS OF QUATERNIONS. [bOOK I.
(11.) As an example of partial differentiation (comp. sub-art. 9), of a vector
function (the word *' vector" being here used as an adjective) of two scalar variables,
let us take the equation,
p = ^(a;,y)=i{a;2a + y2/3 + (a, + y)2^};
in which p (comp. 99, (3.) ) is the vector of a certain cone of the second order; or
more precisely, the vector of one sheet of such a cone, if x and y be supposed to be
real scalars. Here, the two partial derivatives of p are the following :
DarjO = jca + (ar + y) y ; i>yp = y/3 + (a; + .y) y ;
and therefore,
2p = xDxp + !/T>yp ;
so that the three vectors, p, D^p, i>,jp, if drawn (18) from one common origin, are con-
tained (22) in one common plane; which implies that the tangent plane to the sur-
face, at any point p, passes through the origin o : and thereby verifies the conical
character of the locus of that point p, in which the variable vector p, or op, termi-
nates.
(12.) If, in the same example, we make a: = 1, y = — 1, we have the values,
P = l(a-V^), ^xp = ct, Dyp = -/3;
whence it follows that the middle point, say c', of the right line ab, is one of the
points of the conical locus ; and that (comp. again the sub-art. 3 to Art. 99, and the
recent sub-art. 9) the right lines OA and ob are parallel to two of the tangents to the
surface at that point ; so that the cone in question is touched by the plane aob, along
the side (or ray) oc'. And in like manner it may be proved, that the same cone is
touched by the two other planes, BOC and COA, at the middle points a' and b' of the
two other lines BC and CA ; and therefore along the two other sides (or rays), oa'
and ob' : which again agrees with former results.
(13.) It will be found that a vector function of the turn of two scalar variables,
t and (\t, may generally be developed, by an extension of Taylor's Series, under the
form,
0(< + dO = ^(O+d<&(O + id2^(O + ^d'^(O + --
d2 d3
"^^^"^^ 2 + 2:^+--^^^'^=''^^'^'
it being supposed that d'^t= 0, dH = 0, &c. (comp. sub-art. 6). Thus, if <pt=: ^at^,
(as in sub-art. 1), where a is a constant vector, we have d<pt = atdt, d^cpt^adt"^,
d^^t = 0, &c. ; and
(< + dt) = !«(< + dty = laf^ + atdt + |ad^2,
rigoroiisly, without any supposition that dt is small.
(14.) When we thus suppose At = dt, and develope the finite difference, A^{t)
= (< + dt) - ^(t), the first term of the development so obtained, or the term of first
dimension relatively to dt, is hence (by a theorem, which holds good for vector -func-
tions, as well as for scalar functions) the first dfferential d<pt of the function ; but
we do not choose to defi7ie that this Differential is (or means) thsii first term : be-
cause the Formula (100), which we prefer, does not postulate the j9ossJ6i7%, nor even
suppose the conception, of any such development. Many recent remarks will perhaps
appear more clear, when we shall come to connect them, at a later stage, with that
theory of Qnaternions, to which we next proceed.
BOOK II.
ON QUATERNIONS, CONSIDERED AS QUOTIENTS OF VECTORS,
AND AS INVOLVING ANGULAR RELATIONS.
CHAPTER I.
FUNDAMENTAL PRINCIPLES RESPECTING QUOTIENTS OF VECTORS.
Section 1. — Introductory Remarks ; First Principles adopted
from Algebra.
Art. 101. The only angular relations^ considered in the fore-
going Book, have been those of parallelism between vectors
(Art. 2, &c.) ; and the only quotient s,\iii\iQvto employed, have
been of the three following kinds :
I. Scalar quotients ofscalars^ such as the arithmetical frac-
n
tion — in Art. 14;
m
II. Vector quotients^ of vectors divided by scalar s, as — = a
in Art. 16;
III. Scalar quotients of vectors^ with directions Qiih^r simi-
lar or opposite, as — = oj in the last cited Article. But we now
a
propose to treat of other geometric Quotients (or geometric
Fractions, as we shall also call them), such as
— =- = q, with /3wo^ II a (comp. 15);
OA a
for each of which the Divisor (or denominator), a or oa, and
the Dividend (or numerator), /3 or ob, shall not only both be
104 ELEMENTS OF QUATERNIONS. [bOOK II.
Vectors^ but shall also be inclined to eacb other at an Angle,
distinct (in general) from zero, and from two'^ right angles,
102. In introducing this new conception, of a General Quo^
tient of Vectors, with Angular Relations in a given plane, or
in space, it will obviously be necessary to employ some proper-
ties of circles and spheres, which were not wanted for the pur-
pose of the former Book. But, on the other hand, it will be
possible and useful to suppose a much less degree of acquaint-
ance with many important theoriesf of modern geometry, than
that of which the possession was assumed, in several of the
foregoing Sections. Indeed it is hoped that a very moderate
amount of geometrical, algebraical, and trigonometrical prepa-
ration will be found sufficient to render the present Book, as
well as the early parts of the preceding one, fully and easily
intelligible to any attentive reader.
103. It may be proper to premise a few general principles
respecting quotients of vectors, which are indQQ^suggestedhj
algebra, but are here adopted by definition. And 1st, it is
evident that the supposed operation of division (whatever its
full geometrical import may afterwards be found to be), by
which we here conceive ourselves to pass from a given divisor-
line a, and from a given dividend-line j3, to what we have called
(provisionally) their geometric quotient, q, may (or rather
must) be conceived to correspond to some converse act (as yet
not fully known) o^ geometrical multiplication : in which new
act the former quotient, q, becomes a Factor, and operates on
the line a, so as to produce (or generate) the line j3. We shall
therefore write, as in algebra,
(3 = q-a, or simply, j3 = qa, when f5:a = q;
* More generally speaking, from every even multiple of a right angle.
f Such as homology^ homography^ invobition, and generally whatever depends
on anharmonic ratio : although all that is needful to be known respecting such
ratio, for the applications subsequently made, may be learned, without reference to
any other treatise, from the definitions incidentally given, in Art. 25, &c. It was,
perhaps, not strictly necessary to introduce any of these modern geometrical theories,
in any part of the present woik ; but it was thought that it might interest one class,
at least, of students, to see how they could be combined with that fundamental con-
ception of the Vkotob, which the First Book was designed to develope.
CHAP. I.] FIRST PRINCIPLES ADOPTED FROM ALGEBRA. 105
even if the two lines a and j3, or oa and ob, be supposed to
be inclined to each other, as in Fig. 33. And this very sim-
ple and n^iwroi notation (comp. 16) will then allow us to treat
as identities the two following formulae :
P \P ,, qa
a J a a
although we shall, for the present, abstain from writing also
such formulae* as the following :
a a
where a, /3 still denote tivo vectors, and q denotes their geo-
metrical quotient : because we have not yet even begun to con-
sider the multiplication of one vector by another, or the division
of a quotient by a line.
104. As a Ilnd general principle, suggested by algebra,
we shall next lay it down, that if
'—;=-, and a = a, then j3' = j3 ;
a a
or in words, and under a slightly varied form, that unequal
vectors, divided by equal vectors, give unequal quotients. The
importance of this very natural and obvious assumption will
soon be seen in its applications.
105. As a Ilird principle, which indeed may be consi-
dered to pervade the whole of mathematical language, and
without adopting which we could not usefully speak, in any
case, of EQUALITY as existing between any two geometrical
quotients, we shall next assume that two such quotients can
never be equal to the same third] quotient, without being at the
same time equal to each other: or in symbols, that
if q = q, and q" = q, then q" = q'.
* It will be seen, however, at a later stage, that these two formulae are permitted,
and even required, in the development of the Quaternion System,
f It is scarcely necessary to add, what is indeed included in this Ilird principle,
in virtue of the identity q = g, that if q' = q, then q = q' \ or in words, that we shall
never admit that any two geometrical quotients, q and q\ are equal to each other in
one order ^ without at the same time admitting that they are equal^ in the opposite
order also.
P
106 ELEMENTS OF QUATERNIONS. [eOOK II.
106. In the lYth place, as a preparation for operations
on geometrical quotients^ we shall say that any two such quo-
tients, OY fractions (101), which have a common divisor-line, or
(in more familiar words) a common denominator, are added,
subtracted, or divided, among themselves, by adding, subtract-
ing, or dividing their numerators: the common denominator
being retained, in each of the two former of these three cases.
In symbols, we thus define (comp. 14) i\mt^ for any three (ac-
tual) vectors, a, j3, y,
7 I /^ _ 7 + ^ . 7 ^_7-/3.
a a a a a a '
and
a a [5
aiming still at agreement with algebra.
107. Finally, as a Vth principle, designed (like the fore-
going) to assimilate, so far as can be done, the present Calculus
to Algebra, in its operations on geometrical quotients, we shall
define that the following formula holds good :
fi a J j3 a a '
or that if two geometrical fractions, q and^'', he so related, that
the denominator, j3, of the multiplier q (here written towards
the left-hand) is equal to the numerator of the multiplicand q,
then the product, q'-q or q'q, is that third fraction, whereof
the numerator is the numerator y of the multiplier, and the
denominator is the denominator a of the nmltiplicand : all such
denominators, or divisor-lines, being still supposed (16) to be
actual (and not null) vectors.
Section 2. — First Motive fornaming the Quotient of two Vec-
tors a Quaternion.
108. Already we may see grounds for the application of
the name, Quaternion, to such a Quotient of two Vectors as
has been spoken of in recent articles. In the first place, such
a quotient cannot generally be what we have called (17) a Sca-
CHAP. I.] QUOTIENT OF TWO VECTORS A QUATERNION. 107
LAR : or in other words, it cannot generally be equal to any
of the (so-called) reals of algebra^ whether oi ihQ positive or of
the negative kind. For let x denote any such (actual*) scalar,
and let a denote any (actual) vector; then we have seen (15)
that the product xa denotes another (actual) vector, say /3',
which is either similar or opposite in direction to a, according
as the scalar coefficient, or factor, x, is positive or negative ;
in neither case, then, can it represent any vector, such as /3,
which is inclined to a, at any actual angle ^ whether acute, or
right, or obtuse : or in other words (comp. 2), the equation
j3' = j3j or Xa = j3, is impossible, under the conditions here sup-
posed. But we have agreed (16, 103) to write, as in algebra,
'— = a; ; we must, therefore (by the Ilnd principle" of the fore-
a
going Section, stated in Art. 104), abstain fi-om writing also
^ =^x, under the same conditions : x still denoting a scalar.
a
Whatever else a quotient of two inclined vectors may be found
to be, it is thus, at least, a Non-Scalar.
109. Now, in forming the conception of the scalar itself
as the quotient of two parallel] vectors (17), we took into ac-
count not only relative length, or ratio of the usual kind, but
also relative direction, under the form o^ similarity or opposition.
In passing from a to xa, we altered genevaWj (15) the length of
the line a, in the ratio of ± a; to 1 ; and we preserved or reversed
the direction of that line, according as the scalar coefficient x
was positive or negative. And in like manner, in proceeding to
form, more definitely than we have yet done, the conception of
the non-scalar quotient (108), q = (5: a-OB : oa, of two inclined
vectors, which for simplicity may be supposed (18) to be co-
* By an actual scalar, as by an actual vector (comp. 1), we mean here one that
is different from zero. An actual vector, multiplied by a. null scalar, has for product
(15) a null vector ; it is therefore unnecessary to prove that the quotient oitwo actual
vectors cannot be a null scalar, or zero.
f It is to be remembered that we have proposed (15) to extend the use of this
terra parallel, to the case of two vectors which are (in the usual sense of the word)
parallel to one common line, even Avhen they happen to he parts of one and the same
TvAit line.
108 ELEMENTS OF QUATERNIONS. [bOOK II.
initial^ we have 5^2*// to take account both of the relative length,
and of the relative direction^ of the two lines compared. But
while the former element of the complex relation here consi-
dered, between these two lines or vectors, is still represented
by a simple Ratio (of the kind commonly considered in geo-
metry), or by a number* expressing that ratio ; the latter ele-
ment of the same complex relation is now represented by an
Angle, aob : and not simply (as it was before) by an alge-
braical sign, + or -.
110. Again in estimating this angle, for the purpose of
distinguishing one quotient of vectors from another, we must
consider not only its magnitude (or quantity), but also its
Plane : since otherwise, in violation of the principle stated
in Art. 104, we should have ob': oa = ob : oa, if ob and ob'
were two distinct rays or sides of a cone of revolution, with oa
for its axis; in which case (by 2) they would necessarily be
unequal vectors. For a similar reason, we must attend also to
the contrast between two opposite angles, of equal magnitudes,
and in one common plane. In short, for the purpose of know-
ing ^wZ/y the relative direction of two co-initial lines oa, ob in
space, we ought to know not only how many degrees, or other
parts of some angular unit, the angle ^^
aob contains ; but also (comp. Fig. 33)
the direction of the rotation from oa to ^^^^^
ob : including a knowledge of the plane, o-
in lohich the rotation is performed ; and -^'S- 33.
of the hand (as right or left, when viewed from a known side of
the plane), towards ichich the rotation is directed.
111. Or, if we agree to select some one fixed hand (suppose
the right^ hand), and to call all rotations positive when they
* This number^ which we shall presently call the tensor of the quotient, may be
whole or fractional^ or even incommensurable with unity ; but it may always be
equated, in calculation, to a poaitive scalar : although it might perhaps more pro-
perly be said to be a signless number, as being derived solely from comparison of
lengths, without any reference to directions.
t If right-handed rotation be thus considered as positive, then the positive axis
of the rotation aob, in Fig. 33, must be conceived to be directed downward, or below
the plane of the paper.
CHAP. I.] QUOTIENT OF TWO VECTORS A QUATERNIOM. 109
are directed towards this selected hand, but all rotations nega-
tive when they are directed towards the other hand, then, for
any given angle aob, supposed for simplicity to be less than two
right angles, and considered as representing a rotation in a given
plane from oa to ob, we may speak oi one perpendicular oc to
that plane aob as being the positive axis of that rotation ; and
of the opposite perpendicular oc' to the same plane as being the
negative axis thereof; the rotation round the positive axis being
zY^e//" positive, and vice versa. And then the rotation aob may
be considered to be entirely known, if we know, 1st, its quantity,
or the ratio which it bears to a right rotation ; and Ilnd, the
direction of its positive axis, oc : but not without a knowledge
of these two things, or of some data equivalent to them. But
whether we consider the direction of an Axis, or the aspect of
a Plane, we find (as indeed is Avell known) that the determi-
nation of such a direction^ or of such an aspect, depends on two
polar co-ordinates'^ , or other angular elements.
112. It appears, then, from the foregoing discussion, that
for the complete determination, of what we have called the geo-
metrical Quotient of two co-initial Vectors, a System of Four
Elements, admitting each separately of numerical expression,
is generally required. Of these four elements, one serves (109)
to determine the relative length of the two lines compared ;
and the other three are in general necessary, in order to deter-
mine fully their relative direction. Again, of these three latter
elements, one represents the mutual inclination, or elongation,
of the two lines ; or the magnitude (or quantity) of the angle
between them ; while the two others serve to determine the
direction of the axis, perpendicular to their common plane,
round which a rotation through that angle is to be performed,
in a sense previously selected as the positive one (or tow^ards
a fixed and previously selected hand), for the purpose of pass-
ing (in the simplest way, and therefore in the plane of the two
lines) from the direction of the divisor-line, to the direction of
• The actual (or at least the frequent) use of such co -ordinates is foreign to the spirit
of the present System : but the mention of them here seems likely to assist a student,
by suggesting an appeal to results, with which his previous reading can scarcely fail
to have rendered him familiar.
110
ELEMENTS OF QUATERNIONS.
[book II.
the dividend-line. And no more than four numerical elements
are necessary, for our present purpose: because the relative
length of two lines is not changed, when their two lengths are
altered proportionally^ nor is their relative direction changed,
when the angle which they form is merely turned about, in its
own plane. On account, then, of this essential connexion of
that complex relation (109) between two lines, which is com-
pounded of a relation of lengths^ and of a relation of directions,
and to which we have given (by an extension from the theory
of scalar s) the name of a geometrical quotient, with a System
o/'FouR numerical Elements, we have already a motive* for
saying, that '' the Quotient of two Vectors is generally a Qua-
ternion.*'
Section 3. — Additional Illustrations.
113. Some additional light may be thrown, on this first concep-
tion of a Quaternion, by the annexed Figure 34. In that Figure,
the letters cdefg are
designed to indicate
corners of a prisma-
tic desk, resting upon
a horizontal table.
The angle hcd (sup-
posed to be one of
thirty degrees) repre-
sents a (left-handed)
rotation, whereby the
horizontal ledge CD of
the desk is conceived
to be elongated (or
removed) from a given horizontal line ch, which may be imagined to
be an edge of the table. The angle gcf (supposed here to contain
forty degrees) represents the slopej of the desk, or tlie amount of its
inclination to the table. On the face cdef of the desk are drawn two si-
milar and similarly turned triangles, A OB and a'o'b', which are supposed
to be halves of two equilateral triangles ; in such a manner that each
' Several other reasons for thus speaking will offer themselves, in the course of the
present work.
t These two angles, HCD and gcf, may thus be considered to correspond to lonf/i-
tude of node, and inclination of orbit, of a planet or comet in astronomy.
Fig. 34.
CHAP. I.] ADDITIONAL ILLUSTRATIONS. Ill
rotation^ aob or a^o'b' is one of sixty degrees, and is directed towards
one common hand (namely the right hand in the Figure): while if
lengths alone be attended to, the side ob is to the side oa, in one tri-
angle, as the side oV is to the side o'a', in the other; or as the num-
ber two to one.
114. Under these conditions of construction, we consider the two
quotients^ or the two geometric fractions,
OB , o'b'
ob:oa and ob':oa, or — and — — -,
OA o'a'
as being equal to each other; because we regard the two lines, oa and
OB, as having the same relative length, and the same relative direction,
as the two other lines, o'a' and oV. And we consider and speak of
each Quotient, or Fraction^ as a Quaternion: heca,use its complete con-
struction (or determination) depends, for all that is essential to its
conception, and requisite to distinguish it from others, on a system of
four numerical elements (comp. 112); which are, in this Example, the
four numhers,
2, 60, 30, and 40.
115. Of these four eletnents (to recapitulate what has been above supposed), the
1st, namely the number 2, expresses that the length of the dividend-line, ob or
o'b', is double of the length of the divisor-line, OA or o'a'. The Ilnd numerical
element, namely 60, expresses here that the angle aob or a'o'b', is one of sixty de-
grees; while the corresponding rotation, from oa to ob, or from o'a' to o'b', is to-
wards a known hand (in this case the right hand, as seen by a person looking at the
face CDEF of the desk), which hand is the same for both of these two equal angles.
The Ilird element, namely 30, expresses that the horizontal ledge cd of the desk
makes an angle of thirty degrees with a known horizontal line ch, being removed
from it, by that angular quantity, in a known direction (which in this case happens
to be towards the left hand, as seen from above). Finally, the IVth element,
namely 40, expresses here that the desk has an elevation o^ forty degrees as before.
116. Now an alteration in any one of these Four Elements, such as an altera-
tion of the slope or aspect of the desk, would make (in the view here taken) an es-
sential change in the Quaternion, which is (in the same view) fAe Quotient of the two
Zmes compared: although (as the Figure is in part designed to suggest) no such
change is conceived to take place, when the triangle AOB is merely turned about, in
its own plane, without being turned over (comp. Fig. 36) ; or when the sides of that
triangle are lengthened or shortened proportionally, so as to preserve the ratio (in the
old sense of that word), of any one to any other of those sides. We may then briefly
say, in this mode of illustrating the notion of a Quaternion* in geometry, by refe-
* As to the mere word. Quaternion, it signifies primarily (as is well kncwn), like
its Latin original, " Quaternio," or the Greek noun TtTpaKTVQ, a 5c/ of Four : but
it is obviously used here, and elsewhere in the present work, in a technical sense.
112
ELEMENTS OF QUATERNIONS.
[book II.
rence to an angle on a desk, that the Four Elements which it involves are the follow-
ing;
Ratio, Angle, Ledge, and Slope;
although the two latter elements are in fact themselves angles also, but are not im-
mediately obtained as such, from the simple comparison of the two lines, of which the
Quaternion is the Quotient.
Section 4 On Equality of Quaternions; and on the Plane
of a Quaternion.
117. It is an immediate consequence of the foregoing con-
ception of a Quaternion, that two quaternions, or tiuo quotients
of vectors, supposed for simplicity to be all co-initial (\8), are
regarded as being equal to each other, or that the equation,
d Q CD OB
-=— , or — = — ,
y a oc oa
is by us considered and defined to hold good, ivheji the two tri-
angles, AOB and COD, are similar and similarly/ turned, and in
one common plane, as represented in the -^
annexed Fig. 35 : the relative length
(109), and the relative direction
(110), of the two lines, oa, ob, being
then in all respects the same as the re-
lative length and the relative direction
of the two other lines, oc, on,
118. Under the same conditions, we
shall write the following formula of direct similitude,
A AOB a cod;
reserving this other formula,
A AOB oc' aob', or A a'ob a' a'ob',
which we shall call a formula of inverse simili-
tude, to denote that the two triangles, aob and
aob', or a'ob and a'ob', although otherwise simi-
lar (and even, in this case, equal,* on account
of their having a common side, oa or oa'), are
Fig. 35.
Fig. 36.
* That is to say, equal in absolute amount of area, but with opposite algebraic
signs (28). The two quotients OB : OA, and ob' : OA, although not equal (110), will
soon be defined to be conjugate quaternions. Under the same conditions, we shall
write also the formula,
A aob' a 'cod.
CHAP. I.] CONDITIONS OF EQUALITY OF QUATERNIONS. 113
oppositely turned (comp. Fig. 36), as if one were the reflexion
of the other in a mirror ; or as if the one triangle were derived
(or generated) from the other, by a rotation of its plane through
two right angles. We may therefore write,
OB OD .,. ^
— = — , II A AOB ex COD.
OA OC
119. When the vectors are thus all drawn from one com-
mon origin o, i\iQ plane aob oi any two of them maybe called
the Plane of the Quaternion (or of the Quotient), ob : oa ; and
of course also the plane of the inverse (or reciprocal) quater-
nion (or of the inverse quotient), oa : ob. And any two qua-
ternio7is, which have a common plane (through o), may be said
to be Complanar* Quaternions, or complanar quotients, or
fractions ; but any two quaternions (or quotients), which have
different planes {intersecting therefore in a right line through
the origin), may be said, by contrast, to be Diplanar.
120. Any two quaternions, considered as geometric frac-
tions (101), can be reduced to a common denominator without
OB
change of the value^ of either of them, as follows. Let — and
— be the two given fractions, or quaternions ; and if they be
complanar (119), let oe be any linem their common plane; but
if they be diplanar (see again 1 19), then let oe be any assumed
part of the line of intersection of the two planes : so that, in
each case, the line oe is situated at once in the plane aob, and
also in the plane cod. We can then always conceive two other
lines, OF, OG, to be determined so as to satisfy the two condi-
tions of direct similitude (118),
A EOF a aob, Aeoggccod;
* It is, however, convenient to extend the use of this word, complanar^ so as to
inchide the case of quaternions represented by angles in parallel planes. Indeed, as
all rectors which have equal lengths, and similar directions, are equal (2), so the
quaternion, which is a quotient of two such vectors, ought not to be considered as
undergoing any change, when either vector is merely changed in pontion, by a trans-
port without rotation.
•)■ That is to say, the new or transformed quaternions will be respectively equal to
llie old or given ones.
Q
114 ELEMENTS OF QUATERNIONS. [bOOK II.
and therefore also the tioo equations between quotients (117,
118),
OF OB OG OD
OE Oa' OE OC *
and thus the required reduction is effected, oe being the com-
mon denominator sought, while of, og are the new or reduced
numerators. It may be added that if h be a new point in the
plane aob, such that A hoe a aob, we shall have also,
OE ob of
OH ~ OA ~ OE '
and therefore, by 106, 107,
OD OB OG+OF OD OB OG OD OB OG
OC ~ OA OE ' OC*OA~Of' OC OA ~ OH '
whatever tioo geometric quotients (complanar or diplanar) may
be represented by ob : oa and od : oc.
121. If now the two triangles aob, cod are not only com-
planar but directly similar (118), so that A aob oc cod, we shall
evidently have A eof a eog; so that we may write of = og
(or F = G, by 20), the two new lines of, og (or the two new
points F, g) in this case coinciding. The general construction
(120), for the reduction to a common denominator, gives there-
fore here only one new triangle^ eof, and one new quotient^
OF : ok, to which in this case each (comp. 105) of the two given
equal and complanar quotients, ob : oa and od : oc, is equal.
122. But if these two latter symbols (or th^ fractional
forms corresponding) denote two diplanar* quotients, then the
two new numerator lines, of and og, have different directions,
as being situated iii two different planes, drawn through the new
denominator-line oe, without having either the direction of that
line itself or the direction opposite thereto ; they are therefore
(by 2) unequal vectors, even if they should happen to be
equally long; whence it follows (by 104) that the two new
quotients, ^[id therefore also (by 105) that the two old or given
quotients, are unequal, as a consequence of their diplanarity,
* And therefore non scalar (108) ; for a scalar, considered as a quotient (17),
has no determined plane, but must be considered as complanar with every geometric
quotient; since it may be represented (or constructed) by the quotient of two simi-
larly or oppositely directed lines, in any proposed plane whatever.
CHAP. I.] PLANE OF A QUATERNION. 115
It results, then, from this analysis, that diplanar quotients of
vectors, and therefore that Diplanar Quaternions (119), are
always unequal; a new and comparatively technical process
thus confirming the conclusion, to which we had arrived by
general considerations, and in (what might be called) a popular
way before, and which we had sought to illustrate (comp. Fig.
34) by the consideration o^ angles on a desk: namely, that a
Quaternion, considered as the quotient oitwo mutually inclined
lines in space, involves generally a Plane, as an essential part
(comp. 110) of its constitution, and as necessary to the com-
pleteness of its conception.
123. We propose to use the mark
as a Sign of Complanarity, whether of lines or of quotients ;
thus we shall write the formula,
7lll«./3,
to express that the three vectors, a, /3, y, supposed to be (or to
be made) co-initial (18), are situated in one plane ; and the
analogous formula,
q\\\q, or? Ill ^,
y a
to express that the tioo quaternions, denoted here by q and q,
and therefore that i\iQ four vectors, a, /3, 7, S, are complanar
(119). And because we have just found (122) that diplanar
quotients are unequal, we see that one equation of quaternions
includes tivo complanar ities of vectors ; in such a manner that we
may write,
7|||a,/3. and 8|||«,|3. if - =^;
y a.
1 /. . OD OB , . . ... 7 77
the equation oj quotients, — = — , being nupossible, unless all
the four lines from o be in one common plane. We shall also
employ the notation
7 III?-
to express that the vector y is in (or parallel to) the plane of
the quaternion q»
116 ELEMENTS OF QUATERNIONS. [bOOK II.
124. With the same notation for complanarity, we may
write generally,
^a|||a,/3;
a and /3 being any two vectors, and x being ant/ scalar ; be-
cause, if a = OA and j3 = ob as before, then (by 15, 17) aid = oa',
where a' is some point on the indefinite right line through the
points o and a : so that the 'plane aob contains the line oa'.
For a similar reason, we have generally the following formula
oi complanarity of quotients,
whatever two scalars x and y may be ; a and /3 still denoting
any two vectors.
125. It is evident (comp. Fig. 35) that
if A AOB a COD, then A boa a doc, and A aoc a bod ;
whence it is easy to infer that for quaternions, as well as for
ordinary or algebraic quotients,
if - = -, then, inversely, -^=\, and alternately, ~ = t^\
ay p o a JD
it being permitted now to establish the converse of the last for-
mula of 1 18, or to say that
. „ ob od .
II — = — , then A aob a cod.
oa oc
Under the same condition, by combining inversion with alter-
nation, we have also this other equation, - = ^.
126. If the sides, oa, ob, of a triangle aob, or those sides
either way prolonged, be cut (as in
Fig. 37) by dmy parallel, a'b' or a"b",
to the base ab, we have evidently the
relations o^ direct similarity (118), yf^
A a'ob' a AOB, A a"ob" oc aob ;
whence (comp. Art. 13 and Fig. 12)
it follows that we may write, for qua-
ternions as in algebra, the general ^'
equation, or identity, ^'s 2^-
CHAP. I.] AXIS AND ANGLE OF A QUATERNION. 117
xa a '
where x is again ani/ scalar, and o, /3 are ant/ two vectors. It
is easy also to see, that for any quaternion q, and any scalar x,
we have the product (comp. 107),
xp f5 xQ (5 (5 a
p a a xr^a a x^a
so that, in the multiplication of a quaternion by a scalar (as in
the multiplication of a vector by a scalar, 15), the order of the
factors is indifferent.
Section 5 — On the Axis and Angle of a Quaternion ; and on
the Index of a Right Quotient, or Quaternion.
127. From what has been already said (HI, 112), we are
naturally led to define that the Axis, or more fully that the
positive axis, of any quaternion (or geometric quotient) ob ; oa,
is a right line perpendicular to the plane aob of that quaternion ;
and is such that the rotation round this axis, from the divisor-
line OA, to the dividend-line ob, is positive : or (as we shall
henceforth assume) directed towards the right-hand,* like the
motion of the hands of a watch.
128. To render still more definite this conception of the
axis of a quaternion, we may add, 1st, that the rotation, here
spoken of, is supposed (112) to be the simplest possible, and
therefore to be in the plane of the two lines (or of the quater-
nion), being also generally less than a semi-revolution in that
plane ; Ilnd, that the axis shall be usually supposed to be a
line ox drawn ^rom the assumed origin o ; and Ilird, that the
length of this line shall be supposed to be given, ov fixed, and
to be equal to some assumed unit of length : so that the term
X, of this axis ox, is situated (by its construction) on a given
spheric surface described about the origin o as centre, which
surface we may call the surface of the unit-sphere.
129. In this manner, for every given non-scalar quotient
* This is, of course, merely conventional, and the reader may (if lie pleases) sub-
stitute the /e/if-hand throughout.
118 ELEMENTS OF QUATERNIONS. [bOOK II.
(108), or for every given quaternion q which does not reduce
itself (or degenerate) to a mere positive or negative number, the
axis will be an entirely definite vector, which may be called an
UNIT-VECTOR, on account of its assumed length, and which we
shall denote'*, for the present, by the symbol Ax . q. Employ-
ing then the usual sign of perpendicularity, J_ , we may now
write, for any two vectors a, jS, the formula :
Ax.^±a; Ax2j_i3; or briefly, Ax.2± |^.
a a a [o.
130. The Angle of a quaternion, such as ob : oa, shall
simply be, with us, the angle aob between the tivo lines, of
which the quaternion is the quotient ; this angle being sup-
posed here to be one of the usual kind (such as are considered
by Euclid) : and therefore being acute, or right, or obtuse (but
not of any class distinct from these), when the quaternion is a
non-scalar (108). We shall denote this a?igle of a quaternion
q, by the symbol, L q ; and thus shall have, generally, the two
inequalities^ following :
Z5'>0; LqKiT',
where tt is used as a symbol for two right angles.
131. When the general quaternion, q^ degenerates into a
scalar, x, then the axis (like the planeX) becomes entirely in-
determinate in its direction ; and the angle takes, at the same
time, either zero or two right angles for its value, according as
the scalar \& positive ov negative. Denoting then, as above, any
such scalar by x, we have :
* At a later stage, reasons will be assigned for denoting this axis^ Ax .q, of a
quaternion g, by the less arbitrary (or more systematic) symbol, \^Yq ; but for the
present, the notation in the text may suffice.
f In some investigations respecting complanar quaternions, and powers or roots
of quaternions, it is convenient to consider negative angles., and angles greater than
two right angles; but these may then be called amplitudes ; and the word "An-
gle," like the word " Ilatio," may thus be restricted, at least for the present, to its
ordinary geometrical sense.
X Compare the Note to page 114. The angle, as well as the axis, becomes in-
determinate, when the quaternion reduces itself to zero ; unless we happen to know
a law, according to which the dividend-line tends to become null, in the transition
r ^. °
from - to -.
a a
CHAP. I.] CASE OF A RIGHT QUOTIENT, OR QUATERNION. 119
Ax . a; = an indeterminate unit-vector ;
Z :r = 0, if ar > ; z re = tt, if a? < 0.
132. Ot non-scalar quaternions, the most im- b
portant are those of which the angle is right, as in
the annexed Figure 38 ; and when we have thus,
OB , , TT
q= — , and ob_L_oa, or Lq = -,
OA 2i
>-
A
the quaternion q may then be said to be a Right Fig. 38.
Quotient ;* or sometimes, a Right Quaternion.
(1.) If then a = OA and p —op, where o and a are two given (ov fixed) points,
but P is a variable point, the equation
a 2
expresses that the locus of this point p is the plane through o, perpendicular to the
li?ie OA ; for it is equivalent to the formula of perpendicularity p j_ a (129).
(2.) More generally, if /3= ob, b being any third given point, the equation,
p (3
L- = L-
a a
expresses that the locus of p is one sheet of a cone of revolution, with o for vertex,
and OA for axis, and passing through the point b ; because it implies that the angles
AOB and AOP are equal in amount, but not necessarily in one common plane.
(3.) The equation (comp. 128, 129),
Ax.^ = Ax.^,
a a
expresses that the locus of the variable point p is the given plane aob ; or rather the
indefinite half-plane, which contains all the points p that are at once complanar
with the three given points o, A, b, and are also at the same side of the indefinite
right line OA, as the point B.
(4.) The system of the two equations,
a a a a ^
expresses that the point p is situated, either on thej^mVe right linele^^, or on that line
prolonged through ^A, but not through o; so that the locus of p may in this case be
said to be the indefinite half -line, or ray, which sets out from o in the direction of the
vector on or /3 ; and we may write p = .r/3, x> () (x being understood to be a sca-
lar)^ instead of the equations assigned above.
* Reasons will afterwards be assigned, for equating such a quotient, or quater-
nion, to a Vector; namely to the line which will presently (133) be called the Index
of the Bight Quotient.
120 ELEMENTS OF QUATERNIONS. [boOK II.
(5. ) This other system of two equations,
a a a a
expresses that the locus'of p is the opposite ray from o ;
or that p is situated on the prolongation of the revec-
tor BO (1) ; or that p=x(3, x<0; or that p,'''
p = x(3\ x>0, if /3' = ob' = - /3. Fig. 33, bis.
(Comp. Fig. 33, bis.)
(6.) Other notations, for representing these and other geometric loci, will be found
to be supplied, in great abundance, by the Calculus of Quaternions ; but it seemed
proper to point out these, at the present stage, as serving already to show that even
the two symbols of the present Section, Ax. and Z, when considered as Characteris-
tics of Operation on quotients of vectors, enable us to express, very simply and con-
cisely, several useful geometrical conceptions,
133. If a third line, oi, be drawn in the direction of the
axis ox of such a right quotient (and therefore perpendicular,
by 127, 129, to each of the two given rectangular lines, oa,
ob) ; and if the length of this new line oi bear to the length
of that axis ox (and therefore also, by 128, to the assumed
unit of length) the same ratio, which the length of the dividend-
line, OB, bears to the length of the divisor- line, oa; then the
line 01, thus determined, is said to be the Index of the Bight
Quotient. And it is evident, from this definition of such an
Index, combined with our general definition (117, 118) of
Equality between Quaternions, that tivo right quotients are
equal or unequal to each other, according as their two index-
lines (or indices) are equal or unequal vectors.
Section 6 On the Reciprocal, Conjugate, Opposite, and Norm
of a Quaternion; and on Null Quaternions.
134. The Keciprocal {ox ihQ Inverse, comp. 119) of a
quaternion, such as 5' = — , is that other quaternion,
which is formed by interchanging the divisor- line and the divi-
dend-line ; and in thus passing from any non-scalar quater-
nion to its reciprocal, it is evident that the angle (as lately
CHAP. I.] RECIPROCAL OF A QUATERNION. 121
defined in 130) remains unchanged^ but that the axis (127,
1 28) is reversed in direction : so that we may write gene-
rally,
pa p a
135. The product of two reciprocal quaternions is always
equal to positive unity ; and each is equal to the quotient of
unity divided hy the other; because we have, by 106, 107,
1:2 = ":^ « and |.2 = f=l.
a a a p p a a
It is therefore unnecessary to introduce any new or peculiar
notation, to express the mutual relation existing between a
quaternion and its reciprocal; since, if one be denoted by the
symbol q, the other may (in the present System, as in Alge-
bra) be denoted by the connected symbol,* 1 : 5^, or -. We
have thus the two general formulae (comp. 134) :
z-=z<7; Ax.- = -Ax.o'.
9 9
136. Without yet entering on the general i\\QOvy of multi-
plication and division of quaternions, beyond what has been
done in Art. 120, it may be here remarked that if any two
quaternions q and q be (as in 134) reciprocal to each other, so
that q'-q^l (by 135), and if 5'" be any third quaternion, then
(as in algebra), we have the general formula,
. , .1
q :q = q ,q =9'-\
because if (by 120) we reduce q and q' to a common denomina-
tor a, and denote the new numerators by j3 and 7, we shall have
(by the definitions in 106, 107),
„ 7^770 „ ,
137. When two complanar triangles aob, aob', with a com^
* The symbol 5-1, for the reciprocal of a quaternion q, is also permitted in the
present Calculus ; but we defer the use of it, until its legitimacy shall have been
established, in connexion with a general theory of powers of Quaternions.
R
122
ELEMENTS OF QUATERNIONS.
[book II.
^
mon side OA, are (as in Fig. 36) inversely similar (\18), so that
the formula A aob' a' aob holds good, then the iwo unequal
quotients,* — and — , are said to be Conjugate Quater-
^ OA OA
NiONS ; and if the ^rst of them be still denoted by q, then the
second, which is thus the conjugate of that^r^^, or of any other
quaternion which is equal thereto, is denoted by the new sym-
bol, K^ : in which the letter K may be said to be the Charac-
teristic of Conjugation. Thus, with the construction above
supposed (comp. again Fig. 36), we may write,
OB
OA
= <1
OA ^ OA
138. From this definition of conjugate quaternions, it follows,
1st, that if the equation
OB __ OB , - _ , - 17. f ,
— = K — holdffood, then the line ob maybe
OA OA '^ "^
called (118) the reflexion of the lineoB (and conversely, the latter line
the reflexion of the foi^mer), with respect to the line oa ; Ilnd, that, under
the same condition, the line oA (prolonged if necessary) bisects per-
pendicularly the line be', in some point a' (as represented in Fig. 36) ;
and Ilird, that any two conjugate quaternions (like any iv^o reciprocal
quaternions, comp. 1.34, 135) have equal angles, but opposite axes:
so that we may write, geujerally,
L^q=L q\ Ax . K^ = - Ax . q ;
and thereforef (by 135),
Z.K^ = Z.-; Ax.K<7 = Ax.-.
<1 9.
139. The reciprocal of a scalar, x, is simply another scalar,
-, or x'"^, having the same algebraic sign, and in all other re-
X
speCts related to x as in algebra. But the conjugate 'Kx, of a
scalar x, considered as a limit of a quaternion, is equal to that
scalar x itself; as may be seen by supposing the two equalhxxt
opposite angles, aob and aob', in Fig. 36, to tend together to
* Compare the Note to page 112.
t It will soon be seen that these two last equations (138) express, that the con-
jugate and the reciprocal, of any proposed quaternion 5, have always equal versors,
although they have in general unequal tensors.
CHAP. I.] CONJUGATE AND NULL QUATERNIONS. 123
zero, or to two right angles. We may therefore write, gene-
rally,
Kx = x, ifx be any scalar ;
and conversely*,
q = 21, scalar, if Kq = q;
because then (by 104) we must have ob=ob', bb'=0; and
therefore each of the two (now coincident) points, b, b', must
be situated somewhere on the indefinite right line oa.
140. In general, by the construction represented in the
same Figure, the sum (comp. 6) of the two numerators (or di^
vidend-Unes, ob and ob'), of the tivo conjugate fractions (or quo-
tients, or quaternions), q and Kq (137), is equal to the double
of the line oa' ; whence (by 106), the sum of those two conju-
gate quaternions themselves is.
Kg + g = g + Kg = • ;
^ ^ ^ ^ OA
this sum is therefore always scalar, hemg positive if the anple
Z ^ be acute, but negative if that angle be obtuse.
141. In the intermediate case, when the angle aob is right,
the interval oa' between the origin o and the line bb' vanishes ;
and the two lately mentioned numerators, ob, ob', become two
opposite vectors^ of which the sum is null (5). Now, in gene-
ral, it is natural, and will be found useful, or rather necessary
(for consistency \fii\i former definitions), to admit that a null
vector, divided by an actual vector, gives always a Null Qua-
ternion as the quotient; and to denote this null quotient by
the usual symbol for Zero, In fact, we have (by 106) the
equation,
? = fLZf = ^_5. 1.1 = 0;
a a a a
the zero in the numerator of the Z^^-hand fraction represent-
ing here a null line (or a null vector, 1,2); but the zero on the
riyht-hand side of the equation denoting a nidi quotient (or
quaternion). And thus we are entitled to infer that the sum,
* Somewhat later it will be seen that the equation Kq = q may also be written
as V^ = ; and that this last is another mode of expressing that the quaternion, j,
degenerates (131) into a scalar.
124 ELEMENTS OF QUATERNIONS. [bOOK II.
J^q +q, or q + K.q, of a right-angled quaternion, or right quo-
tient (132), and of its conjugate, is always equal to zero,
142. We have, therefore, the three following formulae,
whereof the second exhibits a continuity in the transition from
the j^r5^ to the third :
I. . . ^r + K^r > 0, if Z^ < I ;
11. . . ^ + K^ = 0, if z^=|;
III. . . ^ + K^ < 0, if Lq>~.
And because a quaternion, or geometric quotient, with an ac-
tual and^nite divisor-line (as here oa), cannot become equal to
zero unless its dividend-line vanishes, because (by 104) the
equation
L- = = - requires the equation j3 = 0,
a a
if a be any actual and finite vector, we may infer, conversely, that
the sum q + Kq cannot oanish, without the line oa' also vanish-
ing ; that is, without the lines ob, ob' becoming opposite vectors^
and therefore the quaternion q becoming a right quotient (132),
We are therefore entitled to establish the three following con-
verse formulae (which indeed result from the three former) :
T. , ,if q-V Kq > 0, then Aq <-;
II'. . . if 5' + Kq = 0, then Lq=-',
Iir. . . if 5- + Kq < 0, then Lq> -,
143. When two opposite vectors (1), as j3 and-/iJ, are both
divided by one common (and actual) vector, a, we shall say that
the two quotients, thus obtained are Opposite Quaternions;
so that the opposite of any quaternion q, or of any quotient
/3 : a, may be denoted as follows (comp. 4) :
-p 0-i3 /3 _
a a a a
CHAP. I.] OPPOSITE QUATERNIONS. 125
while the quaternion q itself m2ij, on the same plan, be denoted
(comp. 7) by the symbol + $', ov ■¥ q. The sum of any two
opposite quaternions is zero, and their quotient is negative
unity; so that we may write, as in algebra (comp. again 7),
(-^) + ^ = (+^) + (-^) = 0; (-^):^ = -i; -^ = (-1)^;
because, by 106 and 141,
a a a a a a p
The reciprocals of opposite quaternions are themselves oppo-
site ; or in symbols (comp. 126),
1 1 - a -a a
— = — , because —^ = -77" = - ts*
-q q -(5 (3 j5
Opposite quaternions have opposite axes, and supplementary
angles (comp. Fig. 33, bis) ; so that we may establish (comp.
132, (5.) ) the two following general formulse,
L{-q) = Tr- Lq\ Ax.(- 5-) = - Ax.^'.
144. We may also now write, in full consistency with the
recent formulae II. and 11'. of 142, the equation,
IF. , ,Kq = -q, if ^ ^ = I ;
and conversely* (comp. 138),
ir...ifK^ = -^, then zK^=z^ = ^.
In words, the conjugate of a right quotient, or of a right-angled
(or right) quaternion (132), is the right quotient opposite
thereto ; and conversely, if an actual quaternion (that is, one
which is not null) be opposite to its own conjugate, it must be
a right quotient.
(1.) If then we meet the equation,
Ke = _^, or ^ + K^ = 0,
a a a a
we shall know that p -i_ a ; and therefore (if a = oa, and p = op, as before), that the
* It will be seen at a later stage, that the equation Kq=-q, or g + Kg = 0,
may be transformed to this other equation, Sg = ; and that, under this last form, it
expresses that the scalar part of the quaternion q vanishes : or that this quaternion
is a right quotient (132).
126
ELEMENTS OF QUATERNIONS.
[book II.
locus of the point p is the plane through o, perpendicular to the line OA (as in 132,
(2,) On the other hand, the equation,
K
P-kP =
0,
expresses (by 139) that the quotient p : a is a scalar ; and therefore (by 131) that
its angle I (^p : a) is either or tt ; so that in this case, the locus of p is the indefi-
nite right line through the two points o and A.
145. As the opposite of the opposite, or the reciprocal of the reci-
procal^ so also the conjugate of the conjugate, of any quaternion, is that
quaternion itself; or in symbols,
-(-?) = + ?; l:(l:g) = ^; K% = ^=1^;
so that, by abstracting from the subject of the operation, we may write
briefly,
K2 = KK=1.
It is easy also to prove, that the conjugates of opposite quaternions are
themselves opposite quaternions ; and that the conjugates of reciprocals
are reciprocal: or in symbols, that
I...K(-^) = -K^, or K^+K(-5) = 0;
and
II...Ki=l:K^, or K7.Ki=l.
(1.) The equation K(- g) = — Kg is included (comp. 143) in this more general
formula, Yi(xq') = xKq, where x is any scalar; and this last equation (comp. 126)
may be proved, by simply conceiving that the two lines ob, ob', in Fig. 36, are
multiplied by any common scalar ; or that they are both cut by any parallel to the
line bb'.
(2.) To prove that conjugates of reci-
procals are reciprocal, or that Kg . K - = 1, /
we may conceive that, as in the annexed /
Figure 36, bis, while we have still the f
relation of inverse similitude, \
A aob' (xf AOB (118, 137),
as in the former Figure 36, a new point c
is determined, either on the line OA itself,
or on that line prolonged through A, so as /
to satisfy either of the two following con- ^ig. 36, bis.
nected conditions of direct similitude : ,^
A boc a aob' ; A b'oc oc aob ;
or simply, as a relation between the /our points o, a, b, c, the formula,
A boc a' aob.
•- , P
CHAP. I.] GEOMETRICAL EXAMPLES. 127
For then we shall have the transformations,
1 _ OA _ Ob' _ OB _ OA 1
q OB OC OC Ob' Kq
(3.) The two quotients, ob : OA, and ob : oc, that is to say, the quaternion q
itself, and the conjugate of its reciprocal, or* the reciprocal of its conjugate, have
the same angle, and the same axis ; we may therefore write, generally,
1 1 .
ZK-=Z.o; Ax.K- = Ax.g'.
(4.) Since oa : ob and OA : ob' have thus been proved (by sub-art. 2) to be
a pair of conjugate quotients, we can now infer this theorem, that any two geo-
metric fractions, — and — , which have a common numerator a, are conjugate qua-
ternions, if the denominator jS' of the second be the reflexion of the denominator (3 of
theirs*, with respect to that common numerator (comp. 138, I.) ; whereas it had
only been previously assumed, as a definition (137), that such conjugation exists,
uuder the same geometrical condition, between the two other (or inverse) fractions,
— and — ; the three vectors a, jS, (3' being supposed to be all co-initial (18).
a a
(5.) Conversely, if we meet, in any investigation, the formula
OA : ob' = K (oA : ob),
we shaU know that the point b' is the reflexion of the point b, with respect to the
line OA ; or that this line, OA, prolonged if necessary in either of two opposite direc-
tions, bisects at right angles the line bb', in some point a', as in either of the two
Figures 36 (comp. 138, II.).
(6.) Under the recent conditions of construction, it follows from the most ele-
mentary principles of geometry, that the circle, which passes through the three points
A, B, c, is touched at b, hij the right line OB ; and that this line is, in length, a 7nean
proportional between the lines oa, oc. Let then od be such a geometric mean, and
let it be set off from o in the common direction of the two last mentioned lines, so
that the point d falls between A and c ; also let the vectors oc, od be denoted by the
symbols, y, S', we shall then have expressions of the forms,
d = aa, y=a^a,
where a is some positive scalar, a > ; and the vector /3 of B will be connected
(comp. sub-art. 2) with this scalar a, and with the vector a, by the formula,
OB „ OA oc ,^ OB a^a ^ B
— = K— , or — = K— , or -— = K^.
oc OB OB OA (Ha
(7.) Conversely, if we still suppose that y = a^a, this last formula expresses the in-
verse similitude of triangles, A boc a' aob ; and it expresses nothing more: or in other
* It will be seen afterwards, that the common value of these two equal quater-
nions, K - and — , may be represented by either of the two new symbols, JJq : Tq,
q Kq
or 5 : Nj ; or in words, that it is equal to the versor divided by the tensor; and also
to the quaternion itself divided by the norm.
128 ELEMENTS OF QUATERNIONS. [bOOK II.
words, it is satisfied by the vector (3 of every point b, which gives that inverse simili-
tude. But for this purpose it is only requisite that the length of ob should be (as
above) a geometric mean between the lengths of OA, oc ; or that the two lines, ob,
OD (sub-art. 6), should be equally long: or finally, that b should be situated some-
where on the surface of a sphere, which is described so as to pass through the point D
(in Fig. 36, bis), and to have the origin o for its centre.
(8). If then we meet an equation of the form,
''^=Ki, or eK-P = a^
pa a a
in which a = OA, p — op, and a is a scalar, as before, we shall know that the locus
of the point p is a spheric surface, with its centre at the point O, and with the vector
aa for a radius ; and also that if we determine a point c by the equation oc = a'^a,
this spheric locus of P is a common orthogonal to all the circles apc, which can be
described, so as to pass through the two fixed points, A and c : because every radius
OP of the sphere is a tangent, at the variable point p, to the circle apc, exactly as
OB is to ABC in the recent Figure.
(9.) In the same Fig. 30, ^is, the sinular triangles show (by elementary princi-
ples) that the length of BC is to that of AB in the sub-duplicate ratio of oc to OA ; or
in the simple ratio of OD to OA ; or as the scalar a to 1. If then we meet, in any re-
search, the recent equation in p (sub-art. 8), we shall know that
length of (^p — a^a) = a x length of{p — a) ;
while the recent interpretation of the same equation gives this other relation of the
same kind :
length of p = a x length of a.
(10.) At a subsequent stage, it will be shown that the Calculus of Quaternions
supplies Rules of Transformation, by which we can pass from any one to any other
of these last equations respecting p, without (at the time) constructing any Figure,
or (immediately) appealing to Geometry : but it was thought useful to point out,
already, how much geometrical meanirig* is contained in so simple a fonnula, as that
of the last sub- art. 8.
(11.) The product of two conjugate quaternions is said to be their common
NoRMjt and is denoted thus:
qKq = Ng.
* A student of ancient geometry may recognise, in the two equations of sub-art.
9 a sort of translation, into the language of vectors, of a celebrated local theorem of
Apollonius of Perga, which has been preserved through a citation made by his early
commentator, Eutocius, and may be thus enunciated : Given any two points (as here
A and c) in a plane, and any ratio of inequality (as here that of 1 to a), it is possible
to construct a circle in the plane (as here the circle bdb'), such that the (lengths of
the) two right lines (as here ab and cb, or ap and cp), which are inflected from the
two given points to any common point (as B or p) of the circumference, shall be to
each other in the given ratio. (Avo doOkvTCJv arjutiwv, k. t. X. Page 11 of Halley's
Edition of Apollonius, Oxford, mdccx.)
f This name. Norm, and the corresponding characteristic, N, are here adopted,
as suggestions from the Theory of Numbers ; but, in the present work, they will not
CHAP. I.] RADIAL QUOTIENTS, RIGHT RADIALS. 129
It follows that NK^ = Ngr ; and that the norm of a quaternion is generally a positive
scalar: namely, the square of the quotient of the lengths of the two lines, of which
(as vectors) the quaternion itself is the quotient (112). In fact we have, by sub-art.
6, and by the definition of a norm^ the transformations :
OB Ob' _ OC OB' _ OC OB _ OO _ / OD Y ,
OA~ OA Ob' OA OB OA OA \OA. ]
a a a \length of a J
As a limit, we may say that the norm of a null quaternion is zero; or in symbols,
N0 = 0.
(12.) With this notation, the equation of the spheric locus (sub-art. 8), which
has the point o for its centre, and the vector aa for one of its radii, assumes the
shorter form :
N^ = a2; or N-^=l.
Section 7. — On Radial Quotients; and on the Square of a
Quaternion.
146. It was early seen (comp. Art. 2, and Fig. 4) that ani/
two radii, ab, ac, of any one circle, or sphere, are necessarily
unequal vectors ; because their directions differ. On the other
hand, when we are attending only to relative direction (110),
we may suppose that all the vectors compared are not merely
co-initial (18), but are also equally long; so that if their com-
mon length be taken for the unit, they are all radii, oa, ob, . .
of what we have called the Unit- Sphere ( 1 28), described round
the origin as centre; and may all be
said to be Unit- Vectors (129). And
then the quaternion, which is the
quotient of any one such vector divi-
ded bv any other, or generally the
.- \ i . 77 7 Fig. 39.
quotient oj any two equally long vec-
tors, may be called a Radial Quotient; or sometimes sim-
ply a Kadial. (Compare the annexed Figure 39.)
be often wanted, although it may occasionally be convenient to employ them. For
we shall soon introduce the conception, and the characteristic, of the Tensor, Tq, of
a quaternion, which is of greater geometrical utility than the Norm, but of which it
will be proved that this norm is simply the square,
qKq^-Sq^iTqy.
Compare the Note to sub -art, 3.
S
130
ELEMENTS OF QUATERNIONS.
[book II.
Fig. 40.
147. The two Unit' Scalar s^ namely, Positive and Nega-
tive Unity ^ may be considered as limiting cases of radial quo-
tients, corresponding to the two extreme values, and tt, of the
angle aob, or z §' (131). In the intermediate
case, when aob is a right angle, or Lq = ^,
as in Fig. 40, the resulting quotient, or qua-
ternion, may be called (comp. 132) a Right
Radial Quotient; or simply, a Right Ra-
dial. The consideration of such right radials
will be found to be of great importance, in the whole theory
and practice of Quaternions.
148. The most important general 'property of the quotients
last mentioned is the following : that the Square of every Right
Radial is equal to Negative Unity ; it being understood that
we write generally, as in algebra,
q.q=^qq = q\
and call this product of two equal quaternions the square of
each of them. For if, as in Fig. 41, we
describe a semicircle aba', with o for cen-
tre, and with ob for the bisecting radius,
then the two right quotients, ob : oa,
and oa' : ob, are equal (Qom^. 117); and
therefore their common square is (comp.
107) the product,
^obV oa' ob oa'
^OAy ob oa oa
where oa and ob may represent any
two equally long, but mutually rect- ^
angular lines. More generally, the
Square of every Right Quotient
(132) is equal to a Negative Scalar; namely, to the negative of
the square of the number, which represents the ratio of the
lengths* of the two rectangular lines compared ; or to zero
Fig. 41, bis.
* Hence, by 145, (11.), q^ = -Nq, if Iq-
CHAP.l.] GEOMETRICAL SQUARE ROOTS OF NEGATIVE UNITY. 131
minus the square of the wwm^^r which denotes (comp. 133) the
length of the Index of that Kight Quotient : as appears from
Fig. 41, his^ in which ob is only an ordinate, and not (as be-
fore) a radius, of the semicircle aba' ; for we have thus,
obV oa' (length of obV .r.
— = — = - , ^,; •;. , if OB ± OA.
OAy OA \lengtli oj oaJ
149. Thus everg Might Radial is, in the present System,
one of the Square Roots of Negative Unity ; and may there-
fore be said to be one of the Values of the Symbol \/ - 1 ; which
celebrated symbol has thus a certain degree of vagueness, or at
least 0^ in determination, oi meaning in this theory, on account
of which we shall not often employ it. For although it thus
admits o^ Si. perfectly clear and geometrically real Interpretation,
as denoting what has been above called a Right Radial Quo-
tient, yet the Plane of that Quotient is arbitrary; and therefore
the symbol itself must be considered to have (in the present
system) itidefinitely many values ; or in other words the Equa-
tion,
has (in the Calculus of Quaternions) mc^<?^w2Vc/y many Roots,*
which are all Geometrical Reals : besides any other roots, of
a purely symbolical character, which the same equation may be
conceived to possess, and which may be called Geometrical
Imaginaries.^ Conversely, if q be any real quaternion, which
* It will be subsequently shown, that if x, y, z be ani/ three scalars, of which
the sum of the squares is unity, so that
a:3 + y2+z2 = l;
and if i, j, k be any three right radials, in three mutually rectangular planes; then
the expression,
q = ix+jy + hz,
denotes another right radial, which satisfies {as such, and by symbolical laws to be
assigned) the equation q^ =— i; and is therefore one of the geometrically real values
of the symbol V— 1.
f Stich imaginaries will be found to offer themselves, in the treatment by Qua-
ternions (or rather by what will be called Biquaternions^, of ideal intersections, and
of ideal contacts, in geometry; but we confine our attention, for the present, to ^-ea-
metrical reals alone. Compare the Notes to page 90.
132 ELEMENTS OF QUATERNIONS. [bOOK II.
satisfies the equation q"^ ^-\, it must he a right radial; for if,
as in Fig. 42, we suppose that A aob cx boc,
we shall have
^/ObV OC OB oc
\oAy ~0B OA oa'
and this square of q cannot become equal to
negative unity ^ except by oc being = - oa,
or = oa' in Fig. 4 1 ; that is, by the line ob
being at right angles to the line oa, and
being at the same time equally long^ as in o
Fig. 40.
(1.) If then we meet the equation,
[it-
where a = OA, and p = op, as before, we shall know that the locits of the point p is
the circumference of a circle^ with o for its centre^ and with a radius which has the
same length as the line OA ; while the plane of the circle is perpendicular to that
given line. In other words, the locus of p is a great circle, on a sphere of which the
centre is the origin ; and the given point a, on the same spheric surface, is one of the
poles of that circle.
(2.) In general, the equation 5^ = — a^, where a is any (real) scalar^ requires
that the quaternion q (if real) should be some right quotient (132) ; the number a
denoting the letigth of the index (133), of that right quotient or quaternion (comp.
Art 148, and Fig. 41, 6is). But the plane of 5 is still entirely arbitrary ; and
therefore the equation
g2 = -a2,
like the equation 5'=— 1, which it includes, must be considered to have (in the
present system) indefinitely many geometrically real roots.
(3.) Hence the equation,
t;T
in which we may suppose that a > 0, expresses that the locus of the point p is a
(new) circular circumference, with the line oa for its axis,* and with a radius of
which the length = a x the length of OA.
150. It may be added that the index (133), and the axis (128),
of a right radial (147), are the same; and that its reciprocal (134), its
conjugate (137), and its opposite (143), are all equal to each other. Con-
versely, if the reciprocal of a given quaternion q be equal to the opposite
* It being understood, that the axis of a circle is a right line perpendicular to
the plane of that circle, and passing through its centre.
CHAP. I.] RADIAL QUOTIENTS CONSIDERED AS VERSORS. 133
of that quaternion, then q is a right radial; because its square^ q^,
is then equal (comp. 136) to the quaternion itself, divided hy its op-
posite; and therefore (by 143) to negative unity. But the conjugate
of every radial quotient is equal to the reciprocal of ^Aa^ quotient ;
because if, in Fig. 36, we conceive that the three lines da, ob, ob' are
equally long, or if, in Fig. 39, ^iQ prolong the arc ba, by an equal arc
ab', we have the equation,
^ ob' oa 1
Kg' = — = — = -.
OA ob §-
And conversely,*
if 'Kq- -, or if gK^= 1,
then the quaternion 5' is a radial quotient.
Section 8. — On the Versor of a Quaternion, or of a Vector ;
and on some General Fornfiulce of Transformation.
151. When a quaternion g' = /3 : a is thus a radial quotient
(146), or when the lengths of the two lines a and j3 are equal,
the effect of this quaternion q, considered as a Factor (103),
in the equation qa = jS, is simply the turning of the multipli-
cand-line a, in the plane ofq (119), and towards the hand de-
termined by the direction of the positive axis Ax . q (129),
through the angle denoted hj A q (130) ; so as to bring that
line a (or a revolving line which had coincided therewith) into
a neio direction : namely, into that of the product-line j3. And
with reference to this conceived operation of turning, we shall
now say that every Radial Quotient is a Versor.
152. A Versor has thus, in general, 2i plane, an axis, and
an angle ; namely, those of the Radial (146) to which it cor-
responds, or is equal : the onlg difference between them being
a difference in the points ofview'f from which they are respec-
tively regarded ; namely, the radial as the quotient, q, in the
* Hence, in the notation of norms (145, (11.) ), if l^q= 1, then 5 is a radial ;
and conversely, the norm of a radial quotient is always equal to positive unity.
f In a slightly metaphysical mode of expression it may be said, that the radial
quotient is the result of an analysis, wherein two radii of one sphere (or circle) are
compared, as regards their relative direction ; and that the equal versor is the instru-
ment of a corresponding synthesis, wherein owe radius is conceived to he generated, by
a certain rotation, from the other.
134 ELEMENTS OF QUATERNIONS. [bOOK II.
formula, q = j3: a ; and the versor as the (equal) ^c^or, q, in
the converse formula, f5 = q.a; where it is still supposed that
the two vectors, a and )3j are equally long,
153. A versor, like a radial {} 4^), cannot degenerate into b. scalar,
except by its angle acquiring one or other of the two limit-values^
and TT. In the first case, it becomes positive unity ; and in the second
case, it becomes negative unity : each of these two unit-scalars ( 1 47)
being here regarded as 2, factor (or coefficient^ comp. 12), which ope-
rates on a line, to preserve or to reverse its direction. In this view, we
may say that - 1 is an Inversor ; and that every Right Versor (or ver-
sor with an angle = - is a Semi-inversor :* because it half-inverts the
line on which it operates^ or turns it through half of two right angles
(comp. Fig. 41). For the'same reason, we are led to consider every
right versor (like every right radial, 149, from which indeed we have
just seen, in 152, that it differs only as factor differs from quotient),
as being one of the square-roots of negative unity : or as one of the va-
lues of the symbol y' - 1 .
154. In fact we may observe that the effect of a right versor, con-
sidered as operating on a line (in its own plane), is to turn that line,
towards a given hand, through a right angle. If then q be such a ver-
S07% and if qa = ft, we shall have also (comp. Fig. 41), qP = -a', so
that, if a be any line in the plane of a right versor q, we have the
equation,
q,qa = -a;
whence it is natural to write, under the same condition,
as in 149- On the other hand, no versor, which is not right-angled,
can he a value of y/ -\; or can satisfy the equation q^a --a, as Fig.
42 may serve to illustrate. For it is included in the meaning of this
last equation, as applied to the theory of versors, that a rotation
through 2 Lq, or through the double of the angle of q itself, is equi-
* This word, " semi -inversor," will not be often used ; but the introduction of it
here, in passing, seems adapted to throAV light on the view taken, in the present work,
of the symbol V — 1, when regarded as denoting a certain important class (149) of
Reals in Geometry. There are uses of that symbol, to denote Geometrical Imagi-
naries (comp. again Art. 149, and the Notes to page 90), considered as connected
with ideal intersections, and with ideal contacts ; but with such uses of V - 1 we
have, at present, nothing to do.
CHAP. I.] VERSOR OF A QUATERNION, OR OF A VECTOR. 135
valent to an inversion of direction; and therefore to a rotation through
two right angles.
155. In general, if a be any vector^ and if a be used as a
temporary* symbol for the number expressing its length; so
that a is here a positive scalar, which bears to positive unity,
or to the scalar + 1, the same ratio as that which the length of
the line a bears to the assumed unit of length (comp. 128);
then the quotient a : a denotes generally (comp. 16) a new vec-
tor, which has the same direction as the proposed vector a, but
has its length equal to that assumed unit : so that it is (comp.
146) the Unit- Vector in the direction of a. We shall denote this
unit-vector by the symbol, Ua ; and so shall write, generally,
Ua = -, if a = length of a ;
that is, more fully, if a be, as above supposed, the number
(commensurable or incommensurable, but positive) which re-
presents that length, with reference to some selected standard.
156. Suppose now that 5- = j3 : a is (as at first) 2^ general
quaternion, or the quotient of any two vectors, a and j3, whether
equal or unequal in length. Such a Quaternion will not (gene-
rally) be a Versor (or at least 7iot simply such), according to the
definition lately given ; because its effect, when operating as a
factor (103) on a, will not in general be simply to turn that
line (151) : but will (generally) alter the length,^ as well as the
direction. But if we reduce the two proposed vectors, a and j3,
to the two unit-vectors Ua and Uj3 (155), and ^ovmthQ quotient
of these, we shall then have taken account of relative direction
alone : and the result Avill therefore be a versor, in the sense
lately defined (151). We propose to call the quotient, or the
versor, thus obtained, the versor-element, or briefly, the Yer-
soR, of the Quaternion q ; and shall find it convenient to em-
* "We shall soon propose a general notation for representing the lengths of vectors,
according to which the symbol Ta will denote what has been above called a ; but^
are imwilling to introduce more than one new characteristic of operation, such as K,
or T, or U, &c., at one time.
f By what we shall soon call call an act of tension, which will lead us to the
consideration of the tensor of a quaternion.
136 ELEMENTS OF QUATERNIONS. [bOOK II.
ploy the same* Characteristic, U, to denote the operation of
taking the versor of a quaternion, as that employed above to
denote the operation (155) of reducing a vector to the unit of
length, without any change of its direction. On this plan, the
symbol \]q will denote the versor ofq ; and the foregoing de-
finitions will enable us to establish the General Formula :
a xJa
in which the two unit-vectors, Ua and Uj3, may be called, by
analogy, and for other reasons which will afterwards appear,
the versor s^ of the vectors, a and j3.
157. In thus passing from a given quaternion, q, to its ver-
sor, \Jq, we have only changed (in general) the lengths of the
two lines compared, namely, by reducing each to the assumed
unit of length (155, 156), without making any change in their
directions. Hence \h.Q plane (119), the axis (127, 128), and
the angle (130), of the quaternion, remain unaltered in this
passage ; so that we may establish the two following general
formulae :
L\]q = Lq; Ax . U<7 = Ax . q.
More generally we may write,
* For the moment, this double use of the characteristic U, to assist in denoting
both the unit-vector Ua derived from a given line a, and also the versor Uy derived
from a quaternion q, may be regarded as estabhshed here by arbitrary definition;
but as permitted, because the difference of the symbols, as here a and q, which serve
for the present to denote vectors and quaternions, considered as the subjects of these
two operations U, will prevent Bwch. double use of that characteristic from giving rise
to any confusion. But we shall further find that several important analogies are by
anticipation expressed, or at least suggested, when the proposed notation is employed.
Thus it will be found (comp. the Note to page 119), that every vector a may usefully
be equated to that right quotient, of which it is (133) the index ; and that then the
unit-vector "[] a may be, on the same plan, equated to that right radial (14.7), which
is (in the sense lately defined) the versor of that right quotient. We shall also find
ourselves led to regard every unit-vector as the axis of a quadrantal (or right) rota-
tion, in a plane perpendicular to that axis; which will supply another inducement,
to speak of every such vector as a versor. On the whole, it appears that there will
be no inconvenience, but rather a prospective advantage, in our already reading the
symbol Ua as ^^ versor of a ;" just as we may read the analogous symbol \Jq, as
^^ versor ofq."
t Compare the Note immediately preceding.
CHAP. I.] EQUAL AND RECIPROCAL VERSORS, REVERSORS. 337
Z ^' = Z $', and Ax . ^' = Ax . ^, if \Jq' = JJq ;
the versor of a quaternion depending solely on, but conversely
being sufficient to determine, the relative direction (156) of the
two lines, of which (as vectors) the quaternion itself is the quo-
tient (112); or the axis and angle of the rotation, in the plane
of those two lines, from the divisor to the dividend (128) ; so
that any two quaternions, which have equal versors, must also
have equal angles, and equal (or coincident) axes, as is ex-
pressed by the last written formula. Conversely, from this
dependence of the versor \]q on relative direction'^ alone, it
follows that any two quaternions, of which the angles and the
axes are equal, have also equal versors; or in symbols, that
\]q'==\]q, if Lq'=-Lq, and Ax.^-' = Ax.^'.
For example, we saw (in 138) that the conjugate and the re-
ciprocal of any quaternion have thus their angles and their
axes the same ; it follows, therefore, that the versor of the
conjugate is always equal to the versor of the reciprocal; so
that we are permitted to establish the following general for-
mula,!
q
158. Again, because
it follows that the versor of the reciprocal of any quaternion is,
at the same time, the reciprocal of the versor ; so that we may
write,
* The unit-vector Ucr, which we have recently proposed (156) to call the versor
of the vector a, depends in like manner on the direction of that vector alone; which
exclusive reference^ in each of these two cases, to Direction, may serve as an addi-
tional motive for employing, as we have lately done, one common name^ Veesor,
and one common characteristic, U, to assist in describing or denoting both the Unit-
Vector Ua itself and the Quotient of two such Unit- Vectors, \Jq = U/3 : Ua ; all
danger of confusion being sufficiently guarded against (comp. the Note to Art. 156),
by the difference of the two symbols, a and q, employed to denote the vector and the
quaternion, which are respectively the subjects of the two operations U ; while those
two operations agree in this essential point, that each serves to eliminate the quan-
titative element, of absolute or relative length.
t Compare the Note to Art. 138.
T
138 ELEMENTS OF QUATERNIONS. [bOOK II.
Ui = ^; or JJq.JJ-=l.
q Vq ^ q
Hence, by the recent result (157), we have also, generally,
UK^ = i-; or, U^.UK^ = l.
Also, because the versor XJq is always a radial quotient (151,
152), it is (by 150) the conjugate of its own reciprocal ; and
therefore at the same time (comp. 145), the reciprocal of its
own conjugate; so that the /?roc?wc^ of tic o conjugate versor s,
or what we have called (145, (!!•)) their common Norm, is
always equal io positive unity ; or in symbols (comp. 150),
NU^ = U^.KU^=1.
For the same reason, the conjugate of the versor of any qua-
ternion is equal to the reciprocal of that versor^ or (by what
has just been seen) to the versor of the reciprocal of that qua-
ternion; and therefore also (by 157), to the versor of the con-
jugate; so that we may write generally, as a summary of re-
cent results, the formula :
each of these four symbols denoting a new versor, which has
the same plane, and the same angle, as the old or given versor
\]q, but has an opposite axis, or an opposite direction of rota-
tion-, so that, with respect to that given Versor, it may na-
turally be called a Ke versor.
159. As regards the versor itself, whether of a vector or of
a quaternion, the definition (155) of Ua gives,
UiCo = + Ua, or = - Ua, according as rc> or < ;
because (by 15) the scalar coefjicient x preserves, in the first
ease, but reverses, in the second case, the direction of the vec-
tor a; whence also, by the definition (156) of U^', we have
generally (comp. 126, 143),
U^r^' = + U^', or = - \5q, according as a;> or < 0.
The versor of a scalar, regarded as the limit of a quaternion
(131, 139), is equal to positive or negative unity (comp. 147,
CHAP. I.] GENERAL TRANSFORMATIONS OF A VERSOR. 139
153), according as the scalar itself is positive or negative ; or
in symbols,
Ua; = + 1, or = - 1 , according as a; > or < ;
the plane and axis of each of these two unit scalar s (147), con-
sidered as versors (153), being (as we have already seen) inde-
terminate. The versor of a null quaternion (141) must be re-
garded as wholly arbitrary^ unless we happen to know a Z«i^7,*
according to which the quaternion tends to zero^ before actually
reaching that limit ; in which latter case, the plane^ the axis,
and the angle of the versor] UO may all become determined, as
limits deduced from that law. The versor of a right quotient
(132), or of a right-angled quaternion (141), is always a right
radial (147)) or a right versor (153) ; and therefore is, as such,
one of the square roots of negative unity (149), or one of the
values of the symbol V - 1 5 while (by 150) the axis and the
index of such a versor coincide ; and in like manner its recipro-
cal, its conjugate, and its opposite are all equal to each other.
160. It is evident that if a proposed quaternion q be already
a versor (151), in the sense of being a radial (146), the ope-
ration o^ taking its versor (156) produces no change; and in
like manner that, if a given vector a be already an unit-vector,
it remains the same vector, when it is divided (155) by its own
length; that is, in this case, by the number one. For example,
we have assumed (128, 129), that the axis o^ every quaternion
is an unit-vector ; we may therefore write, generally, in the no-
tation of 155, the equation,
U(Ax./7) = Ax .§'.
A second operation U leaves thus the result of i)iQ first opera-
tion U unchanged, whether the subject of such successive ope-
rations be a line, or a quaternion; we have therefore the two
* Compare the Note to Art. 131.
t When the zero in this symbol^ UO, is considered as denoting a null vector (2),
the symbol itself denotes generally, by the foregoing principles, an indeterminate
unit-vector; although the direction of this unit- vector may, in certain questions, he-
come determined, as a limit resulting from a law.
140 ELEMENTS OF QUATERNIONS. [boOK II.
following general formulae, differing only in the symbols of
that subject :
UUa=Ua; JJUq = Uq;
whence, by abstracting (comp. 145) from the subject of the
operation, we may write, briefly and symbolically,
16 1. Hence, with the help of 145, 158, 159, we easily deduce
the following (among other) transformations of the versor of a qua-
ternion :
K^- q q U^ ^ ^ ^
TJq = Vxq, if £c> ; = - TJxq, if x<0.
We may also write, generally,
the parentheses being here unnecessary, because (as will soon be more
fully seen) the symbol JJq^ denotes one common versor ^ whether we
interpret it as denoting the square of the versor^ or as the versor of
the square^ of q. The present Calculus will be found to abound in
General Transformations of this sort; which all (or nearly all), like
the foregoing, depend ultimately on very simple geometrical concep-
tions ; but which, notwithstanding (or rather, perhaps, on account
of) this extreme simplicity of their origin, are often useful, as elements
of a new kind o^ Symbolical Language in Geometry: and generally,
as instruments of expression, in all those mathematical or physical
researches to which the Calculus of Quaternions can be applied. It
is, however, by no means necessary that a student of the subject,
at the present stage, should make himself familiar with all the
recent transformations of Ug-; although it may be well that he
should satisfy himself of their correctness, in doing which the fol-
lowing remarks will perhaps be found to assist.
(I.) To give &. geometrical illustration^ ■\vhich may also serve asa/3/oo/J of the
recent equation,
CHAP. I.] GEOMETRICAL ILLUSTRATIONS.
141
we may employ Fig. 36, bis ; in which, by 145, (2.), we have
^ Kq OA OB' Ob' \ODJ \ OA j
(2.) As regards the equation, Jj(q^) = (JJqY^ we have only to conceive that the
three lines oa, ob, oc, of Fig. 42, are cut (as in Fig. 42, bis) in
three new points, a', b', c', by an unit-circle (or by a circle with
a radius equal to the unit of length), which is described about
their common origin o as centre, and in their common plane ; for
then if these three lines be called a, ft, y , the three new lines oa',
ob', oc' are (by 155) the three unit-vectors denoted by the sym-
bols, Ua, U/3, Uy; and we have the transformations (comp. 148,
149),
u(,^)=u.(^y=uz=hi=2£;=(-;y=(u,)^.
^^ ^ \a j a Ua OA V<^^ /
(3.) As regards other recent transformations (161), although
we have seen (135) that it is not necessary to invent any new or
peculiar symbol, to represent the reciprocal of a quaternion, yet
if, for the sake of present convenience, and as a merely temporary
notation^ we write
Bq=\,
O A' A
Fig. 42, bis.
employing thus, for a moment, the letter R as a characteristic of reciprocation, or
of the operation of taking the reciproeal, we shall then have the symbolical equations
(comp. 145, 158) :
R2 = K2 = 1; RK = KR; RU = UR = KU=UK;
but we have also (by 160), U2= U ; whence it easily follows that
U = RUR = RKU = RUK = KUR = KRU = KUK
= URK = UKR = UKUR = UKRU = (UK)2 = &c.
(4.) The equation
U
^ -. US or simply, Up = U|3,
a a
expresses that the locus of the point p is the indefinite right line, or ray (comp. 132,
(4.)), which is drawn /rom o in the direction of ob,* but not in the opposite direc-
tion ; because it is equivalent to
^^-
^f-
or (0 = x(3, x>0.
(5.) On the other hand the equation,
or Up=-U/3,
a a
expresses (comp. 132, (5.)) that the locus of p is the opposite ray from o ; or that
it is the indefinite prolongation of the revector bo ; because it may be transformed to
* In 132, (4.), p. 119, OA and a ought to have been ob and b.
142 ELEMENTS OF QUATERNIONS. [bOOK II.
U ^ = - 1 ; or Z ^ = TT ; or p = a:/3, cc < 0.
(6.) If a, j3, y denote (as in sub-art. 2) the three lines oa, ob, oc of Fig. 42 (or
of Fig. 42, his), so that (by 149) we have the equation - = f ^ J , then this other
equation, l^pV^^y^
expresses generally that the locus of p is the system of the two last loci ; or that it is
the whole indefinite right line, both ways prolonged, through the two points o and B
(comp. 144, (2.)).
(7.) But if it happen that the line y, or oc, like oa' in Fig. 41 (or in Fig. 41,
6is), has the direction opposite to that of a, or of oa, so that the last equation takes
the particular form,
I n\2
1,
("fl-
then U- must be (by 154) a right versor ; and reciprocally, every right versor, with
a
a plane containing a, will be (by 153) a value satisfying the equation. In this case,
therefore, the locus of the point p is (as in 132, (1.), or in 144, (1.)) the plane
through o, perpendicular to the line OA ; and the recent equation itself, if supposed
to be satified by a real* vector p, may be put under either of these two earlier but
equivalent /orm* •
Section 9. — On Vector- Arcs, and Vector- Angles, considered
as Representatives of Versors of Quaternions ; and on the
Multiplication and Division of any one such Versor hy
another.
162. Since every unit-vector oa (129), drawn from the
origin o, terminates in some point a on the surface of what we
have called the unit-sphere (128), that term a (1) may be
considered as a Representative Point, of which the position on
that surface determines, and may be said to represent, the
direction of the line oa in space ; or of that line multiplied
(12, 17) by any positive scalar. And then the Quaternion
which is the quotient (112) of any two such unit- vectors, and
which is in one view a Radial (146), and in another view a
Versor (151), may be said to have the arc of a great circle,
AB, upon the unit sphere, which connects the terms of the two
* Compare 149, (2.) ; also the second Note to the same Article ; and the Notes
to page 90.
CHAP. I.] REPRESENTATIVE AND VECTOR ARCS. 143
vectors, for its Representative Arc, We may also call this
arc a Vector Arc, on account of its having a definite direc-
tion (comp. Art. 1), such as is indicated (for example) by a
curved arrow in Fig. 39 ; and as being thus contrasted with
its own opposite, or with what may be called by analogy the
Revector Arc ba (comp. again 1) : this latter arc represent-
ing, on the present plan, at once the reciprocal (134), and the
conjugate (137), of the former versor; because it represents
the corresponding Reversor (158).
163. This mode of representation, of versors of quaternions
by vector arcs, would obviously be very imperfect, unless
equals were to be represented by equals. We shall therefore
define, as it is otherwise natural to do, that a vector arc, ab,
upon the unit sphere, is equal to every other vector arc cd
which can be derived from it, by simply causing (or conceiv-
ing) it to slide* in its own great circle, icithout any change of
length, or reversal of direction. In fact, the two isosceles and
plane triangles aob, cod, which have the origin o for their
common vector, and rest upon the chords of these two arcs as
bases, are thus complanar, similar, and similarly turned ; so
that (by 117, 118) we may here write,
OB CD
A AOB OC COD, — = — ;
OA OC
the condition of the equality of the quotients (that is, here, of
the versors), represented by the two arcs, being thus satisfied.
We shall sometimes denote this sort of equality of two vector
arcs, AB and cd, by the formula,
o AB = /> CD;
and then it is clear (comp. 125, and the ear-
lier Art. 3) that we shall also have, by what
may be called inversion and alternation, j
these two other formulas of arcual equality, oi'-:~_ -'a
Fig. 35, his,
'^BA=/>DC; ^ AC = ^ BD. ^ '
(Compare the annexed Figure 35, his^
* Some aid to the conception may here be derived from the inspection of Fig
34 ; in which two equal angles are supposed to be traced on the suiface of one com-
144 ELEMENTS OF QUATERNIONS. [bOOK II.
164. Conversely, unequal versors ought to be represented
(on the present plan) by unequal vector arcs; and accordingly,
we purpose to regard any two such arcs, as being, for the pre-
sent purpose, unequal (comp. 2), even when they agree in
quantity i or contain the same number of degrees^ provided that
they differ in direction : which may happen in either of two
principal ways, as follows. For, 1st, they may be opposite
arcs oi one great circle; as, for example, a vector arc ab, and
the corresponding revector arc ba ; and so may represent (162)
a versor, OB : oa, and the corresponding reversor, oa : ob, re-
spectively. Or, Ilnd, the two arcs may belong to different
great circles^ like ab and bc in Fig. 43 ; in which latter case,
they represent two radial quotients
( 1 4 6) m different planes ; or (comp .
119) two diplanar versors, ob : oa,
and 00 : OB ; but it has been shown
generally (122), that diplanar qua-
ternions are always unequal: we
consider therefore, here again the
arcs, AB and bc, themselves^ to be
(as has been said) unequal vectors.
165. In this manner, then, we may be led (comp. 122) to
regard the conception of a plane, or o^ the position of a great
circle on the unit sphere, as entering, essentially, in general,*
into the conception of a vector-arc^ considered as the representa-
tive of a versor (162). But even without expressly referring
to versors, we may see that if, in Fig. 43, we suppose that b
is the middle point of an arc aa' of a great circle, so that in a
recent notation (163) we may establish the arcual equation,
we ought then (comp. 105) not to write also,
'^ AB = '^ bc;
mon desk. Or the four lines OA, ob, oc, od, of Fig. 35, may now be conceived to
be equally long; or to be cut by a circle with o for centre, as in the modification of
that Figure, which is given in Article 163, a little lower down.
* We say, in general ; for it will soon be seen that there is a sense in which all
great semicircles, considered as vector arcs, may be said to be eqval to each other.
CHAP. I.] ARCUAL EQUATIONS, CO-ARCUALITY. 145
because the two co-initial arcs, ba and bc, which terminate
differently, must be considered (comp. 2) to be, as vector-arcs y
unequal. On the other hand, if we should refuse to admit (as
in 163) that any two complanar arcs, i^ equally long, and simi-
larly (not oppositely) directed, like ab and cd in the recent
Fig. 35, bis, are equal vectors^ we could not usefully speak of
equality between vector-arcs as existing under any circum-
stances. We are then thus led again to include, generally, the
conception of a plane, or of one great circle as distinguished
from another, as an element in the conception of a Vector-Arc,
And hence an equation between two such arcs must in general
be conceived to include two relations of co-arcuality. For
example, the equation ^ ab = '^ cd, of Art. 163, includes gene-
rally, as apart of its signification, the assertion (comp. 123)
that ihe four points a, b, c, d belong to ouq common great cir-
cle of the unit-sphere ; or that each of the two points, c and d,
is co-arcual Avith the two other points, a and b.
166. There is, however, a remarkable case o1 exception, vav^YiioSx
two vector arcs may be said to be equal, although situated in diffe-
rent planes: namely, when they are both great semicircles. In fact,
upon the present plan, every great semicircle, aa', considered as a
vector arc, represents an inversor (153); or it represents negative
unity (oa' : oa = - a : a = - 1), considered as one limit of a versor;
but we have seen (159) that such a versor has in general an indeter-
minate plane. Accordingly, whereas the initial and final points, or
(comp. 1) the origin a and the term b, of a vector arc ab, are in ge-
neral sufficient to determine the plane of that arc, considered as the
shortest or the most direct path (comp. 112, 128) from the one point
to the other on the sphere; in the particular case when one of the
two given points is diametrically opposite to the other, as a' to A,
the direction of this path becomes, on the contrary, indeterminate.
If then we only attend to the effect produced, in the way of change
of position of a point, by a conceived vection (or motion') upon the
sphere^ we are permitted to say that all great semicircles are equal
vector arcs; each serving simply, in the present view, to transport a
point from one position to the opposite; and thereby to reverse (like
the factor - 1, of which it is here the representative) the direction of
the radius which is drawn to that point of the unit sphere.
u
146 ELEMENTS OF QUATERNIONS. [bOOK II.
(1.) The equation,
r» aa' = o bb',
in which it is here supposed that a' is opposite to a, and b' to b, satisfies evidently
the general conditions of co-arcuality (165); because the /owr points aba'b' are all
on one great circle. It is evident that the same arcual equation admits (as in 163)
of inversion and alternation ; so that
r> a'a = r\ b'b, and n ab = «^ a'b'.
(2.) We may also say (comp. 2) that all null arcs are equal, as producing no
effect on the position of a point upon the sphere ; and thus may write generally,
n AA = n BB = 0,
with the alternate equation, or identity, r> ab = o ab.
(3.) Every such null vector arc AA is a representative, on the present plan, of the
other unit scalar, nsimely positive uniti/, considered as another limit of aversor (153) ;
and its plane is again indeterminate (159), unless some law be given, according to
which the arcual vection may be conceived to begin, from a given point A, to an in-
definitely near point B upon the sphere.
' 167. The principal use of Vector Arcs, in the present
theory, is to assist in representing^ and (so to speak) in con-
structing, by means of a Spherical Triangle, the Multiplica-
tion and Division of any two Diplanar Versors (comp. 119,
164). In fact, any two such versors of quaternions (156),
considered as radial quotients (152), can easily be reduced (by
the general process of Art. 120) to the forms,
$- = j3 :a = OB : OA, g'' = 7 ; j3= oc : ob,
where a, b, c are corners of such a triangle on the unit sphere;
and then (by 107), the former quotient multiplied by the lat-
ter will give for product ;
q\q = ^ : a = OC'. OA.
If then (on the plan of Art. 1) any two successive arcs, as ab
and Bc in Fig. 43, be called (in relation to each other) vector
a^d provector ; while that third arc ac, which is drawn from
the initial point of the first to the final point of the second,
shall be called (on the same plan) the transvector : we may now
say that in the multiplication of any one versor (of a quater-
nion) by any other, if the multiplicand* q he represented (162)
by a vector-arc ab, and if the multiplier q be in like manner
* Here, as in 107, and elsewhere, we write the symbol of the multiplier towards
the left-hand, and that of the multiplicand towards the right.
CHAP. I.] CONSTRUCTION OF MULTIPLICATION OF VERSORS. 147
represented by sl provector-arc bc, which mode of representa-
tion is always possible, by what has been already shown, then
the product q'. q, or q'q, is represented, at the same time, by
the transvector-arc ac corresponding.
168. One of the most remarkable consequences of this con-
struction of the multiplication ofversors is the following : that
the value of the product of two diplanar versors (164) depends
upon the order of tJie factors ; or that q'q and qq are unequal,
unless q be complanar (119) with q. For let aa' and cc' be
any two arcs of great circles, in different planes, bisecting each
other in the point b, as Fig. 43 is designed to suggest; so
that we have the two arcual equations (163),
'^ AB = ^ ba', and '^ bc = '^ c'b ; /^
then one or other of the two following alternatives will hold
good. Either, 1st, the two mutually bisecting arcs will both
be semicircles, in which case the two new arcs, ac and cV, will
indeed both belong to one great circle, namely to that of which
B is a pole, but will have opposite directions therein ; because,
in this case, a' and c' will be diametrically opposite to a and c,
and therefore (by 166, (1.) ) the equation
'^ AC = '^ a'c',
but not the equation
'^ AC = '^ c'a',
will be satisfied. Or, Ilnd, the arcs aa' and cc', which are
supposed to bisect each other in b, will not both be semicircles,
even if one of them happen to be such ; and in this case, the
arcs AC, c'a' will belong to two distinct great circles, so that they
will be diplanar, and therefore unequal, when considered as
vectors. (Compare the 1st and Ilnd cases of Art. 164.) In
each case, therefore, ac and c'a' are unequal vector arcs; but the
former has been seen (167) to represent the product qq-, and
the latter represents, in like manner, the other product, qc[, of
the same two versors taken in the opposite order, because it is
the new transvector arc, when c b (= bc) is treated as the new
vector arc, and ba' (= ab) as the new provector arc, as is indi-
cated by the curved arrows in Fig. 43. The two products,
148 ELEMENTS OF QUATERNIONS. [bOOK II.
(iq and qq^ are therefore themselves unequal, as above asserted,
under the supposed condition of diplanarity,
169. On the other hand, when the two factors, q and q\
are complanar versors^ it is easy to prove, in several different
ways, that their products, q'q and qq\ are equals as in algebra.
Thus we may conceive that the arc cc', in Fig. 43, is made to
turn round its middle point b, until the spherical angle cba'
vanishes; and then the two new transvector-arcs^ ac and cV,
will evidently become not only complanar but equal, in the
sense of Art. 163, as being still equally long, and being now
similarly directed. Or, in Fig. 35, bis, of the last cited Arti-
cle, we may conceive a point e, bisecting the arc bc, and there-
fore also the arc ad, which is commedial therewith (comp.
Art. 2, and the second Figure 3 of that Article) ; and then,,
if we represent the one versor q by either of the two equal
arcs, AE, ED, we may at the same time represent the other
versor q' by either of the two other equal arcs, eg, be ; so that
the one product, q'q, will be represented by the arc ac, and
the other product, qq', by the equal arc bd. Or, without re-
ference to vector arcs, we may suppose that the two factors
are,
q =(3: a = ob: oa, q' <= y : a== oc : OA,
oa, ob, oc being any three complanar and equally long right
lines (see again Fig. 35, bis) ; for thus we have only to deter-
mine a fourth line, S or od, of the same length, and in the same
plane, which shall satisfy the equation S:y=(5:a (117), and
therefore also (by 125) the alternate equation, 01/3 = 7: a;
and it will then immediately follow* (by 107), that
S 13 S S y
q ^q = ^-- = - = -'- = q'q' .
p a a y a
We may therefore infer, for any two versor s of quaternions, q
and q, the two following reciprocal relations :
* It is evident that, in this last process of reasoning, we make no use of the sup-
posed equality of lengths of the four lines compared ; so that we might prove, in ex-
actly the same way, that q'q = qq' if 9' | !| 9 (123), without assuming that these two
complanar factors, or quaternions, q and q', are versors.
CHAP. I.] MULTIPLICATION OF RIGHT VERSORS. 149
l...gq = qq\ if q' \\\ q (123) ;
II. , . i£ q'q =qq\ then 5^' ||| 9- (168) ;
convertibility of factors (as regards -[heiv places in thQ product)
being thus at once a consequence and ?i proof of complanarity.
170. In the 1st case of Art. 168, th^ factors q and q' are both
right versors (153) ; and because we have seen that then their two
products^ q'q and qcf ^ are versors represented by equally long but op-
positely directed arcs of one great circle, as in the 1st case of 164, it
follows (comp. 162), that these two products are at once reciprocal
(134), and conjugate (137), to each other; or that they are related
as versor and reversor (158). We may therefore write, generally,
I. . . qq'=Kq'q, and II. .. m' = -fZ^
if q anc] q be any two right versors; because the multiplication of
any two such versors, in two opposite orders, may always be repre-
sented or constructed by a Figure such as that lately numbered
43, in which the bisecting arcs aa' and cc' are semicircles. The Ilnd
formula may also be thus written (comp. 135, 154):
III. .. if 2'^ = -!, and q'^=-\., then qq-qq=-^^\
and under this form it evidently agrees with ordinary algebra, be-
cause it expresses that, under the supposed conditions.,
q'q.qq'^iKf',
but it will be found that this last equation is not an identity, in the
general theory of quaternions.
171. If the two bisecting semicircles cross each other at riyht
angles., the conjugate products are represented by two quadrants.,
oppositely turned, of one great circle. It follows that if two right
versors, in two mutually rectangular planes, he multiplied together in two
opposite orders, the two resultiiig products will he two opposite right
versors, in a third plane, rectangular to the two former; or in symbols,
that
if ^^ = - 1, 2''^ = - 1, and Ax. q x Ax. q,
then
{qqy=-{qqy^-\, q'q = -qq\
and
Ax. q'q 4- Ax. q. Ax. ^q a. Ax. q\
In this case, therefore, we have what would be in algebra a paradox,
namely the equation,
{q'qy^-q'^.q\
150 ELEMENTS OF QUATERNIONS. [bOOK II.
if q and q' be any two right versors, in two rectangular planes ; but we
see that this result is not more paradoxical, in appearance, than the
equation
qq=-qq,
which exists, under the same conditions. And when we come to ex-
amine what, in the last analysis, may be said to be the meaning of this
last equation, we find it to be simply this : that any two quadrantal or
right rotations^ in planes perpendicular to each other^ compound them-
selves into a third right rotation^ as their resultant^ in a plane perpendi-
cular to each of them: and that this third ox resultant rotation has
one or other of two opposite directions^ according to the order in which
the two component rotations are taken, so that one shall be successive
to the other.
172. We propose to return, in the next Section, to the
consideration of such a System of Right Versors, as that which
we have here briefly touched upon : but desire at present to
remark (comp. 167) that a spherical triangle ABcmay serve to
construct, by means of represeritative arcs (162), not only the
multiplicatioiL, but also the division, of any one of two diplanar
versors (or radial quotients) by the other. In fact, we have
only to conceive (comp. Fig. 43) that the vector arc ab repre-
sents a given divisor, say q, or j3 : a, and that the transvector
arc AC (167) represents a given dividend, suppose q", or y : a;
for then the provector arc bc (comp. again 167) will represent,
on the same plan, the quotient of these two versors, namely
q" : 5', or 7 : j3 (106), or the versor lately called q ; since we
have generally, by 106, 107, 120, for quaternions, as in alge-
bra, the two identities :
(q":q)^q = q"; qq-q^q'-
173. It is however to be observed that, for reasons already as-
signed, we must not employ, for diplanar versors^ such an equation
as q. {q": q) = q" ', because we have found (168) that, for such ver-
sors, the ordinary algebraic identity, qq' — (^q, ceases to he true. In
fact by 169, we may now establish the two converse formulse:
I. . . q{q"'.q)=q'\ if q"\\\q {123);
11. . . iiq\q"'.q) = q", then ^'Mil q.
Accordingly, in Fig. 43, if q, q', q" be still represented by the
arcs AB, BC, AC, the product q {q"'.q), or qq', is not represented by
CHAP. I.] REPRESENTATIVE AND VECTOR ANGLES. 151
AC, but by the different arc c'a^ (168), which as a vector arc has been
seen to be unequal thereto: although it is true that these two last
arcs, AC and c'a', are always equally long^ and therefore subtend
equal angles at the centre o of the unit sphere; so that we may write,
generally, for any two versors (or indeed for any two quaternions)*
q and q" , the formula,
Lq{q":q) = Lq''.
174. Another mode of Representation of Versors, or rather two
such new modes, although intimately connected with each other,
may be briefly noticed here.
1st. We may consider the angle aob, at the centre o of the unit-
sphere, when conceived to have not only a definite quantity, but also
a determined^Zawe (110), and a given direction therein (as indicated
by one of the curved arrows in Fig. 39, or by the arrow in Fig. 33),
as being what may be called by analogy a Vector- Angle ; and may
say that it represents, or that it is the Representative Angle of, the
Versor ob : oa, where oa, ob are radii of the unit- sphere.
Ilnd. Or we may replace this rectilinear angle aob at the centre,
by the equal Spherical Angle ac^b, at what may be *
called the Positive Pole of the representative arc ab ;
so that c^A and c^b are quadrants; and the rotation,
at this pole c', from the first of these two quadrants
to the second (as seen from a point outside the
sphere), has the direction which has been selected
(111, 127) for the positive one, as indicated in the
annexed Figure 44: and then we may consider this
spherical angle as a new Angular Representative of the same versor q,
or ob : OA, as before.
175. Conceive now that after employing ?k first spherical trian-
gle ABC, to construct (as in 167) the multiplication of any one given
versor q, by any other given versor q' , we form a second or polar
triangle, of which the corners a', b', c' shall be respectively (in the
sense just stated) tha positive poles of the three successive sides, bc,
CA, AB, of the former triangle ; and that then we pass to a third tri-
angle A^B^'c', as part of the same lune ^'^" with the second, by tak-
ing for -&" the point diametrically opposite to b' ; so that ^" shall be
* It will soon be seen that several of the formulae of the present Section, respect-
ing the multiplication and division of versors^ considered as radial quotients (151),
require little or no modification, in the passage to the corresponding operations on
quaternions, considered as general quotients of vectors (112).
152
ELEMENTS OF QUATERNIONS.
[book II.
Fi-r. 45.
the negative pole of the arc CA, or the positive pole of what was lately-
called (167) the transvector-arc Ac: also let
c" be, in like manner, the point opposite
to c' on the unit sphere. Then we may not
only write (comp. 129),
Ax. 5' = oc^ Ax. §'' = oa', Ax. q'q = ob'\
but shall also have the equations,
lq = b'^c^a^ Z g' = c' a'b^', Z q'q = C^'b^'a' ;
these three spherical angles^ namely the ivm
base-angles at c' and a\ and the external
vertical angle at b''', of the new or third
triangle a''b''V/, will therefore represent^ re-
spectively, on the plan of 174, II., the mul-
tiplicand^ q, the multiplier^ q\ and the pro-
duct, q'q. (Compare the annexed Figure 45.)
176. Without expressly referring to the former triangle abc,
we can connect this last construction of multiplication of versors (175)
with the general formula (107), as follows.
Let a and y3 be now conceived to be tw^o unit-tangents'^ to the
sphere at c', perpendicular respectively to
the two arcs c^b^' and c'a^ and drawn to-
wards the same sides of those arcs as the
points a' and b' respectively; and let two
other unit-tangents, equal to these, and
denoted by the same letters, be drawn (as
in the annexed Figure 45, his) at the points
B^' and a', so as to be normal there to the
same arcs c'b'^ and c'a', and to fall towards
the same sides of them as before. Let also
two other unit-tangents, equal to each b'/
other, and each denoted by 7, be drawn at
the two last points b" and a', so as to be both perpendicular to the
arc a^b^^ and to fall towards the same side of it as the point c'. Then
(comp. 174,11.) the two quotients, (3 : a and 7 : /3, will be equal to the
two versors, q and q, which were lately represented (in Fig. 45) by the
* By an unit tangent is here meant simply an unit line (or unit vector, 129) so
drawn as to be tangential to the unit-sphere^ and to have its origin, or its initial
point (1), on the surface of that sphere, and not (as we have usually supposed) at
the centre thereof.
Fig. 46, bis.
CHAP.T.] DEPENDENCE OF PRODUCT ON ORDER OF FACTORS. 153
two base angles, at c' and a', of the spherical triangle a'b'^'c'; the pro-
duct, q'q, of these two versors, is therefore (by 107) equal to the third
quotient, 7 : « ; and consequently it is represented, as before, by the
external vertical angle c"b"a.' of the same triangle, which is evidently
equal in quantity to the angle of this third quotient, and has the same
axis ob", and the same direction of rotation, as the arrows in Fig. 45,
his, may assist to show.
177. In each of the two last Figures, the internal vertical angle
at B^' is thus equal to the Supplement, tt - l q'q, of the angle of the
product; and it is important to observe that the corresponding ro-
tation at the vertex b", from the side b^'a' to the side b'^c', or (as we
may briefly express it) from the point k' to the point o', is, positive; a
result which is easily seen to be a general one, by the reasoning of
the foregoing Article.* We may then infer, generally, that when
the multiplication of any two versors is constructed hya spherical trian-
gle, of which the two ba^e angles represent (as in the two last Articles)
t\iQ factors, while the external vertical angle represents t\\Q product,
then the rotation round the axis (ob'O of that product q'q, from the
axis (oa') of the multiplier q', to the axis (oc^) of the multiplicand q, is
positive: whence it follows that the rotation round the axis Ax. q'
of the multiplier, from the axis Ax. q of the multiplicand, to the
axis Ax. q'q of the product, is also positive. Or, to express the
same thing more fully, since the only rotations hitherto considered
have hQQU plane ones (as in 128, &c.), we may say that if the two
latter axes be projected on a plane perpendicular to the former, so as
still to have a common origin o, then the rotation round Ax. q\
from the projection of Ax. q to the projection of Ax. q'q, will be di-
rected (with our conventions) towards the right hand.
178. We have therefore thus a new mode of geometrically
exhibiting the inequality of the two products^ q'q and ^5-', o{two
diplanar versors (168), when taken a3 factors in two different
orders. For this purpose, let
Ax. 5-= OP, Ax.5'=0Q, Ax.qq = OR;
and prolong to some point s the arc PR of a great circle on the
unit sphere. Then, for the spherical triangle pqr, by prin-
* If a person be supposed to stand on the sphere at b", and to look towards the
arc a'c', it would appear to him to have a right-handed direction, which is the one
here adopted as positive (127).
154
ELEMENTS OF QUATERNIONS.
[book II.
ciples lately established, we shall have (comp. 175) the follow-
ing values of the two internal base angles at p and q, and of
the external vertical angle at ii :
RPQ = Lq\ PQR = L q ', SRQ = L q'q ;
and the rotation at q, from the side qp to the side qr will be
right-handed. Let fall an arcual perpendi- g
cular, RT, from the vertex r on the base pq,
and prolong this perpendicular to r', in such
a manner as to have
/^ RT = '^ tr' ;
also prolong pr' to some point s'. We shall
then have a new triangle pqr', which will
be a sort of reflexion (comp. 138) of the old
one with respect to their common base pq ;
and this new triangle will serve to construct
the new product^ qq. For the rotation at p
Fig. 46.
from PQ to pr' will be right-handed, as it ought to be ; and
we shall have the equations,
qpr' = Z^; r'qp = Z5''; qr's' = Z^'^''; on' = Ax.qq \
so that the new external and spherical angle, qr's', will repre-
sent the new versor, qq\ as the old angle srq represented the
old versor, q'q, obtained from a different order of the factors.
And although, no doubt, these two angles, at r and r', are
always equal in quantity, so that we may establish (comp. 1 73)
the general formida,
Lqq^Lqq,
yet as vector angles (174), and therefore as representatives of
versors, they must be considered to be unequal: because they
have different planes, namely, the tangent planes to the sphere
at the two vertices r and r'; or the two planes respectively
parallel to these, which are drawn through the centre o.
179. Division of Versors (comp. 172) can be constructed by
means oi Representative Angles (174), as well as by representative arcs
(162). Thus to divide q" by q, or rather to represent such division
geometrically, on a plan entirely similar to that last employed for
CHAP. I.] CONICAL ROTATION OF AXIS OF VERSOR. 155
multiplication, we have only to determine the two points P and r,
in Fig. 46, by the two conditions,
and then to find a third point q by the two angular equations,
RPQ =Lq, QRP ^tr- L q",
the rotation round p from PR towards pq being positive ; after which
we shall have,
A-K. {q" \ q)=OQ,\ L{q" '.q) = VQ,Vi.
(1.) Instead of conceiving, in Fig. 46, that the dotted line rtk', which connects
the vertices of the two triangles, with pq for their common base (178), is an arc of
a great circle, perpendicularly bisected by that base, we may imagine it to be an arc
of a small circle^ described with the point p for its positive pole (comp. 174, II.).
And then we may say that the passage (comp. 17 B) from the versor q'% or qq, to
the unequal versor q(q" : 9), or qq\ is geometrically performed by a Conical Rota-
tion of the Axis Ax. 5", round the axis Ax. 7, through an angle ~2 Lq^ without
any (jjuantitative) change of the angle Lq"\ so that we have, as before, the general
formula (comp. again 173),
L q (9" : 9) = ^ 9".
(2.) Or if we prefer to employ the construction of multiplication and division by
representative arcs, which Fig. 43 was designed to illustrate, and conceive that a
new point c" is determined in that Figure by the condition ^ a'c" = "^ c'a', we may
then say that in the passage from the versor q'\ which is represented by ac, to the
versor q (5" : 5), represented by c'a' or by a'c", the representative arc of q" is made
to move, without change of length, so as to preserve a constant inclination* to the
representative arc AB ofq, while zYs initial point describes the double of that arc A^,
in passing from a to a'.
(3.) It maybe seen, by these few Examples, that if, even independently of some
new characteristics of operation, such as K and U, new combinations of old symbols,
such as q (q" : q), occur in the present Calculus, which are not wanted in Algebra,
they admit for the most part of geometrical interpretations, of an easy and interest -
ing kind ; and in fact represent conceptions, which cannot well be dispensed with,
and which it is useful to be able to express, with so much simplicity and conciseness.
(Compare the remarks in Art. 161 ; and the sub-articles to 182, 145.)
180. In connexion with the construction indicated by the
two Figures 45, it may be here remarked, that if abc be any
spherical triangle, and if a', b', c' be (as in 175) the positive
poles of its three successive sides, bc, ca, ab, then the rotation
(comp. 177, 179) round a' from b' to c', or that round b' from
* In a manner analogous to the motion of the equator on the ecliptic, by luni-
ioldv precession, in astronomy.
156 ELEMENTS OF QUATERNIONS. [bOOK II.
c' to A, &c., IS positive. The easiest way, perhaps, of seeing
the truth of this assertion, is to conceive that if the rotation
round a from b to c be not already positive, we make it such,
by passing to the diametrically opposite triangle on the sphere,
which will not change the poles a', b', c'. Assuming then that
these poles are thus the near ones to the corresponding corners
of the given triangle, we arrive without any difficulty at the
conclusion stated above : which has been virtually employed
in our construction of multiplication (and division) of versors,
by means of Representative Angles (1 75, 176) ; and which may
be otherwise justified (as before), by the consideration of the
unit-tangents of Fig. 45, Ms.
(1.) Let then a, j3, y be any three given unit vectors, such that the rotation
round the first, from the second to the third, is positive (in the sense of Art. 177);
and let a', /3', y' be three otlier unit vectors, derived from these by the equations,
a'=Ax. (y:/3), /3'= Ax. (a : y), y' = Ax.(/3 : a) ;
then the rotation round a, from /3' to y', will be positive also; and we shall have
the converse formulae,
a = Ax.(y':/5'), ^ = Ax. (a: y'), y = Ax . (/3' : a')-
(2.) If the rotation round a from /3 to y were given to be negative, a', /3', y'
being still deduced from those three vectors by the same three equations as before,
then the signs of a, /3, y would all require to be changed, in the three last (or reci-
procal) formulae ; but the rotaticm round a', from /3' to y', would still be positive.
(3.) Before closing this Section, it may be briefly noticed, that it is sometimes
convenient, from motives of analogy (comp. Art. 5), to speak of the Transvector-
Arc (167), which has been seen to represent a. product of two versors. as being the
Arcual, Sum of the two successive vector-arcs, which represent (on the same plan)
the factors ; Provector being still said to be added to Fector : but the Order of such
Addition of Diplanar Arcs being not now indifferent (168), as the corresponding
order had been early found (in 7) to be, when the vectors to be added were right
lines.
(4.) We may also speak occasionally, by an extension of the same analogy, of
the External Vertical Angle of a spherical triangle, as being the Spherical Sum of
the two Base Angles of that triangle, taken in a suitable order of summation (comp.
Fig. 46); the Angle which represents (174) the Multiplier being then said to be
added (as a sort of Angular Provector) to that other Vector-Angle which represents
the Multiplicand; whilst what is here called the sum of these two angles (and is,
with respect to them, a species of Transvector- Angle) represents, as has been proved,
the Product.
(5.) This conception of angular transvaction becomes perhaps a little more clear,
when (on the plan of 174, I.) we assume the centre o as the common vertex of three
angles aob, boc, aoc, situated generally in three different planes. For then we may
CHAP.
'•]
SYSTEM OF THREE RIGHT VERSORS.
157
conceive a revolving radius to be either carried by two successive angular motions,
frnm OA to OB, and thence to oc ; or to be transported immediately, by one such
motion, from the Ji?-st to the third position.
(6.) Finally, as regards the construction indicated by Fig. 45, bis, in which tan-
gents instead of radii were employed, it may be well to remark distinctly here, that
a'b"c', in that Figure, may be ani/ given spherical triangle, for which the rotation
round b" from a' to c' is positive (177); and that then, if the two factors, q and q',
be defined to be the two versors, of which the internal angles at c' and a' are (in the
sense of 174, II.) the representatives, the reasonings of Art. 176 will prove, without
necessarily referring, even in thought, to any other triangle (such as abc), that the
external angle at b" is (in the same sense) the representative of the product, q'q, as
before.
Section 10. — On a System of Three Right Versors^ in Three
Rectangular Planes ; and on the Laws of the Symbols,
181. Suppose that oi, oj, ok are any three given and co-
initial but rectangular unit-lines, the rotation round the first
from the second to the third being positive ; and let oi', oj,
ok' be the three unit- vectors respectively opposite to these, so
that
Ol' = -OI, Oj'-=-OJ, ok'=-ok.
Let the three new symbols i,j, k denote a system (comp, 172)
of three right versors, in three mutually rectangular planes,
with the three given lines for their respective axes; so that
Ax.i=oj, Ax.j=oj, Ax.k-OK,
and
i = ok:oj, J=oi:ok, A=oj:oi,
as Figure 47 may serve to illustrate.
We shall then have these other expres-
sions for the same three versors :
i = o y : OK = ok'
^ = OK : 01 =01
k = oi : OJ = OJ
Fig. 47.
OJ = OJ : OK ;
ok'= ok: oi' ;
oi' = 01 : oj' ;
while the three respectively opposite versors may be thus ex-
pressed :
- z = oj : OK = OK : OJ = oj : ok
= ok: oj
-j = OK : 01 = oi' : OK = ok' : oi'
= 01 : ok
- A = 01 : OJ = OJ : 01 = oi' : oj'
= OJ : oi'.
/<
/
158 ELEMENTS OF QUATERNIONS. [bOOK II.
And from the comparison of these different expressions seve-
ral important symbolical consequences follow, which it will be
worth while to enunciate separately here, although some of
them are virtually included in the results of former Sections.
182. In \hQjirst place, since
i^ = (oj' : ok) . (OK : oj) = oj' : OJ, &c.,
we deduce (comp. 148) the following equal values for the
squares of the new symbols :
L..z^ = -1; / = -l; k' = -l;
as might indeed have been at once inferred (154), from the
circumstance that the three radial quotients (146), denoted here
by hj, ^3 are all right versors (181).
In the second place, since
ij= (oj:ok') .(ok':oi) = oj : oi, &c.,
we have the following values for the products of the same three
symbols, or versors, when taken iioo hy two, and in a certain
order of succession (comp. 168, 171) :
II. . . ij= k] jk = i; ki =j.
But in the third place (comp. again 171), since
j .i= (ox : ok) . (ok : oj) = oi : oj, &c.,
we have these other and contrasted formulae, for the binary
products of the same three right versors, when taken as fac-
tors with an opposite order :
III. . .ji=-k; kj = -i; ik = -j.
Hence, while the square of each of the three right versors, de-
noted by these three new symbols, ijk, is equal (154) to nega-^
tive unity, the product of any two of them is
equal either to the third itself, or to the oppo-
site (171) of that third versor, according as
the multiplier precedes ov follows the multipli-
cand, in the cyclical succession,
h i, k, i, j\ . . .
which the annexed Figure 47, bis, may give some help towards
remembering.
CHAP. I.] LAWS OF THE SYMBOLS, I, J, K. 159
(1.) To connect such multiplications ofi,j, k with the theory of representative
arcs (162), and of representative angles (174), we may regard any one of the four
quadrantal arcs, JK, Kj', j'k', k'j, in Fig, 47, or any one of the four spherical right
angles, jik, kij', j'ik', k'ij, which those arcs subtend at their common pole i, as re-
presenting the versor i ; and similarly for j and k, with the introduction of the point
i' opposite to I, which is to be conceived as being at the back of the Figure.
(2.) The squaring of i, or the equation i^ = - 1, comes thus to be geometrically
constructed by tbe doubling (comp. Arts. 148, 154, and Figs. 41, 42) of an arc, or of
an angle. Thus, we may conceive the quadrant kj' to be added to the equal arc jk,
their sum being the great semicircle jj', which (by 166) represents an inversor (153),
or negative unity considered as a, factor. Or we may add the right angle kij' to the
equal angle JIK, and so obtain a., rotation through two right angles at the jooZe i, or
at the centre o; which rotation is equivalent (comp. 154, 174) to an inversion of
direction, or to a passage from the radius OJ, to the opposite radius oj'.
(3.) The midtiplication ofj hy i, or the equation ij = k, may in like manner
be arcually constructed, by the addition of k'j, as a provector-arc (167), to ik' as
a vector-arc (162), giving ij, which is a representative of ^, as the transvector-arc,
or arcual-sum (180, (3.) ). Or the same multiplication may be angularly con-
structed, with the help of the spherical triangle ijk ; in which the base-angles at I
and J represent respectively the multiplier, i, and the multiplicand, j, the rotation
round l from j to k being positive : while their spherical sum (180, (4.)), or the ex-
ternal vertical angle at K (comp. 175, 176), represents the same product, k, as
before.
(4.) The contrasted multiplication of i hy j, or of J into* i, may in like manner
be constructed, or geometrically represented, either by the addition of the arc ki, as
a new provector, to the arc jk as a new vector, which new process gives Ji (instead
of ij) as the new transvector ; or with the aid of the new triangle ijk' (comp. Figs.
46, 47), in which the rotation round i from j to the new vertex k' is negative, so
that the angle at i represents now the multiplicand, and the resulting angle at the
new' pole k' represents the new and opposite product, ji = - k.
183. Since we have thus ji = - ij (as we had q'q = - qq in
171), we see that the laws of combination of the neio symbols^
i,j, k, are not in all j^espects the same as the corresponding
laws in algebra; since the Commutative Property of Multipli-
cation, or the convertibility (169) of the places o^ \k\Q factors
without change of value of the product, does not here hold
good: which arises (168) from the circumstance, that the
factors to be combined are here diplanar versor s (181). It is
therefore important to observCj that there is a respect in which
* A multiplicand is said to be multiplied hy the multiplier ; -while, on the other
hand, a multiplier is said to be multiplied into the multiplicand : a distinction of this
sort between the tivo factors being necessary, as we have seen, for quaternions,
although it is not needed for algebra.
160 ELEMENTS OF QUATERNIONS. [bOOK II.
the laws of i, j, k agree with usual and algebraic laws : namely,
in the Associative Property of Multiplication ; or in the pro-
perty that the new symbols always obey the associative for-
mula (comp. 9),
whichever of them may be substituted for z, for ic, and for X ;
in virtue of which equality of values we may omit the pointy in
any such symbol of a ternary product (whether of equal or of
unequal factors), and write it simply as lk\. In particular
we have thus,
i.jk = i,i = i'^ = ~ \ ; ij .k = k.k = k^ = - \ ;
or briefly,
ijk = -l.
We may, therefore, by 182, establish the following important
Formula :
p=f^k^ = ijk = -l ; (A)
to which we shall occasionally refer, as to " Formula A," and
which we shall find to contain (virtually) all the laws of the
symbols ijk, and therefore to be a sufficient symbolical basis
for the whole Calculus of Quaternions i* because it will be
shown that every quaternion can he reduced to the Quadrino-
mial Form,
q=w + ix +jy + kz,
where w, x, y, z compose a system of four scalar s, while 2, j, k
are the same three right versors as above.
(1.) A direct proof of the equation, ijk = — 1, may be derived from the definitions
of the symbols in Art. 181. In fact, we have only to remember that those defini-
tions were seen to give,
* This formula (A) was accordingly made the basis of that Calculus in the first
communication on the subject, by the present writer, to the Royal Irish Academy in
1843 ; and the letters, i, 7', k, continued to be, for some time, the only peculiar sym-
bols of the calculus in question. But it was gradually found to be useful to incor-
porate with these a few other notations (such as K and U, &c.), for representing
Operations on Quaternions. It was also thought to be instructive to establish the
principles of that Calculus, on a more geometrical (or less exclusively symbolicaT)
foundation than at first ; which was accordingly afterwards done, in the volume en-
titled : Lectures on Quaternions (Dublin, 1853) ; and is again attempted in the pre-
sent work, although with many differences in the adopted plan of exposition, and in
the applications brought forward, or suppressed.
CHAP. I.] LAWS OF THE SYMBOLS, I, J, K. 161
t = oj' : OK, j = ok: oi', ^ = oi' : oj ;
and to observe that, by the general fornmla of multiplication (107), whatever four
lines may be denoted by a, /3, y, d, we have always,
y' (3 a y a a ^ a y/^a'
or briefly, as in algebra,
y /3 a a
the point being thus omitted without danger of confusion : so that
ijk = oj' : OJ = — 1, as before.
Similarly, we have these two other ternary products :
jki = (ok' : ot) (oi : oj') (oj' : ok) = ok' : ok = — 1 ;
kij = (oi' : oj) (oj : ok') (ok' : oi) = oi' : oi = - 1 .
(2,) On the other hand,
kji— (oj : oi) (oi : ok) (ok : oj) =oj : oj = + 1 ;
and in like manner,
ikj— + 1, and jik = + 1.
(3.) The equations in 182 give also these other ternary products, in which th»
law of association of factors is Still obeyed :
i . ij = ik = -j = iy = a .j\ iij =-j]
i .ji = i.-k = -ik=j = ki = ij . ?, iji = +j ;
i.jj=i.-l=-i = kj = ij.j, VJ = -i;
with others deducible from these, by mere cyclical permutation of the letters, on the
plan illustrated by Fig. 47, Ms.
(4.) In general, if the Associative Law of Combination exist for ani/ three
symbols whatever of a given class, and for a giiwn mode of combination, as for addi-
tion of lines in Art. 9, or for multiplication of ijk in the present Article, the same law
exists for any fotir (or more) symbols of the same class, and combinations of the same
kind. For example, if each of the four letters t, /c, X, /* denote some one of the three
symbols i, j, k (but not necessarily the same one), we have the formula,
I . (cX/i = t . K . XjLl = tK . X/i = tK . X . /f = ifcX . n = tjcX/A.
(5.) Hence, any multiple (or complex') product of the symbols ijk, in any manner
repeated, but taken in one given order, may be interpreted, with one definite result,
by any mode of association, or of reduction to partial factors, which can be performed
without commutation, or change of place of the given factors. For example, the
symbol ijkkji may be interpreted in either of the two following (among other) ways :
ij.kk.ji = ij.-ji = i.~j'Ki = ii = - 1; ijk.kji=-l. 1=-1.
184. The formula (a) of 183 includes obviously the three equa-
tions (I.) of 182. To show that it includes also the six other
equations, (H.)? (m*)' ^^ ^^^ ^^^^ cited Article, we may observe that
it gives, with the help of the associative principle of multiplication
(which may be suggested to the memory by the absence of the jpomi
in the symbol tjk),
Y
162 ELEMENTS OF QUATERNIONS. [bOOK II.
ij =i-ij .kk = -ijk.k = + k', jk = -i. ijk = + i\
ji =j .jk]=fk = -k', ik = i.ij = i V = -j ;
k' = V • ; = «;'^ = - « ; ^« = - ^^!/ = -P = + J-
And then it is easy to prove, without any reference to geometry/, if the
foregoing laws of the symbols be admitted, that we have also,
jki = kij = - 1 , kji =jik = ikj = + 1 ,
as otherwise and geometrically shown in recent sub-articles. It may-
be added that the mere inspection of the formula (a) is sufficient to
show that the tkree'^ square roots of negative unity j denoted in it by
/, j, k, cannot be subject to all the ordinary rules of algebra : because
that formula gives, at sight,
Pfk'=(-iy^-l=-{ijky;
the non-commutative character (183) , of the multiplication of such roots
among themselves, being thus put in evidence.
Section 11. — On the Tensor of a Vector, or of a Quaternion ;
and on the Product or Quotient of any two Quaternions.
185. Having now sufficiently availed ourselves, in the two
last Sections, of the conceptions (alluded to, so early as in the
First Article of these Elements) of a vector-arc (162), and of
a vector-angle (174), in illustration^ of the laws o^ multiplica-
tion and division of vers or s of quaternions ; we propose to re-
turn to that use of the word. Vector, with which alone the
First Book, and the first eight Sections of this First Chapter
of the Second Book, have been concerned : and shall therefore
henceforth mean again, exclusively^ by that word " vector," a
Directed Right Line (as in 1). And because we have already
considered and expressed the Direction of any such line, by
* It is evident that — i, —j, — k are also, on the same principles, values of the
symbol V — 1; because they also are right versors (153); or because (- gy=q^.
More generally (comp. a Note to page 131), if a:, y, z be any three scalers which sa-
tisfy the condition x^ i- 1/"^ + z"^ = 1, it will be proved, at a later stage, that
(ix-\-jt/ + kzy = -l.
f One of the chief uses of such vectors, in connexion with those laws, has been
to illustrate the non-com>Hutative property (1G8) of multiplication of versors, by ex-
hibiting a corresponding property of what has been called, by analogy to the earlier
operation of the same kind on linear vectors (5), the addition of arcs and angles on
a sphere. Compare 180, (3.), (4.).
CHAP. I.J TENSOR OF A VECTOR. 163
introducing the conception and notation (155) of the Unit-
Vector, Ua, which has the same direction with the line a, and
which we have proposed (156) to call the Versor of that Vec-
tor, a ; we now propose to consider and express the Length of
the same line a, by introducing the new name Tensor, and the
new symbol,* Ta; which latter symbol we shall read, as the
Tensor of the Vector a : and shall define it to be, or to denote,
the Number (comp. again 155) which represents the Length of
that line a, by expressing the Ratio which that length bears
to some assumed standard, or Unit (128).
186. To connect more closely these two conceptions, of
the versor and the tensor of a vector, we may remember that
when we employed (in 155) the letter a as a temporary sym-
bol for the number which thus expresses the length of the line
a, we had the equation, Ua = a : «, as one form of the defini-
tion of the unit-vector denoted by Ua. We might therefore
have written also these two other forms of equation (comp. 15,
16),
a-a.\Ja, a = a'.JJa,
to express the dependence of the vector, a, and of the scalar,
a, on each other, and on what has been called (156) the versor,
Ua. For example, with the construction of Fig. 42, bis (comp.
161, (2.) ), we may write the three equations,
« = OA : oa', b = OB : ob', c = oc : oc',
if «, b, c be thus the three positive scalars, which denote the
lengths of the three lines, oa, ob, oc ; and these three scalars
may then be considered as factors, or as coefficients (12), by
which the three unit-vectors Ua, Uj3, Uy, or oa', ob', oc' (in
the cited Figure), are to be respectively multiplied (15), in
order to change them into the three other vectors a, j3, y, or
OA, OB, oc, by altering their lengths, without any change in
their directions. But such an exclusive Operation, on the
Length (or on the extension) of aline, may be said to be an Act
of Tension ;t as an operation on direction alone may be called
(comp. 151) an act of version. We have then thus a motive
* Compare the Note to Art. 155.
t Compare the Note to Art. 156, in page 135.
164 ELEMENTS OF QUATERNIONS. [bOOK II.
for the introduction of the name, Tensor, as applied to the
positive number which (as above) represents the length of a
line. And when the notation Ta (instead of a) is employed
for such a tensor, we see that we may write generally, for any
vector a, the equations (compare again 15, 16) :
Ua = a : Ta ; Ta = a : Ua ; a ~ Ta . Ua = Ua . Ta.
For example, if a be an unit-vector, so that Ua = a (160),
then Td = 1 ; and therefore, generally, whatever vector may
be denoted by a, we have always,
* TUa=l.
For the same reason, ivhatever quaternion may be denoted by
q, we have always (comp. again 160) the equation,
T(Ax.g)=l.
(1.) Hence the equation
where p = op, expresses that the locus of the variable point p is the surface of the
unit sphere (128).
(2.) The equation Tp = Ta expresses that the locus of p is the spheric surface
with o for centre, which passes through the point a.
(3.) On the other hand, for the sphere through o, which has its centre at A, we
have the equation, ., . 7>
T(p-a) = Ta; ■" /' r. ^^ " "' '^
which expresses that the lengths of the two lines, ap, ao, are equal. , , ' [ ^^H^) '
(4.) More generally, the equation,
T (p - a) = T (/3 - a), 7 (M\ r y.. 4-^- ^
expresses that the locus of p is the spheric surface through b, which has its centre
at A.
(5.) The equation of the Apollonian* Locus, 145, (8.), (9.), may be written
under either of the two following forms :
T(p-a2a)=aT(p-a); Tp=aTa; \^^.^.^ „ ^ '
from each of which we shall find ourselves able to pass to the other, at a later stage,
by general Rules of Transformation, without appealing to geometry (covv^. 145, (10.)),
(6.) The equation,
T(p + a) = T(p-a),
expresses that the locus of p is the plane through o, perpendicular to the line oa ;
because it expresses that if oa' = - oa, then the point p is equally distant from the
two points A and a'. It represents therefore the same locus as the equation,
* Compare the first Note to page 128.
CHAP. I.] GEOMETRICAL EXAMPLES. 165
or as the equation,
Z^=^, of 132, (L);
a i
^ + K^=0, of 144, (L);
a a
or as
f U^Y=-1, of 161, (7.);
or as the simple geometrical formula, p -L a (129). And in fact it will be found
possible, by General Rules of this Calculus, to transform any one of these /ue for-
mulae into any other of them ; or into this sixth form,
a
which expresses that the scalar part* of the quaternion - is ze/o, and therefore that
a
this quaternion is a right quotient (132).
(7.) In like manner, the equation
T(p-/3)=T(p-a)
expresses that the locus of p is the plane which perpendicularly bisects the line ab ;
because it expresses that p is equally distant from the two points A and b.
(8.) The tensor, T«, being generally a positive scalar, but vanishing (as a limit)
with a, we have,
Txa = + xTa, according as x> or < ;
thus, in particular,
T (- a) = Ta ; and TOa = TO = 0.
(9.) That
T(/3 + a) = T/3+Ta, if U/3 = Ua,
but not otherwise (a and fi being any two actual vectors), will be seen, at a later
stage, to be a symbolical consequence from the rules of the present Calculus ; but in
the mean time it may be geometrically proved, by conceiving that while a = OA, as
usual, we make (3+ a = oc, and therefore j3 = oc — OA = ao (4) ; for thus we shall
see that while, iyi general, the three points o, A, c are corners of a triangle, and there-
fore the length of the side oc is less than the sum of the lengths of the two other
sides OA and ac, the former length becomes, on the contrary, equal to the latter sum,
in the particular case when the triangle vanishes, by the point a falling on the finite
line OC ; in which case, OA and AC, or a and /3, have one common direction, as the
equation Ua = U/3 implies.
(10.) If a and (3 be any actual vectors, and if their versors be unequal (Ua not
= U/3), then
T(/3 + a)<T/3 + Ta;
an inequality which results at once from the consideration of the recent triangle oac ;
but which (as it will be found) may also be symbolically proved, by rules of the
calculus of quaternions.
* Compare the Note to page 125 ; and the following Section of the present
Chapter.
166 ELEMENTS OF QUATERNIONS. [bOOK II.
(11.) If U/3 = - Ua, then T(/3 + a) = + (T/3 - Ta), according as T/3 > or < Ta ;
but
T (i3 + a) >+ (T/3 -Ta), if U/3no*=-Ua.
187. The quotient, Uj3 : Ua, of the versors o^ \hQ two vec-
tors, a and j3, has been called (in 156) the Versor of the Quo-
tient, or quaternion, q = ^ : a ; and has been denoted, as such,
by the symbol, \]q. On the same plan, we i3ropose now to
call the quotient, T/3 : Ta, of the tensors of the same two vec-
tors, the Tensor* of the Quaternion q, or (5: a, and to denote
it by the corresponding symbol, Tq. And then, as we have
called the letter U (in 156) the characteristic of the operation
o^ taking the versor, so we may now speak of T as the Cha-
racteristic of the (corresponding) Operation of taking the Ten-
sor^ whether of a Vector, a, or of a Quaternion, q. We shall
thus have, generally,
T(j3 : a) = TjS : Ta, as we had U(/3 : a) = U/3 : Ua (156) ;
and may say that as the versor JJq depended solely on, but
conversely was sufficient to determine, the relative direction
(157), so the tensor Tq depends on and determines the relative
length] (109), of the two vectors, a and /3, of which the qua-
ternion q is the quotient (112).
(1.) Hence the equation T- = l, like T(0 = Ta, to which it is equivalent, ex-
presses that the locus of p is the sphere with o for centre, which passes through the
point A.
* Compare the Note to Art. 109, in page 108; and that to Art. 156, in page
135.
f It has been shown, in Art. 112, and in the Additional Illustrations of the
third Section of the present Chapter (113-116), that Relative Length, as well as
relative direction, enters as an essential element into the very Conception of a Qua-
ternion. Accordingly, in Art. 117, an agreement of relative lengths (as well as an
agreement of relative directions) was made one of the conditions of equality, between
any two quaternions, considered as quotients of vectors : so that we may now say,
that the tensors (as well as the versors) of equal quaternions are equal. Compare
the first Note to page 137, as regards what was there called the quantitative element,
of absolute or relative length, which was eliminated from a, or from q, by means of
the characteristic U ; whereas the new characteristic, T, of the present Section,
serves on the contrary to retain that element alone, and to eliminate what may be
called by contrast the qualitative element, of absolute or relative direction.
CHAP. I.] TENSOR OF A QUATERNION. 167
(2.) The equation comp. 186, (6.) ),
T^i-e = l,
p- a
expresses that the locus of p is the plane through o, perpendicular to the line oa.
(3.) Other examples of the same sort may easily be derived from the sub-arti-
cles to 186, by introducing the notation (187) for the tensor of a quotient, or qua-
ternion, as additional to that for the tensor of a vector (185).
(4.) T(/3 : a) >, =, or < 1, according as T/3 >, =, or < Ta.
(5.) The tensor of a right quotient (132) is always equal to the tensor of its in-
dex (133).
(6.) The tensor of a radial (146) is always positive unity ; thus we haA^e, ge-
nerally, by 156,
TU^ = 1;
and in particular, by 181,
Tt = T; = TA=l.
(7.) Txq = + xHq, according as a; > or < ;
thus, in particular, T(— g') = T5', or the tensors oi opposite quaternions are equal.
(8.) Ta; = + ar, according as x> or < ;
thus, the tensor of a scalar is that scalar taken positively.
(9.) Hence,
TTa = Ta, TTq^Tq;
80 that, by abstracting from the subject of the operation T (comp. 145, 160), we
may establish the symbolical equation,
T^ = TT= T
(10.) Because the tensor of a quaternion is generally a positive scalar, such a
tensor is its own conjugate (139) ; its angle is zero (131) ; and its versor (159) is
positive unity : or in symbols,
KTq^Tq; LTq=Oi VTq=l.
(11.) T(l:5) = T(a:i8) = Ta:T/3 = l:T5;
or in words, the tensor of the reciprocal of a quaternion is equal to the reciprocal of
the tensor,
(12.) Again, since the two lines, ob and ob', in Fig. 36, are equally long, the de-
finition (137) of a conjugate gives
TKq = Tq',
or in words, the tensors of conjugate quaternions are equal.
(13.) It is scarcely necessary to remark, that any two quaternions which have
equal tensors, and equal versors, are themselves equal : or in symbols, that
g' = q, if T:q=Tq, and XJq'^Uq.
188. Since we have, generally,
a Ta.Ua Ta Ua Ua
we may establish the two following general formulae of decom-
15 T^.u^ t/3 uii u^ T^ ^ ,„^ ,_^
168 ELEMENTS OF QUATERNIONS. [bOOK II.
position of a quaternion into two factors, of the tensor and ver-
sor kinds :
I. .,q=Tq.\]qi II. . . ^ = U^.T^ ;
which are exactly analogous to the formulae (186) for the cor-
responding decomposition of a vector, mio factors of the same
two kinds : namely,
r. . .a = Ta.Ua; H'. . . a = Ua . Ta.
To illustrate this last decomposition of a quaternion, q, or
OB : oA, into factors, we may conceive that aa' and bb' are two
concentric and circular, but oppositely directed arcs, which
terminate respectively on the two
lines OB and oa, or rather on the
longer of those two lines itself, and
on the shorter of them prolonged,
as in the annexed Figure 48 ; so
that oa' has the length of oa, but
the direction of ob, while ob', on the
contrary, has the length of ob, but
the direction of oa ; and that therefore we may write, by what
has been defined respecting versors and tensors of vectors (155,
156, 185, 186),
OA' = Ta.U]3; 0B'=Tj3.Ua.
Then, by the definitions in 156, 187, of the versor and tensor
of a quaternion,
JJq = U(oB : oa) = oa' : oa = ob : ob' ;
Tq =T (oB : oa) = ob' : oa = ob : oa' ;
whence, by the general formula of multiplication of quotients
(107),
I. . q = 0b: o\ = (ob : oa') . (oa' : oa) = T^' . Uq ;
and
II. . ^ = ob : oa = (ob : ob') . (ob' : oa) = \Jq . Tq,
as above.
189. In words, if we wish to pass from the vector a to the vec-
tor /3, or from the line oa to the line ob, we are at liberty either,
1st, to begin by turning^ from oa to oa', and then to end by stretching^
CHAP. I.] TENSOR OF A QUATERNION. 169
from oa' to ob, as Fig. 48 may serve to illustrate; or, Ilnd, to begin
by stretching, from oa to ob^, and end by turning, from ob' to ob.
The act of multiplication of a line a by a quaternion q^ considered as
a factor (103), which affects both length and direction (109), may
thus be decomposed into two distinct and partial acts, of the kinds
which we have called Version and Tension ; and these two acts may
be performed, at pleasure, in either of tvjo orders of succession. And
although, if we attended merely to lengths, we might be led to say
that th.Qtensor of a quaternion was a signless number,'^ expressive of
a geometrical ratio of magnitudes, yet when the recent construction
(Fig. 48) is adopted, we see, by either of the two resulting expres-
sions (188) for 1q, that there is b. propriety in treating this tensor
as 2, positive scalar, as we have lately done, and propose systemati-
cally to do,
190. Since TYiq = Tq, by 187, (12.), and UK^=1:U^, by 158,
we may write, generally, for any quaternion and its conjugate, the
two connected expressions:
L. ,q = Tq.\]q', II. .. Kq^Tq'.Uq;
whence, by multiplication and division,
III. . . ^ . K(? = (T^)2 ; IV. . . 2 : K^ = (U^)^
This last formula had occurred before; and we saw (161) that in it
thQ parentheses might be omitted, because (J^qf =^{q^)' In like
manner (comp. 161, (2.) ), we have also
(T?)-^=T(s^) = Tf/,
parentheses being again omitted ; or in words, the tensor of the square
of a quaternion is always equal to the square of the tensor: as ap-
pears (among other ways) from inspection of Fig. 42, his, in which h 1^/
the lengths of oa, ob, oc form a geometrical progression ; whence
obV ^oc T.oc / T.ob V YrpOB"'
oa; ~ oa T.oA~\T.oAy \ oa
At the same time, we see again that the product qKq of two conju-
gate quaternions, which has been called (145, (U.) ) their common
Norm, and denoted by the symbol '^q, represents geometrically the
square of the quotient of the lengths of the two lines, of which (when
considered as vectors) the quaternion q is itself the quotient (112).
We may therefore write generally,!
V. . . qYiq = Tq^ = l^q\ VI. . . T^ = ^/^q^ v/(^/K^).
* Compare the Note in page 108, to Art. 109.
f Compare the Note in page 129.
Z
170 ELEMENTS OF QUATERNIONS. [bOOK II.
(1.) We have also, by II., the following other general transformations for the
tensor of a quateraion :
VII. . . Tg = Kg.U5; VIII. . . Tg^ Ug . % ;
of which the geometrical significations might easily be exhibited by a diagram, but
of which the validity is sufficiently proved by what precedes.
(2.) Also (comp. 158),
(3.) The reciprocal of a quaternion, and the conjugate* of that reciprocal, may
now be thus expressed :
1 _ Kg _ ^_ KUg_ J_ J^ _ ±.J..
g~"f^~K^~ Tq ~ Vq' Tq~ Tq'Vq'
q % Tg2 Tg Kg*
(4.) We may also write, generally,
IX.. . Kg = Tg. KUg = N5:g.
191. In general, let any two quaternions, q and^'', be con-
sidered as multiplicand and multiplier, and let them be re-
duced (by 120) to the forms j3 : a and 7 : j3 ; then the tensor
and versor of that third quaternion, y.a, which is (by 107)
their product q'q^ may be thus expressed :
I...T^'^=T(y:a) = Ty:Ta = (T7:Ti3).(T/3:Ta) = T5'.T^;
Il...U^V = U(7:a) = U7:Ua=(U7:Uj3).(Uj3:Ua) = U^'.U^;
where Tq'q and \Jqq are written, for simplicity, instead of
T{q\q) and U (§''.$'). Hence, in any such multiplication, the
tensor of the product is the product of the tensor; and the ver-
sor of the product is the product of the versors; the order of
the factors being generally retained for the latter (comp. 168,
&c.), although it may be varied for the former^ on account of
the scalar character of a tensor. In like manner, for the divi-
sion of any one quaternion q\ by any other q, we have the
analogous formulae :
III. .. T (?':?) = Tj -.Tq; IV. . . U(?' : q) = \Jq' : JJq ;
or in words, the tensor of the quotient of any two quater-
nions is equal to the quotient of the tensors ; and similarly, the
versor of the quotient is equal to the quotient of the versors.
And because multiplication and division of tensors are per-
formed according to the rules 0^ algebra, or rather of a/^V/^/w^-
* Compare Art. 145, and the Note to page 127.
CHAP. I.] PRODUCT OR QUOTIENT OF TWO QUATERNIONS. l7l
tic (a tensor being always, by what precedes, a positive num-
ber), we see that the difficulty (whatever it may be) of the
general multiplication and division of quaternions is thus re-
duced to that of the corresponding operations on versors : for
which latter operations geometrical constructions have been
assigned, in the ninth Section of the present Chapter.
(1.) The two products, q'q and qq', of any two quaternions taken as factors in
two different orders, are equal or unequal, according as those two factors are compla-
nar or diplanar ; because such equality (169), or inequality (168), has been already
proved to exist, for the case* when each tensor is unity : but we have always
(comp. 178),
Hqq = Tgq\ and lq'q=l qq.
(2.) If Lq = Lq =—i then qq' = Kq'q (170) ', SO that the products of two right
quotients, or right quaternions (132), taken in opposite orders, are always conju-
gate quaternions.
,(3.) If lq = /.g'='~, and Ax.^'-i- Ax.gr, then qq=-q'q,
Lqq'=Lq'q = ^, Ax. q'q -i- Ax . q, Ax . q' q -I- Ax . q' {17 1) ',
so that the product of two right quaternions, in two rectangular planes, is a third
right quaternion, in a plane rectangular to both ; and is changed to its oivn opposite,
when the order of the factors is reversed : as we had ijz=k=-ji (182).
(4.) In general, if q and q' be any two diplanar quaternions, the rotation round
Ax . q', from Ax . 5 to Ax . q'q, is positive (177).
(6.) Under the same condition, q\{q' : g-) is a quaternion with the same tensor,
and same angle, as q', but with a different axis; and this new axis. Ax .g(q' : g),
may be derived (179, (1.)) from the old axis. Ax . q', by a conical rotation (in the
positive direction) round Ax . q, through an angle = 2 Lq.
(6.) The product or quotient of two complanar quaternions is, in general, a third
quaternion complanar with both ; but if they be both scalar, or both right, then this
product or quotient degenerates (131) into a scalar.
(7.) Whether q and q' be complanar or diplanar, we have always as in algebra
(comp. 106, 107, 136) the two identical equations:
V. . . (g' : g) . g = ?' ; VI. . . (9' . ?) : g = q'.
(8.) Also, by 190, V., and 191, I., we have this other general formula :
VII. . .Ng'g = Ng'.Ng;
or in words, the norm of the product is equal to the product of the norms.
192. Let ^ = j3 : a, and 5'' = 7 : j3, as before ; then
1 : ^'^= 1 : (7 : a) =a : 7 = (« : /3) . (/3 : 7) = (1 : g) . (1 :^');
so that the reciprocal of the product of any two quaternions is
* Compare the Notes to pages 148, 151.
172 ELEMENTS OF QUATERNIONS. [bOOK II.
equal to the product of the reciprocals, taken in an inverted
order : or briefly,
I. . . R^''^' = R.^ . ^q\
if R be again used (as in 161, (3.)) as a (temporary) charac-
teristic of reciprocation. And because we have then (by the
same sub-article) the symbolical equation, KU = UR, or in
words, the conjugate of the versor of any quaternion q is equal
(158) to the versor of the reciprocal of that quaternion ; while
the versor of a product is equal (191) to the product of the
versors : we see that
KU^'^ = UR^'^ = UR^ . UR^' = KU^ . KU^'.
But
Kq^Tq. KU(7, by 190, IX. ; and Tq'q = Tq .Tq = T^.T^',
by 191; we arrive then thus at the following other important
and general formula :
II. ..K|7'^ = K^.K^';
or in words, the conjugate of the product of any two quater-
nions is equal to the product of the conjugates, taken (still)
in an inverted order.
(1.) These two results, I., II., may be illustrated, for versors (Tg = T$' = 1), by
the consideration of a spherical triangle abc (comp. Fig. 43) ; in which the sides
AB and BC (comp. 167) may represent q and q', the arc Ac then representing q'q.
For then the new multiplier 'Rq = Kq (158) is represented (162) by ba, and the new
multiplicand Kg' = Kg' by CB ; whence the new product, Rg.Rg'= Kg^.K^', is re-
presented by the inverse arc CA, and is therefore at once the reciprocal Kg'g, and the
conjugate Kq'q, of the old product q'q.
(2.) If q and q' be right quaternions, then Kq = -q, Kg' = — g' (by 144) ; and
the recent formula II. becomes, Kg'g = gg', as in 170.
(3.) In general, that formula II. (of 192) may be thus -written :
III. .. k^ = k^.k2:;
a a /3'
where a, j8, y may denote anj/ three vectors.
(4.) Suppose then that, as in the annexed
Fig. 49, we have the two following relations of in-
verse similitude of triangles (118),
A AOB a' BOC, A BOE a' DOB ;
and therefore (by 137) the two equations,
/3~ a' S l3' Fig. 49.
CHAP. I.] CASE OF TWO RIGHT QUATERNIONS. 173
we shall have, by III.,
^=K-, or ADOCa'AOE;
a
so that this third formula of inverse similitude is a consequence from the other two.
(5.) If then (comp, 145, (6.) ) any two circles, -whether in one plane or in space,
touch one another at a point b ; and if from any point o, on the common tangent bo,
two secants OAC, oed be drawn, to these two circles ; the four points of section,
A, c, D, E, will be on one common circle : for such concircularity is an easy conse-
quence (through equal angles, &c.), from the last inverse similitude.
(6.) The same conclusion (respecting concircularity, &c.) may be otherwise and
geometrically drawn, from the equality of the two rectangles, AOC and doe^ each
being equal to the square of the tangent ob ; which may serve as an instructive «*k^
verification of the recent formula III., and as an example of the consistency of the
results, to which calculations with quaternions conduct.
(7.) It may be noticed that the construction would in general give three circles,
although only one is drawn in the Figure ; but that if the two triangles abc and
DBE be situated m different planes, then these three circles, and of course ih.& five
points ABODE, are situated on one common sphere.
193. An important application of the foregoing general
theory of Multiplication and Division, is to the case of Right
Quaternions (132), taken in connexion with i\iQ\Y Index- Vec-
tors, or Indices (133).
Considering division first, and employing the general for-
mula of 1 06, let /3 and y be each _L a ; and let /3' and -y' be the
respective indices of the two right quotients, q = j3 : a, and
«/' = y : a. We shall thus have the two complanarities, /3' 1 1| /3, 7,
and 7'||| j3, 7 (comp. 123), because the four lines /3, 7, /3', y
are all perpendicular to a ; and within their common plane it
is easy to see, from definitions already given, that these four
lines form a proportion of vectors, in the same sense in which
a, (5, y, d did so, in the fourth Section of the present Chapter :
so that we may write the equation of quotients.
In fact, we have (by 133, 185, 187) the following relations of
length,
TjS' = Tp : Ta, T7' = T7 : Ta, and .-. T (7' : jS') = T (7 : |3) ;
while the relation of directions, expressed by the formula,
U(y:/3') = U(y:j3), or Uy : U/3' = Uy : U/3,
is easily established by means of the equations,
174 ELEMENTS OF QUATERNIONS. [boOK II.
Z(y:y)=Z(/3':i3) = ^; Ax . (y' : 7) = Ax . (/3' :/3) = Ua.
We arrive, then, at this general Theorem (comp. again 133):
that ^^the Quotient of any two Right Quaternions is equal to
the Quotient of their Indices.''*
(1.) For example (comp. 150, 159, 181), the indices of the right versors t, j, k
are the axes of those three versors, namely, the lines 01, oj, ok ; and we have the
equal quotients,
j: » = 01 : oj' = A = OJ : oi, &c.
(2.) In like manner, the indices of - z, —J, —k are 01', oj', ok' ; and
1 : —j = oj' : 01' = A = 01 : Oj', &c.
(3.) In general the quotient of any two right versors is equal to the quotient of
their axes ; as the theory of representative arcs, and of their poles, may easily
serve to illustrate.
1 94. As regards the multiplication of two right quaternions,
in connexion with their indices, it may here suffice to observe
that, by 106 and 107, the product 7 : a = (y : j3) . (j3 : a) is equal
(comp. 136) to the quotient, (7 : i3) : (a : /3) ; whence it is easy
to infer that ''the Product, q'q, of any two Right Quaternions,
is equal to the Quotient of the Index of the Multiplier, q, di-
vided by the Index of the Reciprocal of the Multiplicand, q"
It follows that the plane, whether of the product or of the
quotient of two right quaternions, coincides with the plane of
their indices ; and therefore also with the plane of their axes ;
because we have, generally, by principles already established,
the transformation,
if Z 5' = -, then Index of q = T5' . Ax . q,
* We have thus a new point of agreement, or of connexion, between right qua-
ternions, and their index-vectors, tending to justify the ultimate assumption (not yet
made), of equality betAveen the former and the latter. In fact, we shall soon prove
that the index of the sum (or difference), of any two right quotients (132), is equal to
the sum (or difference) of their indices ; and shall find it convenient subsequently to
interpret ilvQ product (5a of any two vectors, as being the quaternion-product (194)
of the two right quaternions, of which those two lines are the indices (133): after
which, the above-mentioned assumption of equality will appear natural, and be found
to be useful. (Compare the Notes to pages 119, 136.)
CHAP. I.] SUM OF TWO QUATERNIONS. 175
Section 12. — Oii the Sum or Difference of any tico Quater-
nions ; and on the Scalar (or Scalar Part) of a Quater-
nion.
195. The Addition of any given quaternion q^ considered
as a geometrical quotient ov fraction (101), to any other given
quaternion q^ considered also as a fraction, can always be ac-
complished by the first general formula of Art. 106, when these
two fractions have a common denominator ; and if they be not
already ^iven as having such, they can always be reduced so as
to have one, by the process of Art. 120. And because the ad-
dition of any two lines was early seen to be a commutative ope-
ration (7, 9), so that we have always y + /3 = )3 + y, it follow^s
(by 106) that the addition of any two quaternions is likewise a
commutative operation, or in symbols, that
I. . . ^ + ^' = ^' + (7 ;
so that the Sum of any tivo* Quaternions has a Value, which
is independent of their Order : and which (by what precedes)
must be considered to be given, or at least known, or definite,
when the two summand quaternions are given. It is easy also
to see that the conjugate of any such sum is equal to the sum
of the conjugates, or in symbols, that
11. ..K(^'-K7)=%' + K^.
(1 .) The important formula last -written becomes geometrically evident, when it
is presented under the following form. Let obdc be any parallelogram, and let OA
be any right line, drawn from one comer of it, but not generally in its plane. Let
the three other comers, b, c, d, be reflected (in the sense of 145, (5.) ) with respect
to that line OA, into three new points, b', c', d' ; or let the three lines ob, oc, od be
reflected (in the sense of 138) with respect to the same line oa; which thus bisects
at right angles the three joining lines, bb', cc', dd', as it does bb' in Fig. 36. Then
each of the lines OB, oc, od, and therefore also the ^\io\q plane figure ohdc, may be
considered to have simply revolved round the line oa as an axis, by a conical rota-
tion through two right angles ; and consequently the new figure ob'd'c', like that old
one obdc, must be a, parallelogram. Thus (comp. 106, 137), we have
od' = oc' + ob', 5' = -y' + /3', 5': a=(y' : a)+ (/3': a);
and the recent formula II. is justified.
* It will be found that this result admits of being extended to the case of three
(or more) quaternions ; but, for the moment, we content ourselves with two.
176 ELEMENTS OF QUATERNIONS. [bOOK II.
(2.) Simple as this last reasoning is, and unnecessary as it appears to be to draw-
any new Diagram to illustrate it, the reader's attention may be once more invited to
the great simplicity of expression, with which many important ^reome^ncaZ concep-
tions, respecting space of three dimensions, are stated in the present Calculus : and
are thereby kept ready for future application, and for easy combination with other
results of the same kind. Compare the remarks already made in 132, (6.) ; 145,
(10.); 161; 179,(3.); 192,(6.); and some of the shortly following sub-articles to
196, respecting properties of an oblique cone with circular base.
196. One of the most important cases o^ addition y is that
of two conjugate summands^ q and K^ ; of which it has been
seen (in 140) that the sum is always a scalar. We propose
now to denote the ^^Zf of this sum by the symbol.
Is , , ^9'->
hus writing generally,
I. . . ^ + Kg = Ky + ^=2S^;
or defining the new symbol S^' by the formula,
f^ II. ..S^ = i(^ + K^); or briefly, 11'. . . S = i (1 + K).
For reasons which will soon more fully appear, we shall also
call this new quantity, Sg', the scalar part, or simply the Sca-
lar, of the Quaternion, q ; and shall therefore call the letter
S, thus used, the Characteristic of the Operation of taking the
6'caZ«r of a quaternion. (Comp. 132, (6.) ; 137; 156; 187.)
It follows that not only equal quaternions, but also conjugate
quaternions, have equal scalars ; or in symbols,
III. . . S^'=S^, if q^q-, and IV. ..SK9 = S^;
or briefly,
IV'. ..SK=S.
And because we have seen that Kg- = + ^, if 5' be a scalar ( 1 39),
but that li^q^-q, if 5' be a right quotient (144), we find that
the scalar of a scalar (considered as a degenerate quaternion,
131) is equal to that scalar itself, \>\xi that the scalar of a right
quaternion is zero. We may therefore now write (comp. 160):
V. . Sa; = X, if ic be a scalar ; VI. . .SSg = Sg', 8^ = 88 = 8;
and Vll. ..8^ = 0, if z^ = |.
Again, because oa' in Fig. 36 is multiplied by x, when ob is
multiplied thereby, we may write, generally,
CHAP. I.] SCALAR OF A QUATERNION. 177
VIII. • . Sxq = xSq, if oj be any scalar;
and therefore in particular (by 188),
IX. ..S^ = S(T^.U^) = T^.SU^.
Also because SK^=S^, by IV., while KU^ = U-, by 158,
we have the general equation,
X. ..SUq = SJJ-; or X'. . . SU^ = SU ^ ;
whence, by IX.,
XI. . .S^ = T^.Sui; or XI'. . . S^ = T@. SU^ ;
^ ^ q a a p
and therefore also, by 190, (V.), since T^.T- = 1,
XII.. .Sq = TqKS-=^^q.S-; XIF. . . S ^ = N^ • S "
^ ^ q ^ q a a (5
The results of 142, combined with the recent definition I. or
II., enable us to extend the recent formula VII., by writing,
XIII. . . S^' >, =, or < 0, according as Lq <, =, or > - ;
and conversely,
XIV. . . Z ^ <, =, or > -, according as S^- >, =, or <q.
In fact, if we compare that definition I. with the formula of
140, and with Fig. 36, we see at once that because, in that
Figure,
S(ob: oa) = oa': OA,
we may write, generally,
XV. . . S^ = T^.cosz^; or XVI. . . SU^= cos Z ^;
equations which will be found of great importance, as serving
to connect quaternions with trigonometry ; and which show
that
XVII. ,.Lq=^Lq, if SU^' = SU^,
the angle Lq being still taken (as in 130), so as not to fall
outside the limits and tt ; whence also,
2 A
178 ELEMENTS OF QUATERNIONS. [bOOK II.
XVIII. .. Lq'^Lq, if S^' = S^, and Tq' = T^,
the angle of a quaternion being thus given, when the scalar
and the tensor of that quaternion are given, or known. Fi-
nally because, in the same Figure 36 (comp. 15, 103), the
line,
oa' = (oa' : oa) . oa = oa . S (ob : da),
may be said to be the projection of ob on oa, since a' is the
foot of the perpendicular let fall from the point b upon this
latter line oa, we may establish this other general formula :
XIX. . . aS - = S — • a = projection of^ on a ;
a a
a result which will be found to be of great utility, in investi-
gations respecting geometrical loci, and which may be also
written thus :
XX. . . Projection o/ j3 o?2 a = Ua . T/3 . SU ^ ;
with other transformations deducible from principles stated
above. It is scarcely necessary to remark that, on account
of the scalar character of Sq, we have, generally, by 159, and
187, (8.), the expressions,
XXL . . US^ = ±1; XXII. . .TS^ = ±S^;
while, for the same reason, we have always, by 139, the equa-
tion (comp. IV.),
XX III. . . KS^ = S^ ; or XXIII'. . . KS = S ;
and, by 131,
XXIV. . . iSq^O, or = TT, unless Lq = -;
in which last case S^' = 0, by VII., and therefore L Sq is inde-
terminate :* IJSq becoming at the same time indeterminate,
by 159, but TS^ vanishing, by 186, 187.
8-^ = 0,
a
(1.) The equation,
is now seen to be equivalent to the formula, p -^ a ; and therefore to denote the
* Compare the Note in page 118, to Art 131.
CHAP. I.] GEOMETRICAL EXAMPLES. 1^^
same plane locus for p, as that which is represented by any one of the four other
equations of 186, (6.) ; or by the ecjuation,
T^-t^ = l, of 187, (2.).
p~a
(2.) The equation,
S£IJ = 0, or Se=S^,
a a a
expresses that bp j_ oa ; or that the points B and p have the same projection on oa j
or that the locus of p is the plane through b, perpendicular to the line OA.
(3.) The equation,
a a
expresses (comp. 132, (2.) ) that p is on one sheet of a cone of revolution, with o for
vertex, and OA for axis, and passing through the point b.
(4.) The other jsheet of the same cone is represented by this other equation,
a a
and hath sheets jointly by the equation, I
(6.) The equation,
S- = l, or SU^ = T-,
a a p
expresses that the locus of p is the plane through A, perpendicular to the line OA ;
because it expresses (comp. XIX J that the projection of op on oa is the line oa it-
p — a
self; or that the angle oap is right ; or that S =0. P
(6.) On the other hand the equation,
S^=l, or Sug=Tg,. .. \ — '
expresses that the projection of ob on op is op itself ; or that the angle opb is right ;
or that the locus of p is that spheric surface, which has the line ob for a diameter.
(7.) Hence the system of the two equations,
sP = i, S^=l.
a p
represents the circle, in which the sphere (6.), with ob for a diameter, is cut by the
plane (5.), with oa for the perpendicular let fall on it from o.
(8.) And therefore this new equation,
S^.S^ = 1,
a p
obtained by multiplying the two last, represents the Cyclic* Cone (or cone of the
* Historically speaking, the oblique cone with circular base may deserve to be
named the Apollonian Cone, from Apollonius of Perga, in whose great work on Co-
6.
p
A
Ar^-
h'?
180 ELEMENTS OF QUATERNIONS. [bOOK II.
second order, but not generally of revolution), ■which rests on this last circle (7.) as
its lase, and has the point o for its vertex. In fact, the equation (8.) is evidently
satisfied, when the two equations (7.) are so; and therefore every point of the circu-
lar circumference, denoted by those two equations, must be a point of the locus, re-
presented by the equation (8.). But the latter equation remains unchanged, at least
essentially, when p is changed to xp, x being any scalar ; the locus (8.) is, there-
fore, some conical surface, with its vertex at the origin, o ; and consequently it can
be none other than that particular cone (both ways prolonged), which rests (as
above) on the given circular base (7.).
(9.) The system of the two equations,
a p y
(in writing the first of which the point may be omitted,) represents a conic section ;
namely that section, in which the cone (8.) is cut by the new plane, which has oc
for the perpendicular let fall upon it, from the origin of vectors O.
(10.) Conversely, every plane ellipse (or other conic section) in space, of which
the plane does not pass through the origin, may be represented by a system of two
equations, of this last /orm (9.) ; because the cone which rests on any such conic as
its base, and has its vertex at any given point O, is known to be a cyclic cone.
(11.) The curve (or rather the pair of curves), in which an oblique but cyclic
cone (8.) is cut by a concentric sphere (that is to say, a cone resting on a circular
base by a sphere which has its centre at the vertex of that cone), lias come, in mo-
dem times, to be called a Spherical Conic. And an}- such conic may, on the fore-
going plan, be represented by the system of the two equations,
S^ S^=l, Tp=l;
a p
the length of the radius of the sphere being here, for simplicity, supposed to be the
unit of length. But, by writing Tp — a, where a may denote any constant and posi-
tive scalar, we can at once remove this last restriction, if it be thought useful or con-
venient to do so.
(12.) The equation (8.) may be written, by XII. or Xll'., under the form (comp.
191, VII.):
or br' fly,
'I'r^rWv
p a
nics (tc(tJviK'7iv), already referred Lo in a Note to page 128, the properties of such a
cone appear to have been first treated systematically; although the cone of revolu-
tion had been studied by Euclid. But the designation " cyclic cone''' is shorter ; and
it seems more natural, in geometry, to speak of the above-mentioned oblique cone
thus, for the purpose of marking its connexion with the circle, than to call it, as is
now usually done, a cone of the second order, or of the second degree : although
these phrases also have their advantages.
CHAP. I.] GEOMETRICAL EXAMPLES. 181
if a' = /3T^ = Ta.U/3, and /3' = aT^= T^S-Ua ;
so that a and j3' are here the lines oa' and ob', of Art. 188, and Fig. 48.
(13.) Hence the cone (8.) is cut, not only by the plane (5.) in the circle (7.),
which is on the sphere (6.), but also by the (generally) new plane^ S -,= 1, in the
(generally) new circle, in which this new plane cuts the (generally) new sphere,
8'
S — = 1 ; or in the circle which is represented by the system of the two equations,
S-%1, S^'=l.
a p
(14.) In the particular case when (3 \\ a (15), so that the quotient /3 : a is a sca-
lar, Avhich must be positive and greater than unity, in order that the plane (5.) may
(jeally) cut the sphere (6.), and therefore that the circle (7.) and the cone (8.) may
be real, we may write
]3=a2a, a>l, T(|3:a) = a2, „'=„, /3' = /3;
and the circle (13.) coincides with the circle (7.).
(15.) In the same case, the cone is one of revolution ; every point p of its circu-
lar 6a*e (that is, of the circumference thereof) being ai one constant distance from
the vertex o, namely at a distance = aTa. For, in the case supposed, the equations
(7.) give, by XII.,
N^ = S^:S-=l:S- = a2:S^=a2; or To = aTa.
a a p p p
(Compare 145, (12.), and 186, (5.). )
(16.) Conversely, if the cone be one of revolution, the equations (7.) must con-
duct to a result of the form,
a2=N" = S- :S- = S-:S-,or (comp. (2.) ), S' =0:
a a p p p "^ p
which can only be by the line /3 — a'^a vanishing,' or by our having (5= a^a, as in
(] 4.) ; since otherwise we should have, by XIV., p -i- (3- a^a, and all the points of
the base would be situated in one plane passing through the vertex o, which (for any
actual cone) would be absurd.
(17.) Supposing, then, that we have not (3 || a, and therefore not a =a, /3' = (3,
as in (14.), nor even a' \\ a, (3' \\ (3, we see that the cone (8.) is not a cone of revolu-
tion (or what is often called a right cone) ; but that it is, on the contrary, an oblique
(or scalene) cone, although still a cyclic one. And we see that such a cone is cut in
two distinct series* of circular sections, by planes parallel to the two distinct (and
mutually non-parallel) planes, (5.) and (13.) ; or to two new planes, drawn through
the vertex o, which have been calledf the two Cyclic Planes of the cone, namely, the
two following :
* ThGSQ two series o{ sub- contrary (or antiparallel) hut circular sections of a
cyclic cone, appear to have been first discovered by Apollonius : see the Fifth Propo-
sition of his First Book, in which he says, KuXihOuj dk )? Toiavrr} To/xtj v-rrevavria
(page 22 of Halley's Edition).
t By M. Chasles.
182 ELEMENTS OF QUATERNIONS. [bOOK II.
a p
■while the two lines from the vertex, OA and ob, which are perpendicular to these two
planes respectively, may be said to be the two Cyclic Normals.
(18.) Of these two lines, a and /3, the second has been seen to be a diameter of
the sphere (6.)? which may be said to be circumscribed to the cone (8.), when that
cone is considered as having the circle (7.) for its base ; the second cyclic plane (17.)
is therefore the tangent plane at the vertex of the cone, to thatj^rs* circumscribed
sphere (6.).
(19.) The sphere (13.) may in like manner be said to be circumscribed to the
cone, if the latter be considered as resting on the new circle (13.), or as terminated by
that circle as its new base ; and the diameter of this new sphere is the line ob', or j8',
which has by (12.) the direction of the line a, or of thQ first cyclic normal (17.) ; so
that (comp. (18.)) th^. first cyclic plane is the tangent plane at the vertex, to the
second circumscribed sphere (13.).
(20.) Any other sphere through the vertex, which touches the first cyclic plane,
and which therefore has its diameter from the vertex =b'(3% where b' is some scalar
co-efficient, is represented by the equation,
S^'=l,- or S^'=i;
P P
it therefore cut$ the cone in a circle, of which (by (12.) ) the equation of the plane is
S^, = 6', or S-^,= l,
a b a
so that the perpendicular from the vertex is b'a' \\ (3 (comp. (5.) ) ; and consequently
thisp/a«e of section of sphere and cone is parallel to the second cyclic plane (17.).
(21.) In like manner any sphere, such as
S — = 1, where b ia any scalar,
P
w^hich touches the second cyclic plane at the vertex, intersects the cone (8.) in a cir-
cle, of which the plane has for equation,
and is therefore /)araZZeZ to the first cyclic plane.
(22.) The equation of the cone (by IX., X., XVI,) may also be thus written :
SU^.SU^ = T^; or, cos ^^ . cos /|= T ^;
a p (S a (3 (3
it expresses, therefore, that the product of the cosines of the inclinations, of any va-
riable side (p) of an oblique cyclic cone, to two fixed lines (a and f3), namely to the
two cyclic normals (17.), is constant ; or that the product of the sines of the inclina-
tions, of the same variable side (or ray, p) of the cone, to two fixed planes, namely to
the two cyclic planes, is thus a constant quantity.
(23.) The two great circles, in which the concentric sphere Tp = 1 is cut by the two
cyclic planes, have been called the two Cyclic Arcs* of the Spherical Conic (11.), in
Bv M. Cbasles.
CHAP. I.J SCALAR OF A SUM OR DIFFERENCE. 183
which that sphere is cut by the cone. It follows (by (22.) ) that the product of the
sines of the (arcuaV) perpendiculars, let fall from any point v of a given spherical
conic, on its two cyclic arcs, is constant.
(24.) These properties of cyclic cones, and of spherical conies, are not put for-
ward as new ; but they are of importance enough, and have been here deduced with
sufficient facihty, to show that we are already in possession of a Calculus, with its
own Rules* of Transformation, whereby one enunciation of a geometrical theorem, or
problem, or construction, can be translated into several others, of which some may
be clearer, or simpler, or more elegant, than the one first proposed.
197. Let a, /3, 7 be any three co-initial vectors, oa, &c.,
and let 00 = ^ = 74-/3, so that obdc is a parallelogram (6);^
then, if we write i^
[5:a = q, y:a = q', and S : a = q" = q' + q (106)^
and suppose that b', c', d' are the feet of perpendiculars let
fall from the points b, c, d on the line oa, we shall have, by
196, XIX., the expressions,
(ob' =) ^' = aSq, y' = aSq', S' = aSq" = aS (q' + q).
But also OB = CD, and therefore ob'= c'd', the similar projec-
tions of equal lines being equal ; hence (comp. 11) the sum of
the projections of the lines j3, 7 must be equal to the projec-
tion of the sum, or in symbols,
od' = oc'+ob', g' = y-fj3', S': a = (7 :a) + (/3': a).
Hence, generally, for any tioo quaternions, q and q, we have
the formula :
I. . . S(^'+9) = S^' + Sg;
or in words, the scalar of the sum is equal to the sum of the
scalar s. It is easy to extend this result to the case of any three
(or more) quaternions, with their respective scalars ; thus, if
q be a third arbitrary quaternion, we may write
S { ?" + (3' + ?) ) = Sj" + S (j + ?) = S/ + (Sj'+ S?) ;
where, on account of the scalar character of the summands, the
last parentheses may be omitted. We may therefore write,
generally,
II. . . SS^ = 2S^, or briefly, SS = 2S ;
where 2 is used as a sign of Summation : and may say that
* Comp. 146, (10.), &c.
184 ELEMENTS OF QUATERNIONS. [bOOK II.
the Operation of tailing the Scalar of a Quaternion is a Dis-
tributive Operation (comp. 13). As to the general Siibtrac-
tion of any one quaternion from any other, there is no difficulty
in reducing it, by the method of Art. 120, to the second gene-
ral formula of 106 ; nor in proving that the Scalar oftheDiffe-
rence* is always equal to the Difference of the Scalars. In
symbols,
III. . . S(^'-^) = S^'-S^;
or briefly,
IV... SA^ = AS^, SA=AS;
when A is used as the characteristic of the operation of taking
a difference, by subtracting one quaternion, or one scalar, from
another.
(1.) It has not yet been proved (comp. 195), that the Addition oi any number
of Quaternions, q, q\ q" , . . is an associative and a commutative operation (comp. 9).
But we see, already, that the scalar of the sum of any such set of quaternions has
a value, which is independent of their order, and of the mode oi grouping them.
(2.) If the summands be all right quaternions (132), the scalar oieach separately
vanishes, by 196, VI I. ; wherefore the scalar of their sum vanishes also, and that
sum is consequently itself, by 196, XIV., a right quaternion : a result which it is
easy to verify. In fact, if /3 -i- a and y -^ a, then y + /3 -J- a, because a is then per-
pendicular to the plane of /3 and y ; hence, by 106, the sum of any two right qua-
ternions is a right quaternion, and therefore also the sum of any number of such qua-
ternions.
(3.) Whatever two quaternions q and q' may be, we have always, as in algebra,
the two identities (comp. 191, (7.) ) :
V. ..(?'-g) + 5 = g'; VI. ..(9' + 5) -9 = ^'.
198. Without yet entering on the general theory o^ scalars of
products or quotients of quaternions, we may observe here that be-
cause, by 196, XV., the scalar of a quaternion depends only on the
tensor and the angle, and is independent of the axis, we are at liberty
to write generally (comp. 173, 178, and 191, (1.), (5.)),
l...Sqq^=Sq'q; 11. , . S . q (q^: q) = Sq' ;
the two products^ qq' and q'q, having thus always equal scalars,
although they have been seen to have unequal axes, for the general
case of diplanarity (168, 191). It may also be noticed, that in vir-
tue of what was shown in 193, respecting the quotient, and in 194
' Examples have already occurred in 196, (2.), (5.), (16.).
CHAP. I.] SCALAR OF A PRODUCT, QUOTIENT, OR SQUARE. 185
respecting the product, of any two right quaternions (132), in con-
nexion with their indices (133), we may now establish, for any
such quaternions, the formulae :
III. . . S (^' : 5) = S (I^' : I?) = T {q' : q) . cos Z.(Ax. q^ : Ax. q) ;
IV. . . Sq'q =^S{q' .q) = s(lq':l-j = - Tq'q.cos L (Ax. ^': Ax. q)\
where the new symbol \q is used, as a temporary abridgment, to
denote the Index of the quaternion q^ supposed here (as above) to be
a right one. With the same supposition, we have therefore also
these other and shorter formulae :
V. . . SU(g':^)=+ cosz(Ax. ^': Ax.^);
VI. . . SU'^'2' = - cos Z (Ax. q^ : Ax. q) ;
which may, by 196, XVI., be interpreted as expressing that, under
the same condition of rectangularity of q and q\
VII. . . L{q'.q)=^L (Ax. q^: Ax. q) ;
VIII. , . Lq'q = 7r-L (Ax. q' : Ax. q).
In words, the Angle of the Quotient of two Right Quaternions is equal
to the Angle of their Axes; but the Angle of the Product^ of two such
quaternions, is equal to the Supplement of the Angle of the Axes,
There is no difficulty in proving these results otherwise, by con-
structions such as that employed in Art. 193; nor in illustrating
them by the consideration of isosceles quadrantal triangles, upon the
surface of a sphere.
199. Another important case of the scalar of a product, is
the case of the scalar of the square of a quaternion. On refer-
ring to Art. 149, and to Fig. 42, we see that while we have
always T (q') = {Tq)\ as in 190, and \]{q') = U(^)% as in 161,
we have also,
I. . .Z(g)^ = 2z^, and Ax. (q') = Ax. q, if Zg<|; /^^
but, by the adopted definitions of ^^'(130), and of Ax. 5'^^<C[2^
(127, 128), ^^[j
II. ..z(^^) = 2(7r-z^), Ax.(^0=-Ax.^, if z^>^. /
In each case, however, by 196, XVI., we may write, i^
lU. . .S\J(q') = C0SL{q')=C0s2lq;
2 B
186 ELEMENTS OF QUATERNIONS. [bOOK II.
a formula which holds even when z 5^ is 0, or -, or tt, and
which gives,
IV.. . S\](q') = 2{S\]qy-l.
Hence, generally, the scalar of q"^ may be put under either of
the two following forms :
V. . . S(q') = TqKcos2z.q; YL . . S C^^) = 2 (S^)^ - T^'' ;
where we see that it would not be safe to omit the parentheses,
without some convention previously made, and to write simply
Sq\ without first deciding whether this last symbol shall be
understood to signify the scalar of the square, or the square of
the scalar of q: these two things being generally unequal.
The latter of them, however, occurring rather oftener than the
former, it appears convenient to fix on it as that which is to
be understood by Sq^, while the other may occasionally be
written with a point thus, S.q^; and then, with these conven-
tions respecting notation* we may write :
VII. . . Sq' =-. (Sqy ; VIII. . .S.q'=S {q%
But the square of the conjugate of any quaternion is easily seen
to be the conjugate of the square ; so that we have generally
(comp. 190, II.) the formula:
IX. . . K^^ = K {q^) = {KqY = Tq^ : J]q\
(1.) A quaternion, like a positive scalar, may be said to have in general two oppo-
site square roots ; because the squares of opposite quaternions are always equal
(comp. (3.) ). But of these two roots the principal (or simpler') one, and that which
we shall denote by the symbol V9, or Vg-, and shall call by eminence the Square Root
of q^ is that which has its angle acute, and not obtuse. We shall therefore write,
generally, _
^. . . LMq=^ Lq'; Ax. Vg'= Ax. q ;
* As, in the Differential Calculus, it is usual to write da;2 instead of (dx)2 .
while d(x2) is sometimes written as d.x^. But as d^a; denotes a, second differential,
so it seems safest not to denote the square of Sq by the symbol S^q, which properly/
signifies SSg, or Sq, as in 196, VI. ; the second scalar (like the second tensor, 187,
(9,), or the second versor, 160) being equal to the^r*^ Still everj'^ calculator will
of course use his own discretion ; and the employment of the notation S^q for (87)^,
as cos ^x is often written for (cos x)^, may sometimes cause a saving of space.
CHAP. I.] TENSOR AND NORM OF A SUM. 187
with the reservation that, when lq = 0, or = tt, this common axis of q and Vg be-
comes (by 131, 149) an indeterminate unit-line.
(2.) Hence,
XI. ..SVg'>0, if Lq<TT;
while this scalar of the square root of a quaternion may, by VI., be thus trans- ^ ^
formed :
XII. ..SV9 = V{K'r? + Sg)}; \ 4
a formula which holds good, even at the limit Lq—Tr.
(3.) The principle* (1.), that in quaternions, as in algebra, the equation,
XIII... (-9)2 = 92, ^ /^
is an identity^ may be illustrated by conceiving that, in Fig. 42, a point b' is deter-
mined by the equation ob' =bo ; for then we shall have (comp. Fig. 33, his\ L /
(- 0)2 = I — ) = — = 7^, because A aob' a b'oc.
^ ''^ VoA y OA ^ '
200. Another useful connexion between scalars and tensors (or
norms) of quaternions may be derived as follows. In any plane tri-
angle AOB, we havef the relation,
(T. ab)2= (T. oa)2 - 2(T.oa) . (T. ob) . cos aob + (T. ob)2;
in which the symbols T. oa, &c., denote (by 185, 186) the lengths of
the sides oa, &c. ; but if we still write q = 0B: oa, we have q-l
= ab: oa; dividing therefore by (T. oa)^, the formula becomes (by
196, &c.),
I.. . T{q-iy = l-2Tq.SUq + Tq' = Tf-2Sq+l',
or
II.. .N(^-1)=%-2S^+1.
But q is here a perfectly general quaternion; we may therefore
change its sign, and write,
III. ..T (1 + ^)^=1 + 2S?+T^^ IV. ..N(l+j)=I + 2S^ + %.
And since it is easy to prove (by 106, 107) that
+ 1
)^=''"^' .r>-c f
whatever two quaternions q and q^ may be, while .
we easily infer this other general formula,
VII. . . N (^' + ?) =N^' + 2S . qKq' f %;
which gives, if x be any scalar,
VIII. . . N (^ + a;) = N^ + 2x^q -f x\
* Compare the first Note to page 162.
t By the Second Book of Euclid, or by plane trigonometry
188 ELEMENTS OF QUATERNIONS. [bOOK II.
(1.) We are now prepared to effect, hy rules* of transformation^ some other pas-
sages from one mode of expression to another, of the kind which has been alluded to,
and partly exemplified, in former sub-articles. Take, for example, the formula,
T^^!^=l, of 187, (2.);
p-a
or the equivalent formula,
T(p + a) = T(p-a), of 186, (6.) ;
■which has been seen, on geometrical grounds, to represent a certain locus, namely the
plane through o, perpendicular to the line oa ; and therefor the same locus as that
which is represented by the equation,
S- = 0, of 196, (1.).
a
To pass now from the former equations to the latter, by calculation, we have only
to denote the quotient p: ahy q, and to observe that the first or second form, as just
now cited, becomes then,
T(^ + l) = T(g-l); or N(9 + 1) = N (^ - 1) ;
or finally, by II. and IV.,
S^=0,
which gives the third form of equation, as required.
(2.) Conversely, from S - = 0, we can return, by the same general formulas II.
a *
andlV., to the equation n[^-1J=: Nl^+l\ or by I. and III. to Tf^-1 J
= T -+l\ ortoT(p- a") = T(|0 + a), orto T*^ — -= 1, as above; and gene-
\a J p-a
rally,
Sq = gives T(9-1) = T(^+1), or T^=l;
while the latter equations, in turn, involve, as has been seen, the former.
(3.) Again, if we take the Apollonian Locus, 145, (8.), (9.), and employ the Jirst
of the two forms 186, (5.) of its equation, namely,
T(p-a2a)=aT(p-a),
where a is a given positive scalar difierent from unity, we may write it as
T(g-a2) = aT(5-l), or as N (q - a^) = a-^^ {q - 1) ;
or by VIII.,
% - 2a^Sq + a* = a2 (Ng - 2Sq + 1) ;
or, after suppressing - 2a^Sq, transposing, and dividing by a^ _ i^
Ny = a2; or, Np = a2Na; or, Tp = aTa ;
which last is the second form 186, (5.), and is thus deduced from the first, hy calcu-
lation alone, without any immediate appeal to geometry, or the construction of any
dia^am.
* Compare 145, (10.) ; and several subsequent sub-articles.
CHAP. I.] GEOMETRICAL EXAMPLES. 189
(4.) Conversely if we take the equation, , ""^ ' ^ /Vl^
N^ = a2, of 145, (12.), ^
a
which was there seen to represent the same locus, considered as a spheric surface,
with o for centre, and aa for one of its radii, and write it as Ng^ = a2, we can then
hy calculation return to the form
N(g-a2) = a2N(9-l), or T (q-a^) = aT (q -1),
or finally,
T (p - a2a) = aT (p - a), as in 186, (5.) ;
this /rsf/orm of that sub- article being thus deduced from the second, namely from
Tp=aTa, or T- = a. ^
(5.) It is far from being the intention of the foregoing remarks, to discourage
attention to i^xe geometrical interpretation of the various /orws of expression^ and ,^ x
general rules of transformation, which thus offer themselves in working with qua- '^ ^IT
ternions ; on the contrary, one main object of the present Chapter has been to es-
tablish a firm geometrical basis, for all such forms and rules. But when such a. foun-
dation has once been laid, it is, as we see, not necessary that we should continually
recur to the examination of it, in building up the superstructure. That each of the
two forms, in 186, (5.), involves the other, may he proved, as above, by calculation ;
but it is interesting to inquire what is the meaning of this result : and in seeking to
interpret it, we should be led anew to the theorem of the Apollonian Locus.
(6.) The result (4.) of calculation, that
N (g - a2) = a2N (g - 1), if N^ = a2,
may be expressed imder the form of an identity, as follows :
IX. . .N(g-N5) = %.N(g-l);
in which q may be any quaternion.
(7.) Or, by 191, VII., because it will soon be seen that
q(jq-i) = q^ — q, as in algebra,
we may write it as this other identity :
X. . . N(g-Ng) = N(52-5).
(8.) If T (9 - 1) = 1, then S - = - ; and conversely, the former equation follows
q 2
from the latter; because each may be put under the form (comp. 196, XII.),
Ng = 2Sg.
2a
(9.) Hence, if T (p - a) = Ta, then S — = 1, and reciprocally. In fact (comp.
196, (6.) ), each of these two equations expresses that the locus of p is the sphere
which passes through o, and has its centre at a ; or which has on = 2a for a dia-
meter.
(10.) By changing 7 to 7 + 1 in (8), we find that
if Tq=\, then S - — - = 0, and reciprocally.
190 ELEMENTS OF QUATERNIONS. [boOK II.
(11.) Hence if T|0=Trt, then S^ — ^ = 0, and reciprocally ; because (by 106)
|0 + a
-a p—ap+a
p+a
(12.) Each of these two equations (11.) expresses that the locus of pis the
sphere through a, which has its centre at o ; and their proved agreement is a recog-
nition, by quaternions, of the elementary geometrical theorem, that the angle in a
semicircle is a right angle.
^■■'-i'-ii-Mi^^
Section 13. — On the Right Part (or Vector Part) of a Qua-
ternion ; and on the Distributive Property of the Multipli-
cation of Quaternions.
201. A given vector ob can always be decomposed, in one
but in only one way, into two component vectors, of which it
is the sum (6) ; and of which one, as ob' in Fig. 50, is parallel
(15) to another given vector oa, while ,,
the other, as ob" in the same Figure, is i .
perpendicular to that given line oa; j ^^
namely, by letting fall the perpendicu- j ^^^
lar bb' on oa, and drawing ob" = b'b, so \y^
that ob'bb" shall be a rectangle. In p-j g^
other words, if a and j3 be any two given,
actual, and co-initial vectors, it is always possible to deduce
from them, in one definite way, two other co-initial vectors,
/3' and j3", which need not however both be actual (I); and
which shall satisfy (comp. 6, 15, 129) the conditions,
j3' vanishing, when j3 _L a ; and /3" being null, when j3 || a ;
but both being (what we may call) determinate vector func-
tions of a and /3. And of these two functions, it is evident
that j3' is the orthographic projection of j5 on the line a ; and
that j3" is the corresponding j^ro/ec^/ow o/j3 on the plane through
o, which is perpendicular to a.
202. Hence it is easy to infer, that there is always one,
but only one way, of decomposing a given quaternion^
q = 0B : 0A = /3 : a,
into two parts or summands (195), of which one shall be, as in
CHAP. I.] RIGHT PART OF A QUATERNION. 191
196, a scalar, while the other shall be a right quotient (132).
Of these two parts, the. former has been already called (196)
the scalar part, or simply the Scalar of the Quaternion, and
has been denoted by the symbol ^q ; so that, with reference
to the recent Figure 50, we have
I. . . S3' = S(oB : oa) = ob': OA ; or, S (j3 : a) =/3': a.
And we now propose to call the latter part the Eight Part*
of the same quaternion, and to denote it by the new symbol
writing thus, in connexion with the same Figure,
11. . . V^ = V(ob:oa) = ob":oa; or, V(i3 : a) = i3": a.
The System of Notations, peculiar to the present Calculus,
will thus have been completed ; and we shall have the follow-
ing general Formula of Decomposition of a Quaternion into tivo
Summands (comp. 188), of the Scalar and Right kinds :
III. ..^=S^ + V^ = V^ + S^,
or, briefly and symbolically,
IV. . . 1 = S + Y = V+S.
(1.) In connexion with the same Fig. 50, we may write also,
V(ob:
OA) =
b'b : OA,
be(
cause, by construction, b'b = ob".
I
C2.) In like manner, for Fig. 36, we have the equation,
P !>^
V(ob:
OA) =
a'b : OA.
(3.) Under the recent conditions,
V(j3':a)=0,
and
S(/3":a) = 0.
(4.) In general, it is evident that
V. ..g=0, if S^=0,
and
V5'=0; and]
reciprocally.
(5.) More generally,
VI. ..9' =9, if ^q=Sq,
and
Yq = Yq ; with the converse.
(6.) Also VII. ..Y9 = 0,
if
lq = 0, or =
tt;
or
VIII. .. V(^:a) =
= 0, if /3i|a;
the right part of a scalar being zero.
* This Eight Part, Yq, will come to be also called the Vector Part, or simply
the Vector, of the Quaternion ; because it will be found possible and useful to iden-
tify such part with its own Index- Vector (133). Compare the Notes to pages 119,
136, 174.
192 ELEMENTS OF QUATERNIONS. [bOOK II.
(7.) On the other hand,
X...Vn = o. if /« =
IX...Yq = q, if iq^'^;
a right quaternion behig its own right part.
203. We had (196, XIX.) a formula which may now be
written thus,
I. . . ob'= S(oB : oa). OA, or /3' = S--a,
to express the projection o/ob on oa, or of the vector /3 on a ;
and we have evidently, by the definition of the new symbol
V^-, the analogous formula,
II. . . ob" = V (oB : oa) . oA, or /3" = V - • a,
a
to express the projection of (5 on the plane (through o), which
is drawn so as to be perpendicular to a ; and which has been
considered in several former sub-articles (comp. 186, (6.), and
196, (1.) ). It follows (by 186, &c.) that
III. . . Tj^" = TY— Ta= perpendicular distance of a from oa;
this perpendicular being here considered with reference to its
length alone, as the characteristic T of the tensor implies. It
is to be observed that because the factor, V — , in the recent
a
formula II. for the projection jS", is not a scalar, we must write
that factor as a multiplier, and not as a, multiplicand ; althougli
we were at liberty, in consequence of a general convention
(15), respecting the multiplication of vectors and scalars, to
denote the other projection j3' under the form,
r. ..i3' = aS2(196,XIX.).
(1.) The equation,
V^ = 0,
expresses that the locus of p is the indefinite right line oa. , V
(2.) The equation,
ve:i^=o, or ve = v^,
a a a
."■f
'4^
CHAP. I.] GEOMETRICAL EXAMPLES. 193"
expresses that the locus of p is the mdefinite right line bb", in Fig. 60, which is
drawn through the point B, parallel to the line oa.
(3.) The equation
S^Z^ = 0, or S^ = s2, ofl96, (2.),
a a a
has been seen to express that the locus of p is the plane through b, perpendicular
to the line oa ; if then we combine it with the recent equation (2.), we shall express
that the point p is situated at the intersection of the two last mentioned loci ; or that
it coincides with the point b,
(4.) Accordingly, whether we take the two first or the two last of these recent
forms (2.), (3.), namely,
ve^=o, st^=o, or ve=v^, se=s^,
a a a a a a
we can infer this position of the point p: in the first case by inferring, through 202,
v., that = 0, whence p- (3=0, by 142 ; and in the second case by inferring,
a
through 202, VI., that - = — ; so that we have in each case (comp. 104), or as a
consequence from each system, the equality p = /3, or op = on ; or finally (comp. 20)
the coincidence, P = B.
(5.) The equation, ^^ p ^ ^^ ^
a a
expresses that the locus of the point P is the cylindric surface of revolution, which
passes through the point b, and has the line oa for its axis ; for it expresses, by III.,
that the perpendicular distances of P and B, from this latter line, are equal.
(6.) The system of the two equations,
TV^=TV^, S^ = 0,
a a y
expresses that the locus of p is the (generally) elliptic section of the cylinder (5.),
made by the plane through o, which is perpendicular to the line oc.
(7.) If we employ an analogous decomposition of p, by supposing that
p=p' + p", p'\\a, p"-^a,
the three rectilinear or plane loci, (1.), (2.), (8.), may have their equations thus
briefly written :
p" = 0; p" = /3"; p' = /3':
while the combination of the two last of these gives p = |3, as in (4.).
(8.) The equation of the cylindric locus, (5.), takes at the same time the form,
Tp" = T/3";
which last equation expresses that the projection p" of the point p, on the plane through
o perpendicular to OA, falls somewhere on the circumference of a circle, with o for
centre, and ob" for radius : and this circle may 'accordingly be considered as the hast
of the right cylinder, in the sub-article last cited.
204. From the mere circumstance that V^ is always a
right qvotient (132), whenceUV^' is a right versor (153), of
2 c
194 ELEMENTS OF QUATERNIONS. [bOOK II.
which the plane (119), and the axis (127), coincide with those
of §', several general consequences easily follow. Thus we have
generally, by principles already established, the relations :
I. . .ZV^ = ^; 11. . . Ax.V^ = Ax.tJV^ = Ax.ry;
III. . . KV^ = - V^, or KV = - V (144) ;
IV. ..SV^ = 0, or SV=0(196, VII.);
V. . .(UV^)2 = -1 (153,159);
and therefore,
VI. . . (V^)2 = -(TV^)^ = -NV^,*
because, by the general decomposition (188) of a quaternion
mio factors^ we have
VII. .. V^ = TV^.UV^.
We have also (comp. 196, VI.),
VIII. . . VS^ = 0, or VS = (202, VII.) ;
IX. . . VV^ = V^, or V^ = VV = V (202, IX.) ;
and X. .. VK^=-V(?, or VK = - V,
because conjugate quaternions have opposite right parts, by the
definitions in 137, 202, and by the construction of Fig. 36.
For the same reason, we have this other general formula,
XI. . . K^ = S^-V^, or K = S-V;
but we had
^ = S^ + V^, or I = S + V, by 202, III., IV.;
hence not only, by addition,
q + Kq = 2Sqy or 1 + K= 2S, as in 196, I.,
but also, by subtraction,
XIL ..^-K^ = 2V^, or I-K = 2V;
whence the Characteristic, V, of the Operation of taking the
RightPartofa Quaternion (comp. 132, (6.); 137; 156; 187;
196), may be dejined hj either of the two following symbolical
equations :
XIII. .. v = i-S(202, IV.); XIV. . . v = i(i-K);
whereof the former connects it with the characteristic S, and
* Compare the Note to page 130.
CHAP* I.] PROPERTIES OF THE RIGHT (OR VECTOR) PART. 195
the latter with the characteristic K ; while the dependence of
K on S and V is expressed by the recent formula XI. ; and
that of S on K by 196, 11'. Again, if the line ob, in Fig. 50,
be multiplied (15) by any scalar coefficient, the perpendicular
bb' is evidently multiplied by the same ; hence, generally,
XV. . . Nxq = rcV^, if x be any scalar ;
and therefore, by 188, 191,
XVI. .,Yq = Tq . VU^, and XVII. . . TV^ = Tq .TVU^.
But the consideration of the right-angled triangle, ob'b, in the
same Figure, shows that
XVIII. . .TV^ = T^.sinz^,
because, by 202, II., we have
TV^ = T(ob":oa) = T.ob":T.oa,
and
T.ob"= T.ob . sin aob ;
we arrive then thus at the following general and useful for-
mula, connecting quaternions with trigonometry anew :
XIX. . .TVU^ = sinz^;
by combining which with the formula,
SU^ = co3Z^(196, XVL),
we arrive at the general relation :
XX. ..(SU^)2 + (TVUy)2 = l;
which may also (by XVII., and by 196, IX.) be written thus :
XXI. ..(S^)^-f(TV^)^=(T^)^;
and might have been immediately deduced, without sines and
cosines, from the right-angled triangle, by the property of the
square of the hypotenuse, under the form,
(T.ob')2+ (T.b'b)'^ = (T.ob)^
The same important relation may be expressed in various other
ways ; for example, we may write,
XXII. . . % = T^2 = S^^ - Yq\
where it is assumed, as an abridgment oi notation (comp. 199,
VII., VIII.), that
XXIII. . . V^^ = {Yq)\ but that XXIV. . . V. j^ = V(f ),
196 ELEMENTS OF QUATERNIONS. [bOOK II.
the import of this last symbol remaining to be examined.
And because, by the definition of a norm, and by the proper-
ties of S^ and V^',
XXV. . . NS^ = Sf , but XXVI. . . NVy = - Yq\
we may write also,
XXVII. . . % = N(S^ + Yq) = NS^ + NV^ ;
a result which is indeed included in the formula 200, VIII.,
since that equation gives, generally,
XXVIII. . .N(y + rr) = % + Naj, if z^ = ^;
X being, as usual, any scalar. It may be added that because
(by 106, 143) we have, as in algebra, the identity,
XXIX. ..-(?'+?) = -?'- y,
the opposite of the sum of any two quaternions being thus equal
to the sum of the opposites, we may (by XL) establish this
other general formula :
XXX. ..-K^ = V^-S^;
the opposite of the conjugate of any quaternion q having thus
the same right part as that quaternion, but an opposite scalar
part.
(1.) From the last formula it may be inferred, that
if q' = -Kq, then Yq' = + Yg, but Sq' = -Sq;
and therefore that
Tq'=Tg, and Ax. 5'= Ax. g, but L<i =^tt— Lq\
which two last relations might have been deduced from 138 and 143, without the
introduction of the characteristics S and V.
(2.) The equation,
(v^Y=fv^V, or(byXXVL), NV^ = NV^,
\ a \ \ a j a a
like the equation of 203, (5.), expresses that the locus of p is the right cylinder, or
cylinder of revolution, with oa for its axis, which passes through the point b.
(3.) The system of the two equations,
V2
[^'^-[^l\ «^»-
like the corresponding system in 203, (6.), represents generally an elliptic section of
the same right cylinder ; but if it happen that y H a, the section then becomes cir-
cular.
CHAP. I.] GEOMETRICAL EXAMPLES. 197
(4.) The system of the two equations,
S- = x, (v^]=£c2_i, with x>-l, x<J,
represents the circle,* in which the cylinder of revolution, with OAfor axis, and with
(1 - x^)iTa for radius, is perpendicularly cut by a plane at a distance = + xTa from
o ; the vector of the centre of this circular section being xa.
(5.) While the scalar x increases (algebraically) from — 1 to 0, and thence to
+ 1, the connected scalar VCl - x^) at first increases from to 1, and then decreases
from 1 to ; the radius of the circle (4.) at the same time enlarging from zero to a
maximum =Ta, and then again diminishing to zero ; while the position of the centre
of the circle varies continuously, in one constant direction, from ajirst limit-point a',
if oa' = — a, to the point A, as a second limit.
(6.) The locus of all such circles is the sphere, with aa' for a diameter, and there-
fore with o for centre ; namely, the sphere which has already been represented by the
equation Tp = Ta of 186, (2.) ; or by T ^ = 1, of 187, (1.) ; or by
S^^^ = 0, of 200, (11.);
' p + a
but which now presents itself under the new form,
1.
(»;y-('n"-
obtained by eliminating x between the two recent equations (4).
(7.) It is easy, however, to return from the last form to the second, and thence
to the first, or to the third, by rules of calculation already estabhshed, or by the ge-
neral relations between the symbols used. In fact, the last equation (6.) may be
written, by XXII., under the form,
a
whence
T^=l, by 190, VI.;
a
and therefore also Tp = Ta, by 187, and S ^^ = 0, by 200, (11.).
p -\- ci
(8.) Conversely, the sphere through a, with o for centre, might already have
been seen, by the first definition and property of a norm, stated in 145, (ll.)> to ad-
mit (comp. 145, (12.) ) of being represented by the equation N - = 1 ; and there-
a
fore, by XXII., under the recent form (6.) ; in which if we write x to denote the
variable scalar S -, as in the first of the two equations (4.), we recover the second of
those equations : and thus might be led to consider, as in (6.), the sphere in question
* By the word " circle," in these pages, is usually meant a circumference, and
not an area ; and in like manner, the words " sphere," *' cylinder," " cone," &c., are
usually here employed to denote surfaces, and not volumes.
198 ELEMENTS OF QUATERNIONS. [bOOK II.
aa the locus of a variable circle^ which is (as above) the intersection of a variable
cylinder^ with a variable plane perpendicular to its axis.
(9.) The same sphere may also, by XXVII., have its equation written thus.
Nfs^ + V^Vl; or Tfs^-fV^V
(10.) If, in each variable plane represented^by the first equation (4.), we conceive
the radius of the circle, or that of the variable cylinder, to be multiplied by any con-
stant and positive scalar a, the centre of the circle and the axis of the cylinder re -
maining unchanged, we shall pass thus to a new system of circles, represented by this
new system of equations,
se=
"' [^L] -""-'■
(11.) The locus of these new circles will evidently be a Spheroid of Revolution ;
the centre of this new surface being the centre o, and the axis of the same surface
being the diameter aa', of the sphere lately considered : which sphere is therefore
either inscribed or circumscribed to the spheroid, according as the constant a > or
< 1 ; because the radii of the new circles are in the first case greater, but in the se-
cond case less, than the radii of the old circles ; or because the radius of the equator
of the spheroid = aTa, while the radius of the sphere = Ta.
(12.) The equations of the two co-axal cylinders of revolution, which envelope
respectively the sphere and spheroid (or are circumscribed thereto) are :
(v-:y=-- (^£1=-
NV^=l, NV^=i
TV^=1, TV^=a.
a a
(13.) The system of the two equations,
S-=ir, (v|j=a;2_i, with j3 no< II a,
represents (comp. (3.) ) a variable ellipse, if the scalar x be still treated as a va-
riable.
(14.) The result of the elimination of x between the two last equations, namely
this new equation,
or
NS ^ + NV §= 1, by XXV., XXVI. ;
a p
or
Nfs^ + v|Ul, by XXVII.;
or finally,
Tfs^ + V|^=l, by 190, VI.,
CHAP. I.] QUATERNION EQUATION OF THE ELLIPSOID. 199
represents the locus of all such ellipses (13.), and will be found to be an adequate
representation, through quaternions, of the general Ellipsoid (with three unequal
axes) : that celebrated surface being here referred to its centre, as the origin o of
vectors to its points ; and the six scalar (or algebraic) constants, which enter into ^/
the usual algebraic equation (by co-ordinates) of such a central ellipsoid, being here / «-^
virtually included in the two independent vectors, a and (3, which may be called its
two Vector- Constants *
(15.) The equation (comp. (12.) ),
(il=
NV|=1, or TV^=1,
represents a cylinder of revolution, circumscribed to the ellipsoid, and touching it
along the ellipse which answers to the value a: = 0, in (13.) ; so that the plane of
this ellipse of contact is represented by the equation,
a
the normal to this pZane being thus (comp. 196, (17.) ) the vector a, or oa; while
the axis of the lately mentioned enveloping cylinder is (3, or ob.
(16.) Postponing any further discussion of the recent quaternion equation of the
ellipsoid (14.), it may be noted here that we have generally, by XXII., the two fol-
lowing useful transformations for the squares, of the scalar Sq, and of the right part
Yq, of any quaternion q :
XXXI. ..852 = T52 f V52 ; XXXII. ..¥52= Sq^ - Tq^.
(17.) In refei-ring briefly to these, and to the connected formula XXII., upon
occasion, it may be somewhat safer to write,'
(S)2 = (T)2 + (V)2, (Vy = (S)2 - (T)2, (T)2 = (S)2 - (V)2,
than S2 = T2 + V2, &c. ; because these last forms of notation, S2, &c., have been
otherwise interpreted already, in analogy to the known Functional Notation, or No-
tation of the Calculus of Functions, or of Operations (comp. 187, (9.); 196, VI. ;
and 204, IX.).
(18.) In pursuance of the same analogy, any scalar may be denoted by the gene-
ral symbol,
V-'O;
because scalars are the only quaternions of which the right parts vanish.
(19.) In like manner, a right quaternion, generally, maybe denoted by the sym-
bol,
S-'O;
and since this includes (comp. 204, I.) the right part of any quaternion, we may
establish this general symbolic transformation of a Quaternion :
5 = v-io + s-io.
(20.) With this form of notation, we should have generally, at least for realf
quaternions, the inequalities,
• It will be found, however, that other pairs of vector-constants, for the central
ellipsoid, may occasionally be used with advantage.
t Compare Art. 149 ; and the Notes to pages 90, 134.
200 ELEMENTS OF QUATERNIONS. [bOOK II.
(V-i0)2>0; (S-»0)2<0;
so that a (geometrically real) Quaternion is generally of the form :
Square-root of a Positive^ plus Square-root of a Negative.
(21.) The equations 196, XVI. and 204, XIX. give, as a new link between qua-
ternions and trigonometry, the formula :
XXXIII. . . tan Z 5 = TVUg : SUg = TV? : S?.
(22.) It may not be entirely in accordance with the theory of that Functional
(or Operational) Notation, to which allusion has lately been made, but it will be
found to be convenient in practice, to write this last result under one or other of the
abridged forms : *
TV
XXXIV. . . tan z: 9 = — - . 5 ; or XXXIV. . . tan Z 9 = (TV : S) 9 ;
o
which have the advantage oi saving the repetition of the symbol of the quaternion ,
when that symbol happens to be a complex expression, and not, as here, a single let-
ter, q.
(23.) The transformation 194, for the index of a right quotient, gives generally,
by II., for any quaternion q, the formulae :
XXXV. . . IVg = TV? . Ax. ? ; XXXVI. . . IUV9 = Ax. q ;
so that we may establish generally the symbolicalf equation,
xxxvr. . . iuv = Ax.
(24.) And because Ax. (1 : Yq) = - Ax. Vg-, by 135, and therefore = - Ax. q, by
II., we may write also, by XXXV.,
XXXV. . . I (1 : Vg) = - Ax. 5 : TV?.
205. If any parallelogram obdc (comp. 197) be projected
on the plane through o, which is perpendicular to oa, the pro-
jected figure obV'c" (comp. 11) is still a parallelogram; so
that
od" = oc" + ob" (6), or S" = 7" + /3" ;
and therefore, by 106,
g":a=(7":a) + (i3":a).
Hence, by 120, 202, for any two quaternions, q and q\ we have
the general formula,
• Compare the Note to Art. 199.
t At a later stage it will be found possible (comp. the Note to page 174, &c.),
to write, generally,
IV? = V?, lUV? = UV? ;
and then (comp. the Note in page 118 to Art. 129) the recent equations, XXXVI.,
xxxvr., will take these shorter forms :
Ax. ? = UV? ; Ax. = UV.
CHAP. I.] RIGHT PART OF A SUM OR DIFFERENCE. 201
with which it is easy to connect this other,
IL..y(q'-q) = Yq-yq.
Hence also, for any three quaternions, q, q\ q\
V(?"+ (y' + !?)) = Vy"+ V(j' + 5) = V/+(V?' + V?) ;
and similarly for any greater number of summands : so that
we may write generally (comp. 197, II.),
III. . . VS.7 = SV^, or briefly III'. . . VS = SY ;
while the formula II. (comp. 197, IV.) may, in like manner,
be thus written,
IV. ..VA^ = AV^, or IV'. ..VA = AV;
the order of the terms added, and the mode Oti grouping them,
in III., being as yet supposed to remain unaltered, although
both those restrictions will soon be removed. We conclude
then, that the characteristic V, of the operation of taking the
right part (202, 204) of a quaternion, like the characteristic S
of taking the scalar (196, 197), and the characteristic K of
taking the conjugate (137, 195*), is a Distributive Symbol, or
represents a distributive operation: whereas the characteris-
tics, Ax., z, N, U, T, of the operations of taking respectively
theaa;25(128, 129), the«?z^/e(130), the?zorm (145, (11.) ), the
versor (156), and the tensor (187), are not thus distributive
symbols (comp. 186, (10.), and 200, VII.) ; or do not operate
upon a lohole (or sum)^ by operating on its parts (or sum-
mands).
(1.) We may now recover the sjiKibolical equation K^ = 1 (145), under the form
(comp. 196, VI.; 202, IV, ; and 204, IV. VIII. IX. XL):
V. . . K2 = (S-V)2 = S2-SV-VS + V2 = S + V=1.
(2.) In like manner we can recover eacli of the expressions for S^, V^ from the
other, under the forms (comp. again 202, IV.) :
VI. . . S2 = (1-V)2 = 1-2V + V2=1-V = S, as in 196, VI.;
VII.. . V2 = (1-S)3=1-2S+S2 = 1-S = V, as in 204, IX.;
or thus (comp. 196, II'., and 204, XIV.), from the expressions for S and V in terms
ofK:
* Indeed, it has only been proved as yet (comp. 195, (1.)), that KSj = SKj,
for the case of two summands ; but this result will soon be extended.
2 D
202 ELEMENTS OF QUATERNIONS. [bOOK II.
VIII.. .S2 = i(l+K:)2 = i(l + 2K + K2) = i(l + K) = S;
IX. . . V2 = ^(l-K)2=:i(l-2K + K2; = i(l-K) = V.
(3.) Similarly,
X.. . SV = i(l + K)(l-K) = i(l-K2)=0, as in 204, IV.;
and XI. . .VS = K1-K) (1 + K) = i (1-^0=0, as in 204, VIII.
206. As regards the addition {ov subtraction) of such n^^^
parts, Yq, V^-', or generally of any two right quaternions
(132), we may connect it with the addition (or subtraction) of
their indices (133), as follows. Let obdc be again any paral-
lelogram (197, 205), but let oa be now an unit-vector (129)
perpendicular to its plane ; so that
Ta=l, z(/3:a) = Z(7:a) = Z(S:a)=^, S = 7 + /3.
Let ob'd'c' be another parallelogram in the same plane, ob-
tained by a positive rotation of the former, through a right
angle, round oa as an axis ; so that
Z(i3':/3)=A(y:7) = ^(^':S)=|;
Ax. (j3' : ^) = Ax. (y : 7) = Ax. (S' : g) = a.
Then the three right quotients, /3 : a, 7 : a, and ^ : a, may re-
present any two right quaternions, q, q\ and their sum, q -\- q,
w^hich is always (by 197, (2.) ) itself o, right quaternion; and
the indices of these three right quotients are (comp. 133, 193)
the three lines j3', y\ S', so that we may write, under the fore-
going conditions of construction,
/3'=I(i3:a), y = I(7:a), S' = I(g:a).
But this third index is (by the second parallelogram) the sum
of the two former indices, or in symbols, ^' = 7' + /3' ; we may
therefore write,
I. ..!{(][ ^q) = lq +lq, if Z^ = Zg=|;
or in words the Index of the Sum* of any two Right Quater-
nions is equal to the Sum of their Indices, Hence, generally,
for any two quaternions, q and q\ we have the formula,
IL. .\Y{q-^q) = lYq^lYq,
* Compare the Note to page 174.
CHAP. I.] GENERAL ADDITION OF QUATERNIONS. 203
because V^-, Yq are aliuays right quotients (202, 204), and
V {q' + q) is always their sum (205, I.) ; so that the index of
the right part of the sum of any two quaternions is the sum of
the indices of the right parts. In like manner, there is no diffi-
culty in proving that
m...l{q'-q)^lq-lq, if Zj = /y = |;
and generally, that
IV. ..IV(^'-^)=IV^'-IV^;
the Index of the Difference of any two right quotients, or of
the right parts of any two quaternions, being thus equal to the
Difference of the Indices* We may then reduce the addition
or subtraction of any two such quotients, or parts, to the addi-
tion or subtraction of their indices ; a right quaternion being
always (by 133) determined, when its index is given, or
known.
207. We see, then, that as the Multiplication of any
tico Quaternions was (in 191) reduced to (1st) the arithmetical
operation of multiplying their tensors, and (Ilnd) the geometri-
cal operation of multiplying their versors, which latter Avas con^
structed by a certain composition of rotations^ and was repre-
sented (in either of two distinct but connected ways, 167, 175)
by sides or angles of a spherical triangle: so the Addition of
any two Quaternions maybe reduced (by 197, 1., and 206, II.)
to, 1st, the algebraical addition of their scalar parts ^ considered
as two positive or negative numbers (16) ; and, Ilnd, the geo-
metrical addition of the indices of their right parts, considered
as certain vectors (1) : this latter Addition of Lines being per-
formed according to the Rule of the Parallelogram (6.).t In
* Compare again the Note to page 174.
t It does not fall within the plan of these Notes to allude often to the history of
the subject ; but it ought to be distinctly stated that this celebrated Mule, for what
may be called Geometrical Addition of right lines, considered as analogous to compo-
sition of motions (or of forces), had occurred to several writers, before the invention
of the quaternions : although the method adopted, in the present and in a former
■work, of deducing that rule, by algebraical analogies, from the symbol b — A (1)
for the line ab, may possibly not have been anticipated. The reader may com-
pare the Notes to the Preface to the author's Volume of Lectures on Quaternions
(Dublin, 1853).
204 ELEMENTS OF QUATERNIONS. [bOOK II.
like manner, as the general Division of Quaternions was seen (in
191) to admit of being reduced to an arithmetical division of
tensors, and 2^ geometrical division ofversors, so we may now
(by 197, III., and 206, IV.) reduce, generally, the Subtrac-
tion of Quaternions to (1st) an algebraical subtraction of sea-
larsy and (Ilndj Sk geometrical subtraction of vectors: this last
operation being again constructed by a parallelogram, or even
by a plane triangle (comp. Art. 4, and Fig. 2). And because
the sum of any given set of vectors was early seen to have a
value (9), which is independent of their order, and of the mode
of grouping them, we may now infer that the Stim of any num-
ber of given Quaternions has, in like manner, a Value (comp.
197, (l'))» which is independent of the Order, and of the
Grouping of the Summands: or in other words, that the general
Addition of Quaternions is a Commutative* and an Associative
Operation.
(1.) The formula,
Y^q=-2Yq, of 205, III.,
is now seen to hold good, for any number of quaternions, independently of the arrange-
ment of the terms in each of the two sums, and of the manner in which they may be
associated.
(2.) We can infer anew that
K (q' + q) = K^' 4 Kg-, as in 195, II.,
under the form of the equation or identity,
S (7' + 9) - V (q +q)= {Sq - Yq) + QSq - Yq).
(3.) More generally, it may be proved, in the same way, that
K2g = 2 Kg, or briefly, K2 = SK,
whatever the number of the summands may be.
208. As regards the quotient or product of the right paHs, Yq and
Yq', of any two quaternions, let t and f denote the tensors of those
two parts, and let x denote the angle of their indices, or of their axes,
or the mutual inclination of the axes, or of the planes,] .of the two
quaternions q and q' themselves, so that (by 204, XVIII.),
* Compare the Note to page 175.
f Two planes, of course, make with each other, in general, two unequal and sup-
plementary angles ; but we here suppose that these are mutually distinguished, by
taking account of the aspect of each plane, as distinguished from the opposite aspect :
which is most easily done (HI-)) ''}' considering the axes as above.
CHAP. I.] QUOTIENT OR PRODUCT OF RIGHT PARTS. 205
t = TVq = Tq. sin Lq, f = TYq' =Tq\ sin /.q\
and
x = /. {lYq' : lYq) = L (Ax. q' : Ax. q).
Then, by 193, 194, and by 204, XXXV., XXXV'.,
I. . .Yq':Yq = lYq' :lYq = + (TYq' : TYq) . (Ax. q' : Ax. g) ;
II. . . V^^ V^ = IV^' : I ^ = - (T V^' . TYq) . (Ax. q'-.Ax.q)-,
and therefore (comp. 198), with the temporary abridgments pro-
posed above,
III. . . S ( V^' : V^) = ft' cos X ; IV. . . SU (Yq' : V^) = + cos x ;
V. . . S{Yq'.Yq)=-t'tcosx- VI. . . ^U {Yq\Yq) = - cos x;
VII. ..L{Yq':Yq) = x; VIII. . . L{Yq' . Yq)=7r-x.
We have also generally (comp. 204, XVIII., XIX.),
IX. . . TV (Yq' : Yq) = ft' sin a; ; X. . . T VU ( V^' : Yq) = sin a; ;
XI... TV(Vg'.V^)=i'^sina;; XII. . . TY\J (Yq' .Yq)= sin x;
and in particular,
XIII. . . V ( V^' : V^) = 0, and XIV. . . V ( V^' . V^) = 0,
if/|||.i(123);
because (comp. 191, (6.), and 204, VI.) the quotient or product of
the right parts of two complanar quaternions (supposed here to be
both 7ion-scalar (108), so that t audi' are each >0) degenerates (131)
into a scalar, which may be thus expressed :
XV. . . V^' : V^ = + tt\ and XVI. . .Yq\Yq = - t% if a; = ;
but
XVII. ..V^':V^ = -«'<-', and XYIU. . . Yq\Yq = + t% ifx = 7r;
the first case being that of coinciderd, and the second case that of
opposite axes. In the more general case oi diplanarity (119), if we
denote by B the unit-line which is perpendicular to both their axes,
and therefore common to their two planes, or in which those planes
intersect, and which is so directed that the rotation round it from
Ax. q to Ax. q' is positive (comp. 127, 128), the recent formulae I.,
II. give easily,
XIX. . . Ax. (V^': Vg) =+ a; XX. . . Ax. {Yq' ,Yq)=-h',
and therefore (by IX., XI., and by 204, XXXV.), the indices of the
right parts, of the quotient and product of the right parts of any two
diplanar quaternions, may be expressed as follows:
XXI. . . IV ( V^' : V<7) = + a . ft' sin x ;
XXII. . . IV {Yq'. Yq) = -S.fi sin x.
206
ELEMENTS OF QUATERNIONS.
[book ir.
(1.) Let ABC be any triangle upon the unit-sphere (128), of which the spheri-
cal angles and the corners may be denoted by the same letters A, b, c, while the sides
shall as usual be denoted by a^h^ c\ and let it be supposed that the rotation (comp.
177) round A from c to b, and therefore that round b from A to c, &c., U positive,
as in Fig. 43. Then writing, as we have often done,
q = (3: a, and q' = y ■ (3, where a = OA, &c.,
we easily obtain the the following expressions for the three scalars t, t', x, and for
the vector d :
i? = sin c ; if ' = sin a ; a; = tt — b ; d = - (3.
(2.) In fact we have here,
Tq = Tq=l, Lq = c, Lq=a\
whence t and <' are as just stated. Also if a', b', c' be (as in 175) the positive poles
of the three successive sides bc, ca, ab, of the given triangle, and therefore the points
A, b, c the negative poles (comp. 180, (2.)) of the new arcs b'c', c'a', a'b', then
Ax. q = oc'. Ax. q' = Oa' ;
but X and d are the angle and the axis of the quotient of these two axes, or of the
quaternion which is represented (162) by the arc c'a'; therefore x is, as above
stated, the supplement of the angle b, and d is directed to the point upon the sphere,
which is diametrically opposite to the point b.
(3.) Hence, by III. V. VII. VIII. IX. XI., for any triangle abc on the unit-
sphere, with a =OA, &c., we have the formulae:
XXIII.
XXV.
XXIV. . . s
^ V-
^v^
(4.) Also, by XIX. XX. XXI. XXII,
still positive,
XXXL
)
sin a cosec c cos b :
= + sin a sm c <
XXVI. . . L
H-^lh-
XXVII.
XXVIIL. . TV
+ sm a cosec c sm b ;
+ sin a sin c sin b.
if the rotation round b from a to c be
XXIX. . . Ax.
XXX. . . Ax.
('r'a
= + ^;
xxxn. . .ivi v^.v
V — 1 = — /3 sin a cosec c sin b
a j
/3'
+ (3sma sin c sin b.
(5.) If, on the other hand, the rotation round b from a to c were negative, then
writing for a moment ai= — a, /3i = — /?, yi = — y, we should have a new and ojo/jo-
site triangle, AiBiCi, in which the rotation round Bi from Ai to Ci would be positive,
but the angle at bi equal in magnitude to that at b ; so that by treating (as usual)
all the angles of a spherical triangle as positive, we should have Bi = b, as well as
Ci, = c, and ai — a; and therefore, for example, by XXXI.
CHAP. I.] COLLINEAR QUATERNIONS. 207
IV V ^ : V— ) = - /3i sin ai cosec ci sin bi,
V Pi ai I
or IV I V ^ : V - 1 = + j8 sin a cosec c sin b ;
\ (3 a]
the four formulae of (4.) would therefore still subsist, provided that, for this new
direction of rotation in the given triangle, we were to change the sign of [3, in the
second member of each.
(6.) Abridging, generally IVg' : ^q to (IV: S)^, as TVg: Sg- was abridged, in
204, XXXIV'., to (TV: S)*?, we have by (5.), and by XXIV., XXXII., this other
general formula, for any three unit- vectors a, /3, y, considered still as terminating
at the corners of a spherical triangle abc :
XXXIII. .. (IV:S)f v|.V^^ = ±j6tan
the upper or the lower sign being taken, according as the rotation round b from a to
A
c, or that round /3 from a to y, which might perhaps be denoted by the symbol rtj8y,
and which in quantity is equal to the spherical angle b, is positive or negative.
209. When the planes of any three quaternions q, q'^ q'\ consi-
dered as all passing through the origin o (119), contain any co7iimon
line, those three may then be said to be Collinear^- Quaternions ; and
because the axis of each is then perpendicular to that line, it follows
that the Axes of ColUnear Quaternions are Complanar : while con-
versely, the complanarity of the axes insures the collinearity of the
quaternions, because the perpendicular to the plane of the axes is a line
common to the planes of the quaternions.
(1.) Complanar quaternions are always collinear ; but the converse proposition
does not hold good, collinear quaternions being not necessarily complanar.
(2.) Collinear quaternions, considered as fractions (101), can always be reduced
to a common denominator (120) ; and conversely, if three or more quaternions can be
so reduced, as to appear under the form of fractions with a common denominator e,
those quaternions must be collinear : because the line e is then common to all their
planes.
(3.) Any two quaternions are collinear with any scalar ; the plane of a scalar
being indeterminate^ (I'^l)-
(4.) Hence the scalar and right parts, Sg, Sg', Vg, Vg', of any two quaternions,
are always collinear with each other.
(5.) The conjugates of collinear quaternions are themselves collinear.
* Quaternions of which the planes are parallel to any common line may also be
said to be collinear. Compare the first Note to page 113.
t Compare the Note to page 114.
208 ELEMENTS OF QUATERNIONS. [boOK II.
210. Let $', 5', ql' be any three collinear quaternions; and let a
denote a line common to their planes. Then we may determine
(comp. 120) three other lines y8, 7, ^, such that
^ a' ^ "a' ^ a'
and thus may conclude that (as in algebra),
because, by 106, 107,
^y .. ^y _ 7 + /^ g _ 7 + /3 _ 7 ^ /3 ^ 7 « ^ «
a a jh a ^ b S d a d a S
In like manner, at least under the same condition of collinearity,* it
may be proved that
II. . . {q'-q)q" = q'q"-qq''.
Operating by the characteristic K upon these two equations, and
attending to 192, II., and 195, II., we find that
III. . . K2^^(%'+%) = K$'^K/+K^^K^;
IV. . . K^'^(%'-K^) = K2'^K5'-K^'^K^;
where (by 209, (5.) ) the three conjugates of arbitrary collinears,
K5, K(2^ ^q"-> may represent any three collinear quaternions. We
have, therefore, with the same degree of generality as before,
V. . . q" {q' + g) = q"q' + q"q ; VI. . . q'^ {q' -q)= q"q' - q"q.
If, then, q^ q', q", q'"hQ any four collinear quatet-mons, we may esta-
blish the formula (again agreeing with algebra) :
VII. . . (q'^' + q") {q' + q) =- q'"q' + q"q' + q'"q + q'^q ;
and similarly for any greater number, so that we may write briefly,
VIII. .. ^q',^qr=:2q'q,
where
^q' = qy + q2+"-\-qm> ^q' = q'i + q2 + ' •+q'ny
and
-Eq'q = q\q, + . . q^'q^ -Yq'-iqx + . . . + q'^q^^,
m and n being any positive whole numbers. In words (comp. 13),
the Multiplication of Collinearf Quaternions is a Doitbli/ Distributive
Operation.
* It will soon be seen, however, that this condition is unnecessary.
t This distributive property of multiplication will soon be found (compare the last
Note) to extend to the more general case, in which the quaternions are not collie
near.
CHAP. I.] DISTRIBUTIVE MULTIPLICATION OF COLLINEARS. 209
(1.) Hence, by 209, (4.), and 202, III., we have this general transformation,
for the product of any two quaternions :
IX. .. qq = Sq. Sq + Yq\ Sq + Sq'.Yq + Yq'.Yq.
(2.) Hence also, for the square of any quaternion, we have the transformation
(comp. 126 ; 199, VII. ; and 204, XXHI.) :
X. . . q^=Sq^ + 2Sq.Yq + Yq^.
(3.) Separating the scalar and right par^s of this last expression, we find these
other general formulae :
XL . . S . 52 = S52 + Vg3 ; XII. . . V . 92 = 2Sg . V? ;
whence also, dividing by Tq^, we have
XIII. . . SU((?2) = (SU5)2 + (YUg)2; XIV. . . Y\JCq^) = 2S\Jq.YUq.
(4.) By supposing q' = Kq, in IX. , and therefore Sg' = Sg, Vg-' = — Yq, and trans-
posing the two conjugate and therefore complanar factors (corap, 191, (1.) ), we ob'-
tain this general transformation for a norm, or for the square of a tensor (comp. 190,
V. ; 202, III. ; and 204, XI.) :
XV. . . Tg2 = Ng = qKq = (Sg + Vg) (Sg - Vg) = Sg2 - Vg2 ;
which had indeed presented itself before (in 204, XXII.) but is now obtained in a
new way, and without any employment of sines, or cosines, or even of the well-known
theorem respecting the square of the hypotenuse.
(5.) Eliminating Vg2, by XV., from XI., and dividing by Tg2, we find that
XVI. . . S . 92 = 2Sg2 - Tg2 ; XVH. . . SU(g2) = 2 (SUg)2 - 1 ;
agreeing with 199, VI. and IV., but obtained here without any use of the known
formula for the cosine of the double of an angle.
(6.) Taking the scalar and right parts of the expression IX., we obtain these other
general expressions :
XVIII. . . Sg'g = Sg'. Sg + S(Vg'. Vg) ;
XIX. . . Yq'q = Yq'. Sq + Yq.Sq' + Y (Yq'.Yq) ;
in the latter of which we may (by 126) transpose the two factors, Vg', Sg, or Vg,
Sg'. We may also (by 206, 207) write, instead of XIX., this other formula :
XIX'. . . IVg'g = IVg'. Sg + IVg . Sg' + IV(Vg'. Vg).
(7.) If we suppose, in VII., that g" = Kg, g"' = Kg', and transpose (comp. (4.) )
the two complanar (because conjugate) factors, q' + q and K(g'+g), we obtain the
following general expression for the norm of a sum :
(g + g) K (g' + g) = g'Kg' + gKg' + g'Kg + gKg ;
or briefly,
XX. . . N (g' + g) = Ng' + 2S . gKg' + Ng, as in 200, VII. ;
because
g'Kg = K. gKg', by 192, II., and (1 + K).gKg'= 2S.gKg', by 196, II'.
(8.) By changing g' to x in XX., or by forming the product of g + a? and
Kg + X, where x is any scalar, we find that
XXI.. .N(g + a;) = ]Srg + 2a;Sg + a;2, as in 200, VIII. ;
whence, in particular,
XXr. . . N(g - 1) = Ng - 2Sg -|- 1, as in 200, II.
2 E
210 ELEMENTS OF QUATERNIONS. [bOOK II.
(9.) Changing q to fi: a, and multiplying by the square of Ta, we get, for any
two vectors, a and /3, the formula,
XXII. . . T(/3 - a)2 = T/32 - 2T/3 . Ta . SU ^ + Ta\
in which Ta2 denotes* (Ta)2; because (by 190, and by 196, IX.),
N(5-ll = Nt5=(I(^Y, and S^ = ^^Sne
\a J a \ Ta I a la a
(10.) In any plane triangle, abc, with sides of which the lengths are as usual
denoted by a, &, c, let the vertex c be taken as the origin o of vectors ; then
o = CA, /3 = CB, j3-a = AB, Ta = 6, T/3 = a, T(j3-a) = c, SU- = cosc;
a
we recover therefore, from XXII., the fundamental formula of plane trigonometry,
under the form,
XXIIl. . . c2 = a2 - 2ab cos c -i- b^.
(11.) It is important to observe that we have not here been arguing in a circle ;
because although, in Art. 200, we assumed, for the convenience of the student, a pre-
vious knowledge of the last written formula, in order to arrive more rapidly at certain
applications, yet in these recent deductions from the distributive property YIU. of
multiplication of (at least) collinear quaternions, we have founded nothing on the re-
sults of that former Article ; and have made no use of any properties of oblique-an-
gled triangles, or even of right-angled ones, since the theorem of the square of the
hypotenuse has been virtually proved anew in (4.) : nor is it necessary to the argu-
ment, that any properties of trigonometric functions should be known, beyond the
mere definition of a cosine, as a certain projecting factor, from which the formula
196, XVI. was derived, and which justifies us in writing cose in the last equation
(10.). The geometrical Examples, in the sub-articles to 200, may therefore be read
again, and their validity be seen anew, without any appeal to even plane trigonometry
being now supposed.'
(12.) The formula XV. gives Sg2 = T52 + V52, as in 204, XXXI. ; and we know
that V52, as being generally the square of a right quaternion, is equal to a negative
scalar (comp. 204, VI.), so that
XXIV . . Vg2 < 0, unless Lq = 0, or = tt,
in each of which two cases V9 = 0, by 202, (0.), and therefore its square vanishes ;
XXV. . . Sg2 < Tg2, (SU9)2 < 1,
in every other case.
* We are not yet at liberty to interpret the symbol Ta2 as denoting also T(a2) ;
because we have not yet assigned any meaning to the square of a vector, or generally
to the product of two vectors. In the Third Book of these Elements it will be shown,
that such a square or product can be interpreted as being a quaternion : and then it
will be found (comp, 190), that
T(a2) = (Ta)2 = Ta2,
whatever vector a may be.
CHAP. I.] APPLICATIONS TO SPHERICAL TRIGONOMETRY. 211
(13.) It might therefore have been thus proved, without any use of the transfor-
mation SUg = cos Z. 5- (196, XVI.), that (for any real quaternion q) we have the in-
equalities,
XXVI. . . SU9<+1, S>\Jq>-l, and S5<+Tg, S>q>-T:q,
unless it happen that Z g = 0, or = tt ; &\Jq being = + 1, and 85- = + Tg-, in the first
case ; whereas SUg = - 1, and Sg = — Tg, in the second case.
(14.) Since Tg2 = Ng, and Tq . Tq = T. gKg' = T . q'Kq = Ng . T (g' : g), while
S . gKg' = S . g'Kg = Ng . S (g' : g), the formula XX. gives, by XXVI.,
XXVII. . . (Tg' + Tg)2-T(g' + g)2 = 2(T-S)gKg' = 2Ng.(T-S) (g':g)>0,
if we adopt the abridged notation,
XXVIIL . . Tg - Sg = (T - S) g,
and suppose that the quotient g' : g is not a positive scalar ; hence,
XXIX. . . Tg' + Tg>T(g' + g), unless q=xq, and x>0;
in which excepted case, each member of this last inequality becomes = (1 + aj)Tg.
(15.) "Writing g = j3 : a, g'= 7 : a, and multiplying by Ta, the formula XXIX.
becomes
XXX. . . Ty + T/3>T(y + /3), unless y=a;/3, a;>0;
in which latter case, but not in any other, we have Uy = U/3 (155). We therefore
arrive anew at the results of 186, (9.), (10.), but without its having been necessary
to consider any triangle, as was done in those former sub-articles,
(16.) On the other hand, with a corresponding abridgment of notation, we have,
by XXVI.,
XXXI. . . Tg + Sg=(T+S)g>0, unless Z.g=7r;
also, by XX., &c.,
XXXII. . . T(g'+ g)^ - (Tg' -Tg)2= 2(T + S)gKg' = 2Ng.(T + S) (g' : g) ;
hence,
XXXIII. . . T (g' + g) > + (Tg' - Tg), unless g' = - a;g, a: > ;
where either sign may be taken.
(17.) And hence, on the plan of (15.), for any two vectors ]3, y,
XXXIV. . . T (y + 18) > + (Ty - T/3), unless Uy = - Uj3,
whichever sign be adopted ; but, on the contrary,
XXXV. ..T(y + /3) = ±(Ty-T/3), if Uy = -U/3,
the upper or the lower sign being taken, according as Ty > or < T/3 : all which
agrees with what was inferred, in 186, (11.), from ^eome^ncaZ considerations alone,
combined with the definition of Ta. In fact, if we make j3 = ob, y = oc, and - y
= oc', then obc' will be in general a plane triangle, in which the length of the side
BC' exceeds the difference of the lengths of the two other sides ; but if it happen that
the directions of the two lines ob, oc' coincide, or in other words that the lines OB,
oc have opposite directions, then the difference of lengths of these two lines becomes
equal to the length of the line bc'.
(18.) With the representations of g and g', assigned in 208, (1.), by two sides of
a spherical triangle abc, we have the values,
Sg = cosc, Sg' = cosa, Sg'g = S(y : a) = cos t ;
212 ELEMENTS OF QUATERNIONS. [bOOK II.
the equation XVIII. gives therefore, by 208, XXIV., the fundamental formula of
spherical trigonometry (comp. (10.) ), as follows :
XXXVI. . . cos 6 = cos a cos c + sin a sin c cos b.
(19.) To interpret, with reference to the same spherical triangle, the connected
equation XIX., or XIX'., let it be now supposed, as in 208, (6.), that the rotation
round b from c to a is positive, so that b and b' are situated at the same side of the
arc CA, if b' be still, as in 208, (2.), the positive pole of that arc. Then writing
a' = oa', &c., we have
\Yq — y sin c ; IV^'' = a' sin a ; IVg-'^ = — /3' sin 6 ;
and IV (Vg'. Yq) = — /3 sin a sin c sin b (comp. 208, (5.) ),
with the recent values (18.), for Sg and Sj'; thus the formula XIX'. becomes, by
transposition of the two terms last written :
XXXVII. . . j8 sin a sin c sin b = a sin a cos c + /3' sin h-\-y' sin c cos a.
(20.) Let jO =op be any unit-vector; then, dividing each term of the last equa-
tion by jO, and taking the scalar of each of the four quotients, we have, by 196, XVI.,
this new equation :
XXXVIII. . . sin a sin c sin b cos pb = sin a cos c cos pa' + sin h cos pb'
+ sin c cos a cos pc' ;
where a, 6, c are as usual the sides of the spherical triangle abc, and a', b', c' are
still, as in 208, (2.), the positive poles of those sides; but p is an arbitrary point,
upon the surface of the sphere. Also cos pa', cos pb', cos pc', are evidently the sines
of the arcual perpendiculars, let fall from that point upon those sides ; being positive
when p is, relatively to them, in the same hemispheres as the opposite corners of the
triangle, but negative in the contrary case ; so that cos aa', &c., are positive, and
are the sines of the three altitudes of the triangle.
(21.) If we place p at b, two of these perpendiculars vanish, and the last formula
becomes, by 208, XXVIIL,
XXXIX. . . sin6cosBB' = sinasincsinB = TVt V^.V- 1;
\ ^ aj
such then is the quaternion expression for the product of the sine of the side ca, mul-
tiplied by the sine of the perpendicular let fall upon that side, from the opposite ver-
tex B.
(22.) Placing p at A, dividing by sin a cos c, and then interchanging b and c, we
get this other fundamental formula of spherical trigonometry,
XL. . . cos aa'= sin c sin b = sin 6 sin c ;
and we see that this is included in the interpretation of the quaternion equation
XIX., or XIX'., as the formula XXXVI. was seen in (18.) to be the interpretation
of the connected equation XVIII.
(23.) By assigning other positions to p, other formulae of spherical trigonometry
may be deduced, from the recent equation XXXVIII. Thus if we suppose p to co-
incide with b', and observe that (by the supplementary* triangle),
* No previous knowledge of spherical trigonometry, properly so called, is here
supposed ; the supplementary relations of two polar triangles to each other forming
rather a part, and a very elementary one, of spherical geometry.
CHAP. I.] GENERAL DISTRIBUTIVE PROPERTY. 213
b'c' = tt — a, c'a' = tt — b, a'b'= tt — c,
while
cos bb' = sin a sin c = sin c sin a, by XL.,
we easily deduce the formula,
XLI. . . sin a sin c sin A sin b sin c = sin b — cos c cos c sin A - cos a cos A sin c ;
which obviously agrees, at the plane limit, with the elementary relation,
A + B + C = TT.
(24.) Again, by placing p at a', the general equation becomes,
XLII. . . sin a cos c = sin 6 cos c + sin c cos a cos b ;
with the verification that, at the plane limit,
a = 6 cos c + c cos b.
But we cannot here delay on such deductions, or verifications : although it appeared
to be worth while to point out, that the whole of spherical trigonometry may thus be
developed, from the fundamental equation of multiplication of quaternions (107), when
that equation is operated on by the two characteristics S and V, and the results
interpreted as above.
211. It may next be proved, as follows, that the distributive for-
mula I. of the last Article holds good, when the three quaternions,
^, 5', q"^ which enter into it, without being now necessarily colli-
7iem\ are right; in which case \h^\x reciprocals (135), and their swrns
(197, (2.) ), will be right also. Let then
and therefore,
We shall then have, by 106, 194, 206,
W+q')q=^l{q"+qy.lq,
= W:lq,) + W:lq) = q"g + q'q;
and the distributive property in question is proved.
(1.) By taking conjugates, as in 210, it is easy hence to infer, that the oMer dis-
tributive formula, 210, V., holds good for any three right quaternions ; or that
g(iq" + q') = 9q'+qq, if Lq = Lq= Lq'=-.
(2.) For any three quaternions, we have therefore the two equations:
(V^" + Yq') . Yq = Yq" . Yq + Yq' . Yq ;
Yq . (Yq" + Yq') = Yq . Yq" + Yq . Vg'.
(3.) The quaternions g, 7', q" being still arbitrary, we have thus, by 210, IX.,
214 ELEMENTS OF QUATERNIONS. [bOOK II.
{q" +9')'i = (S?" + S?') . S^ + (Vq" + Yq') . Sg + V(? . (Sg" + Sq') + ( Vg" + Yq') . Yq
= (Sq".Sq + Yq".Sq+Yq.Sq"+Yq".Yq) + {Sq'.Sq + Yq'.Sq + Yq.Sq' + Yq'.Yq)
= q"9 + QQ ;
so that the formula 210, I., and therefore also (by conjugates) the formula 210, V.,
is valid generally.
212. The General* Mtiltiplication of Quaternions is there-
fore (comp. 13,210) 2i Doubly Distributive Operation; so that
we may extend, to quaternions generally, the formula (comp,
210, VIII.),
I. . . ^q'.^q^^g'q:
however many the summands of each set may be, and whe-
ther they be, or be not, coUinear (209), or right (211).
(1.) Hence, as an extension of 210, XX., we have now,
11. . . KSg = 2% + 2 2S gKg' ;
where the second sign of summation refers to all possible binary combinations of the
quaternions g, q\ . .
(2.) And, as an extension of 210, XXIX., we have the inequality,
III. . . STg>T2g,
unless all the quaternions g, q', . . bear scalar and positive ratios to each other, in
which case the two members of this inequality become equal : so that the sum of the
tensors, of any set of quaternions, is greater than the tensor of the sum, in every
other case.
(3.) In general, as an extension of 210, XXVII,,
IV. . . (STg)2 - (T2g)2 = 22 (T - S) qKq.
(4.) The formulae, 210, XVIII., XIX., admit easily of analogous extensions.
(5.) We have also (comp. 168) the general equation,
V...(2y)2_2(g2) = 2(gg' + 5'5);
in which, by 210, IX.,
VI. . . qq' + q'q=2(iSq.Sq' + Yq.Sq +Yq'.Sq -^ S(Yq'.Yg));
because, by 208, we have generally
VII. . . Y(Yq'.Yq) = -Y(Yq.Yq);
or VIII. . . Yq'q = - Yqq, if /.q=lq^'^.
(Comp. 191, (2.), and 204, X.)
213. Besides the advantage which the Calculus of Quaternions
gains, from the general establishment (212) oi the Distributive Prin-
ciple, or Distributive Property of Multiplication, by being, so far,
* Compare the Notes to page 208.
CHAP. I.] INTERSECTIONS OF RIGHT LINES AND SPHERES. 215
assimilated to Algebra^ in processes which are of continual occur-
rence, this principle or property will be found to be of great im-
portance, in applications of that calculus to Geometry; and especially
in questions respecting the (real or ideal*) intersections of right
lines ivith spheres^ or other surfaces of the second order, including
contacts (real or ideal), as limits of such intersections. The follow-
ing Examples may serve to give some notion, how the general dis-
tributive principle admits of being applied to such questions : in
some of which however the less general principle (210), respecting
the multiplication of collinear quaternions (209), would be sufficient.
And first we shall take the case of chords of a sphere^ drawn from a
given point upon its surface.
(1.) From a point a, of a sphere with o for centre, let it be required to draw a
chord AP, which shall be parallel to a given
line OB ; or more fully, to assign the vector,
p = OP, of the extremity of the chord so drawn,
as a function of the two given vectors, a = OA,
and /3 = OB ; or rather of a and IJ(3, since it
is evident that the length of the line j3 cannot
affect the result of the construction, which Fig.
51 may serve to illustrate.
(2.) Since AP || ob, or p — a || /3, we may
begin by writing the expression,
p = a + x(3(15),
which may be considered (corap. 23, 99) as a form of the equation of the right line
AP ; and in which it remains to determine the scalar coefficient x, so as to satisfy the
equation of the sphere,
Tp=:Ta(186,(2.)).
In short, we are to seek to satisfy the equation,
T(a + a;/3) = Ta,
by some scalar x which shall be (in general) different from zero ; and then to sub -
Stitute this scalar in the expression p = a + x^, in order to determine the required
vector p. ^/Vo^ -t
(3.) For this purpose, an obvious process is, after dividing both sides by T/3, to
square, and to employ the formula 210, XXI., which had indeed occurred before, as
200, VIII., but not then as a consequence of the distributive property of multiplica-
tion. In this manner we are conducted to a quadratic equation, which admits of
division by x, and gives then,
2xS-. ^^
''*•
fi
2S
/3'
p = a-2(3S-
* Compare the Notes to page 90, &c.
216 ELEMENTS OF QUATERNIONS. [bOOK II.
the problem (1.) being thus resolved, with the verification that /3 may be replaced
by U/3, in the resulting expression for p.
(4.) As a mere exercise of calculation, we may vary the last process (3.), by
dividing the last equation (2.) by Ta, instead of T/3, and then going on as before.
This last procedure gives.
a a
and therefore,
-(-^')=
2S-:N^ = - 2S^ (by 196, XII'.), as before.
a a (5
(5.) In general, by 196, II'.,
1-2S = -K;
hence, by (3.),
and finally,
? = -K?
P = -k|./3,
a new expression for p, in which it is not permitted generally, as it was in (3.), to
treat the vector /3 as the multiplier,* instead of the multiplicand.
(6.) It is now easy to see that the second equation of (2.) is satisfied ; for the
expression (6.) for p gives (by 186, 187, &c.),
Tp = T^.T/3 = Ta,
as was required.
(7.) To interpret the solution (3.), let c in Fig. 51 be the middle point of the
chord AP, and let D be the foot of the perpendicular let fall from a on ob ; then the
expression (3.) for p gives, by 196, XIX.,
CA=i(a-p) = /3s|=OD;
and accordingly, ocad is a parallelogram.
(8.) To interpret the expression (5.), which gives
— P ^« op' ^OA ..
-f = K-, or — =K— , if op' = PO,
(3 (3 OB ob'
we have only to observe (comp. 138) that the angle aop' is bisected internally, or
the supplementary angle aop externally, by the indefinite right line ob (see again
Fig. 51).
(9.) Conversely, the geometrical considerations which have thus served in (7.)
and (8.) to interpret or to verifi/ the two forms of solution (3.), (5.), might have
been employed to deduce those two forms, if we had not seen how to obtain them,
by rules of calculation, from the proposed conditions"^ of the question. (Comp. 145,
(10.), &c.)
(10.) It is evident, from the nature of that question, that a ought to be deduci-
Compare the Note to page 159.
CHAP. I.] IMAGINARY INTERSECTIONS. 2lT
ble from (3 and p, by exactly the same processes as those which have served us to de-
duce p from (3 and a. Accordingly, the form (3.) of p gives,
and the form (5.) gives,
K|=-|, »=-Ke.,.
And since the first form can be recovered from the second, we see that each leads us
back to the parallelism, p — a\\(3 (2.).
(11.) The solution (3.) for x shows that
a; = 0, p = a, p = A, if S(a:/3) = 0, or if /3 -U a.
And the geometrical meaning of this result is obvious ; namely, that a right line
drawn at the extremity of a radius OA of a sphere, so as to be perpendicular to that
radius, does not (in strictness) intersect the sphere, but touches it : its second point
of meeting the surface coinciding, in this case, as a limit, with the first.
(12.) Hence we may infer that the plane represented by the equation,
stZ^^O, or 8^=1,
a a
is the tangent plane (comp. 196, (5.)) to the sphere here considered, at the point a.
(13.) Since /3 may be replaced by any vector parallel thereto, we may substitute
for it y — a, if y = oc be the vector of any given point c upon the chord ap, whether
(as in Fig, 61) the middle point, or not; we may therefore write, by (3.) and (5.),
p = a-2(y-a)S-^ = -K-^.(y-a). :. _, /^ ^ . M
y-a y-a ^ ^^
214. In the Examples of the foregoing Article, there was no
room for the occurrence of imaginary roots of an equation, or for
ideal intersections of line and surface. To give now a case in which
such imaginary intersections may occur, we shall proceed to con-
sider the question of drawing a secant to a sphere, in a given direc-
tion, from a given external point ; the recent Figure 51 still serving
us for illustration.
(1.) Suppose then that 6 is the vector of any given point e, through which it is
required to draw a chord or secant epqPi, parallel to the same given line /3 as before.
We have now, if po = opo,
po = £ + ^oA Ta = Tpo = T (£ + Xq^),
x„2 4-2a;oSi+Ni-N^ = 0,
,. being a new scalar ; and similarly, if |0i = OPi,
2 F
vv - ff- "^I
^18 ELEMENTS OF QUATERNIONS. [boOK II.
by transformations* which will easily occur to any one who has read recent articles
with attention. And the points Po, pi will be together real, or together imaginary^
according as the quantity under the radical sign is positive or negative ; that is, ac-
cording as we have one or other of the two following inequalities,
T|> or <TV|.
(2.) The equation (comp. 203, (6.) ),
represents a cylinder of revolution, with ob for its axis, and with Ta for the radius
of its base. If e be a point of this cylindric surface, the quantity under the radical
sign in (1.) vanishes ; and the two roots xq, x\ of the quadratic become equal. In
this case, then, the line through e, which is parallel to on, touches the given sphere ;
as is otherwise evident geometrically, since the cylinder envelopes the sphere (comp.
204, (12.) ), and the line is one of its generatrices. If e be internal to the cylinder,
the intersections Po, pi are real ; but if E be external to the same surface, those in-
tersections are ideal, or imaginary. ^
(3.) In this last case, if we make, for abridgment.
«i- - '=>/{(--;r-(^'iT}'
9 and t being thus two given and real scalars, we may write,
«ro = a-^V-l; Xi = s+tV -l;
where V — 1 is the old and ordinary imaginary symbol of Algebra, and is not in-
vested here with any sort of Geometrical Intei-pretation.f We merely express thus
the fact of calculation, that (with these meanings of the symbols a, /3, 6, * and t)
the formula Ta = T(e +x(S), (1.), when treated by the rules of quaternions, conducts
to the quadratic equation,
(X - S)2 +(2=0,
which has no real root ; the reason being that the right line through E is, in the
present case, wholly external to the sphere, and therefore does not really intersect it
at all ; although, for the sake of generalization of language, we may agree to say,
as usual, that the line intersects the sphere in two imaginary points.
(4.) We must however agree, then, for consistency of symbolical expression, to
consider these two ideal points as having determinate but imaginary vectors, namely,
the two following :
in which it is easy to prove, 1st, that the real part c + s/3 is the vector t' of the foot
e' of the perpendicular let fall from the centre o on the line through E which is drawn
(as above) parallel to on ; and Ilnd, that the real tensor tT/S of the coefficient of
* It does not seem to be necessary, at the present stage, to supply so many refe-
rences to former Articles, or Sub-articles, as it has hitherto been thought useful ta
give ; but such may still, from time to time, be given.
t Compare again the Notes to page 90, and Art. 149.
CHAP. I.]
CIRCUMSCRIBED CONES.
219
V - 1 in tha ijnaginary part of each expression, represents the length of a tangent
e'e" to the sphere, drawn from that external point, or foot, e'.
(6.) In fact, if we write oe' = «' = £ -f «j3, we shall have
e'e = £ - 6' = - »j3 = /3S — = projection of oe on ob ;
which proves the 1st assertion (4.), whether the points Po, Pi be real or imaginary.
And because
/f(>
+ «^
we have, for the case of imaginary intersections,
«T^ = V(T£'2 - Ta2) = T . E'E",
and the Ilnd assertion (4.) is justified.
(6.) An expression of the form (4.), or of the following,
p' = /3 + V-ly,
in which /3 and y are two real vectors^ while V - 1 is the (scalar) imaginary of al-
gebra, and not a symbol for &. geometrically real right versor (149, 153), may be said
to be a BiVECTOR.
(7.) In like manner, an expression of the form (3.), ora:' = s+<V — 1, where »
and t are two real scalars, but V - 1 is still the ordinary imaginary of algebra, may
be said by analogy to be a Biscalar. Imaginary roofs of algebraic equations aro
thus, in general, biscalars.
(8.) And if a bivector (6.) be divided by a (real) vector, the quotient, such as
H
a a
1 = ?o + ^1 V - 1,
in which go and qi are two real quaternions, but V — 1 is, as before, imaginary, may
be said to be a Biquaternion. *
215. The same distributive principle (212) may be employed in
investigations respecting circumscribed cones^ and the tangents (real
or ideal), which can be drawn to a given sphere from a given point.
(1.) Instead of conceiving that o, a, b are three given points, and that limits of
position of the point e are sought, as in 214, (2.), which shall allow the points of in-
tersection Po, Pi to be real, we may suppose that o, a, e (which may be assumed to
be coUinear, without loss of generality, since a enters only by its tensor) are now the
data of the question ; and that limits of direction of the line ob are to be assigned,
which shall permit the same reality : epoPi being still drawn parallel to ob, as in
214, (1.).
(2.) Dividing the equation Ta = T(€ + xfi) by Tf, and squaring, we have
Compare the second Note to page 131.
220
ELEMENTS OF QUATERNIONS.
[book II.
N" = ^Nfl + a;^'\='jn-2xS^ + a;2N|;
the quarlratic in x may therefore be thua written,
and its roots are real and unequal, or real and equal, or imaginary, according as
TVU^< or= or>T-;
C 6
that is, according as
sinEOB< or = or >T.oa: T.oe.
(3.) If E be interior to the sphere, then Tc < Ta, T(a : e) > 1 ; but TVUg can
never exceed unity (by 204, XIX., or by 210, XV., &c.) ; we have, therefore, in
this case, theirs* of the three recent alternatives, and the two roots of the quadratic
are necessarily real and unequal, whatever the direction of /3 may be. Accordingly
it is evident, geometrically, that every indefinite right line, drawn through an inter-
nal point, must cut the spheric surface in two distinct and real points.
(4.) If the point E be SM/jer/cia?, so that Tf = Ta, T(a:6) = l, then the first
alternative (2.) still exists, except at the limit for which (3 -^ e, and therefore
TVU (j3 : f) = 1, in which case we have the second alternative. One root of the qua-
dratic in a; is now = 0, for every direction of (3 ; and the other root, namely
a: = — 2S(c:/3), is likewise always j-eal, but vanishes for the case when the angle
Eon is right. In short, we have here the same system of chords and of tangents,
from a point upon the surface, as in 213 ; the only difference being, that we noAV
write E for a, or £ for a.
(5.) But finally, if e be an external point, so that Tc >Ta, and T(a : c) < 1,
then TVU (/3 : t) may either fall short of this last tensor, or equal, or exceed it ; so
that any one of the three alternatives (2.) may come to exist, according to the vary-
ing direction of (3.
(6.) To illustrate geometrically
the law of passage from one such
alternative to another, we may ob-
serve that the equation,
TVU^ = T-,
« £
or
sinEOP = T.oA: T.oe,
represents (when e is thus external)
a real cone of revolution, with its
vertex at the centre o of the sphere ;
and according as the line on lies in-
side this cone, or on it, or outside it,
the first or the second or the third of
the three alternatives (2.) is to be ^^S- 52.
adopted ; or in other words, the line
through E, drawn parallel (as before) to on, either cuts the sphere, or touches it, or
does not (really) meet it at all. (Compare the annexed Fig. 52.)
CHAP, I.] POLAR PLANES, CONJUGATE POINTS. 221
(7.) IfEbe still an external point, the cone of tangents which can be drawn
from it to the sphere is real ; and the equation of this enveloping or circumscribed
cone, with its vertex at E, may be obtained from that of the recent cone (6.), by
simply changing p to p — c ; it is, therefore, or at least one form of it is,
TVU^^=T-: or sinoEP = T. oa : T.oe.
€ £
(8.) In general, if q be any quaternion, and x any scalar,
VU(gr + ^) = V5:T(g + £c);
the recent equation (7.) may thcHjfore be thus written :
p-« e '
or
T.p'p:T.ep==T.oa: T.OE,
if p' be the foot of the perpendicular let fall from p on oe ; and in fact the first quo-
tient is evidently = sin oep.
(9.) We may also write,
Tve = T2.T(e-l'j; or = (s^y-N? + N^(Ne- 2Se + , j,
or
as another form of the equation of the circumscribed cone.
(10.) If then we make also
N^ = l, or N^ = N^,
a e e
to express that the point p is on the enveloped sphere, as well as on the enveloping
cone, we find the following equation of the plane of contact, or of what is called the
polar plane of the point b, with respect to the given sphere :
s£-N^Y = Oi or Se-N2 = 0, -^^^ ^ - A^^-^
£
-J=Oi or S.--N. = 0, --^^ , /^^^
while the fact that it is a plane of contact" is exhibited by the occurrence of the ex-
ponent 2, or by its equation entering through its square.
(11.) The vector,
„ jO ^^ a ,
e' = f S - = cN - = OE,
is that of the point e' in which the polar plane (10.) of e cuts perpendicularly the
right line oe ; and we see that
Tc.T6' = Ta2, or T.oe.T.oe' = (T.oa)2,
as was to be expected from elementary theorems, of spherical or even of plane geo-
metry.
* In fact a modern geometer would say, that we have here a case of two coinci-
dent planes of intersection, merged into a single plane of contact.
222 ELEMENTS OF QUATERNIONS. [bOOK II.
(12.) The equation (10.), of the polar plane of e, may easily be thus trans-
formed :
si=[s£.Nl = V" or Si-N^ = 0;
P \ ^ P J P P P
it continues therefore to hold good, when e and p are interchanged. If then we take,
as the vertex of a new enveloping cone, any point o external to the sphere, and
situated on the polar plane ff' . . of the former external point b, the new plane of
contact, or the polar plane dd' . . of the new point c, will pass through the former
vertex e : a geometrical relation of reciprocity, or of conjugation, between the two
points c and e, which is indeed well-known, but which it appeared useful for our pur-
pose to prove by quaternions* anew.
(13.) In general, each of the two connected equations,
P P P' 9
which may also be thus written,
\^ ap a j a a a a
may be said to be a form of the Equation of Conjugation between any two points p and
p' (not those so marked in Fig. 52), of which the vectors satisfy it : because it ex-
presses that those two points ai-e, in a well-known sense, conjugate to each other, with
respect to the given sphere, Tp = Ta.
(14.) If one of the two points, as p', be given by its vector p', while tlie other
point p and vector p are variable, the equation then represents a plane locus;
namely, what is still called the polar plane of the given point, whether that point be
external or internal, or on the surface of the sphere.
(15.) Let P, p' be thus two conjugate points; and let it be proposed to find the
points s, 8', in which the right line pp' intersects the sphere. Assuming (comp. 25)
that
OS = <T = xp+i/p', x + i/ = l, T(T = Ta,
and attending to the equation of conj ugation (13.), we have, by 210, XX., or by
200, VII., the following quadratic equation in y : a;,
(a; + y)2 = N(a;^ + y^'^ = a;2N^-f2a;y + y2N^;
\ a a ) a a
which gives,
(16.) Hence it is evident that, if the points of intersection s, s' are to be real, one
of the two points p, p' must be interior, and the other must be exterior to the sphere ;
because, of the two norms here occurring, one must be greater and the other less than
linity. And because the two roots of the quadratic, or the two values of y : a;, differ
* In fact, it will easily be seen that the investigations in recent sub-articles are
put forward, almost entirely, as exercises in the Language and Calculus of Quaternions,
and not as offering any geometrical novelty of result.
CHAP. I.] EQUATION OF ELLIPSOID, RESUMED. 223
only by their signs, it follows (by 26) that the right line pp' is harmonically divided
(as indeed it is well known to be), at the two points s, s' at which it meets the sphere :
or that in a notation already several times employed (25, 31, &c.), we have the har-
monic formula,
(pspV)=~ 1.
(17.) From a real but internal point p, we can still speak of a cone of tangents,
as bemg drawn to the sphere : but if so, we must say that those tangents are ideal,
or imaginary ;*^ and must consider them as terminating on an imaginary circle of
contact : of which the real but wholly external plane is, by quaternions, as by mo-
dern geometry, recognised as being (comp. (14.) ) the polar plane of the supposed
internal point.
216. Some readers may find it useful, or at least interest-
ing, to see here a few examples of the application of the General
Distributive Principle (212) of multiplication to the Ellipsoid,
of which some forms of the Quaternion Equation were lately
assigned (in 204, (14.) ); especially as those forms have been
found to conductf to a Geometrical Construction, previously
unknown, for that celebrated and important Surface : or ra-
ther to several such constructions. ,In what follows, it will
be supposed that any such reader has made himself already
sufficiently familiar with the chief formulae of the preceding
Articles ; and therefore comparatively few references J will be
given, at least upon the present subject.
(1.) To prove, first, that the locus of the variable ellipse,
I. ..S^=a;, (v^Y=a;2-l, 204,(13.)
« V Pi
which locus is represented by the equation,
the two constant vectors a, /3 being supposed to be real, and to be inclined to each
other at some acute or obtuse (but not right§) angle, is a surface of the second order,
* Compare again the second Note to page 90, and others formerly referred to.
f See the Proceedings of the Royal Irish Academy, for the year 1846.
X Compare the Note to page 218.
§ If /3 -l-a, the system I. represents (not an ellipse but) a pair of right lines,
real or ideal, in which the cylinder of revolution, denoted by the second equation of
that system, is cut by a, plane parallel to its axis, and represented by the first equa-
tion.
224 ELEMENTS OF QUATERNIONS. [dOOK II.
in the sense that it is cut by an arbitrary rectilinear transversal in two (real or ima-
ginary) points, and in no more than two, let us assume two points l, m, or their
vectors \ = ol, /* = om, as given ; and let us seek to determine the points p (real or
imaginary), in which the indefinite right line lm intersects the locus II. ; or rather
the number of such intersections, which will be suflScient for the present purpose.
(2.) Making then p =^-- — (26), we have, for y : 2, the following quadratic
y "I" 2;
equation,
without proceeding to resolve which, we see already, by its mere degree, that the num-
ber sought is two ; and therefore that the locus II. is, as above stated, a surface of
the second order.
(3.) The equation II. remains unchanged, when - p is substituted for p ; the
surface has therefore a centre, and this centre is at the origin o of vectors.
(4.) It has been seen that the equation of the surface may also be thus written :
IV. ..Tfs^-[-V^'\=l; 204,(14.)
it gives therefore, for the reciprocal of the radius vector from the centre, the expres-
sion.
-•i=<4-^)^
and this expression has a real value, which never vanishes,* whatever real value may
be assigned to the versor Up, that is, whatever direction may be assigned to p : the
surface is therefore closed, a,ndi finite.
(5.) Introducing two new constant and auxiliary vectors, determined by the two
expressions,
2/3
6=- • . a,
' /3-i-a
ft-a
which give (by 125) these other expressions,
we have
y V
VII. ..^ + ^ = 2,
a /3
-a-r'--
\r^cMi^
VII'...^+^=:1,
7 ^
and under these conditions, y is said to be the harmonic mean between the two for-
mer vectors, a and /3 ; and in like manner, 5 is the harmonic mean between a and
— /3 ; while 2a is the corresponding mean between y, ^ ; and 2/3 is so, between y
and - d.
* It is to be remembered that we. have excluded in (1.) the case where /3 -t- a
in which case it can be shown that the equation II. represents an elliptic cylinder.
(i
uX^
CHAP. I.] CIRCULAR SECTIONS, CYCLIC PLANES. 225
(6.) Under the same conditions, for any arbitraiy vector p, wo have the trans-
formations,
VIII .e=i('e + eV p=Je_eV ,,f .•, , -7i
ix...e+K|t=se+v£i ^
the equation IV. of the surface may therefore be thus written :
X...T^e + K^]=l; orthus, X'...t(^ + K^') = 1; {ki^ ^
the geometrical meaning of wliich new forms will soon be seen. ' - ' ,
(7.) The system of the two planes through the origin, which are respectively P^f^*''^^^^ f
perpendicular to the new vectors y and 5, is represented by the equation, /^ / tjL^
xi...sese=o, 0. xii...(sey = (sey, ^(^^
combining which with the equation II. we get
XIII...l = (^S^y-(^V^J=N^; or, XIV. . . Tp = T/3.
These two diametral plau«s therefore cut the surface in <«?o circular sections^ with T/3
for their common radius ; and the normals y and ^, to the same two planes, may be
called (comp, 196, (17.) ) the cyclic normals of the surface; while the planes them-
selves may be called its cyclic planes.
(8.) Conversely, if we seek the intersection of the surface with the concentric
sphere XIV., of which the radius is T/3, we are conducted to the equation XII. of
the system of the two cyclic planes, and therefore to the two circular sections (7.) ;
so that every radius vector of the surface, which is not drawn in one or other of these
two planes, has a length either greater or less than the radius T/3 of the sphere.
(9.) By all these marks, it is clear that the locus II., or 204, (14.), is (as above
asserted) an Ellipsoid; its centre being at the origin (3.), and its mean semiazis
being = T/3 ; while U/3 has, by 204, (15.), the direction of the axis of a circum-
scribed cylinder of revolution, of which cylinder the radius is T/3 ; and a is, by the
last cited sub- article, perpendicular to the plane of the ellipse of contact.
(10.) Those who are familiar with modem geometrj^, and who have caught the
notations of quaternions, will easily see that this ellipsoid II., or IV., is a deforma-
tion of what may be called the mean sphere XIV., and is homologous thereto ; the
infinitely distant point in the direction of /3 being a centre of homology, and either
of the two planes XL or XII. being a plane of homology corresponding.
217. The recent form, X. or X'., of the quaternion equa-
tion of the ellipsoid, admits of being interpreted, in such a way
as to conduct (comp. 216) to a simple construction of that sur-
face ; which we shall first investigate by calculation, and then
illustrate by geometry.
2 G
226
ELEMENTS OF QUATERNIONS.
[book II.
(1.) Carrying on the Roman numerals from the sub-articles to 216, and observ-
ing that (by 190, &c.),
t=.K^-.NP, and K?=l4,
y p y p d
the equation X.
takes the form,
Ar
— {U^-^^vl^xW
or
if we make
Xyi...^=T(.H-K^.p),
^^"•••4 = 1 -^ T>7^'
when I and k are two new constant vectors, and < is a new constant scalar, which we
shall suppose to be positive, but of which the value may be chosen at pleasure.
(2.) The comparison of the forms X. and X'. shoAvs that y and 3 may be inter-
changed, or that they enter symmetrically into the equation of the ellipsoid, although
they may not at first seem to do so ; it is therefore allowed to assume that
XVIir. . . Ty > T^, and therefore that XVIII'. . . Tt > Tk ;
for the supposition Ty = T^ would give, by VI.,
T(/3 + a) = T(/3-a), and .'. (by 186, (6.) &c.)
which latter case was excluded in 216, (1.).
(3.) We have thus,
XIX. . . Ut = U5;
XX.
/3'
Tt
XXI.
Tl2 - T/c2
UK: = Uy
(to) ily)
(4.) Let ABC be a plane triangle,
such that
XXII. . . CB = t, CA = k;;
let also
AE = p.
Then if a sphere, which we shall call the
diacentric sphere, be described round the
point c as centre, with a radius = Tk, and
therefore so as to pass through the centre
A (here written instead of o) of the ellip-
soid, and if D be the point in which the
line AE meets this sphere again, we shall
have, by 213, (5.), (18.),
XXIII.
and therefore
CD = -K-.p,
P .
.'btit^
Fig. 53,
xxiir. . . DB
t+K-.p;
P
- rf •
'3
unA.^-<^
/t^fc
jfyi^ff
ty
CHAP. I.] CONSTRUCTION OF THE ELLIPSOID. 227
so that the equation XVI. becomes,
XXIV. . . <2=T.AE.T.DB.
(5.) The point b is external to the diacentric sphere (4.), by the assumption (2.) ;
a real tangent (or rather cone of tangents) to this sphere can therefore be drawn from
that point ; and if we select the length of such a tangent as the value (1.) of the sca-
lar *, that is to say, if we make each member of the formula XXI. equal to unity^
and denote by d' the second intersection of the right line bd with the sphere, as in
Fig. 53, we shall have (by Euclid III.) the elementary relation,
XXV. . .<2=:T.db.T.bd';
whence follows this Geometrical Equation of the Ellipsoid,
XXVI. .. T.AB = T.BD';
or in a somewhat more familiar notation,
XXVII. . . AE = ^;
where ae denotes the length of the line ae, and similarly for bd'.
(6.) The following very simple Rule of Construction (corap. the recent Fig. 53)
results therefore^from our quaternion analysis : —
From a fixed point A, on the surface of a given sphere, draw any chord ad ; let
d' he the second point of intersection of the same spheric surface with the secant bd,
drawn from a fixed external* point b ; and take a radius vector ae, equal in
length to the line bd', and in direction either coincident with, or opposite to, the chord
ad : the locus of the point E will he an ellipsoid, with A for its centre, and with Bfor
a point of its surface.
(7.) Or thus: —
If, of a plane hut variable quadrilateral abed', of which one side ab is given in
length and in position, the two diagonals ae, bd' he equal to each other in length, and
if their intersection D he always situated upon the surface of a given sphere, whereof
the side ad' of the quadrilateral is a chord, then the opposite side be is a chord of
a given ellipsoid,
218. From either of the two foregoing statements, of the
Rule of Construction for the Ellipsoid to wliich quaternions
have conducted, many geometrical consequences can easily be
inferred, a few of which may be mentioned here, with then:
proofs by calculation annexed : the present Calculus being, of
course, still employed.
(1.) That the corner b, of what may be called the Generating Triangle abc, is
in fact a point of the generated surface, with the construction 217, (6.), may be
* It is merely to fix the conceptions, that the point b is here supposed to be exter-
nal(5.) ; the calculations and the construction would be almost the same, if we as-
sumed B to be an internal point, or Ti < T/c, Ty < Td.
228 ELEMENTS OF QUATERNIONS. [bOOK II.
proved, by conceiving the variable chord ad of the given dia centric sphere to take the
position AG; where g is the second intersection of the line ab with that spheric sur-
face.
(2.) Kobe conceived to approach to a (instead ofo), and therefore d' to g
(instead of a), the direction of ae (or of ad) then tends to become tangential to the
sphere at A, while the length of ae (or of bd') tends, by the construction, to become
equal to the length of bg ; the surface has therefore a diametral and circular section,
in a plane which touches the diacentric sphere at A, and with a radius = bg.
(3.) Conceive a circular section of the sphere through A, made by a plane perpen
dicular to bc ; if d move along this circle, d' will move along a parallel circle through
,Sa^^ g, and the length of bd', or that of ae, will again be equal to bg fsuch then is the
radius of a second diametral and circular section of the ellipsoid, made by the lately
f mentioned plane.
(4.) The construction gives us thus two cyclic planes through A ; the perpendi-
culars to which planes, or the two cyclic normals (216, (7.)) of the ellipsoid, are
seen to have the directions of the two sides, ca, cb, of the generating triangle abc
(1.).
(5.) Again, since the rectangle
ba . BG = bd . bd' = bd . ab = double area of triangle abe : sin bde,
we have the equation,
XXVIII. . . perpendicular distance of e from ab = bg • sin bde ;
the third side, ab, of the generating triangle (1.), is therefore the axis of revolution
of a cylinder, which envelopes the ellipsoid, and of which the radius has the same
length, bg, as the radius of each of the two diametral and circular sections.
(6.) For the points of contact of ellipsoid and cylinder, we have the geometrical
relation,
XXIX. . . bdb = a right angle ; or XXIX'. . . adb = a right angle ;
the point d is therefore situated on a second spheric surface, which has the line ab
for a diameter, and intersects the diacentric sphere in a circle, Avhereof the plane passes
through A, and cuts the enveloping cylinder in an ellipse of contact (comp. 204,
(15.), and 216, (9.) ), of that cylinder with the ellipsoid.
(7.) Let AC meet the diacentric sphere again in f, and let bf meet it again in p'
(as in Fig. 53) ; the common plane of the last-mentioned circle and ellipse (6.) can
then be easily proved to cut perpendicularly the plane of the generating triangle abc
in the line af'; so that the line f'b is normal to this plane of contact; and there-
fore also (by conjugate diameters, &c.) to the ellipsoid, at b.
(8.) These geometrical consequences of the construction (217), to which many
others might be added, can all be shoAvn to be consistent with, and confirmed by, the
quaternion analysis from which that construction itself was derived. Thus, the two
circular sections (2.) (3.) had presented themselves in 216, (7.) ; and their two cy-
clic normals (4.), or the sides CA, cb of the triangle, being (by 217, (4.) ) the two
vectors k, t, have (by 217, (1 .) or (8.) ) the directions of the two former vectors y, 5 ;
which again agrees with 216, (7.).
(9.) Again, it will be found that the assumed relations between the three pairs of
constant vectors, a, j3 ; y, d ; and j, *•, any one of which pairs is sufficient to deter-
CHAP. 1.] CONSEQUENCES OF THE CONSTRUCTION. 229
mine the ellipsoid, conduct to the following expressions (of which the investigation is
left to the student, as an exercise) :
XXX. ..a = ~ r = T^ ^=7FT ;U(i + k) = f'b;
XXXI. ../3 = ^y = /-5 = =rP^xU(t-K) = BG;
— y —y i {i- k)
the letters B, f', g referring here to Fig. 53, while a/3y^ retain their former mean-
ings (216), and are not interpreted as vectors of the points abcd in that Figure.
Hence the recent geometrical inferences, that ab (or bg) is the axis of revolution of
an enveloping cylinder (5.), and that f'b is normal to the plane of the ellipse of con-
tact (7.), agree with the former conclusions (216, (9.), or 204, (15.) ), that j3 is
such an axis, and that a is such a normal.
(10.) It is easy to prove, generally, that
c9-i_q (g-i)(Kg+i) ^ %- i g + 1^ yg-1 .
9 + 1 (9+i)(k:3+i) KCz + i)' 9-1 N(^-l)'
whence
t + K T (l + k)* l-K 1 (t - k)2
whatever two vectors t and k may be. But Ave have here,
XXXIII. . . <3 = Ti2 - Tk2, by 217, (5.) ;
the recent expressions (9.) for a and /3 become, therefore,
XXXIV. . . a=;+(i + fc)S*-— ^; i(S = -(i-K:) S— .
1 + K l-K
The last form 204, (14.), of the equation of the ellipsoid, may therefore be now thus
written :
XXXY. ..TiS-^:S'— ^-V-^:S— 1=1
l~K I-
\ i + K 1 +
in which the sign of the right part may be changed. And thus we verify by calcu-
lation the recent result (1.) of the construction, namely that b is a point of the sur-
face ; for we see that the last equation is satisfied, when we suppose
XXXVI. . . p = AB = t-K = /3:s2;
a
a value of p which evidently satisfies also the form 216, IV.
(11.) From the form 216, II., combined with the value XXXIV. of otitis easy
to infer that the plane,
XXXVII. . .s^ = i, or xxxvir. . .S-^ = S^-^,
a 1+ K 1 + K
which corresponds to the value a;= 1 in 216, I., touches the ellipnoid at the point B,
of which the vector p has been thus determined (10) ; the normal to the surface^ at
that point, has therefore the direction of t + ic, or of a, that is, of fb, or of f'b : so
that the last geometrical inference (7.) is thus confirmed, by calculation with quater-
219. A few other consequences of the construction (217) may
be here noted; especially as regards the geometrical determination
230 ELEMENTS OF QUATERNIONS. [bOOK II.
of the three principal semiaxes of the ellipsoid, and the major and
minor semiaxes of any elliptic and diametral section ; together with
the assigning of a certain system of spherical conies^ of -which the
surface may be considered to be the locus.
(1.) Let a, 6, c denote the lengths of the greatest, the mean, and the least semi-
axes of the ellipsoid, respectively ; then if the side bc of the generating triangle cut
the diacentric sphere in the points h and h', the former lying (as in Fig. 53) between
the points b and c, -we have the values,
XXXVIII. ..a = BH'; 6 = bg; c = bh;
so that the lengths of the sides of the triangle abc may be thns expressed, in terms
of these semiaxes,
— a -^ c — a — c — cic
XXXIX. . .BC=Te = -|-; ca = Tk=-— -; ab =T(i - «) = — ;
and we may write,
Ti3 — Tk2
XL. . . a = Ti + T/c; h==—- -; c=Ti-T/c.
T (i - k)
(2.) If, in the respective directions of the two supplementary chords ah, ah' of the
sphere, or in the opposite directions, we set off lines al, an, with the lengths of bh',
BH, the points L, N, thus obtained, will be respectively a major and a minor summit
of the surface. And if we erect, at the centre a of that surface, a perpendicular am
to the plane of the triangle, with a length = bg, the point m (which will be common
to the two circular sections, and will be situated on the enveloping cylinder) will be a
mean summit thereof.
(3.) Conceive that the sphere and ellipsoid are both cut by a plane through a, on
which the points b' and c' shall be supposed to be the projections of b and c ; then c'
will be the centre of the circular section of the sphere ; and if the line b'c' cut this
new circle in the points Di, »2, of which di may be supposed to be the nearer to b',
the two supplementary chords adi, ad2 of the circle have the directions of the major
and minor semiaxes of the elliptic section of the ellipsoid ; while the lengths of those
semiaxes are, respectively, ba.bg: bdi, and ba. bg : BD2; or bd'i and BD'2, if the
secants bdi and BD2 meet the sphere again in Di' and D2'.
(4.) If these two semiaxes of the section be called a, and c„ and if we still de-
note by t the tangent from b to the sphere, we have thus,
XLI. . . BDi = <2 : a = oca -1 ; BD2 = *2 ; c = acc'^ ;
but if we denote by pi and p2 the inclinations of the plane of the section to the two
cyclic planes of the ellipsoid, whereto CA and cb are perpendicular, so that the pro-
jections of these two sides of the triangle are
|o'a = CA . sinpi = ^(a — c) sin pi,
XLII.
[c'b =CB.smp2 = i{a + c)s'mp2,
we have
XLIII. . . BD33 - BDi2 = b'd22 -b'di2 = 4b'c' . c'a = (a^ - c2) sin pi sin p>
whence follows the important formula,
XLIV. . . c,-2 - a, 2 = (c 2 _ a 2) sin pi smpz ;
CHAP. I.] SEMIAXES, SPHERICAL CONICS. 231
or in words, the known and useful theorem, that " the difference of the inverse
squares of the semiaxes, of a plane and diametral section of an ellipsoid, varies as
the product of the sines of the inclinations of the cutting plane, to the two planes of
circular section.
(5.) As verifications, if the plane be that of the generating triangle abc, we
have
pi=p2= -, and a^ = a, c^ = c',
but if the plane be perpendicular to either of the two sides, ca, cb, then either pi or
P2 = 0, and c, = a^.
(6.) If the ellipsoid be cut by any concentric sphere, distinct from the mean
sphere XIV., so that
XLV. . . AE = Tp = r ^ 6, where r is a given positive scalar ;
then
XL VI. . . BD = «2r-i ^ acb-^j that is, ^ ba ;
so that the locus of what may be called the guide-point D, through which, by the
construction, the variable semidiameter ab of the ellipsoid (or one of its prolongations)
passes, and which is still at a constant distance from the given external point b, is
now again a circle of the diacentric sphere, but one of which the plane does not pass
(as it did in 218, (3.) ) through the centre A of the ellipsoid. The point b has there-
fore here, for one locus, the cyclic cone which has A for vertex, and rests on the last-
mentioned circle as its base; and since it is also on the concentric sphere XLV., it
must be on one or other of the two spherical conies, in which (comp. 196, (11.) ) the
cone and sphere last mentioned intersect.
(7.) The intersection of an ellipsoid with a concentric sphere is therefore, gene-
rally, a system of two such conies, varying with the value of the radius r, and be-
coming, as a limit, the system of the two circular sections, for the particular value
r = 6 ; and the ellipsoid itself may be considered as the locu» of all such spherical co-
nies, including those two circles.
(8.) And we see, by (6.), that the two cyclic planes (comp. 196, (17.), &c.) of
any one of the concentric cones, which rest on any such conic, coincide with the two
cyclic planes of the ellipsoid : all this resulting, with the greatest ease, from the con-
struction (217) to which quaternions had conducted.
(9.) With respect to the Figure 53, which was designed to illustrate that con-
struction, the signification of the letters abcdd'efk'ghh'ln has been already ex-
plained. But as regards the other letters we may here add, 1st, that n' is a second
minor summit of the surface, so that an' = na ; Ilnd, that k is a point in which the
chord af', of what we may here call the diacentric circle agf, intersects what may
be called the principal ellipse, * or the section nblen' of the ellipsoid, made by the
plane of the greatest and least axes, that is by the plane of the generating triangle
ABC, so that the lengths of AK and bf are equal; Ilird, that the tangent, vKv', to
this ellipse at this point, is parallel to the side ab of the triangle, or to the axis of
* In the plane of what is called, by many modern geometers, i\\Q focal hyper-
bola of the ellipsoid.
232 ELEMENTS OF QUATERNIONS. [bOOK II.
revolution of the enveloping cylinder 218, (5.), being in fact one fide (or generatrix)
of that cylinder ; IVtb, that ak, ab are thus two conjugate semidiameters of the
ellipse, and therefore the tangent tbt', at the point b of tbat ellipse, is parallel to
the line akf', or perpendicular to the line bff' ; Vth, that this latter line is thus the
normal (comp. 218, (7.), (11.) ) to thesame elliptic section, and therefore also to the
ellipsoid, at b ; Vlth, that the least distance kk' between the parallels ab, kv, being
= the radius b of the cylinder, is equal in length to the line bg, and also to each of
the two semidiameters, as, as', of the ellipse, which are radii of the two circular
sections of the ellipsoid, in planes perpendicular to the plane of the Figure ; Vllth,
that AS touches the circle at A ; and Vlllth, that the point s' is on the chord Ai of
that circle, which is drawn at right angles to the side bc of the triangle.
220. The reader will easily conceive that the quaternion equa-
tion of the ellipsoid admits of being put under several other forms;
among which, however, it may here suffice to mention one, and to
assign its geometrical interpretation.
(I.) For any three vectors, t, k, p, we have the transformations,
XLVIL..N[l + K^UNi-fN^+2S-i^ 0^^ '^
\p p ) p p p p
= NiN- + N-N- + 2S--T-T-
K p >■ p p p I K
\9 I P KJ \p K pi]
Tk ^Vk .Ti\ [JJk.Ti . _Ut.T/c
+ K =N +K
P P } \ 9 P
whence follows this other general transformation :
XLVIir. ..Tfi + K-.p^ = TfuK.Tt + K Hil^!^ . p \
(2.) If then we introduce two new auxiliary and constant vectors, i and k\ de-
fined by the equations,
XLIX. . . t' = - Uk . Ti, K' = -Ut.TK,
which give,
L. . . Tt' = Tt, Tfc' = Tk, T (i' - ^') = T (t - k), Tt'2 - Tk'2 = t\
we may write the equation XVI. (in 217) of the ellipsoid under the following pre-
cisely similar form :
U...il=T(.'.Kl.,)
in which i and k have simply taken the places of t and k.
(3.) Retaining then the centre A of the ellipsoid, construct a new diaceniric
sphere^ with a new centre o', and a new generating triangle ab'c', where b' is a new
fixed external point, but the lengths of the sides are the same, by the conditions,
LII. . Ac' = — k', c'b' = + t', and therefore ab' — i -k \
draw any secant b'd"d"' (instead of bdd'), and set off a line ae in the direction of
CHAP. I.] STANDARD QUADRINOMIAL FORM. 233
ad", or in the opposite direction, with a length equal to that of bd'"; the locus of
the point E will be the same ellipsoid as before.
(4.) The only inference which we shall here* draw from this new construction
is, that there exists (as is known) a second enveloping cylinder of revolution, and that
its axis is the side ab' of the new triangle ab'c' ; but that the radius of this second
cylinder is equal to that of the first, namely to the mean semiaxis, 6, of the ellipsoid ;
and that the major semiaxis, a, or the line al in Fig. 53, bisects the angle bab',
between the two axes of revolution of these two circumscribed cylinders : the plane
of the new ellipse of contact being geometrically determined by a process exactly
similar to that employed in 218, (7.); and being perpendicular to the new vector,
c' + k\ as the old plane of contact was (by 218, (11.)) to t + k.
Section 14. — On the Reduction of the General Quaternion
to a Standard Quadrinomial Form ; icith a First Proof of
the Associative Principle of Multiplication of Quaternions,
221. Retaining the significations (181) of the three rect-
angular unit-lines oi, oj, ok, as the axes, and therefore also
the indices (159), of three given right versors 2, J, k, in three
mutually rectangular planes, we can express the index oq of
any other right quaternion, such as Yq^ under the trinomial
form (comp. 62),
I. . . IV$' = 0Q = a;.oi+y.0J + Z.OK;
where xyz are some three scalar coeflScients, namely, the three
rectangular co-ordinates of the extremity q of the index, with
respect to the three axes oi, oj, ok. Hence we may write
also generally, by 206 and 126,
II. . . \q = xi + yj + zk = ix +jy + kz ;
and this last form, ix +jy + kz^ may be said to be a Standard
Trinomial Form, to which every right quaternion, or the right
part Yq of any proposed quaternion q, can be (as above) re-
duced. If then we denote by w the scalar part, Sq, of the same
general quaternion q, we shall have, by 202, the following
General Reduction of a Quaternion to a Standard Quadri-
nomial Form (183) :
* If room shall allow, a few additional remarks may be made, on the relations
of the constant vectors t, k, &c., to the ellipsoid, and on some other constructions of
that surface, when, in the following Book, its equation shall come to be put under the
new form,
T(tp+pK) = /c2-t2.
2 H
234 ELEMENTS OF QUATERNIONS. [bOOK II.
III. . . 2' = (Sq + V*^ =)w + ix ^jy + kz ;
in which the four scalars, wxyz^ may be said to be the Four
Constituents of the Quaternion. And it is evident (comp. 202,
(5.), and 133), that if we write in like manner,
IV. . . q =w \ ix -vji/ + kz\
where ijk denote the same three given right versors (181) as
before, then the equation
between these two quaternions, q and q\ includes the Jour follow-
ing scalar equations between the constituents :
VI. . . w' = w, x ~x, y "^y^ z' = z\
which is a new justification (comp. 112, 116) of the propriety
of naming, as we have done throughout the present Chapter,
the General Quotient oftioo Vectors (101) a Quaternion.
222. When the Standard Quadrinomial Form (221) is
adopted, we have then not only
1. . . ^q = w, and V^ = ix ^jy + kz,
as before, but also, by 204, XI.,
II. . . K^ = (Sg - Yq =) 10 - ix ~jy - kz.
And because the distributive property of multiplication of qua-
ternions (212), combined with the laws of of the symbols ijk
(182), or with the General and Fundamental Formula of this
whole Calculus (183), namely with the formula,
P=f = k^=^ijk = -\, (A)
gives the transformation,
III. . . {ix +jy 4- kzY = - (a;2 + 2/2 + z%
we have, by 204, &c., the following new expressions :
IV. . . NVg=(TV(?)2 = -V22^a;2 + ?/2+r2.
V. . . TV2= V(^' + 3/' + -2');
VI. . .\]Yq = {ix^jy-\-kz)'. ^/ {x^ -^ y"" ^ z"^) ;
VII. . . % = T^'' = Sy'^-V^2 = w;'^+a;^ + z/2 + 2:^
VIII. . . T^ = V i^o'' + a;2 + 3/2 + z") ;
IX. . . U$' = (w? + ix ^jy -^kz): y/ (w^ + a;^ + z/^ ^ ^2^ .
CHAP. I.] LAW OF THE NORMS. 235
X. . . SU^ = w: s/(w^ + x'^ + 2/2 + z^) ;
XI. . . VU^- = (ix +jy + kz): yj (yo' ^ x' + y''^ z"^) ;
xii...Tvug=) -'rr .
^ \ 2v^ + x^ -\- y^ + z^
(1.) To prove the recent formula III., we may arrange as follows the steps of
the multiplication (comp. again 182) :
Yq = ix ■\-jy + hz,
Yq — ix -\-jy + kz ;
ix .Yq = — x'^-Y kxy —jxz ;
jy-Yq^-y^- kyx + iyz,
kz.Yq = — z^ +jzx — izy ;
Yq^ = Yq.Yq==-x^-y^-z^.
(2.) We have, therefore,
XIII. . . {ix -\-jy + kzy = - 1, if x^+y^+z^ = 1,
a result to which we have already alluded,* in connexion with the partial indeter-
minateness of signification, in the present calculus, of the symbol V — 1, when consi-
dered as denoting a right radial (149), or a right versor (153), of which the plane
or the axis is arbitrary.
(3.) If q" = qq, then N/'=Ng'.%, by 191, (8.); but if g = m; + &c.,
q =z w' ■{ &t,c., (2'"= u;"+ &c,, then
■ w" = w'w — {x'x+y'y + z'z),
x" = (w'x + x'w) + {y'z - z'y),
y" = (w'y + y'w) + (z'^c — a?'*),
z" = (w'z 4- z'w) + {xy — y'x') ;
and conversely these four scalar equations are jointly equivalent to, and may be
summed up in, the quaternion formula,
XV. . . u?" + ix" +J7j" + kz" = (w' + ix' +jy' + kz') (w + ix +jy + kz) ;
we ought therefore, under these conditions XIV., to have the equation,
XVI. . . w"2 + ar"2 + y"2 -I- z"2 = (a,'2 + ^'2 + y'2 + a'2) (^j-i ^ ^'^ + y^ + z^) ',
which can in fact be verified by so easy an algebraical calculation, that its truth
may be said to be obvious upon mere inspection, at least when the terms in the four
quadrinomial expressions w" . . z' are arrangedf as above.
* Compare the first Note to page 131 ; and that to page 162.
f From having somewhat otherwise arranged those terms, the author had some
little trouble at first, in verifying that the twenty-four double products, in the ex-
pansion of w'"^ + &c., destroy each other, leaving only the sixteen /)roc?Mcfs of squares,
or that XVI. follows from XIV,, when he was led to anticipate that result through
quaternions, in the year 1843. He believes, however, that the algebraic theorem
XVI., as distinguished from the quaternion formula XV., with which it is here con-
nected, had been discovered by the celebrated Euler.
XIV.
236 ELEMENTS OF QUATERNIONS. [bOOK II,
223. The principal use which we shall here make of the
standard quadrinomial form (221), is to prove by it the gene-
ral associative property of multiplication of quaternions ; which
can now with great ease be done, the distributive* property
(212) of such multiplication having been already proved. In
fact, if we write, as in 222, (3.)j
[ q = w + ix +jy + kz,
L . . ^ g' = w + ix +jy' + kz\
j^/ = w" + IX ' -^jy" .+ kz%
without now assuming that the relation q" ^qq^ or any other
relation, exists between the three quaternions q^ q\ q\ and
inquire whether it be true that the associative formula^
II. . -qq^q^q-qq,
holds good, we see, by the distributive principle, that we have
only to try whether this last formula is valid when the three
quaternion factors q, 5'', q are replaced, in any one common
order on both sides of the equation, and with or without repe-
tition, by the three given right versors ijk ; but this has al-
ready been proved, in Art. 183. We arrive then, thus, at the
important conclusion, that the GeJieral Multiplication 0/ Qua-
ternions is an Associative Operation^ as it had been previously
seen (2 1 2) to be a Distributive one : although we had also
found (168, 183, 191) that such Multiplication is not (in ge-
neral) Commutative : or that the two products^ q'q and qq\ are
generally unequal. We may therefore omit the point (as in
183), and may denote each member of the equation II. by the
symbol q'q'q-
(1.) Let v = Vq, v' = Yq', v" = Yq" \ SO that v, v', v" are any three right qua-
ternions, and therefore, by 191, (2.), and 196, 204,
f^, Kv'u = vv)\ Sf't? = \ (v'v + vv")j Yv'v = ~ (w'w — vv').
Let this last right quaternion be called w„ and let Sv'v = s„ so that v'v = s^ + v/, we
shall then have the equations,
• At a later stage, a sketch will be given of at least one proof of this Associative
Principle of Multiplication^ which will not pj-esuppose the Distributive Principle.
f- tA^
CHAP. I.] ASSOCIATIVE PRINCIPLE OF MULTIPLICATION. 237
2Vv"w, = v'v, — vv" ; = v"a\ — sv" ;
whence, by addition,
2 V»"t7^ = v". v'v — v'v . v"
— (v"v' + v'v")v - v'{v"v + ry")
= 2wSw'y" — 2©'Su"u ;
and therefore generally, if r, v', t>" be still n^/t<, as above,
in. . . V. v"Yv'v = v^vv" - «'Sr"« ;
a formula with which the student ought to make himself completely familiar, on ac-
count of its extensive utility.
(2.) With the recent notations,
V . v'^v'v = Nv"s^ = v"s^ = v"S«i;';
we have therefore this other very useful formula, ■ / ^
IV. . . V . v"vv = v^v'v"- v'%v"v + v'^vv, ^ ^"/r '
where the point in the first member may often for simplicity be dispensed with ; and
in which it is still supposed that
TT
Lv = Lv = Lv = -.
(3.) The formula IIL gives (by 206),
V. . . IV, v"Yv'v = lv. SvV- lu'. St?"»;
hence this last vector, which is evidently complanar with the two indices Iv and Iw',
is at the same time (by 208) perpendicular to the third index Iv", and therefore (by
(1.) ) complanar with the third quaternion q".
(4.) With the recent notations, the vector,
VI. ..lv, = l\v'v = lV(Vg'.V9),
is (by 208, XXII.) a line perpendicular to both It; and Iw'; or common to the planes
of q and q' ; being also such that the rotation round it from Iv' to \v is positive :
while its length,
TIv,, or Tu,, or TY.v'v, or TV(Vg'.Vg),
hears to the unit of length the same ratio, as that which the parallelogram under the
indices, Iv and Iv', bears to the unit of area.
(6.) To interpret (comp. IV.) the scalar expression,
VII. . . Sv'v'v = Sp"», = S.v"Yv'v,
(because S»"5,= 0), we may employ the formula 208, V. ; which gives the the trans-
formation,
VIII. . . Sv'v'v = Tv". Tw . cos (tt-x);
where Tv" denotes the length of the line Iv", and Tv, represents by (4.) the area
(positively taken) of the parallelogram under Iv' and Iv ; while x is (by 208), the
angle between the two indices Iv", Iv,. Tliis angle will be obtuse, and therefore the
cosine of its supplement will he positive, and equal to the sine of the inclination of
the line Iv' to the plane oflv and Iv, if the rotation round Iv" from Iv' to Iv be
negative, that is, if the rotation round Iv from Iv' to Iv" be positive ; but that cosine
will be equal the negative of this sine, if the direction of this rotation be reversed.
We have therefore the important interpretation :
IX. . . S«"i''v = + volume of parallelepiped under Iv, Iv, \v" ;
238 ELEMENTS OF QUATERNIONS. [bOOK II.
the upper or the lower sign being taken, according as the rotation round Ir, from
\v' to lv\ is positively or negatively directed.
(6.) For example, we saw that the ternary products ijk and kji have scalar va-
lues, namely,
ijk=^-U kji = +l, by 183, (1,), (2.);
and accordingly the /jara^/ff/epipec? of indices becomes, in this case, a.n unit-cube ;
while the rotation round the index oft, from that ofj to that of ^, is positive (181).
(7.) In general, for any three right quaternions vv'v", we have the formula,
X. . . 8vv'v" = — Sv"v'v ;
and when the three indices are complanar, so that the volume mentioned in IX. va-
nishes, then each of these two last scalars becomes zero ; so that we may write, as a
new Formula of Complanarity ;
XI. . . St;"»'« = 0, if Iv" \\\\v', Iv (123) :
while, on the other hand, this scalar cannot vanish in any other case, if the quater-
nions (or their indices) be still supposed to be actual (1, 144); because it then re-
presents an actual volume.
(8.) Hence also we may establish the following Formula of Collinearity, for any
three quaternions :
XII. . . S (Yq" . Yq, Yq) = 0, if lYq" \ \ \ lYq', lYq ;
that is, by 209, if the planes of q, q, q" have any common line.
(9.) In general, if we employ the standard trinomial form 221, II., namely,
v = Yq = ix +jy + kz, v' = ix' + &c. , v"= ix" + &c. ,
the laws (182, 183) of the symbols i,j, k give the transformation,
XIII. . . S^''^'^ = x"{z'y — y'z) + y'\x'z - zx) + z"{rf'x — x'y') \
and accordingly this is the known expression for the volume (with a suitable sign)
of the parallelepiped, which has the three lines op, op', op" for three co-initial
edges, if the rectangular co-ordinates* of the four corners, o, p, p', p" be 000, xyz,
x'y'z', x"y"z".
(10.) Again, as another important consequence of the general associative pro-
perty of multiplication, it may be here observed, that although products oimorethan
two quaternions have not generally equal scalars, for all possible permutations of th«
factors, since we have just seen a case X. in which such a change of arrangement
produces a change of sign in the result, yet cyclical permutation is permitted, under
the sign S ; or in symbols, that for any three quaternions (and the result is easily ex-
tended to any greater number of such factors) the following formula holds good :
XIV. . . Sq'q'q = Bqq'q'.
In fact, to prove this equality, we have only to write it thus,
XIV'...S(9'V-9) = S(g.9'Y),
and to remember that the scalar of the product of any two quaternions remains unal-
tered (198, I.), when the order of those two factors is changed.
* This result may serve as an example of the manner in which quaternions,
although not based on any usual doctrine of co-ordinates, may yet be employed to
deduce, or to recover, and often with great ease, important co-ordinate expressions.
CHAP. l.J COMPLANAR QUATERNIONS. 239
(11.) In like manner, by 192, II., it may be inferred that
XV. . . K'qq'q =^{q". q'q) = Kq'q . Kq" = Kq . Kq' . Kq",
with a corresponding result for any greater number of factors; whence by 192, I.,
if Uq and Il'g' denote the products of any one set of quaternions taken in two op-
posite orders, we may write,
XVI. . . KUq = n'Kq ; XVII. . . RUq = U'Rq.
(12.) But if V be right, as above, then Ku = - v, by 144 ; hence,
XVIII. .. Knc=± n't?; XIX. . . srio = + sn'«; xx. . . vnu =+vn'w;
upper or lower signs being taken, according as the number of the right factors is
even or odd; and under the same conditions,
XXL . . snr = I (uv ± n'v) ;. xxii. . . vn« = i(Uv + Wv) ;
as was lately exemplified (1.), for the c&se where the number is two.
(13.) For the case where that number is three, the four last formulae give,
XXIIT. . . Sv'v'v = — Svv'v" = ~ (v"v'v — vv'v") ;
XXIV. . . Yv'v'v =-\-Yvv'v" = I (y"v'v + vv'v") ;
results which obviously agree with X. and IV.
224. For the case of Complanar Quaternions (119), the power of
reducing each (120) to the form of a fraction (101) which shall have,
at pleasure, for its denominator or for its numerator, any arbitrary
line in the given plane, furnishes some peculiar facilities for proving
the commutative and associative properties oi Addition (207), and the
distributive and associative properties oi Multiplication (212, 223);
while, for this case of multiplication of quaternions, we have already
seen (191, (I-)) *^^^ *^® commutative property also holds good, as
it does in algebraic multiplication. It may therefore be not irrele-
vant nor useless to insert here a short Second Chapter on the subject
oi ^UQh complanars : in treating briefly of which, while assuming as
proved the existence of all the foregoing properties, we shall have an
opportunity to say something of Powers and Roots and Logarithms ;
and of the connexion of Quaternions with Plane Trigonometry, and
with Algebraical Equations. After which, in the Third and last
Chapter of this Second Book, we propose to resume, for a short time,
the consideration oi Diplanar Quaternions; and especially to show
how the Associative Principle of Multiplication can be established,
for them, without* employing the Distributive Principle,
* Compare the Note to page 236.
240 ELEMENTS OF QUATERNIONS. [bOOK II.
CHAPTER II.
ON COMPLANAR QUATERNIONS, OR QUOTIENTS OF VECTORS IN
ONE PLANE ; AND ON POWERS, ROOTS, AND
LOGARITHMS OF QUATERNIONS.
Section 1. — On Complanar Proportion of Vectors; Fourth
Proportional to Three, Third Proportional to Two, Mean
Proportional, Square Root; General Reduction of a Qua-
ternion in a given Plane, to a Standard Binomial Form.
225. The Quaternions of the present Chapter shall all be
supposed to be complanar (119); their common plane being
assumed to coincide.with that of the given right versor t ( 1 8 1 ).
And all the lines, or vectors, such as a, j3, 7, &c., or ao> oi, 02)
&c., to be here employed, shall be conceived to be in that
given plane of 2; so that we may write (by 123), for the pur-
poses of this Chapter, thejbrmulce of complanarity :
?lll?'lll/---llh'; «llh-> /3||iz, ««|||i,&c.
226. Under th^se conditions, we can always (by 103, 117)
interpret any symbol of the form (j3 : a) .7, as denoting a line
8 in the given plane; which line may also be denoted (125)
by the symbol (7 ; a) .j3, but nof^ (comp. 103) by either of the
two apparently equivalent symbols, (J3.7) : a, {y.^):a\ so
that we may write,
I... 8 = ^7 = ^/3,
a a
and may say that this line 8 is the Fourth Proportional to the
* In fact the symbols /3 . y, y . j3, or /3y, y/3, have not as yet received -with us
any interpretation ; and even when they shall come to be interpreted as represent-
ing certain quaternions, it will be found (comp. 168) that the two combinations,
- y and — , have generally different significations.
a a
CHAP. II.] COMPLANAR PROPORTION OF VECTORS. 241
three lines a, P, 7 ; or to the three lines a, 7, /3 ; the two
Means, /3 and 7, of any such Complanar Proportion of Four
Vectors, admitting thus of being interchanged, as in algebra.
Under the same conditions we may write also (by 125),
II...a = -g7 = g0. /3 = -g = -a; 7 = ^a=^S,
so that (still as in algebra) the two Extremes, a and S, of any
such proportion of four lines a, jS, 7, d, may likewise change
places among themselves : while we may also make the means
become the extremes, if we at the same time change the ex-
tremes to means. More generally, if a, /3, 7, ^, e . . . be «wy
odd number of vectors in the given plane, we can always find
another vector p in that plane, which shall satisfy the equa-
tion,
"I Vr-^' - "^'••- •••ii-=i'
and when such a formula holds good, for any 07ie arrangement
of the numerator-lines a, 7, e, . . . and of the denominator-lines
/o, j3, S . . . it can easily be proved to hold good also for any
other arrangement of the numerators, and any other arrange-
ment of the denominators. For example, whatever four (com-
planar) vectors may be denoted by ^yde, we have the trans-
formations,
the two numerators being thus interchanged. Again,
so that the two denominators also may change places.
227. An interesting case of such proportion (226) is that
in which the means coincide; so that only three distinct lines,
such as a, j3, 7, are involved : and that we have (comp. Art.
149, and Fig. 42) an equation of the form,
I. ..7 = ^^, or a=^i3,
a 7
2 I
242 ELEMENTS OF QUATERNIONS. [bOOK II.
but nof^ 7 = ]3j3 : a, nor a = j3/3 : y. In this case, it is said that
the three lines afiy form a Continued Proportion; of which a
and y are now the Extremes, and j3 is the Memi : this line j3
being also said to be af Mea7i Proportional between the two
others, a and y ; while y is the Third Proportional to the two
lines a and j3 ; and d is, at the same time, the third propor-
tional to y and j3. Under the same conditions, we have
1I.../3 = ^, = I„;
SO that this mean, /3, between a and 7, is also the fourth pro-
portional (226) to itself, as first, and to those two other lines.
We have also (comp. again 149),
III. l^\-y fP'
■J
a \y
whence it is natural to write,
and therefore (by 103),
although we are not here to write j3 = (ya)i, nor j3 = (ay)^.
But because we have always, as in algebra (comp. 199, (3.) ),
the equation or identity, (- qy = g\ we are equally well enti-
tled to write.
fi-? -^■e^-e^
the symbol gh denoting thus, in general, either of two opposite
quaternions, whereof however one, namely that one of which
the angle is acute, has been already selectedm 199, (1.), as that
which shall be called by us the Square Root of the quaternion
* Compare the Note to the foregoing Article.
f "We say, a mean proportional ; because we shall shortly see that the opposite
line, — j3, is in the same sense another mean; although a rule will presently be given,
for distinguishing between them, and for selecting one, as that which may be called,
by eminence, the mean proportional.
CHAP. II.] CONTINUED PROPORTION, MEAN PROPORTIONAL. 243
q^ and denoted by 'sj q. We may therefore establish the for-
mula,
if a, jS, 7 form, as above, a continued proportion ; the upper
signs being taken when (as in Fig. 42) the angle aoc, between
the extreme lines a, y, is bisected by the line ob, or /3, itself;
but the lower signs, when that angle is bisected by the opposite
line, -/B, or when j3 bisects the vertically opposite angle (comp.
again 199, (3.) ): but tho, proportion of tensors,
VIII. ..Ty:Tj3 = Tj3:Ta,
and the resulting formula3,
IX. . . T/3^ = Ta .Ty, Tj3 = v/ (Ta .Ty),
in ^aeA case holding good. And when we shall speak simply
of the Mean Proportional between two vectors, a and y, which
make any acute, or right, or obtuse angle with each other, we
shall always henceforth understand the former of these two
bisectors ; namely, the bisector ob of that angle aoc itself, and
not that of the opposite angle : thus taking upper signs, in the
recent formula VII.
(1.) At the limit wheu the angle aoc vanishes, so that Uy = Ua, then U/3 —
each of these two unit-lines; and the mean proportional /3 has the same common
direction as each of the two given extremes. This comes to our agreeing to write,
X. . . VI = + 1, and generally, X'. . . V(a2) =+ a,
if a be any positive scalar.
(2.) At the other limit, when A0C = 7r, or Uy =— Ua, the length of the mean
proportional /3 is still determined by IX., as the geometric mean (in the usual sense)
between the lengths of the two given extremes (comp. the two Figures 41); but,
even with the supposed restriction (225) on the plane in which all the lines are
situated, an ambiguity arises in this case, from the doubt which of the two opposite
perpendiculars at o, to the line AOC, is to be taken as the direction of the mean vec-
tor. To remove this ambiguity, we shall suppose that the rotation round the axis
of i (to which axis all the lines considered in this Chapter are, by 225, perpendicu-
lar), from the first line oa to the second line ob, is in this case positive ; which
supposition is equivalent to writing, for present purposes,
XI.* . . V-l = + i; and XI'. . . V(- a^) = la, if a>0.
* It is to be carefully observed that this square root of negative unity is not, in
any sense, imaginary, nor even ambiguous, in its geometrical interpretation, but
denotes a real and given right versor (181).
244 ELEMENTS OF QUATERNIONS. [bOOK II.
And thus the mean proportional between two vectors (^in the given plane) becomes,
in all cases, determined : at least if their order (as first and third) be given.
(3.) If the restriction (225) on the common plane of the lines, were removed, we
might then, on the recent plan (227), fix definitely the direction, as well as the
length, of the mean OB, in every case hut one: this excepted case being that in
which, as in (2.), the tvio given extremes, OA, oc, have exactly opposite directions ; so
that the angle (aoc = tt) between them has no one definite bisector. In this case, the
sought point b would have no one determined position, but only a locus : namely the
circumference of a circle, with o for centre, and with a radius equal to the geome-
tric mean between oa, oc, while its plane would be perpendicular to the given right
line AOC. (Comp. again the Figures 41 ; and the remarks in 148, 149, 153, 154,
on the square of a right radial, or versor, and on the partially indeterminate cha-
racter of the square root of a negative scalar, when interpreted, on the plan of this
Calculus, as a real in geometry.)
228. The quotient of any two complanar and right quater-
nions has been seen (191, (6.) ) to be a scalar ; since then we
here suppose (225) that q\\\h we are at liberty to write,
I. . . Sg = aj ; V^ 'i=y', y^q - yi = iy ;
and consequently may establish the following Reduction of a
Quaternion in the given Plane (of i) to a Standard Binomial
Form* (comp. 221) :
II. . . q^x^iy, if q\\\i',
X and y being some two scalars, which may be called the two
constituents (comp. again 221) of this binomial. And then an
equation between two quaternions, considered as binomials of
this form, such as the equation,
III, ' ' q' =q, or III'. . , od ■\- iy = x + iy^
breaks up (by 202, (5.) ) into two scalar equations between
their respective constituents^ namely,
IV. . . x=^x, y=y,
notwithstanding the geometrical reality of the right versor, i.
(1.) On comparing the recent equations II., III., IV., with those marked as III.,
v., VI., in 221, we see that, in thus passing from general to com/)7anar quaternions,
we have merely suppressed the coefficients ofj and k, as being for our present purpose,
null ; and have then written x and y, instead of w and x.
* It \& permitted, by 227, XI., to write this expression as aj + y V — 1 ; but the
form a; + ty is shorter, and perhaps less liable to any ambiguity of interpretation.
CHAP. II.] STANDARD BINOMIAL FORM, COUPLE. 245
(2.) As the word " binomial" has other meanings in algebra, it may be conve-
nient to call the form II. a Couple ; and the two constituent scalars x and y, of
which the values serve to distinguish one such couple from another, may not unna-
turally be said to be the Co-ordinates of that Couple, for a reason which it may be
useful to state.
(3.) Conceive, then, that the plane of Fig. 60 coincides with that of i, and that
positive rotation round Ax.i is, in that Figure, directed towards the left-hand;
which may be reconciled with our general convention (127), by imagining that this
axis of i is directed from o towards the back of the Figure ; or below* it, if horizon-
tal. This being assumed, and perpendiculars bb', bb" being let fall (as in the Fi-
gure) on the indefinite line oa itself, and on a normal to that line at o, which nor-
mal we may call oa', and may suppose it to have a length equal to that of oa, with
a left-handed rotation aoa', so that
V. . . 0A' = i.0A, or briefly, V. . . a' = ia,
while j3' = ob', and /3"= ob", as in 201, and q = (3:a, as in 202 ;
then, on whichever side of the indefinite right line oa the point b may be situated,
a comparison of the quaternion q with the binomial form II. will give the two equa-
tions,
VI. . . iK (= S5) = j8' : a ; y (= Yq : i = /3" : ia) = /3" : a ;
so that these two scalars, x and y, are precisely the two rectangular co-ordinates of
the point B, referred to the two lines OA and oa', as ttbo rectangular unit-axes, of
the ordinary (or Cartesian) kind. And since evert/ other quaternion, g'z=x' + iy\
in the given plane, can be reduced to the form y : a, or 00 : OA, where c is a point
in that plane, which can be projected into c' and c" in the same way (comp. 197,
205), we see that the two new scalars, or constituents, x' and y', are simply (for
the same reason) the co-ordinates of the new point c, referred to the same pair of
axes.
(4.) It is evident (from the principles of the foregoing Chapter), that if we thus
express as couples (2.) any two complanar quaternions, q and q, we shall have the
following general transformations for their sum, difference, and product :
Nil.. . q±q = {x'±x) + i(jy'±y);
VIII. . . q,q = (x'x - y'y) + i {x'y + yx).
(6.) Again, for any one such couple, q, we have (comp. 222) not only Sg = x, and
V5 = iy, as above, but also,
IX. . .Kg = a;-z>; X. . . N9 = x2 +y2 . XL . . T5=V(a;2 +y3);
XII... U, = -^,; XIII...i=4^^;&c.
V(-x'2+y^) q a;2-fy2'
(6.) Hence, for the quotient of any two such couples, we have,
f 9' _ x + it/ _ x" + iy'
XIV. . . \'^~ x + iy ~ a;2+y
[_ x" = x'x + y'y, y" = yx - x'y.
2, x" -I- iy = g'K^,
* Compare the second Note to page 108.
246 ELEMENTS OF QUATERNIONS. [bOOK II.
(7.) The law of the norms (191, (8.) ), or the formula, N^'g- = N^' . Nj, is ex-
pressed here (comp. 222, (3.) ) by the well-known algebraic equation, or identity,
XV. . . (af^ + y^) {x^ +y^) = {,xx -y'y)^ ^{x'y + y'xy ',
in which xyx'y' may be any four scalars.
Section 2. — On Continued Proportion of Four or more Vec-
tors ; Whole Powers and Roots of Quaternions ; and Roots
of Unity,
229. The conception of continued proportion {211) may
easily be extended from the case o^ three to that of four or
more (com planar) vectors ; and thus a theory may be formed
oi cubes and higher whole powers of quaternions ^ with a corre-
spondingly extended theory of roots of quaternions, including
roots of scalars^ and in particular of unity. Thus if we sup-
pose that the four vectors a^y^ form a continued proportion,
expressed by the formulae.
I. . . - = 75 = -, whence II. . . - = - ^ ^ ' "^ ^^
7 p a a y p a
(by an obvious extension of usual algebraic notation,) we may
say that the quaternion S : a is the cube^ or the third power, of
j3 : a ; and that the latter quaternion is, conversely, a cube-
root (or third root) of the former ; which last relation may na-
turally be denoted by writing,
III. . . ^ = ('^Y, or Iir. ../3 = ^^Ya(comp.227,IV.,V.).
230. But it is important to observe that as the equation
q"^ = Q, in which «/ is a sought and Q is a given quaternion,
was found to be satisfied by two opposite quaternions q, of the
form ± \/ Q (comp. 227, VII.), so the slightly less simple
equation q^= Q is satisfied by three distinct and real quater-
nions, if Q be actual and real ; whereof each, divided by either
of the other two, gives for quotient a real quaternion, which
is equal to one of the cube-roots of positive unity. In fact, if
we conceive (comp. the annexed Fig. 54) that /3' and /3" are
two other but equally long vectors in the given plane, ob-
CHAP. II.J CUBE-ROOTS OF A QUATERNION, AND OF UNITY. 247
tained from j3 by two successive and positive rotations, each
through the third part of a circumference,
so that
fi' 15" 13'
IV.
or
IV'.
and therefore
V... (|)- = (|)-=,,*„v....f =(!)•, l-d
we shall have
-■.(?)--(fK!)'=!.--.^.(e
SO that we are equally entitled, at this stage, to write, instead
of III. or III'., these other equations :
vii...&'=f^Y, li'M'
or
Yll'...^-J'-l (5"-(^]K.
231. A (real and actual) quaternion Q may thus be said
to have three (real, actual, and) distinct cube-roots ; of which
however only one can have an angle less than sixty degrees ;
while none can have an angle equal to sixty degrees, unless the
proposed quaternion Q degenerates into a negative scalar. In
every other case, one of the three cube-roots of Q, or one of the
three values of the symbol Q^, may be considered as simpler
than either of the other two, because it has a smaller angle
(comp. 199, (!•))» ^^^ ^f w^j for the present, denote this one,
which we shall call the Principal Cube-Hoot of the quaternion
Q, by the symbol ^ Q, we shall thus be enabled to estabhsh
the formula of inequality,
VIII. ..Z^Q<|, if zQ<7r.
232. At the limit, when Q degenerates, as above, into a negative
scalar, one of its cube-roots is itself a negative scalar, and has there-
248 ELEMENTS OF QUATERNIONS. [bOOK II.
fore its angle = w ; while each of the two other roots has its angle
= -. In this case, among these two roots of which the angles are
o
equal to each other, and are less than that of the third, we shall
consider as simpler^ and therefore as principal^ the one which an-
swers (comp. 227, (2.) ) to a positive rotation through sixty degrees ;
and so shall be led to write,
IX...y-l=lii^; and X...^y-l=|;
using thus the positive sign for the radical ^ 3, by which i is multi-
plied in the expression IX. for 2^- 1 ; with the connected for-
mula,
IX'. ..y(-a3) = ^(l4-^V3), if a>0;
although it might at first have seemed more natural to adopt as
principal the scalar value, and to write thus,
3/-l=-l;
which latter is in fact one value of the symbol, (- 1)*.
(1.) "We have, however, on the present plan, as in arithmetic,
XI. ,.^1 = 1; and XI'. . . ^(a3) = a, if a>0.
(2.) The equations,
XII...(^-^] =-1, and XIIL..|^— ^j= + l,
can be verified in calculation^ by actual cubing^ exactly as in algebra ; the only dif-
ference being, as regards the conception of the subject, that although i satisfies the
equation i^ = — 1, it is regarded here as altogether real; namely, as a real right ver-
sor* (181).
233. There is no difficulty in conceiving how the same general
principles may be extended (comp. 229) to a continued proportion
of 71 + 1 complanar vectors,
I. . . a, ai, aa, . . . a„,
* This conception differs fundamentally from one which had occurred to seve-
ral able writers, before the invention of the quaternions ; and according to which
the symbols 1 and V — 1 were interpreted as representing a pair of equally long and
mutually rectangular right lines, in a given plane. In Qtiaternions, no line is repre-
sented by the number, One, except as regards its length ; the reason being, mainly,
that we require, in the present Calculus, to be able to deal with all possible planes ;
and that no one right line is common to all such.
CHAP. II.] FRACTI0NALPOWERS,GENERALR0OTSOFUNITY. 249
when n is a whole number greater than three ; nor in interpreting,
in connexion therewith, the equations,
II...^ = f^'r; III...-'=f2^\^; IV.
a \ a
•••"=(7)""-
Denoting, for the moment, what we shall call the principal n*^ root
of a quaternion Q by the symbol !y/Q, we have, on this plan (comp.
231, VIII.),
V. ..zyQ<-, if za<'^;
VI. . . ,1 (y- 1) = -; VII. . . Y(y- l):e>0;
To
this last condition, namely that there shall be a positive (scalar) co-
efficient y of 2, in the binomial (or couple) form x-\-iy (228), for the
quaternion^- 1, thus serving to complete the determination of
that principal fi*^ root of negative unity ; or of any other negative sca-
lar, since ~ 1 may be changed to -a, if «>0, in each of the two last
formulae. And as to the general n*^ root of a quaternion, we may
write, on the same principles,
VIIL.. Q^=l^. VQ;
where the factor 1», representing the general n*^ root of positive
unity, has n different values, depending on the division of the cir-
cumference of a circle into n equal parts, in the way lately illus-
trated, for the case ?z = 3, by Figure 54 ; and only differing from
ordinary algebra by the reality here attributed to i. In fact, each
of these n*^ roots of unity is with us a real versor; namely the quo-
tient of two radii of a circle, which make with each other an angle,
equal to the n*^ part of some whole number of circumferences.
X
(1.) "We propose, however, to interpret the particular symbol i^, as always de-
noting the principal value of the n*^ root of i ; thus writing,
i n/
IX. . . t« = \/i;
whence it will follow that when this root is expressed under the form of a couple
(228), the two constituents x and y shall both be positive, and the quotient y: x
shall have a smaller value than for any other couple x + iy (with constituents thus
positive), of which the n*^ power equals i.
(2 ) For example, although the equation
52 = (ar + ty)2 = i,
vi satisfied by the two values, ± (1 + : V2, we shall write definitely,
2 K
250 ELEMENTS OF QUATERNIONS. [bOOK II.
x....-.=.v.-=i±i.
(3.) And although the equation,
is satisfied by the three distinct and real couples, (i ± V3) : 2, and - 1, we shall adopt
only the one value,
XI. . . il-V t= — r— .
(4.) In general, we shall thus have the expression,
XII. . . t** = cos -- + 1 sm — - ;
2n 2n
which we shall occasionally abridge to the following :
i TT
Xir. . . i« = cis — :
2n'
and this root^ i", thus interpreted, denotes a versor, which turns any line on which it
operates, through an angle equal to the n*'* part of a right angle, in the positive di-
rection of rotation, round the given axis of i.
234. If m and n be anj/ two positive whole numbers, and q
any quaternion, the definition contained in the formula 233,
II., of the whole power, q^, enables us to write, as in algebra,
the two equations :
I. . . y'"^« = ^»»^ ; II. . . (^")'" = ^™" ;
and we propose to extend the former to the case of mill and
negative whole exponents, writing therefore,
III. . . ^°= 1 ; IV. . . q^ri-n^^m.gn .
and in particular,
Y. . . q-^ = l :q = - = reciprocal* (134) of q.
We shall also extend the formula II., by writing
VI. . . (^")'" = q^,
whether m be positive or negative ; so that this last symbol,
ifm and n be still whole numbers, whereof w may be supposed
to be positive, has as many distinct values as there are units in
the denominator of li^ fractional exponent, when reduced to its
* Compare the Note to page 121.
CHAP. II.J AMPLITUDE OF A QUATERNION. 251
m
least terms ; among which values of q~\ we shall naturally
consider as the principal one, that which is the m^^ power of
the principal n*^ root (233) of q.
(1.) For example, the symbol gi denotes, on this plan, the square of any cube-
root of 9 ; it has therefore three distinct values, namely, the three values of the cube-
root of the square of the same quaternion q ; but among these we regard as principal,
the square of the principal cube-root (231) of that proposed quaternion.
(2.) Again, the symbol q'^ is interpreted, on the same plan, as denoting the
square of any fourth root of 5 ; but because (li)2 =z li = + 1, this square has only
two distinct values, namely those of the square root q^, the fractional exponent |
being thus reduced to its least terms; and among these the principal value is the
square of the principal fourth root, which square is, at the same time, the principal
square root (199, (l.)> ^^ 227) of the quaternion q.
(3.) The symbol q-^ denotes, as in algebra, the reciprocal of a square-root of q ;
while g'2 denotes the reciprocal of the square, &c.
(4.) If the exponent #, in a symbol of the form q^, be still a scalar, but a surd (or
incommensurable), we may consider this surd exponent, t, as a limit, towards which
a variable fraction tends : and the symbol itself may then be interpreted as the corre-
sponding limit oi a, fractional power of a quaternion, which has however (in this case)
indefinitely many values, and can therefore be of little or no use, until a selection
shall have been made, of one value of this surd power &.& principal, according to a law
which will be best understood by the introduction of the conception of the amplitude
of a quaternion, to which in the next Section we shall proceed.
(5.) Meanwhile (comp. 233), (4.) ), we may already definitely interpret the sym-
bol V' as denoting a versor, which turns any line in the given plane, through t right
angles, round Ax.i, in the positive or negative direction, according as this scalar ex-
ponent, t, whether rational or irrational, is itself positive or negative ; and thus may
establish the formula,
-TTxr w 'tt , . tir
VII. . . I* = cos — - -f I sm — ;
2 it I
or briefly (comp. 238, XII'.),
VIII.. . i' = cis— .
2
Section 3.— ^Ow the Amplitudes of Quaternions in a given
Plane; and on Trigonometric Expressions for such Quater-
nions, and for their Powers,
235. Using the binomial or couple form (228) for a qua-
ternion in the plane of/ (225), if we introduce two new and
real scalars, r and z, whereof the former shall be supposed to
be positive, and which are connected with the two former sca-
lars X and y by the equations,
I. . . x-r cos z, y =^r sin ^, r > 0,
252 ELEMENTS OF QUATERNIONS. [bOOK II.
we shall then evidently have the formulae (comp. 228, (5.) ) :
n. . .Tq = T(x + ii/) = r;
III. . . TJq = U (aj + iy) = cos z + i8in.z;
which last expression may be conveniently abridged (comp.
233, Xir., and 234, VIII.) to the following :
IV. . , U<^ =cisz ; so that V. . . g==r cisz.
And the arcual or angular quantity, z, maybe called the Am-
pUtude* of the quaternion q ; this name being here preferred
by us to " Angle" because we have already appropriated
the latter name, and the corresponding symbol Z ^, to denote
(130) an angle of the Euclidean kind, or at least one not ex-
ceeding, in either direction, the limits and tt ; whereas the
amplitude, z, considered as obliged only to satisfy the equa-
tions I., may have any real and scalar value. We shall denote
this amplitude, at least for the present, by XhQ symbol,^ am.y,
or simply, am q ; and thus shall have the following formula,
of connexion between amplitude and angle,
VI. . . (2: =) am . 5^ = 2w7r ± z $» ;
* Compare the Note to Art. 130.
t The symbol V was spoken of, in 202, as completing the system of notations
peculiar to the present Calculus ; and in fact, besides the three letters^ i, j, k, of which
the laws are expressed by thQ fundamental formula (A) of Art. 183, and which were
originally (namely in the year 1843, and in the two following years) the only pecu-
liar symbols of quaternions (see Note to page 160), that Calculus does not habi-
tually employ, with peculiar significations, any more than the^ue characteristics of
operation, K, S, T, U, V, for conjugate, scalar, tensor, versor, and vector (or right
part) : although perhaps the mark N for norm, which in the present work has been
adopted from the Theory of Numbers, will gradually come more into use than
it has yet done, in connexion with quaternions also. As to the marks, Z, Ax., I, R,
and now am . (or am,,), for angle, axis, index, reciprocal, and amplitude, they are to
be considered as chiefly available for the present exposition of the system, and as not
often wanted, nor employed, in the subsequentprac^ice thereof ; and the same remark
applies to the recent abridgment cis, for cos + i sin ; to some notations in the present
Section for powers and roots, serving to express the conception of one «'^ root, &c.,
as distinguished from another ; and to the characteristic P, of what we shall call in the
next section the ponential of a quaternion, though not requiring that notation after-
wards. No apology need be made for employing the purely geometrical signs, -i-,
II, III, for perpendicularity, parallelism, and complanarity : although the last of
them was perhaps first introduced by the present writer, who has found it frequently
useful.
CHAP. II.] ADDITION AND SUBTRACTION OF AMPLITUDES. 253
the upper or the lower sign being taken, according as Ax. q
- ± Ax. i ; and n being any whole number, positive or negative
or null. We may then write also (for any quaternion 5' ||1
the general transformations following :
VII. . . \Jq = cis am q ; VIII. . . 5' = T^ . cis am q.
(1.) Writing q = f3: a, the amplitude am. g', or am (/3 : a), is thus a scalar quan-
tity, expressing (with its proper sign) the amount of rotation^ round Ax. i, from the
line a to the line /3 ; and admitting, in general, of being increased or diminished by
any whole number of circumferences, or oi entire revolutions, when only the direc-
tions of the two lines, a and /3, in the given plane of i, are given.
(2.) But the particular quaternion, or right versor, i itself, shall be considered
as having definitely/, for its amplitude, one right angle; so that we shall establish the
particular formula,
. . '"'
IX. . . am.t = /i 1 = -.
(3.) When, for any other given quaternion q, the generally arbitrary integer
n in VI. receives any one determined value, the corresponding value of the ampli-
tude may be denoted by either of the two following temporary symbols,* which we
here treat as equivalent to each other,
am„ .q, or Zn 9 ;
so that (with the same rule of signs as before) we may write, as a more definite for-
mula than VI., the equation :
X. . . am,, . 9 = Zm 9 = 2«7r ± Z. 9 ;
and may say that this last quantity is the n^^ value of the amplitude of q ; while the
zero-value, amoj, may be called the principal amplitude (or the principal value of
the amplitude).
(4.) With these notations, and with the convention, amo(— l) = + 7r, we may
write,
XI. . . amo q = loq = ±lq',
XII. . . am„ a = am,, 1 = Zn 1 = 2n7r, if a > ;
and
XIII. . . am„ (- a) = am„(- 1) = Z„ (- 1) = (2« + 1) tt,
if a be still a positive scalar.
236. From the foregoing definition of amplitude, and from
the formerly established connexion of multiplication ofversors
with composition of rotations (207), it is obvious that (within
the given plane, and with abstraction made of tensors) multi-
plication and division of quaternions answer respectively to
* Compare the recent Note, respecting the notations employed.
254 ELEMENTS OF QUATERNIONS. [bOOK II.
(algebraical) addition and subtraction of amplitudes : so that,
if the symbol sna.q be interpreted in the general (or indefinite)
sense of the equation 235, VI., we may write :
I. . . am {q'. q) = am q' + am q ; II. . . am (q'l q) = am q'- am q ;
implying hereby that, in each formula, o?ie of the values, of the
first member is among the values of the second member ; but
not here specifying which value. With the same generality
of signification, it follows evidently that, for a product of ani/
number of (complanar) quaternions, and for a whole power of any
one quaternion, we have the analogous formulae :
III. . . am rig = S am 5' ; IV. . . am.qP =p.2Lmq ;
where the exponent p may be any positive or negative integer,
or zero.
(1.) It was proved, in 191, II., that for an7/ two quaternions, the formula Vq'q
= XJq'.Vq holds good; a result which, by the associative principle of multiplication
(223), is easily extended to ani/ number of quaternion factors (complanar or dipla-
nar), with an analogous result for tensors : so that we may write, generally,
V. . . Un^ = U\Jq ; VI. . . TUq = UTq.
(2.) Confining ourselves to the first of these two equations, and combining it with
III., and with 235, VII., we arrive at the important formula :
VII. . . n cis am 5 (= UVq = UII5' = cis am 119) = cis 2 am g ;
whence in particular (corap. IV.),
VIII. . . (cis am q)p=cis(p . am q),
at least if the exponent p be still any whole number.
(3.) In these last formula), the amplitudes am. 5-, am. 5^', &c., may represent a?i^
angular quantities, z, 2', &c. ; we may therefore write them thus,
IX. . . n cis 2 = cis Sz ; X. . . (cis z)p = cis pz ;
including thus, under abridged forms, some known and useful theorems, respecting
cosines and sines of sums and multiples of arcs.
(4.) For example, if the number of factors of the form cis z be two, we have
thus,
IX'. . . cis z' . cis z = cis (z' + 2) ; X'. . . (cis 2)2 = cis 2z ;
whence
cos (z' + z) = S (cis z' . cis 2) = cos s;' cos z - sin z' sin 2; ;
sin(2' + z) = i-iV(cisz'.cisz) = cos 2' sin 2 + sin z' cosz ;
cos 2z = (cos zy — (sin 2)2 ; sin 22 = 2 cos z sin z ;
with similar results for more factors than two.
(5.) Without expressly introducing the conception, or at least the notation of
amplitude, we may derive the recent formula) IX. and X., from the consideration of
the power V (234), as foUoAVS. That pozrer ofi, with a scalar exponent, t, has been
CHAP. II.] POWERS WITH SCALAR EXPONENTS. 255
interpreted in 234, (5.)j as a symbol satisfying an equation which may be written
thus:
XI. . . V — cis z, if z = ^tTT ;
or geometrically as a versor, which turns a line through t right angles, where t may
be any scalar. We see then at once, from this interpretation, that if*' be either the
same or any other scalar, the formula,
XII. . . iHt'= ii^f, or XIII, . . n . i< = i^\
must hold good, as in algebra. And because the number of the factors t* is easily
seen to be arbitrary in this last formula, we may write also,
XIV. . .(it)p=ipf,'
if p be any whole* number. But the two last formulae may be changed by XI., to
the equations IX, and X., which are therefore thus again obtained ; although the
later forms, namely XIII. and XIV., are perhaps somewhat simpler: having in-
deed the appearance of being mere algebraical identities, although we see that their
geometrical interpretations, as given above, are important.
(6.) In connexion with the same interpretation XI. of the same useful symbol i*,
it may be noticed here that
XV. .. K.it=i-i',
and that therefore,
XVI. . . cos — = S. i' = i(z' -f i-t) ;
t'jr
XVII. . . sin — =. i-i V. i* = i i-i (it - i-ty
(7.) Hence, by raising the double of each member of XVI. to any positive whole
power p, halving, and substituting z for ^tir, we get the equation,
XVIII. . . 2p-» (cos z)p= I (it+ i-t)P = | (iP*+ i-p*) + Ip (i(p-2)t + i(2-/>)«) + &c.
= cos pz+p coa(p - 2)z +?-^^-^—^ cos (p - A) z+ 8ic.,
with the usual rule for halving the coefficient of cos Oz, ifp be an even integer ; and
with analogous processes for obtaining the known expansions of 2^"^ (sin z)p, for any
positive whole value, even or odd, of p ; and many other known results of the same
kind.
237. Ifp be still a whole number, we have thus the transforma-
tion,
I. . . qp = (r cis zy = ?'P cis pz = (TqY cis (/> . ato q) ;
in which (comp. 190, 161) the two factors, of the tensor and versor
kinds, may be thus written :
II. . . T (qY = {Tqy = T^'' ; III. . . U (q^) = (U^)^ = Vq^ ;
and any value (235) of the amplitude nm.q may be taken, since all
• It will soon be seen that there is a sense, although one not quite so definite, in
which this formula holds good, even when the exponent p is fractional, or surd ;
namely, that the second member is then one of the values of the first.
256 ELEMENTS OF QUATERNIONS. [bOOK II.'
will conduct to one common value of this whole power q^. And if,
for I., we substitute this slightly different formula (comp. 235,
(3-)),
IV. . . (qP)n = TqP . cis (p . am„ q\ with i? = ~, n'>0,
m^, n', n being whole numbers whereof the first is supposed to be
prime to the second, so that the exponent p is here a fraction in its
least terms, with a positive denominator n\ while the factor Tq^ is
interpreted as expositive scalar (of which the positive or negative
logarithm, in any given system, is equal io px the logarithm of T^-),
then the expression in the second member admits of n' distinct va-
lues, answering to different values of n ; which are precisely the n'
values (comp. 234) of the fractional power q^, on principles already
established : the principal value of that power corresponding to the
value n=0.
(1.) For any value of the integer w, we may say that the symbol (qp),i, defined
by the formula IV., represents the n'^ value of the power qv ; such values, however,
recurring periodically, when p is, as above, o. fraction.
(2.) Abridging (1p)„ to 1^,,, we have thus, generally, by 235, XII.,
V, . . lP„ = cis Ipnir, if /j be any fraction,
a restriction which however we shall soon remove ; and in particular,
VI. . . Principal value oflP= 1Po= 1.
(3.) Thus, making successively jp = |, /> = ^, we have
VII. . . li„ = cis mr, Ik = + 1, l^i = - 1, 1^3 = + 1, &c. ;
-I7TTT -,1 • 2«7r ,. ^ ^. -l + tV3 ^. -l-tV3 ^. ^ -
VIII. . . Un = CIS — , Uo = 1, 1*1 = , 1*3 = , 1*3 = 1, &C.
(4.) Denoting in like manner the n^^ value of (- 1)p by the abridged symbol
(- l^w, we have, on the same plan (comp. 235, XIII.), for any fractional* value
of/?,
IX. . . (- iyn = clsp(2n+ l)7r; whence (comp. 232),
X. ..(-l)io = cis-=+t-, (-l)ii = ci8-2- = -i, (-l)i2 = + t, &c.;
and
XI...(-l)lo = iJ^^ (-1)..=-!, (-l)., = il^%c.,
these three values of (- l)i recurring periodically.
(5.) The formula IV. gives, generally, by V., the transformation,
XII. . . (qp)n = (qP)o cis 2pmr = lP«(gP)o 5
so that the n*'» value of qP is equal to the principal value of that power of y, multi-
As before, this restriction is only a temporary one.
CHAP. II.] PONENTIAL OF A QUATERNION. 257
plied by the corresponding value of the same power of positive unity ; and it may be
remarked, that if the base a be any positive scalar^ the principal p^^ power ^ (^)o)
is simply, by our definitions, the arithmetical value of aP.
(6.) The n*^ value of the p^^ power of any negative scalar, — a, is in like man-
ner equal to the arithmetical p^^^ power of the positive opposite, +a, multiplied by
the corresponding value of the same power of negative unity; or in symbols,
XIII. . . (- a)Pn={- l)Pn (aP)o = (aP)oci8i)(2n+ l)7r.
(7.) The formula IV., with its consequences V. VI. IX. XII. XIII., may be
extended so as to include, as a limit, the case when the exponent p being still scalar,
becomes incommensurable, or surd; and although the number of values of the power
qp becomes thus unlimited (comp. 234, (4.)), yet we can still consider one of them
as the principal value of this (now) surd power : namely the value,
XIV. . . (5^)0 = TqP . cis {p amo q),
which answers to i\xQ principal amplitude (235, (3.) ) of the proposed quaternion q.
238. We may therefore consider the symbol^
^^
in which the base, q^ is any quaternion, while the exponent, p^
is any scalar^ as being now fully interpreted; but no interpre-
tation has been as yet assigned to this other symbol of the
same kind, qq'^
in which both the base q, and the exponent q, are supposed
to be (generally) quaternions, although for the purposes of this
Chapter complanar (225). To do this, in a way which shall
be completely consistent w^th the foregoing conventions and
conclusions, or rather which shall include and reproduce them,
for the case where the new quaternion exponent, q, degenerates
(131) into a scalar, will be one main object of the following
Section : which however will also contain a theory of loga-
rithms of quaternions, and of the connexion of both logarithms
and powers with the properties of a certain function, which
we shall call the ponential of a quaternion, and to consider
which we next proceed.
Section 4. — On the Ponential and Logarithm of a Quater-
ternion; and on Powers of Quaternions, with Quaternions
for their Exponents.
239. If we consider the polynomial function,
I. . . P(^, m)=\^q,^q^^..q,,,,
2 L
258 ELEMENTS OF QUATERNIONS. [bOOK II.
in which q is any quaternion, and m is any positive whole number,
while it is supposed (for conciseness) that
11. ••^-=i.2.3..mV"r(m+l)/
y^^Ai^' *^6n it is not difficult to prove that however great, hut finite and
C ; ^t given, the tensor Tq may be, a finite number m can be assigned, for
^ off ' which the inequality
III. . . T (P(g, m+n)-F (q, m)) < a, if a > 0,
shall be satisfied, however large the (positive whole) number n may
be, and however small the (positive) scalar a, provided that this last
is given. In other words, if we write (comp. 228),
IV. . . q = x + ii/, F(q, m) = X^ + iT^,
a finite value of the number m can always be assigned, such that the
following inequality,
V. . . (X^,,-X„,y + (Y^.„-T,^y<a^
shall hold good, however large the number w, and however small
(but given and > 0) the scalar a may be. It follows evidently that
each of the two scalar series, or succession of scalar functions,
VI...Xo=l, X,= \+x, X,= l+x+''-^,.. X^,..
VII...ro = 0, Yr==7/, T, = y+xy,.. Y^, . . .
converges ultimately to a fixed and finite limit, whereof the one may be
called Xoo, or simply X, and the latter Foo, or F, and of which each
is a certain function of the two scalars, x and y. Writing then
VIII. . . Q = Xoo+iFoo=X+er,
Ave must consider this quaternion Q (namely the limit to which the
following series of quaternions,
IX...P(g,0)=l, P(?, 1) = 1+^, P(^,2)=l + ^ + |',.. P(^,m),...
converges ultimately) as being in like manner a Q,Qiid.m function, which
we shall call the ponential function, or simply the Ponential of q, in
consequence of its possessing certain exponential properties; and
which may be denoted by any one of the three symbols,
P (?» oo), or P {q), or simply P^.
We have therefore the equation,
X. . . Ponential of q=Q==Vq=\-{-qy + q.i-\- . . + qccy
with the signification II. of the term q^.
CHAP. II.] EXPONENTIAL PROPERTY. 259
(1.) In connexion with the convergence of this ponential series, or with the in-
equality pi., it may be remarked that if we write (comp, 235) r = T^', and r^ = Tg-^,
we shall have, by 212, (2.),
XI. . . T (P(gr, m + n) - P (gr, m) ) < P (r, m + n) - P (r, m) ;
it is sufficient then to prove that this last difference, or the sum of the n positive
terms, r»i+i, . • ^w+w, can be made < a. Now if we take a number p>2r -1, we
shall have r^i <|rp, rp+2< |^p+i» &c,, so that a finite number m>p>2r- 1 can
be assigned, such that>»r<ja ; and then, ^ ,^ ft-—
XII. ..P(r,w+«)-P(r,7n)<a(2-i + 2-2 + ..'+2-«)<a; /"^' h
the asserted inequality is therefore proved to exist. ^ ^ ,
(2.) In general, if an ascending series with positive coefficients, such as ^^ %
XIII. . . Ao + Aig' + A2g2 + &c., where Ao> o, Ai>o, &c.,
be convergent when q is changed to a positive scalar, it will ^fortiori converge,
when g- is a quaternion. ' "
^ ^
240. Let q and q^ be any two complanar quaternions, and let q^^
be their sum, so that
I...5" = S' + g, 2"|||2'|ll?;
then, as in algebra, with the signification 239, II. of ^,„, and with
corresponding significations of q'm and q'^^j we have
II. . . qJ' = 1,2.^3!. ^ " ^'"^° "^ ^'""'^' "^ ^''"■'^' "^ • • ^ ^'"^"'
where ^o = ?'o = l- Hence, writing again r = T^, r,„ = T2'„„ and in
like manner r' = T^', r^^=Tq'\ &c., the two differences, _
III. . . P (r', 77z) . P (r, 7/1) - P (r'^ m), *^ ^, (^^ f)^(HUi-^
and s. /^-^' I -vx
IV. ..P(r",2m)-P(r',m).P(r,m), . . ^ ^ < --1-
can be expanded as sums of positive terms of the form r'p..rp (one^^"^^*^ y?
sum containing ^m(m+ 1), and the other containing m(m+ 1) such ^/
terms); but, by 239, HI-, the sum of these two positive differences ^
can be made less than any given small positive scalar a, since s ^*^ (*^ "^
V. . . P (r'^ 2m) - P (r'', m)<a, if a> 0,
provided that the number m is taken large enough ; each difference,
therefore, separately tends to 0, as m tends to 00 ; a tendency which
must exist a fortiori, when the tensors, r, r', r", are replaced by the
quaternions., q, q', q'^. The function Vq is therefore subject to the
Exponential Law,
\l...V{q'^q) = Vq'.Vq:=Vq.Vq\ if q' \\\ q. //
260 ELEMENTS OF QUATERNIONS. [boOK II.
(1.) If we write (comp. 237, (5.) ),
VII. . . PI = c, then VIII. . . Par = (£*)o = arithmetical value oft" ;
where e is the known base of the natural system of logarithms, and x is any scalar.
We shall henceforth write simply £«^ to denote this principal (or arithmetical) value of
the x*^ power of t , and so shall have the simplified equation,
VIII'. . . Pa;;=£*.
(2.) Already we have thus a motive for writing, generally,
IX. . . Vq = i1',
but this formula is here to be considered merely as a definition of the sense in which
we interpret this exponential symbol, (9 ; namely as what we have lately called the
ponential function, Fq, considered as the sum of the infinite but converging series,
239, X. It will however be soon seen to be included in a more general definition
(comp. 238) of the symbol g-?'.
(3.) For any scalar x, we have by VIII. the transformation :
X. . . x = \'Px = natural logarithm of ponential of x.
241. The exponential law (240) gives the following general de-
composition of a ponential into factors,
I. . . P^ = P(a;4-e» = P^.P?>;
in which we have just seen that the factor Vx is a positive scalar.
The other factor, Viy, is easily proved to be a versor, and therefore
to be the versor ofFq, while Fx is the tensor of the same ponen- .
tial; because we have in general,
11. . .P^.P(-g) = PO=I, and III. ..PK^ = KP^,
since IV. . . (K^)- =K(q^) = {say) Kq^ (comp. 199, IX.);
and therefore, in particular (comp. 150, 158), ^ ^J £ "s- ^"^"t '
V. ..l:P^> = P(-^» = KP^3/, or VI.". . NPz> = 1. '"^^^
I ■ jfc
We may therefore write (comp. 240, IX., X.), " ^^,3 ^
VII. . . TFq = VSq = Fx=^; VIII, . . x=Sq = lTFq;
IX. . . UP5 = PVg' = Piy = 6»>=cis?/ (comp. 235, IV.);
this last transformation being obtained from the two series,
X. . . SPz>=l-^ + &c. = cos^;
XI. . . r> VFiy = y - ^ + &c. = sin y.
Hence the ponential P^' may be thus transformed :
XII. . . P^ = P (x + iy) = e'' cis 7/.
CHAP. II.] CONNEXION WITH TRIGONOMETRY. 261
(1) If we had not chosen to assume as known the series for cosine and sine^ nor
to select (at first) any one unit of angle, such as that known one on which their va-
lidity depends, we might then have proceeded as follows. Writing
xiil. ..Piy=/y + %, /(-y)-+/y, 0(-y) = -^y, ^ *^
we should have, by the exponential law (240),
XIV. . ./(y + y') = S(Piy.Piy')=/y./y'-0y.0/;
XV. . .f{y-y)= fy.fy'+<l>y'^y'-.
and then the functional equation, which results, namely,
XVI. . . /(y + y') +/(y -yl = 2/y .//,
would show that
XVII. . . fy = cos,\ - X a right angle
whatever unit of angle may be adopted, provided that we determine the constant c
by the condition,
XVIII. . . c = least positive root of the equation fy(= SFiy) = ;
or nearly,
XVIII'. . . c= 1'5708, as the study of the series* would show.
(2.) A motive would thus arise for representing a right angle by this numerical
constant, c; or for so selecting the angular unit, as to have the equation (tt still de-
noting two right angles),
XIX. . . TT = 2c = least positive root of tke equation fy = — 1 ;
giving nearly,
XIX'. . . 7r = 314159, as usual;
for thus we should reduce XVII. to the simpler form,
XX. . .fy = cosy.
(3.) As to the function (py, since
XXL . . (fyy + (cpyy='Piy-'Pi-iy) = h
it is evident that 0y = + sin y ; and it is easy to prove that the upper sign is to be
taken. In fact, it can be shown (without supposing any previous knowledge of co-
sines or sines) that (pc is positive, and therefore that
XXII. . .<pc = + l, or XXIII. . . P«c= t ;
whence
XXIV. . . (py = S.i-^Fiy = SPi(y-c)=f(y-c),
and
XXV. ..Viy=fy + if{y-c).
If then we replace c by -, we have
* In fact, the value of the constant c may be obtained to this degree of accuracy,
by simple interpolation between the two approximate values of the function/,
- /(l-5)=+ 0-070737, /(l-6) = -0'029200;
and of course there arc artifices, not necessary to be mentioned here, by which a far
more accurate value can be found.
262 ELEMENTS OF QUATERNIONS. [bOOK II.
XXVI. . . 0y = COS [ y - -^ ] = sin 3^ ; and XXVII. . . Viy = cis y, as in IX.
'; (4.) The series X. XL for cosine and sine might thus be deduced^ instead of being
assumed as known : and since we have the limiting value,
XXIX. . . lim. y-i sin y = lim. y-i i^ YFiy = 1,
it follows that the unit of angle, which thus gives Pty = cisy, is (as usual) the angle
subtended at the centre by the arc equal to radius ; or that the number tt (or 2c) is
to 1, as the circumference is to the diameter of a circle.
(5.) If any other angular unit had been, for any reason, chosen, then a right
angle would of course be represented by a different number, and not by 1 '6708 nearly ;
but we should still have the transformation,
XXX. . . Piy = cis ( - X a right angle j,
though not the same series as before, for cos y and sin y.
242. The usual unit being retained, we see, by 241, XII., that
I. . . P. 2m7r = 1, and II. . . P(^ + 2?W) = P^,
if n be any whole number; it follows, then, that the inverse ponen-
tial function, "P'^q, or what we may call the Imponential, of a given
quaternion q, has indefinitely many values, which may all be repre-
sented by the formula,
III. . .P„-'^ = lT^f 2am„^; ^ ~- -^ ^^
and of which eac^ satisfies the equation, i' f ^ -^ P ["^^ ■*
IV. . . PP -1^ = ^- '«^
while the one which corresponds to w = may be called the Princi-
pal Imponential. It will be found that when the exponent p is any
scalar, the definition already given (237, IV., XII.) for the n^^ value
of the p*^ power of q enables us to establish the formula,
v...(20.=P(pP„-V);
and we now propose to extend this last formula, by a new defiiiition,
to the more general case (238), when the exponent is a quaternion q':
thus writing generally, for any two complanar quaternions, q and q,
the General Exponential Formula,
VI...(g^„ = P(2'P„-ff);
the principal value of q'^' being still conceived to correspond to n = 0,
or to the principal amplitude of q (comp. 235, (3.) ).
CHAP. II.] LOGARITHM OF A QUATERNION. 263
(1.) For example,
VII. . . (£9)o = T(qVo-h) = Fq, because Po-if = k = 1 ;
the ponential Fq, which we agreed, in 240, (2.), to denote simply by 6?, is therefore
now seen to be in fact, by our general definition, the principal value of that power,
or exponential.
(2.) With the same notations,
VIII. . . £»y = cis y, cos y = ^ (c'V + e-^v), sin 7/ =— (e'V - £-♦>) ;
these two last only differing from the usual imaginary expressions for cosine and sine,
by the geometrical reality* of the versor i.
(3.) The cosine and sine of a quaternion (in the given plane) may now be defined
by the equations :
IX. . . cos 5 = I (£»■« + £"»■«) ; X. . . sin 5^ = — (£»3 - r'l) ;
and we may write (comp. 241, IX.),
XI. . . cis 5 = £»■« = Fiq.
(4.) With this interpretation of cis q, the exponential properties, 236, IX., X.,
continue to hold good ; and we may write,
XII. . . (59')« = P C^'IT?). P OV amn 5) = (Tq^ cis (5' am,, 5) ;
a formula which evidently includes the corresponding one, 237, IV., for the n*^ value
of the p*^ power of g, when p is scalar.
(5.) The definitions III. and VI., combined with 235, XII., give generally,
XIII. . . 1„5' = (19')« = P . 2in7rq' ; XIV. . . {qi')n = !««'. (q^'^O ;
this last equation including the formula 237, XII.
(6.) The same definitions give,
XV. . . Fo-H = — ; XVI. . . (iOo = £~2- ;
which last equation agrees with a known interpretation of the symbol,
-/-I
considered as denoting in algebra a real quantity.
(7.) The formula VI. may even be extended to the case where the exponent q' is
a quaternion, which is not in the given plane ofi, and therefore not complanar with
the base q ; thus we may write,
XVII.. . (i.> = P(iPo-H-) = P^-^^ = -A;
but it would be foreign (225) to the plan of this Chapter to enter into any further de-
tails, on the subject of the interpretation of the exponential symbol qi', for this case
of diplanar quaternions, though we see that there would be no difficulty in treating
it, after what has been shown respecting complanars.
* Compare 232, (2.), and the Notes to pages 243, 248.
264 ELEMENTS OF QUATERNIONS. [bOOK II.
243. As regards the general logarithm q of a quaternion q (in the
given plane), we may regard it as any quaternion which satisfies the
equation,
I. . . ei' = Vq' = q',
and in this view it is simply the Imponential V'^q, of which the n^^
value is expressed by the formula 242, III. But the principal impo-
nential, which answers (as above) to w = 0, may be said to be the prin-
cipal logarithm^ or simply the Logarithm, of the quaternion q^ and may
be denoted by the symbol,
so that we may write,
I. . . Ig = Po"'2' = ITg' + i amo g';
or still more simply,
II. ..1^ = 1(T2.U^) = 1T^+1U^,
because 1TU2' = 11 =0, and therefore,
III. . . lU^ = i amo q.
We have thus the two general equations,
IV... % = lTg; V. .. V1(? = 1U^;
in which YTq is still the scalar and natural logarithm of the positive
scalar T^'.
(1.) As examples (comp. 235, (2.) and (4.) ),
VI. . . It = ifTT ; VII. . . 1(- 1) = iV.
(2.) The general logarithm of q may be denoted by any one of the symbols,
log . q, or log q, or (log q\,
this last denoting the «*^ value ; and then we shall have,
VIII. . . (log 9)n = 1^ + 2imr.
(3.) The formula,
IX. . . log . 99= log q' + log g-, if q \\\ q,
holds good, in the sense that every value of the first member is one of the values of
the second (comp. 236).
(4.) Principal value ofq'i'= tS'l? ; and one value of log . q9' = q'lq.
(6.) The quotient of two general logarithms,
X...(.og,VK.og,),= '0|^.
may be said to be the ^eweraZ logarithm of the quaternion, q', to the complanar qua-
ternion base, q ; and we see that its expression involves* two arbitrary and indepen-
dent integers, while its principal value may be defined to be Iq' : \q.
As the corresponding expression in algebra, according to Graves and Ohm.
CHAP. II.] EQUATIONS OF ALGEBRAIC FORM. 265
Section 5. — On Finite"^ {or Polynomial) Equations of Alge^
braic Form, involving Complanar Quaternions ; and on the
Existence ofn Real Quaternion Roots, of any such Equa-
tion of the n*'' Degree,
244. We have seen (233) that an equation of the form,
I. . .^"-Q = 0,
where n Is any given positive integer, and Q is anyj given,
real, and actual quaternion (144), has always n real, actual,
and unequal quaternion roots, q, complanar with Q ; namely,
the n distinct and real values of the symbol Q" (233, VIII.),
determined on a plan lately laid down. This result is, how-
ever, included in a much more general Theorem, respecting
Quaternion Equations of A Igebraic Form ; namely, that if
qy, q2i . . qn be any n given, real, and complanar quaternions,
then the equation,
II. . . ^" + q,q^-^ + qiq"-"" -f . . + ^n = 0,
has always n real quaternion roots, q, q", . . q^^\ and no more
in the given plane ; of which roots it is possible however that
some, or all may become equal, in consequence of certain
relations existing between the n given coefficients.
245. As another statement of the same Theorem, if we
write,
I. . . Fnq = q"" + qiq''~' + • -^ qm
the coefficients q^. . qn being as before, we may say that every
such polynomial function, Ynq, is equal to a product ofn real,
complanar, and linear {or binomial) factors, of the form q-q';
or that an equation of the form,
lL..Fnq=={q-q'){q-q")--(q-q'''),
can be proved in all cases to exist : although we may not be
* By saying finite equations, we merely intend to exclude here equations with
infinitely many terms, such as Fq= 1, which has been seen (242) to have infinitely
many roots, represented by the expression q = 2imr, where n may be any whole
number.
t It is true that we have supposed Q ||| t (225) ; but nothing hinders us, in any
other case, from substituting for i the versor UVQ, and then proceeding as before.
2 M
266 ELEMENTS OF QUATERNIONS. [bOOK II.
able, with our present methods, to assign expressions for the
roots, q\ . . q'^^\ in terms of the coefficients ^i, . • . qn-
246. Or we may say that there is always a certain system
ofn real quaternions, q\ &c., ||| 2, which satisfies the system of
equations, of known algebraic form,
Ill . . J qq" + qq" + qq" + . . = + 52 ;
UW"+-- = -^3; &C.
247. Or because the difference f„5' - "Enq is divisible by
q - q, as in algebra, under the supposed conditions of compla-
narity (224), it is sufficient to say that at least one real quater-
nion q always exists (whether we can assign it or not), which
satisfies the equation,
IV. ..F„^' = 0,
with the foregoing form (245, 1.) of the polynomial function f.
248. Or finally, because the theorem is evidently true for
the case n=\, while the case 244, 1., has been considered, and
the case 9'n = is satisfied by the supposition §' = 0, we may,
without essential loss of generality, reduce the enunciation to
the following:
Every equation of the form,*
l>^.q{q-q){q-q")..{q-q^"-'^) = Q,
in which q', q'\ . . and Q are any n real and given quaternions
in the given plane, whereof at least Q and g'' may be supposed
actual (144), is satisfied by at least one real, actual, and com-
planar quaterniDn, q.
* The corresponding ybrm, of the algebraical equation of the n*^ degree, was pro-
posed by Mourey, in his very ingenious and original little work, entitled La vraie
theorie des Quantites Negatives, et des Quant ites pretendues Tmaginaires (Paris,
1828). Suggestions also, towards the ^'eome^ricaZfjroo/ of the theorem in the text
have been taken from the same work ; in which, however, the curve here called (in
251) an oval is not perhaps defined with sufficient precision : the inequality, here
numbered as 251, XII., being not employed. It is to be observed that Mourey's
book contains no hint of the present calculus, being confined, like the Double Alge^
bra of Prof. De Morgan (London, 1849), and like the earher work of Mr. Warren
(Cambridge, 1828), to questions within theplajie : whereas the very conception of the
Quaternion involves, as we have seen, a reference to Tridimensional Space.
CHAP. II.] GEOMETRICAL EXISTENCE OF REAL ROOTS. 267
249. Supposing that the m-l last of the n-l given quater-
nions q' . . g-^""^^ vanish, but that the n-m first of them are actual,
where m may be any whole number from 1 to w - 1, and introduc-
ing a new real, known, complanar, and actual quaternion Qq, which
satisfies the condition,
Q
we may write thus the recent equation I.,
and may (by 187, 159, 235) decompose it into the two following:
IV. ..'17^=1; and Y...Vfq=], or Yl...Simfq-=2p7r',
in which p is some whole number (negatives and zero included).
250. To give a Taoro, geometrical form to the equation, let A, be
any given or assumed line j|| z, and let it be supposed that a, ^, . .
and p, ff, or OA, ob, . . . and op, os, are n - m + 2 other lines in the
same planes, and that ^p is a known scalar function of /o, such that
VII. . . a = 2''X, ^ = q'%.. p = q\ <r = qo\,
and
VIII... <fp=/? ^''^"' "-"-''-^---j'^^y ^^-^'
<Tj a (J y^OSy OA OB
the theorem to be proved may then be said to be, that whatever sys-
tem of real points, o, a, b, . . and s, in a given plane, and whatever
positive whole number m, may be assumed, or given, thei^e is always at
least one real point p, in the same plane, which satisfies the two condi-
tions:
IX. . . T^P = 1 ; X. . . am ^p = 2p7r.
251. Whatever value t\\\i we may assume for the versor (or
unit-vector) JJp, there always exists at least one value of the tensor
T/9, which satisfies the condition IX. ; because the function T^p va-
nishes with T/3, and becomes infinite when T/o = oo, having varied
continuously (although perhaps with fluctuations) in the interval.
Attending then only to the least value (if there be more than one)
of T/>, which thus renders T^p equal to unity, we can conceive a real,
unambiguous, and scalar function Y^t, which shall have the two fol-
lowing properties :
XI. .. T0(tfO = l; XII. . . T^(a;ti^O<l» if a;>0, < 1.
And in this way the equation, or system of equations,
268 ELEMENTS OF QUATERNIONS. [bOOK II.
XIII. ..p = tft, or XIV. . . Up = t, Tp = yjrc,
may be conceived to determine a real, finite^ and plane closed curve,
which we shall call generally an Oval^ and which shall have the two
following properties: 1st, every right line, or ray, drawn/rom the ori-
gin o, in any arbitrary direction within the plane, meets the curve
once, but once only; and Ilnd, no one of the n-m other given points
A, B, . . is on the oval, because ^a = ^/3 = . . = 0.
252. This being laid down, let us conceive a point p to perform
one circiiit of the oval, moving in thepositive direction relatively to the
given interior point O; so that, whatever the given direction of the
line OS may be, the amplitude 2im{p'.a), if supposed to vary conti-
nuously,* will have increased hy four right angles, or by 27r, in the
couTSQ oi this one positive circuit; and consequently, the amplitude
of the left-hand factor (/> : <r)*", of 0p, will have increased, at the same
time, by ^mir. Then, if the point a be also interior to the oval, so
that the line OA must be prolonged to meet that curve, the ray ap will
have likewise made one positive revolution, and the amplitude of the
factor (/> - a)\ a will have increased by 27r. But if a be an exterior
point, so that the finite line oa intersects the curve in a point m, and
therefore never meets it again if prolonged, although the prolonga-
tion of the opposite line ao must meet it once in some point n, then
while thQ point p performs first what we may call the positive half-
circuit from M to N, and afterwards the other positive half-circuit
from N to M again, the ray ap has only oscillated about its initial and
final direction, namely that of the line Ao, without ever attaining the
opposite direction ; in this case, therefore, the amplitude am(AP: oa),
if still supposed to vary continuously, has only fluctuated in lis X2i\\xe,
and has (upon the whole) undergone no change at alh And since
precisely similar remarks apply to the other given points, b, &c.,
it follows that the amplitude, am 0p, of the product (VIJI.) of all
these factors, has (by 236) received a total increment =2{m + t)7r, if
t be the number (perhaps zero) of given internal points, a, b, . . ;
while the number m is (by 249) at least = 1. Thus, while p per-
forms (as above) 07ie positive circuit, the amplitude am >pp has passed
at least m times, and therefore at least once, through a value of the
form 2p'7r; and consequently the condition X. has been at least once
satisfied, Biit the other condition, IX., is satisfied throughout, by the
* That 13, so as not to receive any sudden increment, or decrement, of one or
more whole circumferences (comp. 235, (1.)).
CHAP. II.] GEOMETRICAL ILLUSTRATIONS, QUADRATICS. 269
supposed construction of the oval : there is therefore at least one real
position P, upon that curve, for which <pp or fq = 1 ; so that, /or this
position of that point, the equation 249, HI., and therefore also the
equation 248, I., is satisfied. The theorem of Art. 248, and conse-
quently also, by 247, the theorem of 244, with its transformations
245 and 246, is therefore in this manner proved.
253. This conclusion is so important, that it may be use-
ful to illustrate the general reasoning, by applying it to the
case of a quadratic equation, of the form,
•^^ ^oV^ ; ^^ cr\a J OS OA
We have now to prove (comp. 250, VIII.) that a (real) point p
exists, which renders the fourth
proportional (226) to the three ^
lines OA, op, ap equal to a
given line os, or ab, if this lat- p- ^^
ter be drawn = os ; or which
satisfies the following condition of similarity of triangles
(118),
III. . . A aop a PAB ;
which includes the equation of rectangles,
I V . . . OP.AP = OA-AB. Nt
(Compare the annexed Figures, 55, and
55, bis.) Conceive, then, that a conti-
nuous curve* is described as a locus (or
as part of the locus) of p, by means of this equality IV., with
the additional condition
when necessary, that o
shall be within it; in such
a manner that when (as in
Fig. 56) a right line from
o meets the general or total
locus in several points, m.
Fig. 55, bis.
Fig. 56.
* This curve of the fourth degree is the well-known Cassinian; but when it
breaks up, as in Fig. 56, into two separate ovals, we here retain, as the oval of the
proof only the one round o, rejecting for the present that round A.
270 ELEMENTS OF QUATERNIONS. [bOOK II.
m', n', we reject all but the point m which is nearest to o, as not
belonging (comp. 2.5 1 , XII.) to the oval here considered. Then
while p moves upon that oval, in the positive direction rela-
tively to o, from M to N, and from n to m again, so that the
ray op performs one positive revolution, and the amplitude of
the factor op : os increases continuously by 27r, the ray ap
performs in like manner one positive revolution, or (on the
whole) does not revolve at all, and the amplitude of the factor
AP : OA increases by 27r or by 0, according as the point a is in-
terior or exterior to the oval. In the one case, therefore, the
amplitude am (pp of the product increases by Air (as in Fig. 56^
bis) ; and in the other case, it increases by 27r (as in Fig. 6Q) ;
so that in each case, it passes at least once through a value of
the^r?w 2p7r, whatever its initial value may have been. Hence,
for at least one real position, p, upon the oval, we have
V. . . am 0p = 1 , and therefore VI. . . U^/o = 1 ;
but VII. ..T^^=l,
throughout, by the construction, or by the equation of the locus
IV. ; the geometrical condition (pp=l (II) is therefore satisfied
by at least one real vector p ; and consequently the quadratic
equation fq = 1 (I.) is satisfied by at least one real quaternion
root, q = p:X (250, VII.). But the recent form I. has the same
generality as the earlier form,
VIII. . . Fgg' = q^ + qiq + q2 = (comp. 245),
where qi and q^ are any two given, real, actual, and complanar
quaternions ; thus there is always a real quaternion q' in the
given plane, which satisfies the equation,
Vlir. . . F^q' = q'' + qxq +^2 = (comp. 247) ;
subtracting, therefore, and dividing by g - q^, as in algebra
(comp. 224), we obtain the following depressed or Ihiear equa-
tion q,
IX. . . 5' + 5''+ ^i = 0, or IX. . . 5' = 9'" = - g'-5'i (comp. 246).
The quadratic VIII. has therefore a second real quaternion root,
q, related in this manner to \k\.^ first ; and because the qua-
dratic function Y%q (comp. again 245) is thus decomposable
into two linear factors, or can be put under the form,
CHAP. II.] RELATIONS BETWEEN THE ROOTS. 271
it cannot vanish for any third real quaternion^ q ; so that
(comp. 244) the quadratic equation has no more than two such
real roots.
(1.) The cubic equation may therefore be put under the ybrm (comp. 248),
X. . . Vzq = q^ -]r q\q^ + 9^29' + 53 = 9 (^ - ?') (? -?") + 93= ;
it has therefore one real root, say g', hy ihQ general proof (2b2^i which has been
above illustrated by the case of the quadratic equation ; subtracting therefore (com-
pare 247) the equation 'Fzq^ = 0, and dividing hy q— q\ we can depress the cubic to
a quadratic, which will have two new real roots, 5" and g"' ; and thus the cubic
function may be put under the form,
XI. . . F39' = (? - q) (q - q') {q - q''),
which cannot vanish for any fourth real value of q ; the cubic equation X. has there-
fore no more than three real quaternion roots (comp. 244) : and similarly for equa-
tions of higher degrees.
(2.) The existence of two real roots 9 of the quadratic I., or of two real vectors,
p and p', which satisfy the equation II., might have been geometrically anticipated,
from the recently proved increase = 47r of amplitude ^p, in the course of one circuit,
for the case of Fig. .55, bis, in consequence of which there must be two real positions,
V and p', on the one oval of that Figure, of which each satisfies the condition of si-
milarity III. ; and for the case of Fig. 56, from the consideration that the second (or
lighter^ oval, which in this case exists, although not employed above, is related to A
exactly as HhQ first (or dark) oval of the Figure is related to o ; so that, to the real
position p on the first, there must correspond another real position p', upon the se-
cond.
(3.) As regards the law of this correspondence, if the equation II. be put under
the form,
and if we now write
XIII. ..p = 5a, we may write XIV. . . 9-1 = — 1, q-i = -(T:a,
for comparison with the form VIII. ; and then the recent relation IX'. (or 246) be-
tween the two roots will take the form of the following relation between vectors,
XV. . . p + p' = a ; or XV'. . . op' = p' = a - p = pa ;
so that the point p' completes (as in the cited Figures) the parallelogram opap', and
the line pp' is bisected by the middle point c of OA. Accordingly, with this position
of p', we have (comp. III.) the similarity, and (comp. II. and 226) the equation,
XVI. . . A AOP* a P'AB ; XVII. . . 0p'= ^(a - p) = 0p = 1.
(4.) The other relation between the two roots of the quadratic VIII., namely
(comp. 246),
XVIII . . . q'q" = 92, gives XIX. . . ^ p' = - (7 ;
272 ELEMENTS OF QUATERNIONS. [boOK II.
and accordingly, the line <t, or os, is a fourth proportional to the three lines oa, op,
and AP, or a, p, and - p'.
(5.) The actual solution^ by calculation, of the quadratic eqvationYlll. in com-
planar quaternions, is performed exactly as in algebra ; the formula being,
XK...q^-lq,±V(iqi'-q2),
in which, however, the square root is to be interpreted as a real quaternion, on prin-
ciples already laid down.
(6.) Oubic and biquadratic equations, with quaternion coefficients of the kind
considered in 244, are in like manner reso^red by the known /ormMZ« of algebra;
but we have now (as has been proved) three real (quaternion) roots for the former,
and four such real roots for the latter.
254. The following is another mode of presenting the geometri-
cal reasonings of the foregoing Article, without expressly intro-
ducing the notation or conception of amplitude. The equation
0io= I of 253 being written as follows,
I. . . ^^^p^fi(^p^a\ or II. . . To- = Tx/>, and III. . . Uo- = Ux/>,
a
we may thus regard the vector o- as a known function of the vector /?,
or the point s as di. function of the point p; in the sense that, while o
and A s^TQ fixed, p and s vary together : although it may (and does) hap-
pen, that s may return to a former position without p having similarly
returned. Now the essential property of the oval (253) may be said
to be this: that it is the locus of the points p nearest to o, for which the
tensor Txp has a given value, say h; namely the given value o/To-, or
of OS, when the^om^ s, like o and a, is given. If then we conceive
the point p to move, as before, along the oval, and the point s also to
move, according to the law expressed by the recent formula I., this
latter point must move (by II.) on the circumference of a given circle
(comp. again Fig. 56), with the given origin o for centre ; and the
theorem is, that in so moving, s will pass, at least once, through every
position on that circle, while p performs one circuit of the oval. And
this may be proved by observing that (by III.) the angular motion of
the radius os is equal to the sum of the angular motions of the two rays,
OP and ap; but this latter sum amounts to eight right angles for the
case of Fig. 55-, his,, and to four right angles for the case of Fig. 56;
the radius os, and the point s, must therefore have revolved twice in
the first case, and once in the second case, which proves the theorem
in question.
(1.) In the first of these two cases, namely when a is an interior point, each of
the three angular velocities is positive throughout, and the mean angular velocity of
CHAP. II.] CASSINIAN OVALS, LEMNISCATA. 273
the radius OS is double of that of each of the two rays op, AP. But in the second case,
when A is exterior, the mean angular velocity of the ray ap is zero; and we might
for a moment doubt, whether the sometimes negative velocity of that ray might not,
for parts of the circuit, exceed the always positive velocity of the ray op, and so
cause the radius os to move backwards, for a while. This cannot be, however ; for
if we conceive p to describe, like p', a circuit of the other (or lighter) oval, in Fig. 56,
the point s (if still dependent on it by the law I.) would again traverse the whole of
the same circumference as before ; if then it could ewer fluctuate in its motion, it
would pass more than twice through some given series of real positions on that circle,
during the successive description of the two ovals bj" p ; and thus, within certain
limiting values of the coefficients, the quadratic equation would have more than two
real roots : a result which has been proved to be impossible.
(2.) While 8 thus describes a circle round o, we may conceive the connected point
B to describe an equal circle round a ; and in the case at least of Fig. 56, it is easy
to prove geometrically, from the constant equality (253, IV.) of the rectangles OP'AP
and OA. AB, that these two circles (with t'u and xV as diameters), and the two ovals
(with MN and mV as axes), have two common tangents, parallel to the line OAj
which connects what we may call the two given foci (or focal points), o and a : the
new or third circle, which is described on this focal interval OA as diameter, passing
through the four points of contact on the ovals, as the Figure may serve to exhibit.
(3.) To prove the same things by quaternions, we shall find it convenient to
change the origin (18), for the sake of symmetry, to the central point c; and thus
to denote noio cp by p, and ca by a, writing also CA = Ta = a, and representing still
the radius of each of the two equal circles by b. We shall then have, as the joint
equation of the system of the two ovals, the following :
lY. . .T(p + a).TCp-a)=2ab;
or
V. . . T(52-l)=2c, if q = ^ and c = -.
a, a
But because we h&ve generally (by 199, 204, &c.) the transformations,
VI. . . S . ^2 = 2S52 _ T52 = Tq^ + 2V92 = 2NS9 - % = N^ - 2NV^,
the square of the equation V. may (by 210, (8.) ) be written under either of the tAvo
following forms :
VII. . . (N^ - 1)2 + 4NV5 = 4c2 ; VIII. . . (Ng + 1)2 _ 4NS5 = 4c2 ;
whereof the first shows that the maximum value of TYq is c, at least if 2c < 1, as
happens for this case of Fig. 56; and that this maximum corresponds to the value
Tq=l, or Tp = a : results which, when interpreted, reproduce those of the preceding
sub-article.
(4.) When 2c > 1, it is permitted to suppose S9 = 0, N V9 = Ng = 2c - 1 ; and
then we have only one continuous oval, as in the case of Fig. 55, bis; but if c < 1,
though > I, there exists a certain undulation in the form of the curve (not represented
in that Figure), TYq being a minimum for S^= 0, or for p -i- a, but becoming (as
before) a maximum when Tq = l, and vanishing when 8(72 = 2c + 1, namely at the
two summits M, N, where the oval meets the axis.
(5.) In the intermediate caie, when 2c = 1, the Cassinian curve IV. becomes (as
is known) a lemniscata; of which the quaternion equation may, by V., be written
(comp. 200, (8.) ) under any one of the following forms:
2 N
274
ELEMENTS OF QUATERNIONS.
[book II.
IX. . .T(92-l) =
or finally,
or X. . . N93 = 2S. ^2 . or xi. . . T92 = 2SU . 92
XII. . . Tp2 = 2Ta2 cos 2 ^
Avhich last, when written as
Xir. . . cp2 = 2ca2 . cos 2acp,
agrees evidently with known results.
(6.) This corresponds to the case when
XIII.
(T = -— , and
4
XIV.
P = P
+ -, in 253, XII.,
that quadratic equation having thus its roots equal; and in general, iov all degrees,
cases of equal roots answer to some interesting peculiarities of form of the ovah, on
which we cannot here delay.
(7.) It may, however, be remarked, in passing, that if we remove the restriction
that the vector p, or cp, shall be in a given plane (225), drawn through the line
which connects the two foci, o and a, the recent equation V. will then represent the
surface (or surfaces') generated by the revolution of theora/ (or ovals), orleraniscata,
about that line oa as an axis.
255. If we look back, for a moment, on the formula oi similarity,
253, III., we shall see that it involves not merely an equality of rect-
angles, 253, IV., but also an equality of angles, aop and pab; so that
the angle oab represents (in the Figures 55) a given difference of the
base angles aop, pao of the triangle oap: but to construct a triangle^
by means of such a given difference, combined with a given base, and
a given rectangle of sides, is a known problem of elementary geome-
try. To solve it briefly, as an exercise, by quaternions^ let the given
base be the line aa', with for its middle point, as in the annexed
Figure 57 ; let baa' represent the given diffe-
rence of base angles, paa' - aa'p ; and let oa . ab
be equal to the given rectangle of sides, ap • a^.
We shall then have the similarity and equa-
tion,
p +a /3- a
A OA'P a PAB ;
II.
a p — a
whence it follows by the simplest calculations,
that
III.
-i'-.lf-
1+1 =
^
+ 1 =
or that /> is a mean proportional (227) between a and /3. Draw,
therefore, a line op, which shall be in length a geometric mean be-
tween the two given lines, oa, ob, and shall also bisect their angle
CHAP. II.] IMAGINARY QUATERNION ROOTS. 275
AOB ; its extremity will be the required vertex, p, of the sought tri-
angle aa'p: a result of the quaternion analysis, vflaioh geometrical syn-
thesis* easily confirms.
(1.) The equation III. is however satisfied also (comp. 227) by the opposite vec-
tor, op' = PC, or p' = - (0 ; and because /3 = (p : a) . p, we have
lV...t±i=.t=^ = t^ or IV'. ..^=-= - = ^.
p-\^ a a p d^ p'a oa op oa' '
so that the^bttr following triangles are similar (the two first of them indeed being
equal) :
V. . . A a'op' a AOP <x FOB aAP'B ;
as geometry again would confirm.
(2.) The angles ap'b, bpa, are therefore supplementary, their sum being equal to
the sum of the angles in the triangle oap ; whence it follows that the four points A,
P, b, p' are concircular :f or in other words, the quadrilateral Apbp' is inscriptible
in a circle, which (we may add) passes through the centre c of the circle oab (see
again Fig. 57), because the angle aob is double of the angle ap'b, by what has been
already proved.
(3.) Quadratic equations in quaternions may also be employed in the solution
of many other geometrical problems; for example, to decompose a given vector into
two others, which shall have a given geometrical mean, &c.
Section 6 — On the n^-n Imaginary {or Symbolical) Roots
of a Quaternion Equation of the n'^' Degree^ with Coeffi-
cients of the kind considered in the foregoing Section,
256. The polynomial function F,,q (245), like the quaternions
q, qi, . . qn on which it depends, may always be reduced to the form of
a couple (228) ; and thus we may establish the transformation (comp.
239),
I. . . Fnq = F„ (x + iy) = X„ + i F,, = Gu {x, y) + iH,, {x, y),
Xn and y„, or Gn and Hn, being two known, real, finite, and scalar
functions of the two sought scalars, x and y\ which functions, rela-
* In fact, the two triangles I. are similar, as required, because their angles at o
and p are equal, and the sides about them are proportional.
t Geometrically, the construction gives at once the similarity,
A AOP oc fob, whence L bpa = opa + pad = poa' ;
and if we complete the parallelogram apa'p', the new similarity,
A oa'p a op'b, gives L ap'b = oa'p + a'po = aop ;
thus the opposite angles bpa, ap'b are supplementary, and the quadrilateral ai'bp' is
inscriptible. It will be shown, in a shortly subsequent Section, that these four
points, A, p, P, p', form a harmonic group upon their common circle.
276 ELEMENTS OF QUATERNIONS. [boOK II.
tively to them, are each of the w^'' dimension, but which involve also,
though only in ihe first dimension, the 2n given and real scalarsy
rci, 2/i, . . . X,,, y„. And since the one quaternion (or couple) equation,
F,,q = 0, is equivalent (by 228, IV.) to the system of the two scalar
equations,
II. . . J:„=0, F„ = 0, or III. . . . Gn{x,y) = 0, Hn{x,y)=0,
we see (by what has been stated in 244, and proved in 252) that
suxih a system, of two equations of the n"* dimension, can always be
satisfied by n systems (or pairs) of real scalars, and by not more than
n, such as
IV. . .x^,y'\ x", y" ; . . icW, y('»^ ;
although it may happen that two or more of these systems shall coin-
cide with (or become equal to) each other.
(1.) \ix and y be treated as co-ordinates (comp. 228, (3.) ), the two equations
II. or III. represent a system of two curves, in the given plane ; and then the theo-
rem is, that these two curves intersect each other {generally*^ in n real points^ and
in no more : although two or more of these n points may happen to coincide with
each other.
(2.) Let h denote, as a temporary abridgment, the old or ordinary imaginary,
V— 1, oi algebra, considered as an uninterpreted symbol, and as not equal to any
real versor, such as t (comp. 181, and 214, (3.) ), but as following the rules ofsca-
lars, especially as regards the commutative property oi mvXil^WcaXXon (\2Q) ; so that
V. . . ^2 + 1 = 0, and VI. . . W = ih, but VII. . . A «o* = + i.
(3.) Let q denote still a real quaternion, or real couple, x + iy ; and with the
meaning just now proposed of h, let [(j\ denote the connected but imaginary alge-
braic quantity, or bi-scalar (214, (7.) ), x + hy ; so that
Ylll. q=x + iy, but IX. . . lq'] = x + hy',
and let any biqiiaternion (214), (8.), or (as we may here call it) bi-couple, of the
form [7'] + i\_q"'\i be said to be complanar with »; with the old notation (123) of
complanarity.
(4.) Then, for the polynomial equation in real and complanar quaternions,
JJ'^g = (244, 245), we may be led to substitute the following connected algebraical
equation, of the same degree, n, and involving real scalars similarly :
X. . . [F„g] = lq-]» + [91] [qy + • • + [^n] = ;
* Cases of equal roots may cause points of intersection, which are generally ima-
ginary, to become real, but coincident with each other, and vi'ith. former real roots :
for instance the hyperbola, x^ -y"^ = a, is intersected in two real and distinct points,
by the pair of right lines xy = 0, if the scalar a > or < ; but for the case a = 0, the
two pairs of lines, x^ — y"- =■ and xy = 0, may be considered to havofour coincident
intersections at the origin.
CHAP. II.] NEW SYMBOLICAL ROOTS OF UNITY. 277
which, after the reductions depending on the substitution V. of - 1 for h^, receives
the form,
where Xn and Vn are the same real and scalar Junctions as in I.
(5.) But we have seen in II., that these two real functions can be made to va-
nish together, by selecting any one ofn real pairs IV. of scalar values, x and y ; the
General Algebraical Equation X., of the n*^ Degree, has therefore n Real or Imagi-
nary/ Roots,* of the Form ar + 2/ V — 1 ; and it has no more than n such roots.
(6.) Elimination of y, between the two equations IT. or III., conducts generally
to an algebraic equation in x, of the degree n^ ; which equation has therefore n^ alge-
braic roots (5.), real or imaginary ; namely, by what has been lately proved, n real
and scalar roots, x', . . a;("), with real and scalar values y , . .y(") (comp. IV.) of y
to correspond; and «(«—!) other roots, with the same number of corresponding
values of y, which may be thus denoted,
XII. . . [x(«+i), . . [a;(«'^] ; XIII. . . [yf"+i)], . . [y(«2)] ;
and which are either themselves imaginary (or bi-scalar, 214, (7.)), or at least cor-
respond, by the supposed elimination, to imaginary or bi-scalar values ofy; since if
a;(w+i) and y("+^), for example, could both be real, the quaternion equation Fnq=0
would then have an (w 4-l)st real root, of the form, ^(w+i) = a;(w+i) + ^^(n+i)^ contraiy
to what has been proved (252).
257- On the whole, then, it results that the equation F„q = in
complanar quaternions, of the w^'' degree, with real coefficients,
while it admits of only n real quaternion roots,
L..^^^^..2W(244,&c.),
is symbolically satisfied also (corap. 214, (3.)) by n(n- \) imaginary
quaternion roots, or hy n^ -n bi-quaternions (214, (8.) ), or bi-couples
(256, (3.) ), which may be thus denoted,
and of which the first, for example, has the /orm,
III. . . [^'"^^^] = [a;("^^)] + 2[?/^"^^T = a;/"^») + ^a;//«^') + 2(y/^"'^^ + %//^"''0 ;
where a;/"^'\ XjI''*^\ y/''^^\ and y,/"^'^^ are four real scalars, but h is
the imaginary of algebra (256, (2.) ).
(L) There must, for instance, be n(n - 1) imaginary n*^ roots of unity, in the
given plane of i (comp. 256, (3.) ), besides the n real roots already determined (233,
* This celebrated Theorem of Algebra has long been known, and has been proved
in other ways ; but it seemed necessary, or at least useful, for the purpose of the pre-
sent work, to prove it anew, in connexion with Quaternions : or rather to establish
the theorem (244, 252), to which in the present Calculus it corresponds. Compare
the Note to page 266.
278 ELEMENTS OF QUATERNIONS. [bOOK II.
237); and accordingly in the case n = 2, we have the four foWowmg sqvare-roois
ofl \\\i, two real and two imaginary :
IV, . . +1, -1 ; +hi, -hi;
for, by 256, (2.), we have
V. . . (± hiy = hH^ = (- 1) (- 1 ) = + 1.
And the two imaginary roots of the quadratic equation F^q = 0, which generally
exist, at least as symbols (214, (3.) ), may be obtained by multiplying the square-
root in the formula 253, XX. by hi ; so that in the particular case, when that radi-
cal vanishes, the four roots of the equation become real and equal : zero having thus
only itself for a square-root.
(2.) Again, if we write (comp. 237, (3.)),
-1 + »V3 -l-iV3
Yl...q=lh= , g^ = \h= ,
so that 1, q, qi are the three real cube-roots of positive unity, in the given plane ;
and if we write also,
v.i..,e=M=zi±i^, e^ = hp = zizA^,
so that 9 and 02 are (as usual) the two ordinary (or algebraical) imaginary cube-
roots of unity ; then the nine cube-roots o/ 1 (| 1 1 1) are the following :
VIII. . . 1 ; 9, 52 ; 0, 02 ; Qq^ e^ ; 9^q, Q^q^ ;
whereof the first is a real scalar ; the two next are real couples, or quaternions \\\i ;
the two following are imaginary scalars, or biscalars; and the four that remain are
imaginary couples, or bi-couples, or biquaternions.
(3.) The sixteen fourth roots of unity (|[| i) are:
IX. ..+1; ±i; +/*; ±hi; ±|(1±/0(1±0;
the three ambiguous signs in the last expression being all independent of each other.
(4.) Imaginary roots, of this sort, are sometimes useful, or rather necessary, in
calculations respecting ideal intersections,* and ideal contacts, in geometry: although
in what remains of the present Volume, we shall have little or no occasion to employ
them.
(5.) We may, however, here observe, that when the restriction (225) on the
plane of the quaternion q is removed, the General Quaternion Equation of the n*^
Degree admits, by the foregoing principles, no fewer than «* Hoots, real or imagi-
nary : because, when that general equation is reduced, by 221, to the Standard
Quadrinomial Form,
X...Fnq= Wn + iXn +j Vn + hZn = 0,
it breaks up (comp. 221, VI.) into a System of Four Scalar Equations, each (gene-
rally) of the «*'» dimension, in w, x, y, z-, namely,
XI. ..r,»=0, X„=0, Yn=0, Zn = 0;
and if x, y, z be eliminated between these four, the restilt is (generally) a scalar (or
algebraical) equation of the degree n*, relatively to the remaining constituent, w;
Comp. Art. 214, and the Notes there referred to.
CHAP. II.] RECIPROCAL OF A VECTOR. 279
which therefore has n^ (algebraical) values, real or imaginary : and similarly for the
three other constituents, x, y, z, of the sought quaternion q.
(6.) It may even happen, when no plane is given, that the number of roots (or
solutions) of a ^raiYe* equation in quaternions shall become infinite; as has been
seen to be the case for the equation q^ =—1 (149, 154), even when we confine our-
selves to what we have considered as real roots. li imaginary roots he admitted,
we may write, still more generally, besides the two biscalar values, + h, the expres-
sion,
XII. . . (-l)i = «+ Ar', S«=Sw'=S«w' = 0, Nw-Nzj'=l;
V and v' being thus any two real and right quaternions, in rectangular planes, pro-
vided that the norm of ih.Q first exceeds that of the secondhy unity.
(7.) And in like manner, besides the two real and scalar values, + 1, we have
this general symbolical expression for a square root of positive unity, with merely
the difference of the norms reversed :
XIII. . . li=y + Ay', Sy=S«' = Sw' = 0, N«'-Nr = l.
Section 7. — On the Reciprocal of a Vector^ and on Harmo-
nic Means of Vectors; with Remarks on the Anharmonic
Quaternion of a Group of Four Points, and on Conditions
of Concircularity.
258. When two vectors, a and a', are so related that
I. . . a = - Ua : Ta, and therefore 11. . . a = - Ua : Ta,
or that
III. . . Ta . Ta' = 1, and IV. . . Ud + Ua = 0,
we shall say that each of these two vectors is the Reciprocal^
of the other ; and shall (at least for the present) denote this
relation between them, by writing
V. ..a=Ea, or VI. ..a = Ea';
so that for every vector a, and every right quotient v,
VII.. .Ra = -Ua:To; VIII. . .R^a = RRa = a;
and
IX. . . EIv = IRi; (comp. 161, (3.), and 204, XXXV'.).
259. One of the most important properties of such reci-
procals is contained in the following theorem :
* Compare the Note to page 265.
t Accordingly, under these conditions, we shall afterwards denote this recipro-
cal of a vector a by the symbol a"' ; but we postpone the use of this notation, until
we shall be prepared to connect it with a general theory of products and poivers of
vectors. Compare 234, V., and the Note to page 121. And as regards the tempo-
rary use of the characteristic R, compare the second Note to page 252.
280 ELEMENTS OF QUATERNIONS. [bOOK II.
If any two vectors oa, ob, have oa', ob' for their recipro-
cals, then (comp. Fig. 58) the right line a b'
is parallel to the tangent od, at the origin o,
to ' the circle oab ; and the two triangles,
gab, obV, are inversely similar (118). Or
in symbols,
I. . . if oa =R.OA, and ob' = R.ob,
then
A oab a' ob'a'.
(1.) Of course, under the same conditions, the tangent at o to the circle oa'b' is
parallel to the line ab.
(2.) The angles bao and ob'a' or bod being equal, the fourth proportional (226)
to AB, AO, and ob, or to ba, oa, and ob, has the direction of od, or the direction op-
posite to that of a'b' ; and its length is easily proved to be the reciprocal (or inverse)
of the length of the same line a'b', because the similar triangles give,
II. . . (oa : ba) . ob = (ob' : a'b'). ob = 1 : aV,
it being remembered that
III. . . OA . oa' = OB . ob' = 1 ;
we may therefore write,
IV. . .(oa:ba).ob = R.a'b', or V. . . — ^i3 = R(Ri3 - Ra),
a — p
whatever two vectors a and /3 may be.
(3.) Changing a and /3 to their reciprocals, the last formula becomes,
VI. . . R(/3-a) = - — ^.R/3; or VII. . . (oa': b'a').ob' = R. ab.
Ka —Up
yiii...K2=5?.
(4.) The inverse similarity I. gives also, generally, the relation,
Ra
R^*
(5.) Since, then, by 195, II., or 207, (2.),
IX, . . K-+1 = K'-^^, we have X... - ^-
a ' R/3 R()3±a)'
the lower signs agreeing with VI.
(6.) In general, the reciprocals of opposite vectors are themselves opposite ; or
in symbols,
XI. . . R(-a) = -Ra.
(7.) More generally,
XII. . . Rxa = x-^B,a,
if X be any scalar.
(8.) Taking lower signs in X., changing a to y, dividing, and taking conjugates,
we find for any three vectors a, /3, y (complanar or diplanar') the formula :
Ry:^^,J_Ry_ R(/3-a) \ a r^_oA bc
■^^"••^ Ra-R/3 VK(i8-y)" Ra / /3- a ' - y "ab* co'
if a = OA, j3 = OB, and y = oc, as usual.
CHAP. II.] ANHARMONIC AND EVOLUTIONARY QUATERNIONS. 281
(9.) If then we extend, to any four points ofspace^ the notation (25),
,,,,, . ^ AB CD
XIV. . . Cabcd) = — .— ,
^ ' BC DA
interpreting esich. of these two factor-quotients as a quaternion, and defining that
t\\eiv product (in this order^ is the anharmonic quaternion function, or simply the
Anharmonic, of the Group of four points A, B, C, D, or oi the (^plane or gauche^ Quw
drilateral ABCD, we shall have the following general and useful ^rmw/a of transfor-
mation :
XV.. (0ABc) = KgI-^ = K_„
where oa', ob', ob' are supposed to be reciprocals of oa, ob, oc.
(10.) With this notation XIV., we have generally, and not merely for coUinear
groups (35), the relations :
XVI. . . (abcd) + (acbd) = 1 ; XVII. . . (abcd). (adcb) = 1.
(11.) Let o, A, B, c, D be any five points, and oa', . . od' the reciprocals of OA, . .
od ; we shall then have, by XV.,
XVIII. . . ^ = K (OCBA), ^ = K (oadc) ;
bo' ^ ^ DA ^
and therefore,
XIX. . . K (a'b'c'd') = (oADc) (ocba) = - (oadcba),
if we agree to write generally, for any six points, the formula,*
,„ , ^ AB CD EF
XX. . . Cabcdef) = — . — . — .
EC DE fa
(12.) If then the five points o . . d be complanar (225), we have, by 226, and
by XIV.,
XXI. . . K (a'b'c'd') = (abcd), or XXI'. . . (a'b'c'd') = K (abcd) ;
the anharmonic quaternion (abcd) being thus changed to its conjugate, when the
four rays OA, . . od are changed to their reciprocals,
260. Another very important consequence from the defi-
nition (258) of reciprocals of vectors, or from the recent theo-
rem (259), may be expressed as follows:
If any three coinitial vectors^ oa, ob, oc, be chords of one
common circle^ then (see again Fig. 58) their three coinitial re-
* There is a convenience in calling, generally, this /iroc/Mc^ of three quotients,
(abcdef), the evolutionary quaternion, or simply the Evolutionary, of the Group
of Six Points, A . . F, or (if they be not collinear) of the plane or gauche Hexagon
abcdef : because the equation,
(abca'b'c') = - 1,
expresses either 1st, that the three pairs of points, aa', bb', cc', form a collinear in-
volution (26) of a well-known kind ; or Ilnd, that those threepairs, or the three cor-
responding diagonals of the hexagon, compose a complanar or a homospheric Involu-
tion, of a new kind suggested by quaternions (comp. 261, (11.) ).
2
282 ELEMENTS OF QUATERNIONS. [BOOK 11.
ciprocals, oa', ob', oc', are terminO'ColUnear (24) : of, in other
words, \S ihe four points o, a, b, c be concircular, then the three
points a', b'j c' are situated on one right line.
And conversely, if three coinitial vectors^ oa', ob', oc', thus
terminate on one right line, then their three coinitial recipro-
cals, oa, ob, oc, are chords of one circle; the tangent to which
circle, at the origin, is parallel to the right line; while the
anharmonic function (259, (9.) )? of the inscribed quadrilateral
OABC, reduces itself to a scalar quotient of segments of that line
(which therefore is its own conjugate, by 139) : namely,
I. . . (oABc) = b'c' : bV = (oo a'b'c') = (o . oabc),
if the symbol oo be used here to denote the point at infinity on
the right line a'b'c' ; and if, in thus employing the notation
(35) for the anharmonic of a plane pencil, we consider the null
chord, 00, as having the direction^ of the tangent, od.
(1.) If p = OP be the variable vector of a point p upon the circle oab, the qua-
ternion equation of that circle may be thus written :
II. . . Ep = E/3 + a;(Ea - Ej3), where III. . . a; = (oabp) ;
the coefficient x being thus a variable scalar (comp. 99, I.), which depends on the
variable position of the point p on the circumference.
(2.) Or we may write,
IV...Ep = 2^±i^,
^ t+u ^
as another form of the equation of the same circle oab ; with which may usefully be
contrasted the earlier form (comp. 25), of the equation of the line ab,
^ t+u
(3.) Or, dividing the second member of IV. by the first, and taking conjugates,
we have for the circle,
to up ... _^_T ta uj3
VI. . .-i-+ -^=< + «; while VII. . . - + -^ = f + m,
a (i P ' P
for the right line.
(4.) Or we may write, by II.,
this latter symbol, by 204, (18.), denoting any scalar.
* Compare the remarks in the second Note to page 139, respecting the possible
determinateness of signification of the symbol UO, when the zero denotes a line,
which vanishes according to a law.
CHAP. II.] CIRCULAR AND HARMONIC GROUP. 283
(5.) Or still more briefly,
IX. . . V(OABP) = ; or IX'. . . (oabp) = V-i 0.
(6.) If the four points o, A, b, o be still concircular, and if p be any fifth point
in their plane, while POi, . . PCi are the reciprocals of po, . . PO, thea by 259, XXI.,
we have the relation,
X. , . (OiAiBiCi) = K(OABC) = (OABC) = V"! ;
the^wr new points Oi. . Ci are therefore generally concircular.
(7.) If, however, the point p be again placed on the circle oabc, those four new
points are (by the present Article) collinear; being the intersections of i\iQ pencil
p.oabo with a, parallel to the tangent at p. In this case, therefore, we have the
equation,
XI. . . (p. oabc) = (oiAiBiCi) = (oabc) ;
so that the constant anharmonic of the pencil (35) is thus seen to be equal to what
we have defined (259, (9.) ) to be the anharmonic of the group.
(8.) And because the anharmonic of a circular group is a scalar, it Is equal (by
187, (8.) ) to its own tensor, either positively or negatively taken : we may therefore
write, for any inscribed quadrilateral oABC, the formula,
XII. . . (OABc) = + T (OABc) = + (OA . BC) : (aB . CO),
= + & quotient of rectangles of opposite sides; the upper or the lower sign being
taken, according as the point b' falls, or does not fall, between the points a' and c' :
that is, according as the quadrilateral oabc is an uncrossed or a crossed one.
I; (9.) Hence it is easy to infer that /or any circular group o, A, b, c, we have the
equation,
XIII...U^ = + U^;
AB - CB
the upper sign being taken when the succession oabo is a direct one, that is, when
the quadrilateral oabc is uncrossed; and the lower sign, in the contrary case,
namely, when the succession is (what may be called) indirect, or when the quadri-
lateral is crossed: while conversely this equation XIII, is sufficient to prove, when-
ever it occurs, that the anharmonic (oabc) is a negative or a positive scalar, and
therefore by (5.) that the gro^ip is circular (if not linear^, as above.
(10.) If A, b, c, d, e be any five homospheric points (or points upon the surface
of owe sphere), and if o be any sixth point of space, while oa', . . oe' are the reciprocals
of OA, . . OE, then the five new points a'. . e' are generally homospheric (with each
other) ; but if o happens to be on the sphere abcde, then a' . . e' are complanar,
their common plane being parallel to the tangent plane to the given sphere at o :
with resulting anharmonic relations, on which we cannot here delay.
26 1 . An interesting case of the foregoing theory is that
when the generally scalar anharmonic of a circular group be-
comes equal to negative unity ; in which case (comp. 26), the
group is said to be harmonic, A few remarks upon such czV-
ctdar and harmonic groups may here be briefly made : the stu-
284 ELEMENTS OF QUATERNIONS. [bOOK II.
dent being left to fill up hints for himself, as what must be
now to him an easy exercise of calculation.
(1.) For such a group (comp. again Fig. 58), we have thus the equation,
I. . . (oABc) = - 1 ; and therefore II. . . a'b' = b'c' ;
or III. ..R/3=KKa + R7);
and under this condition, we shall say (comp. 216, (5.) that the Vector /3 is the Har-
monic Mean between the two vectors, a and y.
(2.) Dividing, and taking conjugates (comp. 260, (3.), and 216, (5.) ), we thus
obtain the equation,
IV... ^ + ^=2; or V. . . /3 = -?i- y = ^a;
or
VI. . ./3 = -y = ^a, if VII. . . £ = |(y-f a);
£ thus denoting here the vector oe (Fig. 68) of the middle point of the chord ao.
We may then say that the harmonic mean between any two lines is (as in algebra)
the fourth proportional to their semisum^ and to themselves.
(3.) Geometrically, we have thus the similar triangles,
VIII. . . A AOB a EOC ; VIII'. . . A aoe a boc ;
whence, either because the angles oba and oca, or because the angles oac and obc
are equal, we may infer (comp. 260, (5.) ) that, when the equation I. is satisfied,
the four points o, a, b, c, if not coUinear^ are coneircular.
(4.) We have also the similarities,
IX. . . A OEC a ceb, and IX'. . . A oea a aeb ;
or the equations,
X...^ = I^', and X'.,.t.'=2Zi,
y — c -c a- 1 — c
in fact we have, by VI. and VII.,
XI.
£
.1-.; xn...^(=iL&". = x-I«] = (.-^y
(5.) Hence the line ec, in Fig. 58, is the mean proportional (227) between the
lines EO and eb ; or in words, the semisum (oe), the semidifference (ec), and the
excess (be) of the semisum over the harmonic mean (ob), form (as in algebra) a
continued proportion (227).
(6.) Conversely, if any three coinitial vectors, eo, ec, eb, form thus a continued
proportion, and if we take ea = ce, then the four points oabc will compose a circu-
lar and harmonic group ; for example, the points apbp' of Fig. 67 are arranged so
as to form such a group.*
(7.) It is easy to prove that, for the inscribed quadrilateral oabc of Fig. 58,
the rectangles under opposite sides are each equal to half of the rectangle under the
* Compare the Note to 255, (2.). In that sub-article, the text should have run
thus : of which (we may add) the centre c is on the circle oab, &c. In Fig. 68, the
centre of the circle oabc is coneircular with the three points o, E, b.
CHAP. II.] INVOLUTION IN A PLANE, OR IN SPACE. 285
diagonals; which geometrical relation answers to either of the two anharmonic
equations (comp. 259, (10.)) :
XIII. . . (0BAC) = + 2; Xlir. . . (ocab) =+ ^.
(8.) Hence, or in other ways, it may be inferred that these diagonals, ob, ac, are
conjugate chords of the circle to which they belong : in the sense that each passes
through the pole of the other^ and that thus the line db is the second tangent from
the point d, in which the chord ac prolonged intersects the tangent at o.
(9.) Under the same conditions, it is easy to prove, either by quaternions or by
geometry, that we have the harmonic equations :
XIV. . . (abco) = (bcoa) = (coab) = - 1 ;
so that AC is the harmonic mean between ab and ao ; bo is such a mean between
BC and BA ; and ca between co and cb.
(10.) In any such group, any two opposite points (or opposite corners of the qua-
drilateral), as for example o and b, may be said to be harmonically conjugate to each
other, with respect to the two other points^ a and c ; and we see that when these two
points A and c are given, then to every third point o (whether in a given plane, or
in space) there always corresponds a. fourth point b, which is in this sense conju-
gate to that third point : this fourth point being always complanar with the three
points A, c, o, and being even concircular with them, unless they happen to be colli-
near with each other ; in which extreme (or limiting') case, the fourth point b is still
determined, but is now coUinear with the others (as in 26, &c.).
(11.) When, after thus selecting two* points, A and c, or treating them as given
or fixed, we determine (10.) the harmonic conjugates b, b', b", with respect to them,
of any three assumed points, o, o', o", then the three pairs of points, O, B ; o', b' ;
o", b", may be said to form an Involution,f either on the right line AC, (in which
case it will only be one of an already well-known kind), or zw a plane through that
line, or even generally in space : and the two points A, c may in all these cases be
said to be the two Double Points (or Foci^ of this Involution. But the field thus
opened, for geometrical investigation by Quaternions, is far too extensive to be more
than mentioned here.
(12.) We shall therefore only at present add, that the conception of the Aarmonic
mean between two vectors may easily be extended to any number of such, and need
not be limited to the plane : since we may define that ij is the harmonic mean of the
n arbitrary vectors ai, . . an, when it satisfies the equation,
XV. . . Rj; = i (Rai + . . + Ra„) ; or XVI. . . nB.r) = SRa.
n
(13.) Finally, as regards the notation Ra, and the definition (258) of the recipro-
cal of a vector, it may be observed that if we had chosen to define reciprocal vectors as
having similar (instead of opposite') directions, we should indeed have had the posi-
tive sign in the equation 258, VII. ; but should have been obliged to write, instead of
258, IX., the much less simple formula,
RIt> = -IRr.
* There is a sense in which the geometrical process here spoken of can be applied,
even when the two fixed points, or foci, are imaginary. Compare the Geomctrie
Superieure of M. Chasles, page 136.
t Compare the Note to 259, (11.).
286 ELEMENTS OF QUATERNIONS. [bOOK II.
CHAPTER III.
ON DIPLANAR QUATERNIONS, OR QUOTIENTS OF VECTORS IN
SPACE : AND ESPECIALLY ON THE ASSOCIATIVE PRINCI-
PLE OF MULTIPLICATION OF SUCH QUATERNIONS.
Section 1. — On some Enunciatio7is of the Associative Pro-
perty, or Principle, of Multiplication of Diplanar Quater-
nions.
262. In the preceding Chapter we have confined ourselves
almost entirely, as had been proposed (224, 225), to the con-
sideration of quaternions in a y iv en plane (that of i) ; alluding
only, in some instances, to possible extensions* of results so
obtained. But we must now return to consider, as in the
First Chapter of this Second Book, the subject of General
Quotients of Vectors : and especially their Associative Multi-
plication (223), which has hitherto been only proved in con-
nexion with the Distributive Principle (212), and with the
Laws of the Symbols i,j\ k (183j. And first we shall give a
iQW geometrical enunciations of that associative principle, which
shall be independent of the distributive one, and in which it
will be sufficient to consider (corap. 191) the multiplication of
versors; because the multiplication of tensors is, evidently an
associative operation, as corresponding simply to arithmetical
multiplication, or to the composition of ratios in geometry.f
We shall therefore suppose, throughout the present Chapter,
that </, r, s are some three given but arbitrary versors, in three
given and distinct planes ;% and our object will be to throw
* As in 227, (3.); 242, (7.); 254, (7.); 257, (6.) and (7.) ; 259, (8.), (9.),
(10.), (11-); 2G0, (10.); and 2G1, (11.) and (12.).
f Or, move generally, for any tliree pairs of magnitudes, each pair separately-
being lioraogeneous.
X If the factors q, r, a were complanar, we could always (by 120) put them
CHAP. HI.] ASSOCIATIVE PRINCIPLE, SYSTEM OF SIX PLANES. 287
some additional light, by new enunciations in this Section,
and by new demonstrations in the next, on the very impor-
tant, although very simple, Associative Formula (223, II.),
w^hich may be written thus :
I. . . sr.g = s.rq;
or thus, more fully,
II. ■ ' qg = t, if q' - 5r, s' = rq, and t = ss' ;
q\ s\ and t being here three new and derived versors, in three
neio and derived planes.
263. Already we may see that this Associative Theorem
of Multiplication^ in all its forms, has an essential reference to
a System of Six Planes, namely the planes of these six ver-
sors,
IV. . . q, r, s, rq, sr, srq, or IV. . . q, r, s, s, q', t;
on the judicious selection and arrangement of which, the clear-
ness and elegance of every geometrical statement or proof of
the theorem must very much depend : while the versor cha-
racter of the factors (in the only part of the theorem for which
proof is required) suggests a reference to a Sphere, namely to
what we have called the unit-sphere (128). And the three
following arrangements of the six planes appear to be the most
natural and simple that can be considered : namely, 1st, the
arrangement in which the planes all pass through the centre of
the sphere ; Ilnd, that in which they all touch its surface ;
and IlIrd, that in which they are the six faces of an inscribed
solid. We proceed to consider successively these three ar-
rangements.
264. When the Jirst arrangement (263) is adopted, it is natural
to employ a7'cs of great circles, as representatives of the versors, on the
under the forms,
(3 y d
and then should have (comp. 183, (1.) ) the two equal ternart/ products,
d 3 d dy
^ (3 a a ya ^'
so that in this case (comp. 224) the associative property would be proved without any
difficulty.
288 ELEMENTS OF QUATERNIONS. [bOOK II.
plan of Art. 162. Representing thus the factor q by the arc ab,
and r by the successive arc bc, we represent (167) their product rq^
or 5^, by AC; or by any equal arc (165), such as de, in Fig. 59, may
be supposed to be. Again, representing s by ef, we shall have df
as the representative of the ternary iU /v
product s.rq, or ss^, or t. taken in ^^"--^ — J^ r-
one order of association. To repre- y^ /' /\ "\ //^^
sent the other ternary product, \^ ( ( \ J^ y
sr. q, or q'q, we may first determine ^\^ Jzic=<r^^^$^ ^^^^
three new points, g, h, i, by arcual c'^^^-~\_^/l-----g'''^
B A
equations (165), between gh, bc, Fig. 69.
and between hi, ef, so that bc, ef
intersect in h, as the arcs representing &' and s had intersected in e;
and then, after thus finding an arc Gi which represents 5r, or q^, may
determine three other points, k, l, m, by equations between kl, ab,
and between lm, gi, so that these two new arcs, kl, lm, represent q
and g-', and that ab, gi intersect in l ; for in this way we shall have
an arc, namely km, which represents q^q as required. And the theo-
rem then is, that this last arc km is equal to the former arc df, in the
full sense of Art. 165; or that when (as under the foregoing condi-
tions of construction) the five arcual equations,
I. . . n AB = '^ KL, nBC = '^GH, <^ EF = n HI, '^ AC = O DE, nGI = '^LM,
exists then this sixth equation of the same kind is satisfied also,
II. . . '^ DF = '^ KM :
the two points, K and m, being both on the same great circle as the two
previously determined points, d and f; or d and m being on the
great circle through f and k: and the two arcs, df and km, of that
great circle, or the two dotted arcs, dk, fm in the Figure, being
equally long, and similarly directed (165).
(1.) Or, after determining the nine points a . . i so as to satisfy the three middle
equations I., we might determine the three other points, k, l, m, without any other
arcual equations^ as intersections of the three pairs of arcs ab, df ; ab, gi ; df, gi ;
and then the theorem would be, that (if these three last points be suitably distin-
guished from their own opposites upon the sphere) the two extreme equations I., and
the equation II., are satisfied.
(2.) The same geometrical theorem may also be thus enunciated : If the first,
third, and fifth sides (kl, gh, ed) of a spherical hexagon klghed be respectively
and arcunlhj equal (165)^0 the first, second, and third sides (ab, bc, CA) of a. sphe-
rical triangle ABC, then the second, fourth, and sixth sides (lg, he, dk) of the same
hexagon are equal to the three successive sides (mi, if, fm) of another spherical tri-
angle, MIF.
CHAP. III.] FIRST AND SECOND ARRANGEMENTS OF PLANES. 289
(3.) It may also be said, that if five successive sides (kl, . . ed) of one spherical
hexagon be respectively and arcually equal to the^t^e successive diagonals (ab, mi,
BC, IF, ca) of another such hexagon (ambicf), then the sixth side (dk) of the^rs^
is equal to the sixth diagonal (km) of the second.
(4.) Or, if we adopt the conception mentioned in 180, (3.), of an arcualsum, and
denote such a sum by inserting + between the symbols of the two summands, that of
the added arc being written to the left-hand, we may state the theorem, in connexion
with the recent Fig. 59, by the formula :
III... '^DF + '^BA=nEF+ OBO, if " DA = o EC ;
where b and f may denote any two points upon the sphere.
(5.) We may also express* the same principle, although somewhat less simply
as follows (see again Fig. 69, and compare sub-art. (2.) ) :
IV. . . if <-> ED + n GH + " KL= 0, then o DK + « HE + -^ LG= 0.
(6.) If, for a moment, we agree to write (comp. Art. 1),
V. . . '^ ab = B - A,
we may then express the recent statement IV. a little more lucidly thus :
VI. ..ifD-E + H-G + L-K = 0, then k-d + e-h + g-l, = 0.
(7 ) Or still more simply, if '^, o', r," be supposed to denote any three dipla-
nar arcs, which are to be added according to the rule (180, (3.) ) above referred to,
the theorem may be said to be, that
VII.. .(o"+o')+^ = n" + (n'+o);
or in words, that Addition ofArcsi on a Sphere is an Associative Operation.
(8.) Conversely, if any independent demonstration be given, of the truth of any
one of the foregoing statements, considered as expressing a theorem of spherical geo-
metry, f a new proof Avill thereby be furnished, of the associative property of multi-
plication of quaternions.
265. In the second arrangement (263) of the six planes, instead
of representing the three given versors, and their partial or total
products, by arcs, it is natural to represent them (174, 11.) by an-
gles on the sphere. Conceive then that the two versors, q and r,
are represented, in Fig. 60, by the two spherical angles, eab and
ABE; and therefore (175) that their product, rq or s% is represented
by the external vertical angle at e, of the triangle abe. Let the
* Some of these formulae and figures, in connexion with the associative principle,
are taken, though for the most part with modifications, from the author's Sixth Lec-
ture on Quaternions, in which that whole subject is very fully treated. Comp. the
Note to page 160.
t Such a demonstration, namely a deduction of the equation II. from the five
equations I., by known properties of spherical conies, will be briefly given in the en-
suing Section.
2 p
290
ELEMENTS OF QUATERNIONS.
[book II.
second versor r be also represented by the angle fbc, and the third
versor s by bcf; then the
other binary product, sr or
3', will be represented by
the external angle at f, of
the new triangle bcf. Again,
to represent the^r^^ ternary
product, t=ss' = s.rq, we have
only to take the external an-
gle at D of the triangle ecd,
if D be a point determined
Fig. 60.
by the two conditions, that the angle ecd shall be equal to bcf,
and DEC supplementary to bea. On the other hand, if we conceive
a point d' determined by the conditions that d'af shall be equal to eab,
and afd' supplementary tocFB, then the external angle at t>\ of the
triangle afd^, will represent the second ternary product, q^q = sr. q,
•which (by the associative principle) must be equal to the first.
Conceiving then that ed is prolonged to G, and fd' to h, the
two spherical angles, gdc and ad'h, must be equal in all respects ; their
vertices d and d' coinciding, and the rotations (174, IT 7) which they
represent being not only equal in amount, but also similarly/ directed.
Or, to express the same thing otherwise, we may enunciate (262) the
Associative Principle by saying, that when the three angular equations,
I. . . ABE = FBC, BCP = ECD, DEC = TT - BEA,
are satisfied, then these three other equations^
II, . . DAF = EAB, FDA = CDE, AFD - TT - CFB,
are satisfied also. For not only is this theorem of spherical geometry a
consequence of the associative principle oi multiplication of quaternions ,
but conversely any independent demonstration* of the theorem is,
at the same time, a proof of the principle.
266. The third arrangement (263) of
the six planes may be illustrated by con-
ceiving a gauche hexagon, ab'ca^bc^ to be
inscribed in a sphere, in such a manner that
the intersection d of the three planes, c'ab',
b'ca', a'bc', is on the surface; and there-
fore that the three small circles, denoted by
these three last triliteral symbols, concur -p. g.
* Such as we shall sketch, in the following Section, with the help of the known
properties of the spherical conies. Compare the Note to the foregoing Article.
CHAP. III.] THIRD ARRANGEMENT, SPHERICAL HEXAGON. 291
in one point d ; while the second intersection of the two other small
circles, ab'c, ca'b, may be denoted by the letter d', as in the annexed
Fig. 61. Let it be also for simplicity at first supposed, that (as in
the Figure) the Jive circular successions^
I. . . c'ab'd, ab'cd', b'ca'd, ca'bd', a'bc'd,
are all direct ; or that the Jive iTiscrihed quadrilaterals, denoted by
these symbols I., are all uncrossed ones. Then (by 260, (9.) ) it is
allowed to introduce three versors, q, r, 5, each having two expres-
sions, as follows :
__ _._b'd __ab' -^da' „ca'
^ DC' AC/ B'D Cb'
^^ CD' „ BD'
ca' a'b
although (by the cited sub-article) the last members of these three
formulae should receive the negative sign, if the first, third, and
fourth of the successions I. were to become indirect, or if the corre-
sponding quadrilaterals were crossed ones. We have thus (by 191)
the derived expressions,
III. . . s' = rq = TJ — • =U — ;; o' = 5r=U — - = U — ■;
^ DC' BC' ^ cb' AB'
whereof, however, the two versors in the first formula would differ
in their signs, if the fifth succession I. were indirect; and those in
the second formula, if the second succession were such. Hence,
IV.. .t = ss^ = s.rq = V — ', q'q = sr.q = \J — ;
and since, by the associative principle, these two last versors are to
be equal, it follows that, under the supposed conditions of construc-
tion, the four points, b, c', a, d', compose a circular and dij'ect suc-
cession ; or that the quadrilateral, bc'ad', is plane, inscriptible* and
uncrossed.
267. It is easy, by suitable changes of sign, to adapt
the recent reasoning to the case where some or all of the suc-
cessions I. are indirect ; and thus to infer, from the associa-
tive principle, this theorem of spherical geometry : 7/*ab'ca'bc'
* Of course, siuce the four points bc'ad' are known to be homospheric (comp.
260. (10.)), the inseriptihility of the quadrilateral in a circle would follow from its
being plane, if the latter were otherwise proved : but it is here deduced from the
equality of the two versors IV., on the plan of 260, (9.J.
292 ELEMENTS OF QUATERNIONS. [bOOK II.
he a spherical hexagon, such that the three small circles c'ab',
b'ca', a'bc' concur in one point d, then, 1st, the three other small
circles, ab'c, ca'b, bc'k, concur in another point, d'; and Ilnd,
of the six circular successions, 266, I., and bc'ad', the number
of those which are indirect is always even (including zero).
And conversely, any independent demonstration* of this geo-
metrical theorem will be a new proof oi the associative prin-
ciple.
268. The same fertile principle of associative multiplication may
be enunciated in other ways, without limiting the factors to be ver-
sors, and without introducing the conception of a sphere. Thus we
may say (comp. 264, (2.) ), that if o . abcdef (comp. 35) be any
pencil of six rays in space, and o.a'b^c' any pencil of three rays, and
if the three angles aob, cod, eof of the first pencil be respectively
equal to the angles b'oc', c'oa', a'ob^ of the second, then another
pencil of three rays, o . a'^b^'o''', can be assigned, such that the three
other angles boc, doe, foa oith.Q first pencil shall be equal to the
angles b'^oc''', c'^oa'', a'^ob'^ of the third: equality of angles (with
one vertex) being here understood (comp. 165) to include complana'
rity, and similarity of direction of rotations.
(1.) Again (comp. 264, (4.)), we may establish the following formula, in which
the four vectors a/3y5 form a complanar proportion (226), but e and Z, are any two
lines in space :
T ^^-^^ if ^_^.
ye at 7 «
for, under this last condition, we have (comp. 125),
II £? = ?^ ? = ? ^?
* " y e aye. a' d e'
(2.) Another enunciation of the associative principle is the following :
III. . . if -- = -, then -- = -;
y a e ay o
for if we determine (120) six new vectors, r]9i, and kX/i, so that
= -, - = — , whence - = -,
y I a It
IV. . . ^ and
I ^_« f _/^
I. K a fi y
* An elementary proof, by stereographic projection, will be proposed in the fol-
lowing Section.
CHAP. III.] PBOOFS BY SPHERICAL CONICS. 293
we shall have the transformations,
V - = -- = ^ -1 = -L ^ = -l = f^ or VI - = ^
(3.) Conversely, the assertion that this last equation or proportion VI. is true,
•whenever the twelve vectors a . . fx are connected by the five proportions IV., is a
form of enunciation of the associative principle ; for it conducts (comp. IV. and V.)
to the equation,
VII. , .-.ij = --.^, atleastif e\\\i,0;
but, even with this last restriction, the three factor-quotients in VII. may represent
any three quaternions.
Section 2. — On some Geometrical Proofs of the Associative
Property of Multiplication of Quaternions, which are inde-
pendent of the Distributive* Principle.
269. We propose, in this Section, to furnish three geome-
trical Demonstrations of the Associative Principle, in con-
nexion with the three Figures (59-61) which were employed
in the last Section for its Enunciation ; and with the three ar-
rangements oi six planes, which were described in Art. 263.
The two first of these proofs will suppose the knowledge of a
few properties oi spherical conies (196, (11.)); but the third
will only employ the doctrine of stereographic projection, and
will therefore be of a more strictly elementary character. The
Principle itself is, however, of such great importance in this
Calculus, that its nature and its evidence can scarcely be put
in too many different points of view.
270. The only properties of a spherical conic, which we shall in
this Article assume as known, f are the three following: 1st, that
through any three given points on a given sphere, which are not on a
great circle, a conic can be described (consisting generally oitwo oppo-
site ovals), which shall have a given great circle for one of its two cyclic
arcs; Ilnd, that if a transversal arc cut hath these arcs, and the conic,
the intercepts (suitably measured) on this transversal are equal; and
Ilird, that if the vertex of a spherical angle move along the conic,
while its legs pass always through two fixed points thereof, those legs
* Compare 224 and 262 ; and the Note to page 236.
t The reader may consult the Translation (Dublin, 1841, pp. 46, 50, 55) by the
present Dean Graves, of two Memoirs by M. Chasles, on Cones of the Second De-
gree^ and Spherical Conies.
294 ELEMENTS OF QUATERNIONS. [bOOK II.
intercept a constant interval^ upon each cyclic arc, separately taken.
Admitting these three properties, we see that if, in Fig. 59, we con-
ceive a spherical conic to be described, so as to pass through the
three points b, f, h, and to have the great circle daec for one cyclic
arc, the second and third equations I. of 264 will prove that the arc
GLIM is the other cyclic arc for this conic; the first equation I. proves
next that the conic passes through k ; and if the arcual chord fk be
drawn and prolonged, the two remaining equations prove that it
meets the cyclic arcs in d and m ; after which, the equation 11. of
the same Art. 264 immediately results, at least with the arrange-
ment* adopted in the Figure.
(1.) The 1st property is easily seen to correspond to the possibility of circum-
scribing a circle about a given plane triangle, namely that of which the comers are
the intersections of a plane parallel to the plane of the given cyclic arc, with the
three radii drawn to the three given points upon the sphere : but it may be worth
while, as an exercise, to prove here the Ilnd property by quaternions.
(2.) Take then the equation of a cyclic cone, 196, (8.), which may (by 196,
XII.) be written thus :
I...S^S^ = N^; andlet II. . . S^' S^' = K^',
p and p' being thus two rays (or sides) of the cone, which may also be considered to
be the vectors of two points p and p' of a spherical conic, by supposing that their
lengths are each unity. Let r and r' be the vectors of the two points t and t' on
the two cyclic arcs, in which the arcual chord pp' of the conic cuts them ; so that
III. ..S- = 0, S^=0, and IV. . . Tr = Tr' = 1.
a (5
The theorem may then be stated thus : that
V. . . if jO = a;r + xt', then VI. . . p' = aV + xt ;
or that this expression VI. satisfies II., if the equations I. III. IV. V. be satisfied.
Now, by III. V. VI., we have
a a X a j3 ^ x' (i
whence it follows that the first members of I. and II. are equal, and it only remains
to prove that their second members are equal also, or that Tp' = Tp, if Tr' = Tr.
Accordingly we have, by V. and VI.,
VIII. . . ^-Ili' = ^^.^^^ = S-iO, by 200, (11.), and 204, (19.);
p' + p X +X T+T ^ ^ '^ ^'
and the property in question is proved.
* Modifications of that arrangement may be conceived, to which however it would
be easy to adapt the reasoning.
CHAP. III.] PROOF BY STEREOGRAPHIC PROJECTION. 295
271. To prove the associative principle, with the help of Fig. 60,
three other properties of a spherical conic shall be supposed known :*
1st, that for every such curve two focal points exist, ipossessing seve-
ral important relations to it, one of which is, that if these two foci
and one tangent arc he given, the conic can be constructed; Ilnd,
that if, from any point upon the sphere, two tangents be drawn to the
conic, and also two arcs to the foci, then one focal arc makes with one
tangent the same angle as the other focal arc with the other tangent ;
and Ilird, that if a spherical quadrilateral be circumscribed to such
a conic (supposed here for simplicity to be a spherical ellipse, or the
opposite ellipse being neglected), opposite sides subtend supplementary/
angles, at either of the two (interior) foci. Admitting these known
properties, and supposing the arrangement to be as in Fig. 60, we
may conceive a conic described, which shall have e and f for its two
focal points, and shall touch the arc bc ; and then the two first of the
equations I., in 265, will prove that it touches also the arcs ab and
CD, while the third of those equations proves that it touches ad, so
that ABCD is a circumscribedf quadrilateral: after which the three
equations II., of the same article, are consequences of the same pro-
perties of the curve.
272. Finally, to prove the same important Principle in a
more completely elementary way, by means of the arrangement
represented in Fig. 61, or to prove the theorem of spherical
geometry enunciated in Art. 267, we may assume the point d
as the pole of a stereograpjhic projection^ in which the three
small circles through that point shall be represented by right
lines ^hui the three others by czVc/ei",
iall being in one common plane. And
then (interchanging accents) the
theorem comes to be thus stated :
7/* a', b', c' be any three points
(comp. Fig. 62) on the sides bc,
CA, AB of any plane triangle^ or on
those sides prolonged, then^ 1st, ^ ^Y\si. 62
the three circles^
* The reader may again consult pages 46 and 50 of the Translation lately cited.
In strictness, there are of course /owr /oa, opposite two by two.
t The writer has elsewhere proposed the notation, ef(. .) abcd, to denote the
relation of the focal points e, f to this circumscribed quadrilateral.
296 ELEMENTS OF QUATERNIONS. [bOOK II.
I. . . c'ab', a'bc', b'ca',
will meet in one point d ; and Ilnd, an even number (if any)
of the six (linear or circular) successions,
II. . . ab'c, bc'a, cab, and 11'. . . c'ab'd, a'bcd, b'ca'd,
will be direct; an even number therefore also (if any) being
indirect. But, under i\\\Qform* the theorem can be proved
by very elementary considerations, and still without any em-
ployment of the distributive principle (224, 262).
(1.) 1h.B first part of the theorem, as thus stated, is evident from the Third Book
of Euclid ; but to prove both parts together, it may be useful to proceed as follows,
admitting the conception (235) oi amplitudes, or of angles as representing ro«a«ions,
■which may have any values, positive or negative, and are to be added with attention
to their signs.
(2.) We may thus write the three equations,
III. . . ab'c = nTT, bc'a = w'tt, ca'b = n"7r,
to express the three coUineations, ab'c, &c. of Fig. 62 ; the integer, n, being odd or
even, according as the point b' is on the finite line AC, or on a prolongation of that
line ; or in other words, according as the first succession II. is direct or indirect :
and similarly for the two other coefficients, n' and n".
(3.) Again, if opqr be any four points in one plane, we may establish the for-
mula,
IV. . . POQ 4- QOR = POR + 2m7r,
with the same conception of addition of amplitudes ; if then d be any point in the
plane of the triangle abc, we may write,
V. . . ab'd + db'c = n7r, bc'd + dc'a = nV, ca'd + da'b = w'V ;
and therefore,
VI. . . (ab'd + dc'a) + (bc'd + da'b) 4 (ca'd + db'c) — (» + w' 4 »") TT.
(4.) Again, if any four points opqr bo not merely complanar but concircular,
we have the general formula,
VII. . . CPQ4QRO=/J7r,
the integer p being odd or even, according as the succession opqr is direct or indi'
* The Associative Principle of Multiplication was stated nearly under this ^brm,
and was illustrated by the same simple diagram, in paragraph XXII. of a commu-
nication by the present author, which was entitled Letters on Quaternions, and has
been printed in the First and Second Editions of the late Dr. Nichol's Cyclopcedia of
the Physical Sciences (London and Glasgow, 1857 and 1860). The same commu-
nication contained other illustrations and consequences of the same principle, which it
has not been thought necessary here to reproduce (compare however Note C) ; and
others may be found in the Sixth of the author's already cited Lectures on Quater-
nions (Dublin, 1863), from which (as already observed) some of the formulae and
figures of this Chapter have been taken.
CHAP. III.] ADDITIONAL FORMULA, NORM OF A VECTOR. 297
red ; if then we denote by d the second intersection of the first and second circles I. ,
whereof c' is & first intersection, we shall have
VIII. . . ab'd + dc'a =/>7r, bc'd + da'b =p'7r,
p and p' being odd, when the two first successions II'. are direct, but even in the con-
trary case.
(5.) Hence, by VI., we have,
IX. . . ca'd + db'c =/>"7r, where X. . . jo + />' + p" = « + »' f n" ;
the third succession II'. is therefore always circular, or the third circle I. passes
through the intersection D o^ ih.Q two first ; and it is direct or indirect, that is to
say, p" is odd or even, according as the number of even coefficients, among thej^re
previously considered, is itself even or odd ; or in other words, according as the
number of indirect successions, among the five previously considered, is even (includ-
ing zero), or odd.
(6.) In every case, therefore, the total number of successions of each kind is even,
and both parts of the theorem are proved : the importance of the second part of it
(respecting the even partition, if any, of the six successions II. 11'.) arising from
the necessity of proving that we have always, as in algebra,
XI . . sr.q = -\-s.rq, and never Xll. . , sr. q = — s.rq,
if q, r, s be any three actual quaternions.
(7.) The associative principle of multiplication may also be proved, without the
distributive principle, by certain considerations of rotations of a system, on which we
cannot enter here.
Section 3. — On some Additional FormulcB.
273. Before concluding the Second Book, a few additional re-
marks may be made, as regards some of the notations and transfor-
mations which have already occurred, or others analogous to them.
And first as to notation, although we have reserved for the Third
Book the interpretation of such expressions as /3a, or a^ yet we have
agreed, in 210, (9.), to abridge the frequently occurring symbol (Ta)^
to Ta^; and we now propose to abridge it still further to Na, and to
call this square of the tensor (or of the length) of a vector, a, the Norm
of that Vector: as we had (in 190, &c.), the equation Tg'^ = N5', and
called N^- the norm of the quaternion q (in 145, (11.) ). We shall
therefore now write generally, for any vector a, the formula,
I. . .(Ta)2 = Ta2 = Na.
(1.) The equations (comp. 186, (1.) (2.) (3.) (4.) ),
II. ..Np = l; III. ..Np = Na; IV- . . N(p -«) = Na ;
V. ..N(p-a) = N(/3-a),
represent, respectively, the unit-sphere; the sphere through a, with o for centre ;
the sphere through o, with a for centre ; and the sphere through b, with the same
centre a.
2q
298 ELEMENTS OF QUATERNIONS. [bOOK II.
(2.) The equations (comp. 186, (6.) (7.) ),
VI. ..N(p + a) = N(p-a); VII. . . N(p-i8) = N(p- a),
represent, respective!}'', the plane through o, perpendicular to the line oa ; and the
plane which perpendicularly bisects the line Ab.
274, As regards transformations, the few following may here be
added, which relate partly to the quaternion forms (204, 216, &c.)
of the Equation"^ of the Ellipsoid.
(1.) Changing K(k: p) to Ep : Rk, by 259, VIII., in the equation 217, XVI.
of the ellipsoid, and observing that the three vectors p, Rp, and Rk are complanar,
while 1 : Tp = TRp by 258, that equation becomes, when divided by TRp, and when
the value 217, (5.) for t^ is taken, and the notation 273 is employed :
I. .. Tf-i-+-?-VNt-m-;
V Rp ^ Rk /
of which the first member will soon be seen to admit of being written f asT(ip + p^),
and the second member as /c^ - i^.
(2.) If, in connexion with the earlier forms (204, 216) of the equation of the
same surface, we introduce a new auxiliary vector^ a or os, such that (comp. 2 1 6,
VIII.) •
the equation may, by 204, (14.), be reduced to the following extremely simple form :
III. .. T(T=T/3;
which expresses that the locus of the new auxiliary point s is what we have called
the mean sphere, 216, XIV. ; while the line PS, or (t — p., which connects any two
corresponding points, p and s, on the ellipsoid and sphere, is seen to be parallel to
the fixed line /3; which is one element of the homology, mentioned in 216, (10.).
(3.) It is easy to prove that
IV. . .S^ = S^ S?, and therefore V. . . S ^': S^ = S^' : S^,
a c 6
if p' and <t' be the vectors of two new but corresponding points, p' and s', on the
ellipsoid and sphere ; whence it is easy to infer this other element of the homology,
that any two corresponding chords, pp' and ss', of the two surfaces, intersect each
other on the cyclic plane which has d for its cyclic normal (comp. 216, (7.) ) : in
fact, they intersect in the point t of which the vector is,
.,,_ iPp + X'p' X(T + x'o' p' p
VI. . . r = -^--— 7- = — r-, if x = S^, and a;' = -S^;
x + x x-\-x d d
* In the verification 216, (2.) of the equation 216, (1.), considered as repre-
senting a surface of the second order, V— and V^ ought to have been printed, in-
stead of V - and V - : but this does not affect the reasoning.
a a
t Compare the Note to page 233.
CHAP. III.] HOMOLOGIES OF ELLIPSOID AND SPHERE. 299
and this point is on the plane just mentioned (comp. 216, XI.), because
VII. . . S^=0.
(4.) Quite similar results would have followed, if we had assumed
VIII. . . cr = (- S^-f V^^/3 = p-2i8S^,
\ a (5 j y
which would have given again, as in III.,
IX. ..T<T = T^, but with X. ..S-=-S^ S^;
r « y
the other cyclic plane, with y instead of d for its normal, might therefore have been
taken (as asserted in 216, (10.) ), as another plane of homology of ellipsoid and
sphere, with the same centre of homology as before : namely, the poin^ at infinity 07i
the line /3, or on the axis (204, (15.) ) of one of the two circumscribed cylinders of
revolution (comp. 220, (4.)).
(5.) The same ellipsoid is, in two other ways, homologous to the same mean
sphere, with the same two cyclic planes as /> /an es of homology, but with a new centre
of homology, which is the infinitely distant point on the axis of the second circum-
scribed cylinder (or on the line ab' of the sub-article last cited).
(6.) Although not specially connected with the ellipsoid, the following general
transformations may be noted here (comp. 199, XII., and 204, XXXIV.) :
XL..TVV7=V{KTry-S7)}; XIL . • tan iZ(? = (TV: S) V7 = ^I|^^.
(7.) The equations 204, XVI. and XXXV., give easily,
XIII. . . UYq = UVU«7 ; XIV. . . VlYq = AK.q; XV. . . TlYq = TVq ;
or the more symbolical forms,
Xlir. . . UVU = UV ; XIV'. . . UIV = Ax. ; XV'. . . TIV = TV ;
and the identity 200, IX. becomes more evident, when we observe that
XVI. . .5-N"5=7(l-K5).
(8.) We have also generally (comp. 200, (10.) and 218, (10.)),
XVII ^^ = (g-l)(Kg4l) ^ Ng-1 + 2V(7
'"q + 1 (q + l)(Kq+l) Nj + 1 + 28? '
(9.) The formula,*
XVIII. . .V(rq + Kqr) = U(Sr. S^ + Yr.Yq) = r"! (r^-i^ q-\
in which q and r may be any two quaternions, is not perhaps of any great importance
in itself, but will be found to furnish a student Avith several useful exercises in trans-
formation.
(10.) When it was said, in 257, (1.), that zero had only itself iox a square-root,
the meaning was (comp. 225), that no binomial expression of the form x-\- »y (228)
could satisfy the equation,
XIX. . . = 52 = (x + ty)8 = (x^ - y2) 4- 2ixy,
* This formula was given, but in like manner without proof, in page 587 of the
author's Lectures on Quaternions.
300 ELEMENTS OF QUATERNIONS. [bOOK II.
for any real or imaginary values of the two scalar coefficients x and y, diflFerent
from zero ;* for if biquaternions (214, (8.) ) be admitted, and if h again denote, as
in 256, (2.), the imaginary of algebra, then (comp. 257, (6.) and (7.)) we may
write, generally, besides the real value Qi =■ 0, the imaginary expression^
XX. . . Qi=v-{ hv', if S» = S»' = SW=:Ntj'-N» = 0;
V and v' being thus any two real right quaternions, with equal norms (or with equal
tensors), in planes perperpendicular to each other.
(11.) For example, by 256, (2.) andby the laws (183) of y A, we have the trans-
formations,
XXI. . . {i+hj)i=i^-f -Vh{ij^ji) = + A0 = 0;
so that the bi-quaternion i 4- hj is one of the imaginary values of the symbol 0^.
(12.) In general, when bi-quaternions are admitted into calculation, not only the
square of one, but the product of two such factors may vanish, without either of them
separately vanishing : a circumstance which may throw some light on the existence
of those imaginary (or symbolical) roots of equations, which were treated of in 257.
(13.) For example, although the equation
XXII. . . g2-l = (g-l) (9-t-l) =
has no real roots except ± 1, and therefore cannot be verified by the substitution of
any other real scalar, or real quaternion, for q, yet if we substitute for q the bi-qua-
ternionf v + hv', with the conditions 257, XIII., this equation XXII. is verified.
(14.) It will be found, however, that when two imaginary but non-evanescent
factors give thus a null product, the norm of each is zero; provided that we agree
to extend to bi-quaternions the formula Ng^= Sq'^—Yq^ (204, XXII.) ; or to define
that the Norm of a Biquaternion (like that of an ordinary or real quaternion) is
equal to the Square of the Scalar Fait, minus the Square of the Right Part : each
of these two parts being generally imaginary, and the former being what we have
called a Bi-scalar.
(15.) With this definition, if q and q' be any two real quaternions, and if h be,
as above, the ordinary imaginaiy of algebra, we may establish the formula :
XXIII. . . N(9 + hq) = (Sq + hSqy - (Vq + hYq'^ ;
or (comp. 200, VII., and 210, XX.),
XXIV. . . N(9 + A5') = N5-Ng'+2^S.5K9'.
(16.) As regards the norm of the sum of any two real quaternions, or real vec-
tors (273), the following transformations are occasionally useful (comp. 220, (2.) );
XXV. . . N (5' + g) = N (Tq. Vq + Tq . Vq') ;
XXVI. . . N(/3 + a)=N(T/3.Ua + Ta.U/3);
in each of which it is permitted to change the norms to the tensors of which they are
the squares, or to write T for N.
* Compare the Note to page 276.
t This includes the expression + hi, of 257, (1.), for a symbolical square-root of
positive unity. Other such roots are + hJ, and + hk.
BOOK III.
ON QUATERNIONS, CONSIDERED AS PRODUCTS OR POWERS OF
VECTORS ; AND ON SOME APPLICATIONS OF QUATERNIONS.
CHAPTER I.
ON THE INTERPRETATION OF A PRODUCT OF VECTORS, OR
POWER OF A VECTOR, AS A QUATERNION.
Section 1. — On a First Method of interpreting a Product of
Two Vectors as a Quaternion.
Art. 275. In the First Book of these Elements we inter-
preted, 1st, the difference of any two directed right lines in
space (4) ; Ilnd, the sum of two or more such lines (5-9) ; Ilird,
the product of one such line, multiplied by or into a positive
or negative number (15) ; IVth, the quotient of such a line,
divided by such a number (16), or by what we have called
generally a Scalar (17); and Vth, the sum of a system of
such lines, each affected (97) with a scalar coefficient (99), as
being in each case zY^e//" (generally) o. Directed Line'^ in Space ^
or what we have called a Vector (1).
276. In the Second Book, the fundamental principle or
pervading conception has been, that the Quotient of two such
Vectors is, generally, a Quaternion (112, 116). It is how-
ever to be remembered, that we have included under this ge-
neral conception, which usually relates to what may be called
an Oblique Quotient, or the quotient of two lines in space
making either an acute or an obtuse angle with each other
* The Fourth Proportional to any three complanar lines has also been iaince in-
terpreted (226), as being another line in the same plane.
302 ELEMENTS OF QUATERNIONS. [bOOK III.
(130), the three following particular cases: Ist, the limiting
case, when the angle becomes null, or when the two lines are
similarly directed, in which case the quotient degenerates (131)
into 2i positive scalar; Ilnd, the other limiting case, w^henthe
angle is equal to two right angles, or when the lines are oppo-
sitely directed, and when in consequence the quotient again
degenerates, but now into a negative scalar ; and Ilird, the
intermediate case, when the angle is right, or when the two
lines are perpendicular (132), instead of being parallel (15),
and when therefore their quotient becomes what we have
called (132) a Right Quotient, or a Eight Quaternion:
which has been seen to be a case not less important than the
two former ones.
277. But no Interpretation has been assigned, in either of
the two foregoing Books, for a Product of two or more Vec-
tors ; or for the Square, or other Power of a Vector: so that
the Symbols,
I. . . /3a, 7j3aj . . and II. . . a% a^ . . a"S ... a*,
in which a, j3, 7 . . denote vectors, but t denotes a scalar, re-
main as yet entirely uninterpreted; and we are therefore /re^
to assign, at this stage, any meanings to these new symbols, or
new combinations of symbols, which shall not contradict each
othei\ and shall appear to be consistent with convenience and
analogy. And to do so will be the chief object of this First
Chapter of the Third (and last) Book oi' these Elements : which
is designed to be a much shorter one than either of the fore-
going.
278. As a commencement o£ such. Interpretation we shall
here define, that a vector a is multiplied by another vector j3,
or that the latter vector is multiplied into* the former, or
that the product j3a is obtained, ivhen the multiplier-line j3
is divided by the reciprocal^a (258) of the multiplicand-line a ;
as we had proved ( 1 36) that one quaternion is multiplied into
another, when it is divided by the reciprocal thereof. In sym-
bols, we shall therefore write, as a first definition, the for-
mula:
* Compare the Notes to pages 14G, 159.
CHAP. I.J INTERPRETATION OF A PRODUCT OF TWO VECTORS. 303
I. . ./3a=j3:Ra; where II. . . Ra = - Ua : Ta (258, VII.).
And we proceed to consider, in the following Section, some of
the general consequences of this definition, or interpretation, of
a Product of two Vectors, as being equal to a certain Quotient^
or Quaternion.
Section 2. — On some Consequences of the foregoing Inter-
pretation,
279. The definition (278) gives the formula :
I. . . |3a = :^ ; and similarly, T. . . a/3 = ^ ;
it gives therefore, by 259, VIII., the general relation,
II. . . /3a = Ka/3 ; or 11'. . . a/3 = Kj3a.
The Products of two Vectors, taken in two opposite orders, are
therefore Conjugate Quaternions; and the Multiplication of
Vectors, like that of Quaternions (168), is (generally) a Non-
Commutative Operation.
(1.) It follows from II. (by 196, comp. 223, (1.) ), that
III. . . S/3a = + Sa/3 = i(/3a + a/3).
(2.) It follows also (by 204, corap. again 223, (I.) ), that
IV. . . V^a = - Ya^ =^\(pa- a(3).
280. Again, by the same general formula 259, VI II., we
have the transformations,
' R{a-va) K/3 E/3 R/3 Ra IV
it follows, then, from the definition (278), that
II. . ./3(a + a')=/3a+/3a';
whence also, by taking conjugates (279), we have this other
general equation,
III. . . (a + a) /3 = a/3 -f a'/3.
Multiplication of Vectors is, therefore, like that of Quaternions
(212), a Doubly Distributive Operation.
281. As we have not yet assigned any signification for a
ternary product of vectors, such as yfia, Ave are not yet pre-
304 ELEMENTS OF QUATERNIONS. [bOOK III.
pared to pronounce, whether the Associative Principle (223)
o^ Multiplication of Quaternions does or does not extend to
Vector-Multiplication. But we can already derive several other
consequences from the definition (278) ofsibinari/ product, j3a ;
among which, attention may be called to the Scalar character
of a Product of two Parallel Vectors; and to the Right cha-
racter of a Product of two Perpendicular Vectors, or of two
lines at right angles with each other.
(1.) The definition (278) may be thus written,
I. ../3a = -T/3.Ta.U(/3:a);
it gives, therefore,
II. ..T/3a=T/3.Ta; III.. . U/3a = -U(i3 : a) = U/3.Ua ;
the tensor and versor of the product of two vectors being thus equal (as for quater-
nions, 191) to the product of the tensors^ and to the product of the versors, re-
spectively.
{2.) Writing for abridgment (comp. 208),
IV. ..a = Ta, 6 = T/3, y=Ax.(/3:a), a; = Z(/3:a),
we have thus,
V. . . T(3a = ba ; VI. . . S(3a = Saj3 = - 6a cos a; ;
VII. ..SU/3a = SUai3 = -cos^; VIII... L(3a = 7r-x;
so that (comp. 198) the angle of the product of any two vectors is the supplement of
the angle of the quotient.
(3.) We have next the transformations (comp. again 208),
IX. . . TV/3a = TVaj3 = 6a sin a; ; X. . . TVUj3a = TVUaj3 = sin a: ;
XI. . . I Vj3a = - y6a sin x ; XI'. . . I Va/3 = + yab sin x ;
XII. . . IUV/3a=Ax.|3a = -y; XII'. . . IUVa/3 = Ax. a/3 = + y ;
so that the rotation round the axis of a product of two vectors, from the multiplier to
the multiplicand, is positive.
(4.) It follows also, by IX., that the tensor of the right part of such a product,
(3a, is equal to the parallelogram under the factors; or to the double of the area of
the triangle OAB, whereof those two factors a, (3, or OA, OB, are two coinitial sides :
so that if we denote here this last-mentioned area by the symbol
A OAB,
we may write the equation,
XIII. . . TY(3a = parallelogram under a, (3, = 2A OAB ;
and the index, lY (3a, is a right line perpendicular to the plane of this parallelogram,
of which line the length represents its area, in the sense that they bear equal ratios
to their respective units (of length and of area).
(6.) Hence, by 279, IV.,
XIV. . . T((3a - a (3) = 2 X parallelogram = 4 A oab.
(6.) For any two vectors, «, (3,
CHAP. I.] PRODUCTS OF PARALLELS AND PERPENDICULARS. 305
XV. ..S/3a = -Na.S(|3:a); XVI. . . V^a=-Na . V(|3 : a);
or briefly,*
XVII. ../3a = -Na.(/3:a),
with the signification (273) of Na, as denoting (Ta)2.
(7.) If the two factor-lines be perpendicular to each other, so that a; is a right
angle, then the parallelogram (4.) becomes a rectangle, and the product (3a becomes
a right quaternion (132) ; so that we may write,
XVIII. . . S(3a = Sa/3 = 0, if /3 -J- a, and reciprocally.
(8.) Under the same condition of perpendicularity,
XIX. . . Z)3a=Za/3 = |; XX. . . I^a = - y6a ; XXI. . . la(3 = + yab.
(9.) On the other hand, if the two factor-lines he parallel, theright part of their
product vanishes, or that product reduces itself to a scalar, which is negative or po^
sitive according as the two vectors multiplied have similar or opposite directions ; for
we may establish the formula,
XXII. . . if /3 II a, then V/3a = 0, Va/3 - ;
and, under the same condition oi parallelism,
XXin. ., pa=a^ = S(3a = Sa(3 = + ba,
the upper or the lower sign being taken, according as a; = 0, or = tt.
(10.) We may also write (by 279, (1.) and (2.) ) the following ybrmM?a of per-
pendicularity, and formula of parallelism :
XXIV. . . if /3 4- a, then (3a =- a(3f and reciprocally ;
XXV. . . if j8 II a, then /3a = + a/3, with the converse.
(11.) If a, (3, y be any three unit-lines, considered as vectors of the comers
A, B, c of a spherical triangle, with sides equal to three new positive scalars, a, b, c,
then because, by XVII,, (3a = - (3: a, and y/B = - y : /3, the sub-articles to 208 allow
us to write,
XXVI. . . S (Vy/3 . V/3a) = sin a sin c cos b ;
XXVII. . . IV(Vy(3.V/3a) = ±/3sinasincsinB;
XXVIII. . . (IV: S) (Vy/3.V/3a) = + ^3 tan b ;
upper or lower signs being taken, in the two last formulae, according as the rotation
round (3 from a to y, or that round b from A to c, is positive or negative.
(12.) The equation 274, I., of the Ellipsoid, may now be written thus :
XXIX. . . T(«p + pfc) = Ti2-TK2; or XXX. . . T(tp + pK)=Nt-N'K.
282. Under the general head o£ sl product of two parallel
vectors, two interesting cases occur, which furnish two first
examples of Powers of Vectors : namely, 1st, the case when
* All the consequences of the interpretation (278), of the product (3a of two vec-
tors, might be deduced from this formula XVII. ; which, however, it would not have
been so natural to have assumed for a definition of that symbol, as it was to assume
the formula 278, I.
2 R
306 ELEMENTS OF QUATERNIONS. [bOOK III.
the two factors are equal, which gives this remarkable result,
that the Square of a Vector is always equal to a Negative Sca-
lar; and Ilnd, the case when the factors are (in the sense
already defined, 258) reciprocal to each other, in which case
it follows from the definition (278) that i\iQ\v product is equal
to Positive Unity : so that each may, in this case, be consi-
dered as equal to unity divided by the other, or to the Potver
of that other which has Negative Unity for its Exponent,
(I.) When (5 = a, the product (3a reduces itself to what we may call the square
of a, and may denote by a^; and thus we may write, as a particular but important
case of 281, XXIIL, the formula (comp. 273),
I. . . a2=-a2 = -(Ta)2 = -Na;
so that the square of any vector a is equal to the negative of the norm (273) of that
vector; or to the negative of the square of the number Ta, which expresses (185)
the length of the same vector.
(2.) More immediately, the definition (278) gives,
II. .. a2 = aa = a : Ra = - (Ta)« = - Na, as before.
(3.) Hence (compare the notations 161, 190, 199, 204),
III. . . S.a2 = -Na; IV. ..V.a2=0;
and
V. . . T.a2 = T(a2) = + Na = (Ta)2 = Ta2;
the omission of i\ie parentheses, or of the point, in this last symbol of a tensor,* for
the square of a vector, as well as for the square of a quaternion (190), being thus
justified : and in like manner we may write,
VI. ..U.a2 = U(a3) = -l=(Ua)2 = Ua2;
the square of an unit-vector (129) being always equal to negative unity, and paren-
theses (or points) being again omitted.
(4.) The equation
VII. . . p2 = a\ gives VII'. . . Np = Na, or VII". . . Tp = Ta ;
it represents therefore, by 186, (2.), the sphere with o for centre, which passes
through the point a.
(6.) The more general equation,
VIII. . . (p - a)2 = ((3 - a)«, (comp.f 186, (4.), )
represents the sphere with a for centre, which passes through the point b.
(6.) For example, the equation,
IX. . . (p - a)2 = a2, (comp. 186, (3.), )
represents the sphere with a for centre, which passes through the origin o.
* Compare the Note to page 210.
t Compare also the sub-articles to 275.
CHAP. I.] SQUARE AND RECIPROCAL OF A VECTOR. 307
(7.) The equations (comp. 18G, (6.), (7.)),
X. . . (p + a)2 = (p-a)2; XI. . . (p - /3)2 = (p- a)^,
represent, respectively, the plane through o, perpendicular to the line OA ; and the
plane which perpendicularly bisects the line ab.
(8.) The distributive principle oi veetor'tnultiplication (280), and the formula
279, III., enable us to establish generally (comp. 210, (9.) ) the formula,
XII. . . (|3±a)2 = /3-2+2S/3a + a3;
the recent equations IX. and X. may therefore be thus transformed :
IX'. . . p^- = 2Sap ; and X', . . Sap = 0.
(9.) The equations,
XIII. . . p2+a2 = 0; XIV. .. p2 + 1=10,
represent the spheres with o for centre, which have a and 1 for their respective radii ;
so that this very simple formula, p'+ 1 = 0, is (comp. 186, (1.) ) a form of the Equa-
tion of the Unit- Sphere (128), and is, as such, of great importance in the present
Calculus.
(10.) The equation,
XV. . . p«-2Sap + c = 0,
may be transformed to the following,
XVI. . . N(p-a) = -(p-a)2 = c-a2 = c + Na;
or XVr. . . T(p-a) = V(c-a2) = V(c + Na);
it represents therefore a (real or imaginary) sphere, with a for centre, and with this
last radical (if real) for radius.
(11.) This sphere is therefore necessarily real, if c be a positive scalar ; or if this
scalar constant, c, though negative^ be (algebraically) greater than a*, or than — Na :
but it becomes imaginary, if c + Na < 0.
(12.) The radical plane of the two spheres,
XVII. . . p2 - 2Sap + c = 0, p2 - 2Sa'p + c' = 0,
has for equation,
XVIII. . . 2S(a'-a)p = c'-c;
it is therefore always real, if the given vectors a, a and the given scalars c, c be
such, even if one or both of the spheres themselves be imaginary.
(13.) The equation 281, XXIX., or XXX., of the Central Ellipsoid {ox of the
ellipsoid with its centre taken for the origin of vectors), may now be still further sim-
plified,* as follows :
XIX.. .T(tp + pK:)=/c^~i2.
(14.) The definition (278) gives also,
XX. . . aRa = a : a = 1 ; or XX'. . . Ra . a = Ra : Ra = 1 ;
whence it is natural to write, f
* Compare the Note to page 233.
t Compare the second Note to page 279.
308 ELEMENTS OF QUATERNIONS. [bOOK III.
XXL . . Ra = l:i = a-S
a
if we so far anticipate here the general theory oi powers of vectors^ above alluded to
(277), as to use this last symbol to denote the quotient^ of unity divided by the vector
a ; so as to have identically, or for every vector, the equation,
XXII. . . a.a-i = a-'.a=l.
(15.) It follows, by 258, VII., that
XXIII. . . a-i = - Ua : Ta ; and XXIV. . . (Sa = ft : aK
(16.) If we had adopted the equation XXIII. as a definition* otthesymbol a"',
then the formula XXIV. might have been used, as a formula of interpretation for
the symbol (3a. But we proceed to consider an entirely different method, of arriving
at the same (or an equivalent) Interpretation of this latter symbol : or of a Binary
Product of Vectors, considered as equal to a Quaternion.
Section 3. — On a Second Method of arriving at the same In-
terpretation, of a Binary Product of Vectors.
283. It cannot fail to have been observed by any attentive
reader of the Second Book, how close and intimate a connexion'\
has been found to exist, between a Right Quaternion (132), and
its Index, or Index- Vector (133). Thus, if u and v' denote (as
in 223, (1.), &c.) any two right quaternions, andif lu, Iv de-
note, as usual, their indices, we have already seen that
I. . . Iv' = Iv, if v'=v, and conversely (133);
IL . . l(v'±v)=^Iv'±lv (206);
111. . . Iv: lv=v:v (193);
to which may be added the more recent formula,
IV. . .EI?;=mi;(258, IX.).
284. It could not therefore have appeared strange, if we
had proposed to establish this new formula of the same kind,
I. . . lv',Iv = v'.v = vv,
as a definition (supposing that the recent definition 278 had
not occurred to us), whereby to interpret the product of any two
indices of right quaternions, as being equal to thQ product of
those tivo quaternions themselves. And then, to interpret the
product /3a, of any two given vectors, taken in a given order,
* Compare the Note to page 305.
t Compare the Note to page 174.
CHAP. I.] SECOND INTERPRETATION OF A PRODUCT. 309
we should only have had to conceive (as we always may), that
the two proposed ^c^6>r5, a and j3j are the indices of two right
quaternions, v and v, and to multiply these latter, in the same
order. For thus we should have been led to establish the for-
mula,
II. . . j3a = vv, if u = Iv, and /3 = Iv ;
or we should have this slightly more symbolical equation,
III. . .j3a = j3.a = r^i3.Fa;
in which the symbols,
I'a and T^jS,
are understood to denote the two right quaternions, whereof
the two lines a and /3 are the indices.
(1.) To establish now the substantial fc?ew^zVy of these two interpretations, 278 and
284, of a binary product of vectors (3a, notwithstanding the difference of form of
the definitional equations by which they have been expressed, we have only to ob-
serve that it has been found, as a theorem (194), that
IV. . .v'v = It)': I (1 : i;) = Iv: IRr ;
but the definition (258) of Ra gave us the lately cited equation, RIu = IRv ; we have
therefore, by the recent formula II., the equation,
V. . . Iy'.Iy = Iw':RIt?; or VI. . . i3.a = j3 : Ra,
as in 278, I. ; a and /3 still denoting any two vectors. The two interpretations
therefore coincide, at least in their results, although they have been obtained by dif-
ferent processes, or suggestions, and are expressed by two different /br?wwte.
(2.) The result 279, II., respecting conjugate products of vectors, corresponds
thus to the result 191, (2.), or to the first formula of 223, (1.)-
(3.) The two formulae of 279, (1.) and (2.), respecting the scalar and right
parts of the product (3a, answer to the two other formulas of the same sub-article,
223, (1.), respecting the corresponding parts ofv'v.
(4.) The doubly distributive property (280), oi vector-multiplication, is on this
plan seen to be included in the corresponding but more general property (212), of
multiplication of quaternions.
(5.) By changing YVq, YVq', t, t' , and o, to a, (3, a, b, and y, in those formula)
of Art. 208 which are previous to its sub-articles, we should obtain, with the recent
definition (or interpretation) II. of (3a, several of the consequences lately given (in
sub-arts, to 281), as resulting from the former definition, 278, I. Thus, the equa-
tions,
VI., VII., VIII,, IX., X., XL, XII., XXIL, and XXIII.,
of 281, correspond to, and may (with our last definition) be deduced from, the for-
mulaa,
v., VI., VIII., XL, XIL, XXII., XX., XIV., and XVI., XVIIL,
of 208. (Some of the consequences from the sub- articles to 208 have been already
considered, in 281, (11.) )
310 ELEMENTS OF QUATERNIONS. [bOOK III.
(6.) T\\& geometrical properties of the line IV/3a, deduced from ihQ first defini-
tion (278) of /3a in 281, (3.) and (4.), (namely, t\xQ positive rotation round that line,
from /3 to a ; '\U perpendicularity to their plane ; and the representation by the same
line of the paralellogram under those two factors^ regard being had to units oi length
and of area,^ might also have been deduced from 223, (4.), by means of the second
definition (284), of the same product^ (3a.
Section 4. — On the Symbolical Identification of a Right Qua-
ternion with its own Index: and on the Construction of a
Product of Two Rectangular Lines, by a Third Line, rect-
angular to both.
285. It has been seen, then, that the recent formula 284,
II. or III., mag replace the formula 278, 1., as a .second definition
of a product of two vectors, which conducts to the same conse-
quences, and therefore ultimately to the same interpretation
of such a product, as the^r^^. Now, in the ^ecowc? formula,
we have interpreted that product, /3a, by changing the two fac-
tor-lines, a and j3, to the two right quaternions, v and v, or
r^a and I"^j3, of which they are the indices; and by then de-
fining that the sought product j3a is equal to the product v'v,
of those two right quaternions. It becomes, therefore, impor-
tant to inquire, at this stage, how far such substitution, of I"^a
for a, or of v for lu, together with the converse substitution, is
permitted in this Calculus, consistently with principles already
established. For it is evident that if such substitutions can
be shown to be generally legitimate, or allowable, we shall
thereby be enabled to enlarge greatly the existing field of inter-
pretation: and to treat, in «// cases, Functions of Vectors, as
being, at the same time. Functions of Right Quaternions.
286. We have first, by 133 (comp. 283, I.), the equality,
L..r>/3 = rx if ^=a.
In the next place, by 206 (comp. 283, II.), we have the formula of
addition or subtraction,
11. . . r'()3±a)=I-'^ir'a;
with these more general results of the same kind (comp. 207
and 99),
III. . . I^2a = sr'fl : IV. . . l-'2xa = 2a;r^a.
CHAP. I.] RIGHT QUATERNION EQUAL TO ITS INDEX. 311
In the third place, by 193 (comp. 283, III.), we have, for division,
the formula,
V... r»/3:r'a = ^:a;
while the second definition (284) oi multiplication of vectors, which has
been proved to be consistent with the first definition (278), has given
us the analogous equation,
VI. . . I-'^.I-'a = ^.a = /3a.
It would seem, then, that we might at once proceed to define, for the
purpose of interpreting any proposed Function of Vectors as a Quater-
ternion, that the following general Equation exists :
VII. ..ria=:a; or VIII. . . I«7 = V, if V = -;
or still more briefly and symholically, if it be understood that the
subject of the operation I is always a right quaternion,
IX. ..1=1.
But, before finally adopting this conclusion, there is a case (or rather
a class of cases), which it is necessary to examine, in order to be cer-
tain that no contradiction to former results can ever be thereby caused.
287. The most general form of a vector -function, or of a vector
regarded as a function of other vectors and of scalars, which was
considered in the First Book, was the form (99, comp. 275),
1. . . p = l^xa ;
and we have seen that if we change, in this form, each vector a to the
corresponding right quaternion I'^a, and then take the index of the
new right quaternion which results, we shall thus be conducted to
precisely the same vector p, as that which had been otherwise ob-
tained before; or in symbols, that
II. . . -Ixa^l^xl-'a (comp. 286, IV.).
But another form of a vector-function has been considered in the Se-
cond Book ; namely, the form,
III. ..^ = ...^^a(226,III.);
in which o, /3, 7, ^, e . . . are any odd number of complanar vectors.
And before we accept, as general, the equation VII. or VIII. or IX.
of 286, we must inquire whether we are at liberty to write, under
the same conditions of complanarity, and with the same signification
of the vector p, the equation,
312 ELEMENTS OF QUATERNIONS. [BOOK III.
■—(••■S-B-'-)
288. To examine this, let there be at first only three given com-
planar vectors, 7|||a, /3; in which case there will always be (by
226) 2, fourth vector />, in the same plane, which will represent or
construct the function (7: /3).a; namely, thQ fourth proportional to
/3, 7, a. Taking then what we may call the Inverse Index- Functions^
or operating on these four vectors a, y3, 7, p by the characteristic I"^
we obtain/owr collinear and right quaternions (209), which may be
denoted by v, v'^ v'\ v'" ; and we shall have the equation,
V. . . v"'\v--{p\a=r^\^^)v"\v'\
or VI. . . v'"--{v"'.v').v\
which proves what was required. Or, more symbolically,
VII ^=^=:^=Ii)f.
viiL..^.a = />=i(i-V)=i(J;Jj.rH
And it is so easy to extend this reasoning to the case of any greater
odd number of given vectors in one plane, that we may now consi-
der the recent formula IV. as proved.
289. We shall therefore adopts as general^ the symbolical
equations VII. VIII. IX. of 286; and shall thus be enabled,
in a shortly subsequent Section, to interpret ternary (and other)
products of vectors, as well as powers and other Functions of
Vectors, as hQing generally Quaternions; although they may,
in particular cases, degenerate (131) into scalars, or may be-
come right quaternions ( 132) : in which latter event they may,
in virtue of the same principle, be represented by, and equated
to, their own indices (133), and so be treated as vectors. In
symbols, we shall wnte generally, for any set of vectors a, j3,
y, . . . and any function f the equation,
I. ../(a,p,7,...)=/a-^«»I"/3,I-^y,--) = ?,
q being some quaternion; while in the particular case when
this quaternion is right, or when
q = v=S-^0 = l-'p,
CHAP. I.] PRODUCT OF TWO RECTANGULAR LINES A LINK. 313
we shall write also, and usually by preference (for that case),
the formula,
n. . ./(a, /3, r, . . .) = i/(i-'<«. i-'/3. i-'7> • • •) =P.
jO being a vector.
290. For example, instead of saying (as in 281) that the
Product of any two Rectangular Vectors is a Right Quaternion,
with certain properties of its Index^ already pointed out (284,
(6.) ), we may now say that such a product is equal to that in-
dex. And hence will follow the important consequence, that
the Product of any Two Rectangular Lines in Space is equal
to (or may be constructed by) a Third Line, rectangular to
both ; the Rotation round this Product-Line, from the Multi-
plier-Line to the Multiplicand' Line, being Positive : and the
Length of the Product being equal to the Product of the
Lengths of the Factors, or representing (with a suitable refe-
rence to units) the Area of the Rectangle under them. And
generally we may now, for all purposes of calculation and ex-
pression, identify* a Right Quaternion with its own Index.
Section 5 On some Simplifications of Notation, or of Ex-
pression, resulting from this Identification ; and on the Con-
ception of an Unit-Line as a Right Versor.
29 1 . An immediate consequence of the symbolical equa-
tion 286, IX., is that we may now suppress the Characteristic
I, of the Index of a Right Quaternion, in all the formulas into
which it has entered ; and so may simplify the Notation. Thus,
instead of writing,
Ax. q = lUV^, or Ax. = lUV, as in 204, (23.),
or Ax. q = JJlYq, Ax. = UIV, as in 274, (7.),
we may now Avrite simplyj,
L..Ax.^=UV^; or II. ..Ax.= UV.
The Characteristic Ax., of the Operation of taking the Axis of
a Quaternion (132, (6.) ), may therefore henceforth be replaced
* Compare the Notes to pages 119, 136, 174, 191, 200.
t Compare tbe first Note to page 118, and the second Note to page 200.
2 s
314 ELEMENTS OF QUATERNIONS. [bOOK III.
whenever we may think fit to dispense with it, by this combina-
tion of two other characteristics, U and V, which are of greater
and more ^ew^r«/ utility, and indeed cannot'* be dispensed with,
in the practice of the present Calculus.
292. We are now enabled also to diminish, to some extent,
the number of technical terms^ which have been employed in
the foregoing Book. Thus, whereas we defined, in 202, that
the right quaternion V^ was the Right Part of the Quater-
nion g, or of the sum Sq + Yq, we may now, by 290, identify
that part with its own index-vector lYq, and so may be led to
call it the vector part, or simply ^Ae Vector,-}- of that Quater-
nion q, without henceforth speaking of the right part: although
the plan of exposition, adopted in the Second Book, required
that we should do so for some time. And thus an enuncia-
tion, which was put forward at an early stage of the present
work, namely, at the end of the First Chapter of the First
Book, or the assertion (17) that
^^ Scalar plus Vector equals Quaternion"
becomes entirely intelligible, and acquires a perfectly definite
signification. For we are in this manner led to conceive a
Number (positive or negative) as being added to a Li7ie,%
when it is added (according to rules already established) to
that right quotient (132), of which the line is the Index, In
symbols, we are thus led to establish the formula,
1. . . q = a-^a, when II. . . </ = a + I'^a ;
* Of course, any one who chooses may invent new symbols^ to denote the same
operations on qvaternions, as those which are denoted in these Elements, and in the
elsewhere cited Lectures, by the letters U and V ; but, under some form, such sym-
bols must be used: and it appears to have been hitherto thought expedient, by other
writers, not hastily to innovate on notations which have been already employed in
several published researches, and have been found to answer their purpose. As to the
type used for these, and for the analogous characteristics K, S, T, that must evidently
be a mere affair of taste and convenience : and in fact they have all been printed
as small italic capitals, in some examination-papers by the author.
f Compare the Note to page 191.
X On account of this possibility of conceiving a quaternion to be the sum of a
number and a line, it was at one time suggested by the present author, that a Qua-
ternion might also be called a Grammarithm, by a combination of the two Greek
words, ypafifit] and dpiOfiog, vhich signify respectively a Line and a Number.
CHAP. I.] CONCEPTIONOF ANUNIT-LINE ASA RIGHT VERSOR. 315
lohatever scalar^ and whatever vector, may be denoted by a
and a. And because either of these two parts, or summands,
may vanish separately, we are entitled to say, that both Sca-
lars and Vectors, or Numbers and Lines, are included in the
Conception of a Quaternion, as now enlarged or modified.
293. Again, the same symbolical identification of Iv with
v (286, VIII.) leads to the forming of a new conception of an
Unit-Line, or Unit-Vector (129), as being also a Riyht Versor
(153) ; or an Operator, of which the effect is to tur7i a line, in
a plane perpendicular to itself, through a positive quadrant of
rotation : and thereby to oblige the Operand-Line to take a
neiv direction^ at right angles to its old direction, but without
any change of length. And then the remarks (154) on the
equation q'^==-\, where q was a right versor in \\iQ former
sense (which is still a permitted one) of its being a right ra-
dial quotient (147), or the quotient of two equally long hut mu-
tually rectangular lines, become immediately applicable to the
interpretation of the equation,
10^ = - I, or ^2 + 1 ^ (282, XIV.) ;
where p is still an unit-vector,
(1.) Thus (comp. Fig. 41), if a be any line perpendicular to such a vector p,
we have the equations,
I. ..pa = |8; II. . . |02a = p/3 = a'=-a;
j8 being another line perpendicular to jO, which is, at the same time, at right angles
to a, and of the same length with it ; and from which a third line a\ or — a, oppo-
site to the line a, but still equally long, is formed by a repetition of the operation,
denoted by (what we may here call) the characteristic p ; or having that unit-vec-
tor p for the operator, or instrument employed, as a sort oi handle, or axis* of ro-
tation.
(2.) More generally (comp. 290), if a, (3, y beany three lines at right angles to
each other, and if the length of y be numerically equal to the product of the lengths
of a and (3, then (by what precedes) the line y represents, or constructs, or is equal
to, the product of the two other lines, at least if a certain order of the factors
(comp. 279) be observed: so that we may write the equation (comp. 281, XXI.),
III. ..a/3 = y, if IV. . . /8 -J- a, y J- rt, y -i- f3, and V. . . Ta, T/3 = Ty,
* Compare the first Note to page 136.
316 ELEMENTS OF QUATERNIONS. [bOOK III.
provided that the rotation round a, from /3 to y, or that round y from a to (3, &c.,
has the direction taken as the positive one.
(3.) In this more general case, we may still conceive that the multiplier- line
a has operated on the multiplicand-line j3, so as to produce (or generate} the pro-
duct-line y ; hut not now by an operation of version alone, since the tensor of j3 is
(generally) multiplied by that of a, in order to form, by V., the tensor of the pro-
duct y.
(4.) And if (comp. Fig. 41, his, in which a was first changed to (3, and then to
a') we repeat this compound operation, of tension and version combined (comp. 189),
or if we multiply again hy a, we obtain a, fourth line (3', in the plane of /3, y, but
with a direction opposite to that of /3, and with a length generally different : namely
the line,
VI. . . ay=aaP = a'^j3=^' = - a"^^, if a = Ta.
(5.) The operator a^, or aa, is therefore equivalent, in its effect on (3, to the ne-
gative scalar, — a?, or — (Ta)2, or — Na, considered as a coefficient, or as a (scalar)
multiplier (15) : whence the equation,
VII. .. a2 = -Na(282, L),
may be again deduced, but now with a new interpretation, which is, however, as we
see, completely consistent, in all its consequences, with the one first proposed (282).
Section 6. — (^w the Interpretation of a Product of Three or
more Vectors, as a Quaternion.
294. There is now no difficulty in interpreting a ternary
product of vectors (comp. 277, I.), or a product of more vec-
tors than three, taken always in some piven order ; namely, as
the result (289, I.) of the substitution of the corresponding
right quaternions in that product: which result is generally
what we have lately called (276) an Oblique Quotient, or a
Quaternion with either an acute or an obtuse angle (130) ; but
maj degenerate (131) into a scalar, or may become itself a
right quaternion (132), and so be constructed (289, II.) by a
new vector. It follows (comp. 28 1), that Multiplication of Vec-
tors, like that of Quatetmions (223), in which indeed we now
see that it is included, is an Associative Operation: or that
we may write generally (comp. 223, II.), for ang three vec-
tors, a, j3, 7, the Formula,
I. . . yj5a = y »(5a.
(1.) The formulae 223, III. and IV., are now replaced by the following :
II. . . V.yV/3a = aS/3y -/3Sya;
III. . . Vy/3a= aS)3y ~^Sya + ySa/3 ;
CHAP. I.] TERNARY PRODUCTS OF VECTORS. 317
in which Yy (3a is written, for simplicity, instead ofV(yj3a), or V. yj3a; and with
which, as with the earlier equations referred to, a student of this Calculus will find
it useful to render himself verj/ familiar.
(2.) Another useful form of the equation II. is the following :
IV. . . V(Va/5.y) = aS/3y-/3Sya.
(3.) The equations IX. X. XIV. of 223 enable us now to write, for any three
vectors, the formula :
V. . . Sy/Sa = - Sa|3y = Say (3 = - S/3ya = S(3ay = - Sya/3
= + volume of parallelepiped under a, /3, y,
= + 6 X volume of pyramid oabc ;
upper or lower signs being taken, according as the rotation round a from /3 to y is
positive or negative : or in other words, the scalar Sy/3a, of the ternary product of
vectors yj3a, being positive in the first case, but negative in the second.
(4.) The condition of complanarity of three vectors, a, (3, y, is therefore ex-
pressed by the equation (comp. 223, XI.) :
VI. ..Sy/3a = 0; or VI'. . . Sa/3y = ; &c.
(5.) If a, (3, y be any three vectors, complanar or diplanar, the expression,
VII. .. 5 = aS/3y-/3Sya,
gives ^VIII. . . Sy5=0, and IX. . . Sa/3^ = 0;
it represents therefore (comp. II. and IV.) a. fourth vector S, which is perpendicular
to y, but complanar with a and (3: or in symbols,
X. ..^_Ly, and XL . . d \\\ a, (3.
(Compare the notations 123, 129.)
(6.) For any four vectors, we have by II. and IV. the transformations,
XII. . . V(Vaj3 . Vy5) = dSa^y - ySa[3d ;
XIII. . . V (Ya(3 . Vy ^) = asl3yd- /3Say 5 ;
and each of these three equivalent expressions represents a. fifth vector t, which is at
once complanar with a, (3, and with y, ^; or a line oe, which is in the intersection
of the two planes, OAB and ocd.
(7.) Comparing them, we see that any arbitrary vector p may be expressed as
a linear function of any three given diplanar vectors, a, (3, y, by the formula :
XIV. . . pSajSy = aS(3yp + /SSyap + ySa/3p ;
which is found to be one of extensive utility.
(8.) Another very useful formula, of the same kind, is the following:
XV. . . pSa/3y=V/3y.Sap+Vya.S/3p+Vaj3.Sy|0;
in the second member of which, the points may be omitted.
(9.) One mode of proving the correctness of this last formula XV., is to operate
on both members of it, by the three symbols, or characteristics of operation,
XVI. ..S. a, S./3, S.y;
the common results on both sides being respectively the three scalar products,
XVII. . .Sap. Sa(3y, S(3p . Sa(3y, Syp . Sa/3y ;
where again the points may be omitted.
318 ELEMENTS OF QUATERNIONS. [bOOK III.
(10.) We here employ the principle, that if the three vectors a, /3, y he actual
and diplanar, then no actual vector \ can satisfy at once the three scalar equations^
XVIII. . . SaX = 0, S/3X = 0, SyX = ;
because it cannot he perpendicular at once to those three diplanar vectors.
(11.) If, then, in any investigation with quaternions, we meet a system of this
form XVIII., we can at once infer that
XIX. ..X = 0, if XX. . . Sa;3y^0;
while, conversely, if X he an actual vector, then a, /3, y must be complanar vectors,
or Sa/3y = 0, as in VI'.
(12.) Hence also, under the same condition XX., the three scalar equations,
XXI. . . SaX = Saju, S/3X = S/3/«, SyX = Sy/z,
give XXII. . . X = /ii.
(13.) Operating (comp. (9.)) on the equation XV. by the symbol, or charac-
teristic, S . ^, in which d is any new vector, we find a result which may be written
thus (with or without the points) :
XXIII. . . = Sap . S/3y^ - S/3p . Sy ^a + Syp . S^a/3 - S^p . Sa/3y ;
where a, /3, y, ^, p may denote any five vectors.
(14.) In drawing this last inference, we assume that the equation XV. holds
good, even when the three vectors a, /3, y are complanar : which in fact must be true,
as a limit, since the equation has been proved, by (9.) and (12.), to be valid, if y be
ever so little out of the plane of a and /3.
(15.) We have therefore this new formula :
XXIV. . . V/3y Sap + Vy a S,3p + Va/3Syp = 0, if Sa/3y = ;
in which p may denote any fourth vector, whether in, or out of, the common plane
of a, /3, y.
(16.) If p ha perpendicular to that plane, the last formula is evidently true, each
term of the first member vanishing separate!}^, by 281, (7.) ; and if we change p to
a vector d in the plane of a, /3, y, we are conducted to the following equation, as an
interpretation of the same formula XXIV., which expresses a known theorem of
plane trigonometry, including several others under it ;
XXV. . . sin Boc . cos aod + sin coa . cos bod + sin aob . cos cod = 0,
for any four complanar and co-initial lines, OA, OB, oc, OD.
(17.) By passing from od to a line perpendicular thereto, but in their common
plane, we have this other known* equation :
XXVI. . . sin BOC sin aod + sin coA sin bod + sin aob sin cod = ;
which, like the former, admits of many transformations, but is only mentioned here
as offering itself naturally to our notice, when we seek to interpret the formula
XXIV. obtained as above by quaternions.
(18.) Operating on that formula by S.^, and changing p to c, we have this new
equation :
* Compare page 20 of the Oeometrie Snperieure of M. Chasles.
CHAP. 1.] ELIMINATION OF A VECTOR. 319
XXVII. . . = SaeSfiyd + Si^eSyaS + SyeBal3d, if Sa/3y = ;
which might indeed have been at once deduced from XXIII.
(19.) The equation XIV., as well as XV., must hold good at the limit, when o,
/3, y are complanar ; hence
XXVIII. . . aS/3yp + (3Syap + ySafSp = 0, if Sa(3y = 0.
(20.) This last formula is evidently true, by (4.), if p be in the common plane
of the three other vectors ; and if we suppose it to be perpendicular to that plane,
so that
XXIX. . . p II Yj3y 11 Yya \\ YajS,
and therefore, by 281, (9.), since S (S/Sy. p) = 0,
XXX. . . S/3yp = S(V/3y.p) = V/3y.p, &c.,
we may divide each term hy p, and so obtain this other formula,
XXXI. . . aV/3y + /3Vya + y VajS = 0, if Sa^Sy = 0.
(21.) In general, the vec/or (292) of this last expression vanishes by II. ; the
expression is therefore equal to its own scalar, and we may write,
XXXTI. . . aV/3y + /3Vya + y Va/3 = 3Sa/3y,
whatever three vectors may be denoted by a, /3, y.
(22.) For the case of complanar ity, if we suppose that the three vectors are
equally long, we have the proportion,
XXXIII. . . V]3y : Vya : Va/3 = sin boc : sin COA : sin aob ;
and the formula XXXI. becomes thus,
XXXIV. . . OA . sin BOC 4- ob . sin coa + oc . sin aob = ;
where oa, ob, oc are any three radii of one circle, and the equation is interpreted as
in Articles 10, 11, &c.
(23.) The equation XXIII. might have been deduced from XIV., instead of
XV., by first operating with S.^, and then interchanging d and p.
(24.) A vector p may in general be considered (221) as depending on three sca-
lars (the co-ordinates of its term) ; it cannot then be determined hy fewer than three
scalar equations ; nor can it be eliminated between /ewer than four.
(25.) As an example of such determination of a vector, let a, P, y be again any
three given and diplanar vectors ; and let the three given equations be,
XXXV. . . Sap = a, S/3p = ^ Syp = c;
in which a, b, c are supposed to denote three given scalars. Then the sought vector
p has for its expression, by XV.,
^XXXVI. . . p = e-i(aV/3y + 6Vya + cVa/3), if XXXVII. . . e = Sa^y.
(26.) As another example, let the three equations be,
XXXVIII. . . S/3yp = a, Syap = 6', Sa/3p = c ;
then, with the same signification of the scalar e, we have, by XIV,
XXXIX. . . p = e-i (aa + 6'/3 + c'y).
(27.) As an example of elimination of a vector, let there be the four scalar
equations,
XL. ..Sap=a, S/3p = &, Syp=r, S^p = d;
320 ELEMENTS OF QUATERNIONS. [bOOK III.
then, by XXIII., we have this resulting equation^ into which p does not enter, but
only the /o«r vectors, a . . d, and the^wr scalars, a ..d:
XLI. . . a . SjSyd -b.SySa+c. SSafi -d. Sa/3y = 0.
(28.) This last equation may therefore be considered as the condition of concur-
rence of the four planes, represented by the four scalar equations XL., in one com-
mon point; for, although it has not been expressly stated before, it follows evidently
from thQ definition 278 of a binary product of vectors, combined with 196, (5.),
that every scalar equation of the linear form (comp. 282, XVIII.),
XLII. . . Sap = a, or Spa = a,
in which a = OA, and p = op, as usual, represents a plane locus of the point P ; the
vector of the foot s, of i\xQ perpendicular on that plane from the origin, being
XLIIL . . OS = <T=aRa = aa-i (282, XXL).
(29.) If we conceive a pyramidal volume (68) as having an algebraical (or sca-
lar^ character, so as to be capable of bearing either a positive or a negative ratio to
the volume of a given pyramid, with a given order of its points, we may then omit
the ambiguous sign, in the last expression (3.) for the scalar of a ternary product of
vectors : and so may write, generally, oabc denoting such a volume, tbe formula,
XLIV. . . Sa/3y = 6 . OABC,
= a positive or a negative scalar, according as the rotation round OA from ob to oc is
negative or positive.
(30.) More generally, changing o to d, and oa or a to a - d, &c., we have thus
the formula :
XLV. . . 6 . DABC = S(a - ^) (i3 - 5) (y - ^) = Sa/3y - S(3yd + Sy^a - S^a/3 ;
in which it may be observed, that the expression is changed to its own opposite, or
negative, or is multiplied by — 1, when any two of the four vectors, a, (3, y, d, or when
any two of the four points, A, B, c, D, change places with each other; and therefore
is restored to its former value, by a second such binary interchange.
(31.) Denoting then the new origin of a, (3, y, d by E, we have first, by XLIV.,
XLV., the equation,
XLVI. . . DABC = EABC — EBCD + ECDA — EDAB ;
and may then write the result (comp. 68) under the more symmetric form (because
— EBCD = BECD = &C.) :
XLVII. . . BCDE 4 CDEA + DEAB + EABC + ABCD = ;
in which A, B, c, d, e may denote any five points of space.
(32.) And an analogous formula (69, III.) of the First Book, for any six points
OABCDE, namely the equation (comp. 65, 70),
XL VIII. . . OA.BCDE + OB.CDEA+ OC. DEAB + OD. EABC + OB. ABCD = 0,
in which the additions are performed according to the rules of vectors, the volumes
being treated as scalar coefficients, is easily recovered from the foregoing principles
and results. In fact, by XLVII., this last formula may be written as
XLIX. . . ED. EABC = EA . EBCD + EB . ECAD + EC, EABD ;
or, substituting a, ft, y, S for ea, eb, ec, ed, as
CHAP. I.] STANDARD TRINOMIAL FORM FOR A VECTOR. 321
L. . . SSaBy = aSjSyd + (5Syad + ySa/3^ ;
which is only another form of XIY., and onght to hQ familiar to the student.
(33.) The formula 69, II. may be deduced from XXXI., by observing that, when
the three vectors a, j8, y are complanar, we have the proportion,
LI. . . Y(3y : Vya : Ya(S : V (/3y + ya + a/3) = OBC : oca : oab : abc,
i{ signs {or algebraic ox scalar ratios) of areas be attended to (28, 63); and the
formula 69, I., for the case of three collinear points A, b, c, may now be written as
follows :
LII. . . a (|3 - y) + /3 (y - a) 4- y (a - /3) = 2 V(iSy + y a + a/3)
= 2V(/S-a)(y-a) = 0.
if the three coinitial vectors a, /3, y be termino-collinear (24).
(84.) The case when four coinitial vectors a, (3, y, d are termino-complanar (64)^
or when they terminate in /owr complanar points A, b, c, p, is expressed by equating
to zero the second or the third member of the formula XLV.
(35 ) Finally, for ternary products of vectors in general, we have the formula:
LIIL . . a2/32y2 + (Sai8y)2 = (Va/3y)2 = (aS;8y - /3Sya + ySa/3)2
= a? (S/3y)2 + /33 (Sya)2 + y2 (Sa/3)2 - 2S|3y Sya Sa/3.
295. The identity (290) of a right quaternion with its in-
dex, and the conception (293) of an unit-line as a ?'/^A^ versor,
allow us now to treat the three important versors, i,j, k, as
constructed by, and even as (in our present view) identical
with, their own axes ; or with the three lines ox, oj, ok of 181,
considered as being each a certain instrument, or operator, or
agent in a right rotation (293, (1.) ), which causes any line, in
a plane perpendicular to itself, to turn in that plane, through
a positive quadrant, without any change of its length. With
this conception, or construction, the Laxcs of the Symbols ijk
are still included in the Fundamental Formula of 183, namely,
i^=f = k'^=ijk = - 1; (A)
and if we now, in conformity with the same conception, transfer
the Standard Trinomial Form (221) from Right Quaternions
to Vectors, so as to write generally an expression of the form,
I. . , p =ix +jy + kz, or T. . . a = ia +jb + he, &c.,
where xyz and abc are scalars (namely, rectangular co-ordi-
nates), w^e can recover many of the foregoing results with ease :
and can, if we think fit, connect them with co-ordinates,
(1.) As to the laws (182), included in the Fundamental Formula A, the law
j2 __ 1^ &c., may be interpreted on the plan of 293, (1.), as representing the rever-
sal which results from two successive quadrantal rotations.
2 T
322 ELEMENTS OF QUATERNIONS. [bOOK III.
(2.) The two contrasted laws, or formulag,
ij = Jrk, ji = - k, (182, II. and III.)
may now be interpreted as expressing, that although a positive rotation through a
right angle, round the line i as an axis, brings a revolving line from the position j to
the position k, or + k, yet, on the contrary, a positive quadrantal rotation round the
line j, as a new axis, brings a new revolving line from a new initial position, i, to a
new final position, denoted by — k, or opposite* to the old final position, + k.
(3.) Finally, the law ijk = — 1 (183) may be interpreted by conceiving, that we
operate on a line a, which has at first the direction of +j, by the three lines, k,j, i,
in succession ; which gives three new but equally long lines, (3, y, d, in the direc-
tions of - i, + k, —j, and so conducts at last to a line — a, which has a direction op-
posite to the initial one.
(4.) The foregoing laws of ijk, which are all (as has been said) included (184)
in the Formula A, when combined with the recent expression I. for p, give (comp.
222, (1.) ) for the square of that vector the value :
1 1. . . p2 = (ia; + j> + Ar)« = - (.r2 + y' + 22) ;
this square of the line p is therefore equal to the negative of the square of its length
Tp (185), or to the negative of its norm Np (273), which agrees with the former
resultf 282, (1.) or (2.).
(5.) The condition of perpendicularity of the two lines p and a, when they are
represented by the two trinomials I. and I'., may be expressed (281, XVIII.) by the
fonnula,
III. . . = Sap = -(^ax + bt/+ cz) ;
which agrees with a well-known theorem of rectangular co-ordinates.
(6.) The condition of complanarity of three lines, p, p', p", represented by the
trinomial forms,
IV. . . p = ix +jy + kz, p' = ix' + &c., p" = ix" + &c.,
is (by 294, VI.) expressed by the formula (comp. 223, XIII.),
V. . . = Sp'V'p = x" (z'y - y'z) + y'\x'z - zx') + z'ijy'x - x'y) ;
agreeing again with known results.
(7.) "When the three lines p, p', p", or op, op', op", are not in one plane, the
recent expression for Sp"p'p gives, by 294, (3.), the volume of the parallelepiped
* In the Lectures, the three rectangular unit-lines, i, j, k, were supposed (in
order to fix the conceptions, and with a reference to northern latitudes) to be directed,
respectively, towards the south, the west, and the zenith ; and then the contrast of
the two formulae, ij = -\- k,ji = — k, came to be illustrated by conceiving, that we at
one time turn a moveable line, which is at first directed westward, round an axis
(or handle) directed towards the south, with a right-handed (or screwing) motion,
through a right angle, which causes the line to take an upward position, as its fnal
one ; and that at another time we operate, in a precisely similar manner, on a line
directed at first southward, with an axis directed to the west, which obliges this new
line to take finally a downward (instead of, as before, an upward) direction.
t Compare also 222, IV.
CHAP. I.] PRODUCT OF ANY NUMBER OF VECTORS. 323
(comp. 223, (9.) ) of which they are edges ; and this volume, thus expressed, is a
positive or a negative scalar, according as the rotation round p from p' to p" is itself
positive or negative : that is, according as it has the same direction as that round
+ X from +y to +z (or round i from j to k), or the direction opposite thereto.
(8.) It may be noticed here (comp. 223, (13.) ), that if a, (3, y be ang three
vectors, then (by 294, III. and V.) we have :
VI. . . SaySy = - 8y(3a = i (a/^y - yfta) ;
VII. . . Va/3y = + V7|3a = |(a/3y + y|8a).
(9.) More generally (comp. 223, (12.) ), since a vector, considered as represent-
ing a right quaternion (290), is always (by 144) the opposite of its own conjugate, so
that we have the important formula, *
VIII. . . Ka = - a, and therefore IX. . . KTIa = + Wa,
we may write for ang number of vectors, the transformations,
X. . . sna = + sn'a=Kri«±n'a),
XI. . . vna = + vn'a = |(na +n'a),
upper or lower signs being taken, according as that number is even or odd : it being
understood that
XII. . . n'a = ...yj3a, if Ua = a(3y...
(10.) The relations of rectangularity,
XIII. . . Ax. i-i- Ax.j; Ax.y -i- Ax. A ; Ax. A 4- Ax. i,
which result at once from the definitions (181), may now be written more briefly, as
follows :
XIV. . . i-i-y-, j-i-k, A-i-i;
and similarly in other cases, where the axes, or the planes, of any two right quater-
nions are at right angles to each other.
(11.) But, with the notations of the Second Book, we might also have writtten,
by 123, 181, such formulae oi complanarity as the following, Ax.^ \\\i, to express
(comp. 225) that the axis of j was a line in the plane of i ; and it might cause some
confusion, if we were now to abridge that formula tojT ||| i. In general, it seems
convenient that we should not henceforth employ the sign \\\, except as connecting
either symbols of three lines, considered still as complanar ; or else symbols of three
right quaternions, considered as being collinear (209), because their indices (or axes')
are complanar : or finally, any two complanar quaternions (123).
(12.) On the other hand, no inconvenience will result, if we now insert the sign of
parallelism, between the symbols of two right quaternions which are, in the former
sense (123), complanar : for example, we may write, on our present plan,
XY...xi\\i, yjWj, zk\\k,
if xyz be any three scalars.
* If, in like manner, we interpret, on our present plan, the symbols Ua, Ta, Na
as equivalent to Ul"ia, Tl'a, NI''a, we are reconducted (compare the Notes to
page 136) to the same significations of those symbols as before (155, 185, 273) ; and
it is evident that on the same plan we have now,
Sa = 0, Va = a.
324 ELEMENTS OF QUATERNIONS. [boOK III.
296. There are a few particular but remarkable cases^ of ternary
and oihQx products of vectors^ which it may be well to mention here,
and of which some may be worth a student's while to remember:
especially as regards the products of successive sides of closed polygons ^
inscribed in circles, or in spheres.
(1.) If A, B, c, D be any four concircular points, we know, by the sub-articles to
260, that their anharmonic function (abcd), as defined in 259, (9.), \s scalar; being
a\m positive or negative, according to a law of arrangement of those four points,
which has been already stated.
(2.) But, by that definition, and by the scalar (though negative) character of the
square of a vector (282), we have generally, for any plane or gauche quadrilateral
ABCD, the formula :
I. . . e2(ABCD) = AB.BC.CD.DA= </ie continued product of the four sides;
in which the coefficient e^ is a positive scalar, namely the product of two negative
or of two positive squares, as follows :
II. . . e3 = BC2 . DA2 = BC2. DA^ > 0.
(3.) If then abcd be deplane and inscribed quadrilateral, we have, by 260, (8.),
the formula,
III. . . ab.bc.cd.da = a positive or negative scalar,
according as this quadrilateral in a circle is a crossed or an uncrossed one.
(4.) The product a(3y of any three complanar vectors is a vector, because its
scalar part Sa(3y vanishes, by 294, (3.) and (4.); and if the factors be three suc-
cessive sides AB, BC, CD of a quadrilateral thus inscribed in a circle, their product has
either the direction of the fourth successive side, DA, or else the opposite direction,
or in symbols,
IV. . . AB.BC.CD : DA > or < 0,
according as the quadrilateral abcd is an uncrossed or a crossed one.
(5.) By conceiving the fourth point d to approach, continuously and indefinitely,
to the first point A, we find that the product of the
three successive sides of any plane triangle, abc, is /""'^ ^^\C
given by an equation of the form : / ^--'""''"^iX
V. . . AB . BC . CA = AT ; -^Lc^::^— —— -4p
at being a line (comp. Fig. 63) which touches the \ \ / /' /
circumscribed circle, or (more fully) which touches \ \ //'V,/^
the segment ABC of that circle, at the point A ; or re- \,.J\^^^> ;^X'^
presents the initial direction of motion, along the cir- ^ IJ A
cumference, from A through B to C : while the length ^^S- ^^•
of this tangential product-line, AT, is equal to, or
represents, with the usual reference to an unit of length, the product of the lengths
of the three sides, of the same inscribed triangle abc
(6.) Conversely, if this theorem respecting the product of the sides of an inscribed
triangle be supposed to have been otherwise proved, and if it be remembered, then
since it will give in like manner the equation,
CHAP. I.] PRODUCTS OF SIDES OF INSCRIBED POLYGONS. 325
A
Fig. 63, bis.
VI. . . AC.CD.DA=AU,
if D be any fourth pointy concircular with A, B, c, -while AU is, as in the annexed
Figures 63, a tangent to the new segment ACD, we can
recover easily the theorem (3.), respecting the product j.
of the sides of an inscribed quadrilateral ; and thence
can return to the corresponding theorem (260, (8.) ),
respecting the anharmonic function of any such figure gl
abcd: for we shall thus have, by V. and VI., the
equation,
VII. . . AB.BC.CD.DA= (at. Au) : (CA.Ac),
in which the divisor CA, AC or N. Ao, or Jc^ is always
positive (282, (1.) ), but the dividend at. AU is nega-
tive (281, (9.)) for the case of an ttwcrosse<i quadrilateral (Fig. 63), being on the
contrary posiiife for the other case of a crossed one (Fig. 63, bis),
(7.) If P be any point on the circle through a given point A, which touches at a
given origin o a given line OT = r, as represented in Fig. 64, we shall then have by
(5.) an equation of the form,
VIII. . . OA.AP.PO = a;.OT,
in which x is some scalar coetficient, which
varies with the position of p. Making then
OA= a, and op= p, as usual, we shall have
IX. . . a(p — a)p = ~ XT,
or
IX'. . . p-^ - a-^ = XT : a^p^,
or
IX". . . Vrp-i = Vra-i ;
and any one of these may be considered as a 'S*
form of the equation of the circle, determined by the given conditions.
(8.) Geometrically, the last formula IX." expresses, that the line p-i-a-\ or
Kp - Ra, or a'p' (see again Fig. 64), if oa' = a"' = Ra = R. OA, and op' = p-i = R. op,
is parallel to the given tangent t at o ', which agrees with Fig. 58, and with Art.
260.
(9.) If B be the point opposite to o upon the circle, then the diameter ob, or (3,
as being J- r, so that t(3-^ is a vector, is given by the formula,
X. . . rj3-i = Vra-i ; or X'. . . )3 = - r : Vra'i;
in which the tangent r admits, as it ought to do, of being multiplied by any scalar,
without the value of /3 being changed,
(10.) As another verification, the last formula gives,
XI. . . OB = T^ = Ta : TVUra"! = OA : sin act.
(11.) If a quadrilateral oabc be not inscriptihle in a circle, then, whether it be
plane or gauche^ we can always circumscribe (as in Fig. 65) two circles, cab and obc
about the two triangles, formed by drawing the diagonal OB; and then, on the plan
of (6.), we can draw two tangents or, ou, to the two segments CAB, obc, so as to repre-
sent the two ternary products.
326 ELEMENTS OF QUATERNIONS. [bOOK III.
OA.AB.BO, and ob.bc.co;
after which we shall have the quaternary product^
XII. . . OA.AB.BC.CO = OT.OU : 0B« ;
where the divisor, oB^, or bo . ob, or N . ob, is a
positive scalar, but the dividend OT.ov, and there-
fore also the quotient in the second member, or the
product in the first member, is a quaternion.
(12.) The axis of this quaternion is perpen-
dicular to the plane Tou of the two tangents ; and
therefore to the plane itself of the quadrilateral
oabc, if that be a plane figure ; but if it be gauche,
then the axis is normal to the circumscribed sphere
at the point o : being also in all cases such, that the rotation round it, from ox to
OU, is positive.
(13.) The angle of the same quaternion is the supplement of the angle tou be-
tween the two tangents above mentioned ; it is therefore equal to the angle u'ot, if
ou' touch the new segment ocb, or proceed in a new and opposite direction from o
(see again Fig. 65) ; it may therefore be said to be the angle between the two arcs,
oab and ocb, along which a point should move, in order to go from o, on the two
circumferences, to the opposite corner b of the quadrilateral OABO, through the two
other corners, A and c, respectively : or the angle between the arcs ocb, oab.
(14.) These results, respecting the axis and angle of the product of the four suc-
cessive sides, of any quadrilateral oabc, or abcd, apply without any modification to
the anharmonic quaternion (259, (9.)) of the same quadrilateral; and although,
for the case of a quadrilateral in a circle, the axis becomes indeterminate, because
the quaternary product and the anharmonic function degenerate together into sca-
lars, or because the figure may then be conceived to be inscribed, in indefinitely many
spheres, yet the angle may still be determined by the same rule as in the general
case : this angle being ■= tt, for the inscribed and uncrossed quadrilateral (Fig. 63) ;
but =0, for the inscribed and crossed one (Fig. 63, bis).
(15.) For the gauche quadrilateral oabc, which may ahvays be conceived to be
inscribed in a determined sphere, we may say, by (13.), that the angle of the qua-
ternion product, /. (oA. AB.BC.co), is equal to the angle of the lunule, bounded
(generally) by the two arcs of small circles oab, ocb ; with the same construction
for the equal angle of the anharmonic^
L (oabc), or L (oa : ab. bc : co).
(16.) It is evident that the general principle 223, (10.), of the permissibility of
cyclical permutation of quaternion factors under the sign S, must hold good for
the case when those quaternions degenerate (294) into vectors ; and it is still more
obvious, that every permutation of factors is allowed, under the sign T : whence
cyclical permutation is again allowed, under this other sign SU ; and consequently
also (comp. 196, XVI.) under the sign L.
(17.) Hence generally, for any four vectors, we have the three equations,
XIII. . . SajSy^ = SiSy^a ; XIV. . . SUa/Syo = SU/3y^a ;
XV. . . Z. a/3y^ = L (3ySa ;
CHAP. I.] PENTAGON IN A SPHERE. 327
and in particular, for the successive sides of any plane or gauche quadrilateral abcd,
we have ih.efour equal angles^
XVI. . . L (ab . bc . CD . da) = Z. (bc . CD . da . ab) =r &c. ;
with the corresponding equality of the angles of the four anharmonics,
XVII. . . L (abcd) = L (bcda) = L (cdab) = L (dabc) ;
or of those of the four reciprocal anharmonics (259, XVII.),
XVII'. . . L (adcb) = L (badc) = L (cbad) = L (dcba).
*■ (18.) Interpreting now, by (13.) and (15.), these last equations, we derive from
them the following theorem, for the plane, or for space : —
Let abcd be any four points, connected hy four circles, each
passing through three of the points : then, not only is the angle
at A, between the arcs abc, adc, equal to the angle at c, be-
tween CDA and cba, but also it is equal (comp. Fig. 66) to the
angle at B, between the two other arcs BCD and bad, and to
the angle at D, between the arcs dab, dcb.
(19.) Again, let abode be any pentagon, inscribed in a
sphere ; and conceive that the two diagonals AC, ad are drawn.
We shall then have three equations, of the forms,
XVIII. . . ab.bc.ca = at; ac.cd.da = au;
AD.DE.EA=AV;
where at, au, av are three tangents to the sphere at a, so that their product is a
fourth tangent at that point. But the equations XVIII. give
XIX. - . AB.BC . CD . DE . EA = (at . AU . Av) : (ac^ . AD^)
= AW = a new vector, which touches the sphere at A.
We have therefore this Theorem, which includes several others'under it :-^
" The product of the five successive sides, of any {generally gauche) pentagon
inscribed in a sphere, is equal to a tangential vector, drawn from the point at which
the pentagon begins and ends^
(20.) Let then p be a point on the sphere which passes through o, and through
three given points A, b, c ; we shall have the equation,
XX. .. = S(oA.AB.BC.CP.Po) = Sa(|3-a)(y-/8) (p-y)(_p)
= a2S)3yp + /32Syap + y^^a^p - p2Sa/3y.
(21.) Comparing with 294, XIV., we see that the condition for the four co-ini-
tial vectors a, (3, y, p thus terminating on one spheric surface, which passes through
their common origin o, may be thus expressed :
XXL . .if p = xa+yj3 + zy, then p^ = xa^ + y(3^ + zy^.
(22.) If then y^e project (comp. 62) the variable point p into points a', b\ c' on
the three given chords OA, OB, oc, by three planes through that point p, respectively
parallel to the planes BOC, COA, aob, we shall have the equation :
XXII. . . op2 = OA . oa' 4- OB . ob' + oc . oc\
(23.) That the equation XX. does in fact represent a spheric locus for the point
p, is evident from its mere /orm (comp. 282, (10.)); and that this sphere passes
328 ELEMENTS OF QUATERNIONS. [bOOK III.
through the four given points, O, A, B, c, may be proved by observing that the equa-
tion is satisfied, when we change p to any one of the four vectors, 0, a, j3, y.
(24.) Introducing an auxiliary vector, OD or ^, determined by tlie equation,
XXIII. . . ^Sa/3y = a«Vi3y + /32Vya+7^Va/3,
or by the system of the three scalar equations (comp. 294, (25.) ),
XXIV. . . a2 = S^a, (S^ = S^/3, y2 = S^y,
or XXIV. . . S^a-» = S^/3-i = Soy-i = 1,
the equation XX. of the sphere becomes simply,
XXV. . . p2 = s^p, or XXV'. . . S^p-i = 1 ;
so that D is the point of the sphere opposite to o, and 5 is a diameter (comp. 282,
IX'.; and 196, (6.)).
(25.) The formula XXIII., which determines this diameter, may be written, in
this other way :
XXVI. . . ^Sa/3y = Va (;S - a) (y - /3) y ;
or XXVr. . . 6.0ABC.0D = - V(OA.AB.BC.CO) ;
where the symbol oabc, considered as a coefficient, is interpreted as in 294, XLIV. ;
namely, as denoting the volume of the pyramid oabc, which is here an inscribed
one.
(26.) This result of calculation, so far as it regards the direction of the axis of
the quaternion OA. ab.bc.co, agrees with, and may be used to confirm, the theorem
(12.), respecting theproduct of the successive sides of a gauche quadrilateral, oabc ;
including the rule of rotation, which distinguishes that axis from its opposite.
(27.) The formula XXIII. for the diameter S may also be thus written :
XXVII. .. o.Sa-i/3-iy-i = V(/3-iy-i + y-'a-i+a-i/3-0
= V(/3-i-a-i)(y-i-a-i);
and the equation XX. of the sphere may be transformed to the following :
XXVIII. . . = S (|S-1 - a-l) (y-i - a-i) (p'l - a"') ;
which expresses (by 294, (34.), comp. 260, (10.) ), that the four reciprocal vec-
tors,
XXIX. . . oa' = a' = a-J, ob' = ^' = /3-i, oc' = y' = y"i, of' = p'=p~^,
are termino-complanar (64) ; the plane a'b'cV, in which they all terminate, being
parallel to the tangent plane to the sphere at o : because the perpendicular let fall
on this plane from o is
XXX. . .d' = S'i,
as appears from the three scalar equations,
XXXI. . . Sa'd = s(5'5 = sys = 1.
(28.) In general, if d be the foot of the perpendicular from o, on the plane abc,
then
XXXII. . . 5 = Sa(3y :Y((3y + ya + a(3) ;
because this expression satisfies, and may be deduced from, the three equations,
XXXIII. . . Sa^-i = S/3^l = Sy^-i = 1.
As a verification, the formula shows that the length TS, of this perpendicular, or
altitude, OD, is equal to the sextuple volume of the pyramid oabc, divided by the dou-
ble area of the triangular base ABC. (Compare 281, (4.), and 294, (3.), (33.).)
CHAP. I.] EQUATION OF HOMOSPHERICITY. 329
(29.) The equation XX., of the sphere oabc, might have been obtained by the
elimination of the vector ^, between the four scalar equations XXIV. and XXV., on
the plan of 294, (27.).
(30.) And another form of equation of the same sphere, answering to the deve-
lopment of XXVIII., may be obtained by the analogous elimination of the same vec-
tor ^, between the four other equations , XXIV. and XXV'.
(31.) The product of any even number of complanar vectors is generally a qua-
ternion with an axis perpendicular to their plane ; but the product of the successive
sides of a hexagon abcdep, or any other even-sided figure, inscribed in a circle, is
a scalar : because by drawing diagonals AC, ad, ae from the first (or last) point a
of the polygon, we find Us in (6.) that it differs only by a scalar coefficient, or divisor,
from the product of an e^>en number of tangents, at the first point.
(32.) On the other hand, the product oi any odd number of complanar vectors is
always a line, in the same plane; and in particular (comp. (19.)), the product of
the successive sides of a pentagon, or heptagon, &c., inscribed in a circle, is equal to
a tangential vector, drawn from the first point of that inscribed and odd-sided poly-
gon : because it differs only by a scalar coefficient from the product of an odd num-
ber of such tangents.
(33.) The product of any number oi lines in space is generally a quaternion
(289) ; and if they be the successive sides of a hexagon, or other even-sided polygon,
inscribed in a sphere, the axis of this quaternion (comp. (12.) ) is normal to that
sphere, at the initial (or final) point of the polygon.
(34.) But the product of the successive sides of a heptagon, or other odd-sided
polygon in a sphere, is equal (comp. (19.) ) to a vector, which touches the sphere at
the initial or final point ; because it bears a scalar ratio to the product of an odd
number of vectors, in the tangent plane at that point.
(35.) The equation XX., or its transformation XXVIII., may be called the con-
dition or equation of homo sphericity (comp. 260, (10.)) oi the five points o. A, B,
c, P ; and the analogous equation for the five points abode, with vectors afiydt
from any arbitrary origin o, may be written thus :
XXXIV.. . = S(a-/3) {(3-y) (y- 5) (5- f) (t - a);
or thus, XXXV. . . = aa* + 6/32 + cy2 + dd^ + ee^,
six times the second member of this last formula being found to be equal to the se-
cond member of the one i)receding it, if
XXXVI. .. a = BODE, 6 = CDEA, C = DEAB, rf = EABC, e = ABCD,
or more fully,
XXXVII. . . 6a = S (y - 18) (^ - /3) (€ - /3) = S {yh - Stf5 + sjSy - (Syd), &c. ;
so that, by 294, XLVIII. and XLVII., we have also (comp. 65, 70) the equation,
XXXVIII. . . = aa + bl3 + cy + d8 + ee,
with the relation between the coefficients,
XXXIX. . . = a + b + c + d + e,
which allows (as above) the origin of vectors to be arbitrary.
(36.) The equation or condition XXXV. may be obtained as the result of an
elimination (294, (27.) ), of a vector k, and of a scalar g, between ^ve scalar equa-
tions of the form 282, (10.), namely the five following,
2 u
330 ELEMENTS OF QUATERNIONS. [bOOK III.
XL. . . a2-2SKa + ^ = 0, /32- 2Sk/3 + ^ = 0, . . f2_2SK£4^=0;
K being the vector of the centre K of the sphere Abcd, of which the equation may be
written as
XLI. . . p2_2S/cp + 5' = 0,
ff being some scalar constant ; and on which, by the condition referred to, the Jifth
point E is situated.
(37.) By treating this fifth point, or its vector e, as arbitrary, we recover the
condition or equation of concircularily (3.), of the four points A, B, c, D ; or the
formula,
XLII. .. = V(a- /3)(i3-y)(y-^)(^-a).
(38.) The equation of the circle ABC, and the equation o^the sphere abcd, may
in general be written thus :
XLIII. ..0 = V(a-^)(/3-y)(y-p)(p-«);
XLIV. ..0 = S(a-/3)(/3-y)(y-^)((^-p)(p-a);
p being as usual the vector of a variable point p, on the one or the other locus.
(39.) The equations of the tangent to the circle abc, and of the tangent plane
to the sphere abcd, at the point A, are respectively,
XLV...O=V(a-^)(^-y)(y-«)(p-a),
and XLVI. . . = S(a -^8) (/3-y) (y-^) (^-a) (p- a).
(40.) Accordingly, whether we combine the two equations XLIII. and XLV.,
or XLIV. and XLVI., we find in each case the equation,
XLVIL . . (p - a)2 = 0, giving p = «, or p = a(20);
it being supposed that the three points a, b, c are not collinear, and that the four
points, a, b, c, d are not complanar.
(41.) If the centre of the sphere abcd be taken for the origin o, so that
XLVIIL . . a2=/32 = y2=^2 = _r2, or XLIX. . . Ta = T/3 = Ty = T^ = r,
the positive scalar r denoting the radius, then after some reductions we obtain the
transformation,
L...V(a-/3)(/3-y)(y-^)(5-a) = 2aS(/3-a)(y-a)(^-«).
(42.) Hence, generally, if k be, as in (36.), the centre of the sphere, we have the
equation (comp. XXV I'.),
LI. . . V(ab.bc.cd.da) = 12ka.aecd.
(43.) "We may therefore enunciate this theorem : —
" The vector part of the product of four successive sides, of a gauche quadrila-
teral inscribed in a sphere, is equal to the diameter drawn to the initial point of the
polygon, multiplied by the sextuple volume of the pyramid, which its four points de-
termine.^^
(44.) In effecting the reductions (41.), the following general formulce of trans-
formation have been employed, which may be useful on other occasions :
LIL . . aq + qa = '2{a^q + Sga) ; LII'. . . aqa = a^Kq + "la^qa ;
where a may be any vector, and q may be any quaternion.
CHAP. I.] FOURTH PROPORTIONAL TO DIPLANAR VECTORS. 31^1
Section 7. — On the Fourth Proportional to Three Diplanar
Vectors,
297. In general, when a.nj four quaternions, q, q', q"^ q"\ satisfy
the equation of quotients,
I. . . q"':q"=^q':q,
or the equivalent formula,
II. . . q'"={q':q).q" = q'q-'q",
we shall say that they form a Proportion ; and that the fourth,
namely q'", is the Fourth Proportional to iho, first, second, and third
quaternions, namely to q, q', and q", taken in this given order.
This definition will include (by 288) the one which was assigned in
226, for the fourth proportional to three complanar vectors, a, yS, 7,
namely ih^i fourth vector in the same plane, 8= I3a^<y, which has been
already considered; and it will enable us to interpret (comp. 289)
the symbol
III. . . ;3a-i7, when ^ not\\\a, {3,
as denoting not indeed a Vector, in this new case, but at least a Qua-
tej-nion, which may be called (on the present general plan) the Fourth
Proportional to these Three Diplanar Vectors, a, /3, 7. Such fourth
proportionals possess some interesting properties, especially with re-
ference to their vector parts, which it will be useful briefly to consi-
der, and to illustrate by showing their connexion with spherical
trigonometry, and generally with spherical geometry.
(1.) Let a, (3, y be (as in 208, (1.), &c.) the vectors of the corners of a triangle
ABC on the unit-sphere, whereof the sides are a, b, c ; and let us write,
(I = cos a = Sy/3-i = - S^Sy,
IV. . . I m = cos 6 = Say"^ = — Sya,
[n = cos c = S/3a~^ = - Sa/3;
where it is understood that
V. .. a2 = /32 = 72^-1, or VI. . . Ta = T/3 = Ty = l;
it being also at first supposed, for the sake of fixing the conceptions, that each of these
three cosines, /, m, n, is greater than zero, or that each side of the triangle abc is
less than a quadrant.
(2.) Then, introducing three new vectors, S, i, ^, defined by the equations,
VII. . . jc =Vy/3-ia = Va|3-iy = ny + la - m(3^
(^ =Vay- 1/3 = V]8y-' a = Zrt +m[3-ny,
332 ELEMENTS OF QUATERKIONS. [bOOK III.
we find that these three derived vectors have all one common lengthy say r, because
they have one common norm ; namely,
VIII. . . N^ = N£ = N?=:^2^.;„2^„2_2Zmn = r2;
so that IX. . . T^ = Te = T? = r = V(/2 + m^ + n^ - 2lmn').
(S.) This common length, r, is less than uniiy ; for if we write,
X. . . Sa)3y = S^a-V = e,
we shall have the relation,
XL . . e2 + r2=:N/3a-^y = l;
and the scalar e is different from zero, because the vectors a, (3, y are diplanar.
(4.) Dividing the three lines ^, £, ? by their lengthy r, we change them to their
versors (155, 156); and so obtain a new triangle, def, on the unit-sphere, of which
the corners are determined by the three new unit-vectors,
XII. . . OD = U5 = r->^ ; OE = Ue = r-h ;
(5.) The sides opposite to d, e, f, in this new or de-
rived triangle, are bisected, as in Fig. 67, by the corners
A, B, c of the old or given triangle ; because we have the d~
three equations,
XIII. . .c + ^ = 2Za; ^ + ^=2»i/3; ^+e = 2«y.
(G.) Denoting the halves of the new sides by a', b', c' (so that the arc Er = 2a',
&c.), the equations XIII. show also, by IV. and IX., that
XIV. . . cos a = r cos a', cos b — r cos b', cos c = r cos e •
the cosines of the half-sides of the new (or bisected) triangle, def, are therefore /jro -
portional to the cosines of the sides of the old (or bisecting) triangle ABC.
(7.) The equations IV. give, by 279, (1.),
XV. .. 2Z = -(^y + y/3), 2m = -(ya + ay), 2n = - (a/3 + /3a) ;
we have therefore, by VII., the three following equations between quaternions,
XVI. . . af = ^a, f3K = S(3, yd = ey;
which may also be. thus written,
XVr. . . ea = aK, K(3 = ^d, dy = yf ,
and express in a new way the relations of bisection (5.).
(8.) We have therefore the equations between vectors,
XVII. . . c = a?a-i, K = /3^/3-i, d = yty^^ ; .
or XVir. . . ^ = a£a-i, d = l30-\ £ = y^y-i.
(9.) Hence also, by V., or because a, j3, y are unit-vectors,
XVIII... c = -a^a, K = ~I3^P, ^ = -y£y;
or XVIir. . . ? = - asa, d = - /3^/3, e = - y ^y.
(10.) In general, whatever the length of the vector a mag be, the first equation
XVII. expresses that the line s is (comp. 138) thereflexion of the line ^, with respect
to that vector a ; because it may be put (comp. 279) under the form,
XIX. . . ^a-»=a-»£ = K£a-i, or XIX'. . . fa-i =K^a-'.
(11.) Another mode of arriving at the same interpretation of the equation
CHAP. I.] EXPRESSIONS FOR CONICAL ROTATION. 333
£ = rt^a-J, is to conceive ^ decomposed into two suramand vectors, ^' and ^", one pa-
rallel and the other perpendicular to a, in such a manner that
XX. ..^=r+r, riia, r^a;
for then we shall have, by 281, (10.), the transformations,
XXI. . . £ = a^'a-i + aCa-^ = I'aa-^ - V'aa-^ = ^' - Z," ;
the parallel part of Z, being thus preserved^ but thQ perpendicular part being reversed,
hy the operation a (^ )a-^
(12.) Or we may return from e = a^a"' to the form ea — a?, that is, to the first
equation XVI'. ; and then this equation between quaternions will show, as suggested
in (7.), that whatever may be the length of a, we must have,
XXII. ..T£ = T?, Ax.*£a = Ax.a^, Lta==Lal-,
so that the two lines s, ^ are equally long, and the rotation from £ to a is equal to
that from a to ^ ; these two rotations being similarly directed, and in one common
plane.
(13.) We may also write the equations XVII. XVII'. under the forms,
XXIII. . . e=a-Ka, Sec, XXIII'. . . Z=a-ha, &c.
(14.) Substituting this last expression for ^ in the second equation XVII'., we
derive this new equation,
XXIV. . .d = /3a-^f aj3-i ; or XXIV. . . t = a/3-i^/3a-i ;
that is, more briefly,
XXY. ..d = qeq-\ and XXY'. . . e = q-^dq, if XXYl. . . q = (3a-K
(15.) .A.n expression of this form, namely one with such a symbol as
XXVII. . . 9 ( ) g-i
for an operator, occurred before, in 179, (1.), and in 191, (5.) ; and was seen to in-
dicate a conical rotation of the axis of the operand quaternion (of which the symbol
is to be conceived as being written within the parentheses'), round the axis of q,
through an angle =2 Lq, without any change of the angle, or of the tensor, of that
operand; so that a vector must remain a vector, after any operation of this sort, as
bting still a right-angled quaternion (290) ; or (comp. 223, (10.) ) because
XXVIII. . . S9P5-1 = S9-I5P = Sjo = 0.
(1 6.) If then we conceive two opposite points, p' and p, to be determined on the
unit-sphere, by the conditions of being respectively ihe positive poles of the two op-
posite arcs, ab and ba, so that
XXIX. . . op' = Ax. /3a-' = Ax. g, and op = p'o = Ax. a/3-' = Ax. 9-',
we can infer from XXIV. that the line od may be derived from the tine OE, by a co-
nical rotation round the line op' as an axis, through an angle equal to the double of
the angle aob (if o be still the centre of the sphere).
(17.) And in like manner we can infer from XXIV'., that the line oe admits
* It was remarked in 291, that this characteristic Ax. can be dispensed with,
because it admits of being replaced by UV ; but there may still be a convenience in
employing it occasionally.
334
ELEMENTS OF QUATERNIONS.
[book III.
of being derived from od, by an equal but opposite conical rotation, round the line
OP as a new positive axis, through an angle equal to twice the angle boa.
(18.) To illustrate these and other connected results, the annexed Figure 68 ia
drawn ; in which p represents, as above,
the positive pole of the arc ba, and arcs are
drawn from it to D, e, f, meeting the great
circle through A and b in the points R, s, T.
(The other letters in the Figure are not, for
the moment, required, but their significa-
tions will soon be explained.)
(19.) This being understood, we see,
first, that because the arcs ef and fd are
bisected (5.) at A and b, the three arcual
perpendiculars, Es, FT, dr, let fall from E,
F, D, on the great circle through A and b,
are equally long; and that therefore the
point P is the interior pole of the small cir-
cle def', if f' be the point diametrically op-
posite ioF: so that a conical rotation round
this pole p, or round the axis op, would in fact bring the point D, or the line OD, to
the position E, or OE, which is one part of the theorem (17.).
(20.) Again, the quantity of this conical rotation, is evidently measured by the
arc RS of the great circle with p for pole ; but the bisections above mentioned give
(comp. 165) the two arcual equations,
XXX. . . r, rb= « bt, r,ix = ^ as; whcnce XXXI. . . '^ rs = 2 <-> ba,
and the other part of the same theorem (17.) is proved.
(21.) The point F may be said to be the reflexion, on the sphere, of the point D,
with respect to the point b, which Insects the interval between them ; and thus we
may say that two successive reflexions of an arbitrary point upon a sphere (as here
fromD to F, and then from f to e), with respect to two given points (b and a) of a
given great circle, are jointly equivalent to one conical rotation, round the pole (p) of
that great circle ; or to the description of an arc of a small circle, round that j9o/e, or
parallel to that great circle : and that the angular quantity (dpe) of this rotation
is double of that represented by the arc (ba) connecting the two given points ; or is
the double of the angle (bpa), which that given arc subtends, at the same pole (p)^
(22.) There is, as we see, no difficulty in geometrically proving this theorem of
rotation : but it is remarkable how simply quaternions express it : namely by the
formula,
XXXII. . . a. i8- V|3. a- i=a|3V p. j3rt-i,
in which a, j3, p may denote any three vectors ; and which, as we see by the points^
involves essentially the associative principle of multiplication.
(23.) Instead of conceiving that the point d, or the v/' ""\
line OD, has been reflected into the position f, or of, /'' /fx.
with respect to the point b, or to the line ob, with a simi- / r b/ I ^XA S >
lar successive reflexion from F to E, we may conceive that \ /
a point has moved along a small semicircle, with B for
pole, from d to f, as indicated in Fig. 69, and then along
Fig. 09.
CHAP. I.] CONSTIIUCTION OF A FOURTH PROPORTIONAL. 335
another small semicircle, with A for pole, from f to e ; and we see that the result, or
effect, of these two successive and semicirctdar motions is equivalent to a motion along
an arc de of a third small circle, which is parallel (as before) to the great circle
through B and A, and has a projection rs thereon, which (still as before) is double of
the given arc ba.
(24.) And instead of thus conceiving two successive arcual motions of a point D
upon a sphere, or two successive conical rotations of a radius OD, considered as cotn-
ponnding themselves into one resultant motion of that point , or rotation of that ra-
dius, we may conceive an analogous composition of two successive rotations of a
solid body (or rigid system^, round axes passing through a point o, which \& fixed in
space (and in the body) : and so obtain a theorem respecting such rotation, which
easily suggests itself from what precedes, and on which we may perhaps return.
(25.) But to draw some additional consequences from the equations VII., &c., and
from the recent Fig. 68, especially as regards the Construction of the Fourth Pro-
portional to three diplanar vectors, let us first remark, generally, that when we have
(as in 62) a linear equation, of the form
aa -f 6/3 -r cy 4 rf^ = 0,
connecting /oMr co-initial vectors a . . d, whereof no three are complanar, then this
fifth vector,
e=aai bl3= - cy - dS,
is evidently complanar (22) with a, (3, and also with y, d (comp. 294, (6.) ) ; it is
therefore part of the indefinite liiie of intersection of the plane aob, cod, of these
two pairs of vectors.
(26.) And if we divide this fifth vector e by the two (generally unequal) sca-
lars,
a + 6, and — c ~ d,
the two (generally unequal) vectors,
(aa + */3) : (a + 6), and {cy + rf^) : (c + d),
which are obtained as the quotients of these two divisions, are (comp. 25, 64) the
vectors of two (generally distinct) points of intersection, oilines yf'iih planes, namely
the two following :
ABOCD, and cdoab.
(27.) When the two lines, ab and cd, happen to intersect each other, the two
last-mentioned points coincide ; and thus we recover, in a new way, the condition
(63), for the complanarity of thQ four points o, A, b, c, or for the termino-compla-
narity of the four vectors a, j3, y, d ; namely the equation
ai-b + c + d=0,
which may be compared with 294, XLV. and L.
(28.) Resuming now the recent equations VII., and introducing the new vector,
XXXIII. . . X = Za-m/3-^(c-5),
which gives,
XXXIV. . . SyX = 0, and XXXV. . . T\ = V(r« -n^)=r sin c\
we see that the two arcs ba, de, prolonged, meet in a point l (comp. Fig. 68), for
which OL= UX, and which is distant by a quadrant from o : a result which may be
confirmed by elementary considerations, because (by a well-kno fy-n theorem respect-
336 ELEMENTS OF QUATERNIONS. [bOOK III.
iiig transversal arcs) the common bitector ba of the two sides, de and ef, must meet
the third side in a point i^, for which
sinDL= sin el,
(29.) To prove by quaternions this last equality of sines, and to assign their
common value, we have only to observe that by XXXIII.,
XXXVL . . Va = Vf \ = AVc^« ;
in which,
T5\ = TfX = r2 sin c', and TV^t = r' sin 2c' ;
the sines in question are therefore (by 204, XIX.),
XXXVr. . . TVUa = TVU6X = ^r-i sin 2c : r' sin c' = cos c'.
(30.) On similar principles, we may interpret the two vector-equations y
XXXVII. . . V/3\ = lY(ia, YaX = mY(3a,
in which
XXXVIII. . . TX : TV/3a = r sin c' : sin c = tan c': tan c,
an equivalent to the trigonometric equations,
tan CD cosBC cos AC
XXXIX.
tan AB sm bl sin al
(31.) Accordingly, if we let fall the perpendicular OQ on ab (see again Fig. 68),
so that Q bisects rs, and if we determine two new points m, n by the arcual equa-
tions,
XL. . . rt I.M = -^ ab = '^ QR, r> LN = r> CD,
the arcs mr, kd will be quadrants ; and because the angle at r is right by construc-
tion (18.), M is the pole of dr, and dm is a quadrant ; whence d is the pole of mn
and the angle lnm is right : conceiving then that the arcs CA and cb are drawn, we
have three triangles, right-angled at Q and n, which show, by elementary principles,
that the three trigonometric quotients in XXXIX. have in fact a common value,
namely cos cq, or cos l.
(32.) To prove this last result by quaternions, and without employing the auxi-
liary points M, N, Q, R, we have the transformations,
XLI. . . COSL=bU -— — =SU — r- = i :^jr^- b — = 1 — —
Yde yX Y(3a yX V/3a
because
XLII. . . ^ = ny-X, e = ny+\, Ydt=2ny\, UV^e = UyX,
and
XLIII. . . S^ = ?^=-S/3a-'yX-' =-S5X-» =1,
yX (yX)2
it being remembered that X -J- y, whence
VyX = yX = - Xy, (yX)2 = - y2X« = X2, SyX'l = 0.
(33.) At the same time we see that if P be (as before) the positive pole of ba,
and if k, k' be the negative and positive poles of de, while l' is the negative (as l.
is the positive) pole of cq, whereby all the letters in Fig. 68 have their Bignification*
determined, we may write,
XL! V. . . OP = TJYfSa ; ok' = yUX ; ok = - yUX ; ol' = - UX ;
while oi< = + UX, as before.
CHAP. I.] ANGLE OF A FOURTH PROPORTIONAL. 337
(34.) Writing also,
XLV. . . K = - y\, or \ = yK, and fi = (3a-^ X,
so that XLV. . . OK = U/c, and om = U/z,
we have XLVI. . . /3a-i.y = /t\-».\«;-» =/m»c-i ;
this fourth proportional, to the three equally long hut diplanar vectors, a, /3, y, ia
therefore a versor, of which the representative arc (162) is km, and the representa-
tive angle (174) is kdm, or l'dr, or edp 5 and we may write for this versor, or qua-
ternion, the expression :
XLVII. . /3a"iy = cos l'dr + od . sin l'dr.
(35.) The double of this representative angle is the sum of the two base-angles of
the isosceles triangle dpe ; and because the two other triangles, epf', f'pd, are also
isosceles (19.), the lune ff' shows that this sum is what remains, when we subtract
the vertical angle F, of the triangle def, from the sum of the supplements of the two
base-angles d and e of that triangle ; or when we subtract the sum of the three an-
gles of the same triangle /row four right angles. We have therefore this very simple
expression for the Angle of the Fourth Proportional :
XL VIII. . . L /3a-iy = l'dr = 7r - |(d + e + f).
(36.) Or, if we introduce the area, or the spherical excess, say 2, of the triangle
def, writing thus
XLIX. . . 2 = d + e+f- TT,
we have these other expressions :
L. . . Z./3a-^y = i7r-|S; LI. . . /3a-»y = sin|2 1- r'^o cos i2 ;
because
OD = U^ = r-io, by XIL
(37.) Having thus expressed (3a-^y, we require no new appeal to the Figure, in
order to express this other fourth proportional, ya' 1/3, which is the negative of its
conjugate, or has an opposite scalar, but an eqiial vector part (comp. 2U4, (1.), and
295, (9.) ) : the geometrical diflference being merely this, that because the rotation
round a from /3 to y has been supposed to be negative, the rotation round a from y
to j3 must be, on the contrary, positive.
(38.) We may thus write, at once,
LIL . . ya-i/3 = - K/3a-i y = - sin |2 + ri^ cos |2 ;
and we have, for the angle of this new fourth proportional, to the same three vectors
a, (3, y, of which the second and third have merely changed places with each other,
the formula :
LIII, . . Z.ya-ij3 = RDL = :i(D + E + F) = i7r + i2.
(39.) But the common vector part of these <t<JO fourth proportionals is d, by VII ;
we have therefore, by XI.,
LIV. . . r = cos|2; c = ±sini2;
the upper sign being taken, when the rotation round a from ^ to y is negative, as
above supposed.
(40.) It follows by (6.) that when the sides 2a', 1b\ 2c', of a spherical triangle
2 X
338 ELEMENTS OF QUATERNIONS. [bOOK IH.
DEF, of which the area is 2, are bisected by the corners A, E, c of another spherical
triangle, of which the sides* are a, b, c, then.
LV. . . cos a : cos a' = cosb : cos b' = cos c ; cos c' = cos i^S.
(41.) It follows also, from what has been recently shown, that the angle rdk, or
MDN, or the arc mx in Fig. 68, represents the semi-area of the bisected triangle def;
■whence, by the right-angled triangle lmn, we can infer that the sine of this semi-area
is equal to the sine of a side of the bisecting triangle abc, multiplied into the sine of
the perpendicular, let fall upon that side from the opposite corner of the latter trian-
gle ; because we have
LVI. . . sin IS = sin mn = sin lm . sin l = sin ab . sin CQ.
(42.) Tlie same conclusion can be drawn immediately, by quaternions, from the
expression,
LVII. . . sin IS = e = Sa/3y = S(V/3a. y->) = TV/3a. SU(V/3a : y);
in which one factor is the sine of ab, and tlie other factor is the cosine of op, or the
sine of cq.
(43.) Under the same conditions, since
LVIII. . . a = U(£ + = F*(c + 0, &c.,
■we may write also,
LIX. . .8iniS=SU(« + ^)(?+^) (^ + = S^6? : 4//jm ;
in which, by IV. and XIII.,
LX. . . 4Zm« =- 8(5 + (e + = »•= -S(t? +KS + St).
(44.) Hence also, by LIV,,
LXI. . . cos is = r = (r3 - rS (e^ + ?5 + St) ) : Umn ;
TYTT t.niT=i= S^^^ ^ SU5.^
^'^ ^ r r3_rS^£$ + ^5+50 1 - SUf^- SU^5 - Smc '
and under this last form, we have & general expression for the tangent of half the
spherical opening at o, of any triangular pyramid odef, whatever the lengths Td,
Tf, T^ of the edges at o may be.
(45.) As a verification, we have
LXIII. . . (4/mn)3 = -i.(f + ^2 (^4 ^)2 (a + e)»
= 2 (r2 - SfO (^2 - SS5) (r2 - Sdt) ;
but the elimination of ^S between LIX. LXI. gives,
LXIV. . . (Almny = (SdeKy + (rS - r(StK + S^5 + Sds) )2 ;
•we ought then to find that
LXV. . . {SSeK)^ = r^-r^(SeK)^ + {BZSy+iSSey'}-2StKSKSSSe,
if 5* = «2 = ^3 = — r2 ; and in fact this equality results immediately from the general
formula 294, LIU.
(46.) Under the same condition, respecting the equal lengths of S, f, ^, we have
also the formula,
* These sides abc, of the bisecting triangle ABC, have been hitherto supposed for
simplicity (1.) to be each less than a quadrant, but it will be found that the for-
mula LV. holds good, without any such restriction.
CHAP. I.] CONNEXION WITH SPHERICAL AREA. 339
LXVI. . . - V(^ + £) (£ + (^ + ^) = 25 (r2 - SeK - S^d - SSe) = SlmnS ;
whence other verifications may be derived.
(47.) If (7 denote the area* of the bisecting triangle ABC, the general principle
LXII. enables us to infer that
LXVII. . . tan ^ = ^-^^ = !
2 1 - S/3y - Sya - Sa/3 l-^Z+m + w
sin c sin p ,
1 + cos a -t- cos 6 + cos c
if p denote the perpendicular cq from c on ab, so that
e = sin c sin/> = sin b sine sin a = &c. (comp. 210, (21.) ).
(48.) But, by (IX.) and (XL),
LXVIII. . . e2 + (H-/ + m + «)2=2(l + (1 + m) (l+n)
I . a b c
= 1 4 cos - cos - cos -
\ 2 2 2
hence the cosine and sine of the 7iew semi-area are,
<7 1 + cos a + cos b 4 cos c
LXIX.
2 a b c
4cos - cos - cos -
2 2 2
a b
siu - sin - sin c
Tvv • '^ 2 2 ,
LXX. . . sm - = ————— = &c.
2 c
cos -
2
(49,) Returning to the bisected triangle^ def, the last formula gives,
^^^^T^ . 1^ sin a' sin i' sin F . , .
LXXI. . . sm ^2 = '. = sui » sm c sec c ,
^ cose '
if />' denote the perpendicular from F on the bisecting arc ab, or ft in Fig. 68;
but cos ^2 = cos c sec c, by LV. ; hence
LXXII. . . tan 1 2 = sinp' tan c = sin ft . tan ab.
Accordingly, in Fig. 68, we have, by spherical trigonometry,
sin FT = sin es = sin le sin l = cos ln sin mn cosec lm = tan mn cot ab.
(50.) The arc MX, which thus represents in quantity the semiarea of def, has its
pole at the point d, and may be considered as the representative arc (162) of a certain
new quaternion^ Q, or of its versor, of which the axis is the radius OD, or U^ ; and
this new quaternion may be thus expressed :
LXXIII. .. Q = dya(3 = -S^+ dSaiSy = r^-^ ed;
its tensor and versor being, respectively,
LXXIV. . . TQ = r = cos|2; LXXV. . . UQ = cos^2 +0D.sin^2.
(51.) An important transformation of this last versor maybe obtained as fol-
lows :
* The reader will observe that the more usual symbol 2, for this area of abc,
in here employed (36.) to denote the area of the exscribed triangle def.
340 ELEMENTS OF QUATERNIONS. [bOOK ill.
LXXVI. . . UQ = U(V.ar».?j3-0=(^OK«^0K^^-'>i
so that
LXXVII. . . iS = A Q= A dya[3=L (^£->> (f^O' (^^0* ?
these powers of quaternions, with exponents each = |, being interpreted as square
roots (199, (1.) ), or as equivalent to the symbols V(^£-i), &c.
(52.) The conjugate (or reciprocal) versor, UQ"i, which has nm for its repre-
tentative arc, may be deduced from UQ by simply interchanging /3 and y, or c and
^ ; the corresponding quaternion is,
LXXVIII. . . Of = KQ=S(3ay = r« - e^ ;
and we have
LXXIX. . . UQ' = cos IS - OD . sin 12 = (5^i> (^f-i)' (f ^')* ;
the rotation round d, from e to f, being still supposed to be negative.
(53.) Let H be any other point upon the sphere, and let oh = rj; also let 2' be
the area of the new spherical triangle, dfh ; then the same reasoning shows that
LXXX. . . cos |S' + OD.sin p'= (^^-i)' (.W^y^ (»?5'0s
if the rotation round d from f to h be negative ; and therefore, by multiplication of
the two co-axal versors, LXXVI. and LXXX., we have by LXXV. the analogous
formula :
LXXXL . . cos 1(2 + 2') + oD.sin |(2 + 20 = (^£"0' (f^O^' iKrj-^y {no'')';
where 2 + 2' denotes the area of the spherical quadrilateral, defh.
(54.) It is easy to extend this result to the area of ang spherical polygon, or to
the spherical opening (44.) oi any pyramid; and we may even conceive an exten-
sion of it, as a limit, to the area of any closed curve upon the sphere, considered as
decomposed into an indefinite number of indefinitely small triangles, with some cofn-
mon vertex, such as the point d, on the spheric surftice, and with indefinitely small
arcs EP, FH, . . of the curve, for their respective bases : or to the spherical opening
of any cone, expressed thus as the Angle of a Quaternion, which is the limit* of the
product of indefinitely many factors, each equal to the square-root of a quaternion,
lohich differs indefinitely little from unity.
(55.) To assist the recollection of this result, it may be stated as follows (comp.
180, (3.) for the definition of an arcual sum) : —
" The Arcual Sum of the Halves of the successive Sides, of any Spherical Poly-
gon, is equal to an arc of a Great Circle, which has the Initial {or Final) Point of
* This Limit is closely analogous to a definite integral, of the ordinary kind ; or
rather, we may say that it is a Definite Integral, but one of a new kind, which could
not easily have been introduced without Quaternions. In fact, if we did not employ
the non-commutative property (168) of quaternion multiplication, the Products here
considered would evidently become each equal to imity : so that they would fur-
nish no expressions for spherical or other areas, and in short, it would be useless to
speak of them. On the contrary, when that property or principle of multiplication
is introduced, these expressions of product-form are found, as above, to have ex-
tremely useful significations in spherical geometry ; and it will be seen that they sug-
gest and embody a remarkable <Aeorem, respecting ihQ resultant of rotations of a sys-
tem, round any number of successive axes, all passing through one fixed point, but in
other respects succeeding each other with any gradual or sudden changes.
CHAP. I.] AREA OF POLYGON OR CURVE ON SPHERE. 341
the Polygon for its Pole^ and represents the Semi-area of the Figure;'' it being un-
derstood tliat this resultant arc is reversed in direction, when the half-sides are (ar-
cually) added in an opposite order.
(56.) As regards the order thus referred to, it may be observed that in the arcual
addition, which corresponds to the quaternion multiplication in LXXVI., we con-
ceive a point to move, first, from b to F, through half i\iQ arc df ; which half-side
of the triangle def answers to the right-hand factor, or square-root, (^5~0^- ^^
tlien conceive the same point to move next from f to A, through half the arc fe,
which answers to the factor placed immediately to the left of the former ; having
thus moved, on the whole, so far, through the resultant arc ba (as a transvec-
tor, 180, (3.))j or through any equal arc (163), such as ml in Fig. 68. And
finally, we conceive a motion through half the arc ed, or through any arc equal to
that half, such as the arc ln in the same Figure, to correspond to the extreme left-
hand factor in the formula ; the final resultant (or total transvector arc), which
answers to ihQ product of the three square roots, as arranged in the formula, being
thus represented by i\\Q final arc mn, which has the point d for its positive pole, and
the half-area, ^S, for the angle (51.) of the quaternion (or versor) product which
it represents.
(57.) Now the direction o^ positive rotation on the sphere has been supposed to
be that round d, from f to e; and therefore along the perimeter, in the order dfe,
as seen* from any point of the surface within the triangle : that is, in the order in
which the successive sides df, fe, ed have been taken, before adding (or compound-
ing) their halves. And accordingly, in the conjugate (or reciprocal) formula
LXXIX., we took the opposite order, def, in proceeding as usual from right-hand
to left-hand factors, whereof the former are supposed to be multiplied hgf the latter;
while the result was, as we saw in (52.), a new versor^ in the expression for which,
the area S of the triangle was simply changed to its own negative.
(58.) To give an example of the reduction of the area to zero, we have only to
conceive that the three points D, e, f are co-arcwaZ (165), or situated on one great
circle ; or that the three lines d, e, K are complanar. For this case, by the laws+
of complanar quaternions, we have the formula,
LXXXII. . . (^ri)i {sK-^)i (?^')* = h if S^£?= ;
thus cos iS = l, and 2 = 0.
* In this and other cases of the sort, the spectator is imagined to stand on the
point of the sphere, round which the rotation on the surface is conceived to be per-
formed ; his body being outside the sphere. And similarly when we say, for exam-
ple, that the rotation round the line, or radius, OA, from the line OB to the line oc,
is negative (or left-handed), as in the recent Figures, we mean that such would ap-
pear to be the direction of that rotation, to a person standing thus with h\s feet on
A, and with his body in the direction of OA prolonged : or else standing on the centre
(or origin) o, with his head at the point A. Compare 174, II. ; 177; and the Note
to page 153.
t Compare the Notes to pages 146, 159.
X Compare the Second Chapter of the Second Book.
342 ELEMENTS OF QUATERNIONS. [bOOK HI.
(59.) Again, in (_53.) let the point H be co-arcual with d and f, or let Sd^rj = ;
then, because
LXXXir. . . {KT^)i (»j^-i> = (^^i>, if S^^7/ = 0,
the product of four factors LXXXI. reduces itself to the product of three factors
'LXXVI. ; the geometrical reason being evidently that in this case the added area
2' vanishes ; so that the quadrilateral defh has only the same area as the triangle
DEF.
(60.) But this added area (53.) may even have a negative* effect^ as for exam-
ple when the new point H falls on the old side de. Accordingly, if we write
LXXXIII. . . Qi=:(t^J)^ {W)' (»?£-')*.
and denote the product LXXXI. of four square-roots by Qi, we shall have the trans-
formation,
LXXXIV. . . Q2 = (^£-' )i Q) (£5-» )i, if ^^tr, = ;
which shows (comp. (15.) ) that in this case the angle of the quaternary/ product Qz
is that of the ternary product Qi, or the half-area of the triangle efh (= def — dhf),
although the axis of Qz is transferred from the position of the axis of Qi, by a ro-
tation round the pole of the arc ed, which brings it from oe to od.
(Gl.) From this example, it may be considered to be sufficiently evident, how the
formula LXXXL may be applied and extended, so as to represent (comp. (54.) ) the
area of any closed figure on the sphere, with any assumed point D on the surface as
a sort of spherical origin ; even when this auxiliary point is not situated on the pe-
rimeter, but is either external or internal thereto.
(62 ) A new quaternion Qo, with the same axis od as the quaternion Q of (50,),
but with a double angle, and with a tensor equal to unity, may be formed by simply
squaring the versor UQ ; and although this squaring cannot be effected by removing
the fractional exponents,^ in the formula LXXVI., yet it can easily be accomplished
in other ways. For example we have, by LXXIII. LXXIV., and by VII. IX. X.,
the transformations :|
LXXXV. . . Qo = UQ2 = r-2(5yo/3)2 = - ^^ ya/3^.^yo/3
= - (y«/5)2 = - (e - (5)2 = r2 - e« + 2ed ;
and in fact, because S — r. od, by XII., the trigonometric values LIV. for r and e
enable us to write this last result under the form,
LXXXVI. . . Qo = - (7a/3)2 = cos S + od . sin 2.
(63.) To show its geometrical signification, let us conceive that abc and lmn
* In some investigations respecting areas on a sphere, it may be convenient to
distinguish (comp. 28, 63) between the two symbols def and dfe, and to consider
them as denoting two opposite triangles, of which the sum is zero. But for the pre-
sent, we are content to express this distinction, by means of the two conjugate qua-
ternion products, (51.) and (52.).
t Compare the Note to (54.).
X The equation 5ya/3 = ya/?^ is no< valid generally ; butwehave/jere d=~y-/aj3;
and in general, qp ■= pq, if p || Yq.
CHAP. I.] CASE OF SIDES GREATER THAN QUADRANTS. 343
have the same meanings in tlie new Fig. 70, as in Fig. 68 ; and that AiBiMi are
three new points, determined by the three arcual equations (163),
LXXXVII. OAC = '>CAi, <^BC='^CBi,
r> MN = n NMi ;
which easily conduct to this fourth equation of
the same kind,
LXXXVir. . . n LMi = " BiAi.
This new arc LMi represents thus (comp. 167, and
Fig. 43) the product aiy-*.y/3rJ = ya-i./3y-i ;
while the old arc ml, or its equal ba (31.), represents afl-^ ; whence the arc mmi,
which has its pole at d, and is numerically equal to the whole area S of def (be-
cause MN was seen to be equal (50.) to half that area), represents the product
ya-i]3y-i. a(3-\ or - (ya/3)2, or Qq. The formula LXXXVI. has therefore been
interpreted^ and may be said to have been proved anew, by these simple geometri-
cal considerations.
(64.) We see, at the same time, how to interpret the symbol^
LXXXVIII. . . Qo=--^;
a y /3
namely as denoting a versor, of which the axis is directed to, or from, the corner d
of a certain auxiliary spherical triangle def, whereof the sides, respectively o/>/)osj7e
to D, E, F, are bisected (5.) by the given points A, b, o, according as the rotation round
a from /3 to y is negative or positive; and of which the angle represents, or is numeri-
cally equal to, the area S of that auxiliary triangle : at least if we still suppose, as
we have hitherto for simplicity done (1.), that the sides of the^'it'ew triangle abc are
each less than a quadrant.
298. The case when the sides of the given triangle are all greater,
instead of being all less, than quadrants, may deserve next to be
(although more briefly) considered; the case when they are all
equal to quadrants, being reserved for a short subsequent Article:
and other cases being easily referred to these, by limits, or by passing
from a given line to its opposite,
(1.) Supposing now that
I. . , / < 0, m<0, n < 0,
or that II. ..a>-, o>—, c>— ,
we may still retain the recent equations lY. to XI. ; XIII. ; and XV. to XXVI., of
297 ; but we must change the sign of the radical, r, in the equations XII. and XIV.,
and also the signs of the versors JJd, Ue, U^ in XII., if we desire that the sides of
the auxiliary triangle, def, may still be bisected (as in Figures 67, 68) by the cor-
ners of the given triangle ABC, of which the sides a, 6, c are now each greater than
a quadrant. Thus, r being still the common tensor of d, i, ^, and therefore being still
supposed to be itself >0, we must write now, under these new conditions I. or II.,
the new equations.
344 ELEMENTS OF QUATERNIONS. [bOOK III.
III. . . OD = -m = -r-i5; OE=-U£=-r-'6; OF = -U^ = -r-»^;
I V. . . cos a = — r cos a', cos b =—r cos 6', cos c = — r cos c'.
(2.) The equations IV. and VIII. of 297 still holding good, we may now write,
V. . . + 2r cos a cos b' cos c = cos a'2 + cos 6'2 + cos c'^—l,
according as we adopt positive values (297), or negative values (298), for the co-
sines I, m, n of the sides of the bisecting triangle ; the value of r being still supposed
to be positive.
(3.) It is not difficult to prove (comp. 297, LIV., LXIX.), that
VI. . . r=4:C0S |S, according as ^>0, &c., or l<0,&c.;
the recent formula V. may therefore be written unambiguously as follows :
VII. . . 2 cos a cos b' cos c' cos ^2 = cos a'2 -|- cos 6'2 + cos c'^ — 1 ;
and the formula 297, LV. continues to hold good.
(4.) In like manner, we may write, without an ambiguous sign (comp. 297, LI.),
the following expression for the fourth proportional /3a"iy to three unit-vectors a, /3,
y, the rotation round the first from the second to the third being negative :
VIII. . . j3a-Jy = sin AS + 0D. cos IS;
where the scalar part changes sign, when the rotation is reversed.
(6.) It is, however, to be observed, that although this ^rmwZa VIII. holds good,
not only in the cases of the last article and of the present, but also in that which has
been reserved for the next, namely when Z= 0, &c. ; yet because, in the present case
(298) we have the area S> tt, the radius on is no longer the (positive) axis XJd of
the fourth proportional jSa-^y ; nor is Att — iS any longer, as in 297, L., the (posi-
tive) angle of that versor. On the contrary we have noWy for this axis and angle,
the expressions :
IX. . . Ax. /3rt-Jy = DO=-OD; X. . . Z./3a-iy = i(2-7r).
(6.) To illustrate these results by a construction, we may remark that if, in Fig.
67, the bisecting arcs bc, ca, ab be supposed each greater than a quadrant, and if
we proceed to form from it a new Figure, analogous to 68, the perpendicular CQ will
also exceed a quadrant, and the poles p and k will fall between the points c and Q ;
also M and k will fall on the arcs lq and ql' prolonged: and although the arc km,
or the angle kdm, or l'dr, or edp, may still be considered, as in 297, (34.), to re-
present the versor /3a"' y, yet the corresponding rotation round the point d is now o'
a negative character.
(7.) And as regards the quantity of this rotation, or the magnitude of the angle
at D, it is again, as in Fig. 68, a base-angle of one p
of three isosceles triangles, with p for their common , /-'^l^^v^ ; /
vertex ; but we have now, as in Fig. 71, a new ar- \>^, y^ \ ^s^/'
range7nent, in virtue of which this angle is to be B^>^~ C \ ~^^
found by halving what remains, when the sum of """^^^TTrrrr^^^^'
the supplements of the angles at d and e, in tlietri- Yig. 71.
angle def, is subtracted /ro?» the angle at f, instead
of our subtracting (as in 297, (35.) ) the latter angle from the former sum ; it i^i
therefore now, in agreement with the recent expression X.,
XL . . Z. /3a-«y = ^(d f e 1 f) - tt.
CHAP. I.] MODIFICATIONS OF THE CONSTRUCTION. 345
(8.) The negative of the conjugate of the formula VIII. gives,
XII. . . ya-^j3 = - sin IS + OD . cos iS ;
and by taking the negative of the square of this equation, we are conducted to the
following :
XIII. . . ^ 5 ^ = _ (y a-i/3)3 = cos S + OD . sin S ;
ay 13
a result which had only been proved before (comp. 297, (62.), (64.)) for the case
2 < TT ; and in which it is still supposed that the rotation round a from /3 to y is
negative.
(9.) With the same direction of rotation, we have also the conjugate or recipro-
cal formula,
XIV. . . ^^- = -(/3a-»y)2 = cos2-OD.sin2.
a(3y
(10.) If it happened that only one side, as ab, of the given triangle abc, was
greater, while each of the two others was less than a quadrant, or that we had Z > 0,
tn > 0, but n < ; and if we wished to represent the fourth proportional to a, /?, y by
means of the foregoing constructions ; we should only have to introduce the point c'
opposite to c, or to change y to y' = — y ; for thus the new triangle abc' Avould have
each side greater than a quadrant, and so would fall under the case of the present
Article; after employing the construction for which, we should only have to change
the resulting versor to its negative.
(11.) And in like manner, if we had I and m negative, but n positive, we might
again substitute for c its opposite point c', and so fall back on the construction of
Art. 297: and similarly in other cases.
(12.) In general, if we begin with the equations 297, XII., attributing any arbi-
trary (but positive) value to the common tensor, r, of the three co-initial vectors
^, f, ^, of which the versorsy or the unit-vectors Vd, &c., terminate at the corners of
a given or assumed triangle def, with sides = 2a', 26', 2c', we may then suppose
(comp. Fig. 67) that another triangle abc, with sides denoted by a, 6, c, and with
their cosines denoted by /, m, n, is derived from this one, by the condition of bisect-
ing its sides ; and therefore by the equations (comp. 297, LVIII.),
XV. ..OA=a = U(€ + 0, OB = ^=U(^+5), oc = y = U(5 + e),
with the relations 297, IV. V. VI., as before; or by these other equations (comp.
297, XIII. XIV.),
XVI. . . 6 + ^ = 2mco3a', <^ + S=2rl3 cos b', d+€=2ry cose'.
(13.) When this simple construction is adopted, we have at once (comp. 297,
LX.), by merely taking scalars of products of vectors, and without any reference to
areas (compare however 297, LXIX., and 298, VII.), the equations,
XVII. . . 4 cos a cos 6' cos c' = 4 cos b cos c cos a' = 4 cos c cos a' cos b'
= - r-2S (? + 6) (5 + f) = &c. = 1 + cos 2a' + cos 26' + cos 2c' ;
or
cos a _ cos6 _ cose _ cos a'^ + cos b'^ + cos c'^ ~ 1
cos a' cos b' cos c 2 cos a' cos b' cos c' '
which can indeed be otherwise deduced, by the known formulae of spherical trigo-
nometry.
2 Y
346 ELEMENTS OF QUATERNIONS. [BOOK III.
(14.) We see, then, that according as the sum of the squares of the cosines of
the half-sides, of a given or assumed spherical triangle, def, is greater than unity,
or equal to unity, or less than unity, the sides of the inscribed and bisecting triangle^
ABC, are together less than quadrants, or together equal to quadrants, or together
greater than quadrants.
(15.) Conversely, t/the sides of a given spherical triangle abc be thus all less,
or all greater than quadrants, a triangle def, but only one* such triangle, can be
exscrihed to it, so as to have its sides bisected, as above : the simplest process being
to let fall a perpendicular, such as CQ in Fig. 68, from c on ab, &c. ; and then to draw
new arcs, through c, &c., perpendicular to these perpendiculars, and therefore coin-
ciding in position with the sought sides de, &c., of def.
(16.) The trigonometrical results of recent sub-articles, especially as regards the
area\ of a spherical triangle, are probably all well known, as certainly some of them
are ; but they are here brought forward only in connexion with quaternion formulcB ;
and as one of that class, which is not irrelevant to the present subject, and includes
the formula 294, LIIL, the following may be mentioned, wherein a, (3, y denote any
three vectors, but the order of the factors is important :
XIX . . (a/3y)2 = 2a2^2y9 + a2 (/3y)^ + /32 (ay)2 + y2 (a/3 ^a _ Any Sa/3 S/3y .
(17.) And if, as in 297, (1.), &c., we suppose that a, (3, y are three unit-vec-
tors, OA, OB, oc, and denote, as in 297, (47.), by a the area of the triangle abc,
the principle expressed by the recent formula XIII. may be stated under this appa-
rently different, but essentially equivalent form :
^v n + /3y-I-aj3+y
XX. . . . . - — - = cos 0- + a sin (T ;
/8 + 7 a + /3 y + a
which admits of several verifications.
(18.) We may, for instance, transform it as follows (comp. 297, LXVII.) :
XXI -(« + ig)(<3+y)(y + «) ^ -2e+2a(l + ^+m + n)
' • • K(a-l-/3)(i3f y) (y + a) + 2^+ 2a(l + ^+ w + n)
. , - , , 1 + a tan - cos - + a sm -
_l4-/ + w + n-fca_ 2 2 2
l + / + m + » — ea _ a a , a
\ ~ a tan ~ cos - — a sm -
2 2 2
-[
i- + a sm - = cos (T + a sm <T, as above.
* In the next Article, we shall consider a case of indeterminateness, or of the ex-
istence of indefinitely many exscribed triangles def : namely, when the sides of abc
are all equal to quadrants.
t This opportunity may be taken of referring to an interesting Note, to pages
96, 97 of Luby's Trigonometry (Dublin, 1852); in which an elegant construction,
connected with the area of a spherical triangle, is acknowledged as having been men-
tioned to Dr. Luby, by a since deceased and lamented friend, the Rev. William Digby
Sadleir, F.T.C.D. A construction nearly the same, described in the sub-articles to
297, was suggested to tlie present writer by quaternions, several years ago.
CHAP. I.] CASE OF SIDES EQUAL TO QUADRANTS. 347
(19.) This seems to be a natural place for observing (comp. (16.) ), that if a, j8,
y, d be any four vectors^ the lately cited equation 294, LIII., and the square of the
equation 294, XV., with S written in it instead of p, conduct easily to the following
very general and symmetric formula :
XXII. . . a2/32y252 + (S,8ySa^)2+ (SyaS/3^)H (Sa(3Sydy
+ 2a^SPyS(35Sy5 + 2/3^SyaSy^Sa5 + 2y^Sa[5SaSS(5d + 2S^Sa(5S^ySya
= 2SyaSal3Sl3SSyd + 28a(3SI3ySydSa8 + 2S[3ySyaSadS(3d
+ ^2y2(Sa^)3 + y2a2 (8/3^)2+ a2/32(Sy^)2
+ a2^2(S/3y)2 + /32^2(Sya)2 + y2^2(Sa/3)2.
(20.) If then we take anr/ spherical quadrilateral abcd, and write
XXIII. . . r = cos AD = — SVad, m' = cos bd = - SU/3^, «' = cos cd = &c.,
treating a, (3, y as the unit-vectors of the points A, b, c, and /, m, n as the cosines
of the arcs bc, ca, ab, as in 297, (1.), we have the equation,
XXIV. . . 1 + M'2 + m^ra"- + n^n'a + 2Zm'»'+ 2mnl' + 2nl'm\ 2lmn
= 2mnm'n' + 2nln'l' + 2lml'm
+ Z34 m2+ n2 + Z'2 + m'2 + n'2 ;
which can be confirmed by elementary considerations,* but is here given merely as
an interpretation of the quaternion formula XXII.
(21.) In squaring the lately cited equation 294, XV., we have used the two
following formulae of transformation (comp. 204, XXIL, and 210, XVIII.), in
which a, /3, y may be any thr^e vectors^ and which are often found to be useful :
XXV. . . (Va/3)2 = (Sa/3)2 - a2/32 ; XXVI. .. S (V/3y . Vy a) = y2Sa/3 - S/3ySya.
299- The two cases, for which the three sides «, b^ c, of the given
triangle abc, are all less, or all greater, than quadrants, having been
considered in the two foregoing Articles, with a reduction, in 298,
(10.) and (11.), of certain other cases to these, it only remains to
consider that third principal case, for which the sides of that given
triangle are all equal to quadrants : or to inquire what is, on our
general principles, the Fourth Proportional to Three Rectangular
Vectors. And we shall find, not only that tJiis fourth proportional
is not itself a Vector, but that it does not even contain any vector
part (292) different from zero : although, as being found to be equal
to a Scalar, it is still included (131, 276) in the general conception
of a Quaternion.
(1.) In fact, if we suppose, in 297, (1.), that
I. . . Z = 0, TO = 0, n = 0, or that II. . . a = 5 = «
* A formula equivalent to this last equation of seventeen terms, connecting the
six cosines of the arcs which join, two by two, the corners of a spherical quadrilateral
abcd, is given at page 407 of Carnot's Geometric de Position (Paris, 1803).
348 ELEMENTS OF QUATERNIONS. [bOOK III.
or III. . . S/37 = Sya = Sa/3 = 0, while IV. . . Ta = T/3=:Ty = 1,
the formulae 297, VII. give,
V. ..a = 0, £ = 0, ^=0;
but these are the vector parts of the three pairs of fourth proportionals to the three
rectangular unit-lines, a, (3, y, taken in all possible orders ; and the same evane-
scence of vector parts must evidently take place, if the three given lines be only at
right angles to each other, without being equally long.
(2.) Continuing, however, for simplicity, to suppose that they are unit lines, and
that the rotation round a from /3 to y is negative, as before, we see that we have now
r=0, and e=l, in 297, (3.); and that thus the six fourth proportionals reduce
themselves to their scalar parts, namely (here) to positive or negative unit?/. In
this manner we find, under the supposed conditions, the values :
VI. . . j3a-'y = y/3-Ja = ay-i/3 = +l; VI'. . . ya-^(3 =al3-^y = l3y-^a = -l.
(3.) For example (comp. 295) we have, by the laws (182) of i, j, A, the values,
VII. . . ij-^k =jk-H = ki-^j = + 1 ; VII'. . . Jcj-H = ik-^j =ji-^k = - 1.
In fact, the two fourth proportionals, ij'^k and kj-^i, are respectively equal to the
two ternary products, — ijk and - kji, and therefore to + 1 and - 1, by the laws in-
cluded in the Fundamental Formula A (183).
(4.) To connect this important result with the constructions of the two last Ar-
ticles, we may observe that when we seek, on the general plan of 298, (15.), to
exscrihe a spherical triangle, def, to a given tri-quadrantal (or tri-rectangular)
triangle, ABC, as for instance to the triangle ijk (or jik) of 181, in such a manner
that the sides of the new triangle shall be bisected by the corners of the old, the
problem is found to admit of indefinitely many solutions. Any point p may be as-
sumed, in the interior of the given triangle abc ; and then, if its reflexions D, E, f
be taken, with respect to the three sides a, b, c, so that (comp. Fig. 72) the arcs
PD, PE, PF are perpendicularly bisected by those
three sides, the three other arcs ef, fd, de will be
bisected by the points A, b, c, as required : because
the arcs ae, ap have each the same length as ap,
and the angles subtended at a by pe and pf are to-
gether equal to two right angles, &c.
(5.) The positions of the auxiliary points, d, e,
F, are therefore, in the present case, indeterminate,
or variable ; but the sum of the angles at those three
points is constant, and equal to four right angles ;
because, by the six isosceles triangles on pd, pe, pf as bases, that sum of the
three angles d, e, f is equal to the sum of the angles subtended by the sides of the
given triangle abc, at the assumed interior point p. The spherical excess of the
triangle def is therefore equal to two right angles, and its area 2 = tt ; as may be
otherwise seen from the same Figure 72, and might have been inferred from the for-
mula 297, LV., or LVI.
(6.) The radius od, in the formula 297, XLVII., for the fourth proportional
/3a-'y, becomes therefore, in the present case, indeterminate ; but because the angle
i/dp., or ^ (tt - S), in the same equation, vanishes, the formula becomes simply
CHAP. I.] OTHER VIEW OF A FOURTH PROPORTIONAL. 349
/3a-iy = 1, as in the recent equations VI. ; and similarly in other examples, of the
class here considered.
(7.) The conclusion, that the Fourth Proportional to Three Rectangular Lines
is a Scalar, may in several other ways be deduced, from the principles of the present
Book. For example, with the recent suppositions, we may write,
Vlir. ..^a-i = -y, y(3-^ = -a, ay-i=-/3;
Vlir. . . ya-J= + /3, a/3-»= + y, /3y-» = + a;
the three fourth proportionals VI. are therefore equal, respectively, to — y^, -a^,
- (3^, and consequently to + 1 ; while the corresponding expressions VI'. are equal
to + /3*, + y2, + a2, and therefore to - 1.
(8.) Or (comp. (3.) ) we may write generally the transformation (comp. 282,
XXI.*")
IX. . . /3a-iy = a-2./3ay, if a-3=l: aS
in which the factor a'^ is always a scalar, whatever vector a may be ; while the
vector part of the ternary product j3ay vanishes, by 294, III., when the recent con-
ditions of rectangularity III. are satisfied.
(9.) Conversely, this terna?-y product jSay, and this fourth proportional (3a-^y,
can never reduce themselves to scalars, unless the three vectors a, (3, y (supposed to
be all actual (Art. 1)) are perpendicular each to each.
Section 8. — On an equivalent Interpretation of the Fourth
Proportional to Three Diplanar Vectors^ deduced from the
Principles of the Second Book.
300. In the foregoing Section, we naturally employed the results
of preceding Sections of the present Book, to assist ourselves in at-
taching a definite signification to the Fourth Proportional (297)
to Three Diplanar Vectors ; and thus, in order to interpret the sym-
bol /3a"^7, we availed ourselves of the interpretations previously ob-
tained, in this Third Book, of a"' as a line, and of a/3, u^^ as quater-
nions. But it may be interesting, and not uninstructive, to inquire
how the equivalent symbol,
I. . . {^\a),^i, or -7, with 7 not \\\ a, /3,
might have been interpreted, on the principles of the Second Booh, with-
out at first assuming as known, or even seeking to discover, any in-
terpretation of the three lately mentioned symbols,
II. . . a-', up, 0^7.
It will be found that the inquiry conducts to an expression of the
form,
* The formula here referred to should have been printed as Ra = 1 : a = a-*.
350 ELEMENTS OF QUATERNIONS. [bOOK HI.
III. . . (i3:a).7=^^-ew;
where S is the same vector, and e is the same scalar, as in the recent
sub-articles to 297; while u is employed as a temporary symbol, to
denote a certain Fourth Proportional to Three Rectangular Unit
Lines, namely, to the three lines oq, ol', and op in Fig. 68; so
that, with reference to the construction represented by that Figure,
we should be led, by the principles of the Second Book, to write the
equation :
IV. . . (oB : oa) . oc = CD . cos JS + (ol' : oq) . op. sin ^2.
And when we proceed to consider what signification should be at-
tached, on the principles of the same Second Book, to that particular
fourth proportional, which is here the coefficient of sin 12, and has
been provisionally denoted by u, we find that although it may be
regarded as being in one sense a Line, or at least homogeneous with a
line, yet it must not he equated to any Vector: being v2ii\mT analogo^is,
in Geometry, to the Scalar Unit of Algebra, so that it may be naturally
and conveniently denoted by the usual symbol 1, or + 1, or be equated
to Positive Unity. But when we thus write u=\, the last term
of the formula III. or IV., of the present Article, becomes simply
e, or sin -^2 ; and while this term (or part) of the result comes to be
considered as a species of Geometrical Scalar, the complete Expres-
sion for the General Fourth Proportional to Three Diplanar Vectors
takes the Form of a Geometrical Quaternion: and thus the fortnula
297, XL VII., or 298, VIIL, is reproduced, at least if we substitute
in it, for the present, (/3: a).7 for ^a^r^, to avoid the necessity of
interpreting here the recent symbols II.
(1.) The construction of Fig. 68 being retained, but no principles peculiar to the
Third Book being employed, we may write, with the same significations of c, jo, &c.,
as before,
V. . . OB : OA = OR : OQ = cos c + (ol' : oq) sin c ;
VI. . . oc = OQ . cos/> + OP . sin/) .
(2.) Admitting then, as is natural, for the purposes of the sought interpretation,
that distributive property which has been proved (212) to hold good for the multi-
plication of quaternions (as it does for multiplication in algebra); and writing for
abridgment,
VII. . . M = (ol' : oq) . op;
we have the quadrinomial expression :
VIII. . . (oB : oa). oc = ol'. sin c cos/)i- OQ . cos ccos/>
+ OP . cos c sin j3 + « . sin c sin j9 ;
in which it may be observed that the sum of the squares of the four coefficients of the
CHAP. I.] SCALAR UNIT IN SPACE. 351
three rectangular unit-vectors, oq, ol', op, and of their fourth proportional, m, is
equal to unity,
(3.) But the coefficient of this fourth proportional, which may be regarded as a
species oi fourth unit, is
IX, . . sin c sin p = sin mn = sin 12 = e ;
we must therefore expect to find that the three other coefficients in VIII., when di-
vided by cos 12, or by r, give quotients which are the cosines of the arcual distances
of some point x upon the unit-sphere, from the three points i!, Q, p ; or that a point
X can be assigned, for which
X. . . sin c cosp = /' cos l'x ; cos c cos /? = r cos Qx ; cos c sin p = r cos px.
(4.) Accordingly it is found that these three last equations are satisfied, when we
substitute d for x ; and therefore that we have the transformation,
XI. . . OL,'. sinccos/j + OQ .cose cos/? + OP.coscsin/j = OD . cosiS = ^,
whence follow the equations IV. and III. ; and it only remains to study and interpret
t\\Q fourth unit, u, which enters as a factor into the remaining part of the quadrlno-
mial expression VIII., without employing any principles except those of the Second
Booh : and therefore without using the Interpretations 278, 284, of (3a, &c.
301. In general, when two sets of three vectors, a, /3, 7, and
«'» ^\ 7'* are connected by the relation,
1. ..--—= 1, or 11... ,= -7^
a Y P a ^ a'
it is natural to write this other equation,
III. . .-7 = - 7 ;
a ft
and to say that these two fourth proportionals (297), to a, /3, 7, and
to a', [i\ 7^ are equal to each other: whatever the /wZ^ signification oi
each of these two last symbols III., supposed for the moment to be
not yet fully known, may be afterwards found to be. In short, we
may propose to make it a condition of the sought Interpretation, on
the principles of the Second Book, of the phrase,
^''Fourth Proportional to Three Vectors,'^
and of either of the two equivalent Symbols 300, I., that the recent
Equation III. sha.\\ follow from I. or II.; just as, at the commence-
ment of that Second Book, and before concluding (112) that the ge-
neral Geometric Quotient /3: a of any two lines in space is a Quaternion,
we made it a condition (103) of the interpretation of such a quotient,
that the equation {fi:a).a = jS should be satisfied.
302. There are however two tests (comp. 287), to which the re-
cent equation III. must be submitted, before its final adoption; in
352 ELEMENTS OF QUATERNIONS. [bOOK 111.
order that we may be sure of its consistency^ 1st, with the previous
interpretation (226) of a Fourth Proportional to Three Complanar
Vectors, as a Line in their common plane; and Ilnd, with the gene^
ral principle of all mathematical language (105), that things equal to
the same thing, are to be considered as equal to each other. And it
is found, on trial, that both these tests are home : so that they form
no objection to our adopting the equation 301, III., as true hy defini-
tion^ whenever the preceding equation II., or I., is satisfied.
(1.) It may happen that the first member of that equation III. is equal to a.line
df as in 226 ; namely, when a, j3, y are complanar. In this case, we have by II.
the equation,
y y Y a a a '
so that a', /3', y are also complanar (among themselves), and the line B is their
fourth proportional likewise : and the equation III, is satisfied, both members being
symbols for one common line, ^, which is in general situated in the intersection of
the two planes, ajSy and a'jS'y' ; although those planes may happen to coincide,
without disturbing the truth of the equation.
(2.) Again, for the more general case oi diplanarity of a, /3, y, we may con-
ceive that the equation* II. co-exists with this other of the same form,
V. . . ^ 1- = ^ ; which gives VI. . . ^ y =Cr",
a y a a a
if the definition 301 be adopted. If then that definition be consistent with general
principles of equality, we ought to find, by III. and VI., that this third equation be-
tween two fourth proportionals holds good :
VII. . . ^'y' = ^'y" ; or that VIII. . . ^L = ^
a a a y a
when the equations II. and V. are satisfied. And accordingly, those two equations
give, by the general principles of the Second Book, respecting quaternions considered
as quotients of vectors, the transformation,
(B'y' /3 y y' /3 y jS"
— -i^ = C: ± . _L_ = c: _i- = '--^ as required.
a' y ay y' ay" a
303. It is then permitted to interpret the equation 301, III., on
the principles of the Second Book, as being simply a transformation
(as it is in algebra) of the immediately preceding equation II., or I.;
and therefore to write, generally,
I. . . 57 = 2V> if II. . . 5(7:7')=?';
* In this and other cases of reference, the numeral cited is always supposed to be
the one which (with the same number) has last occurred before, although perhaps
it may have been in connexion with a shortly preceding Article. Compare 217, (1.).
CHA1\ T.] FOURTH PROPORTIONAL RESUMED. 353
where 7, 7' are any two vectors^ and q, q' are any tivo quaternions^
which satisfy this last condition. Now, if v and v be any two right
quaternions^ we have (by 193, comp. 283) the equation,
III, . . Iv'.lv' - v'.v' = vv'^ ;
or
IV. . . v~^{l.v: Iv') - «j'"^ ; whence V. . . v^ . Iv = v'-^. lv\
by the principle which has just been enunciated. It follows, then,
that '■''if a right Line (Iv) he multiplied hy the Reciprocal (v") of the
Right Quaternion (v), of which it is the Index, the Product {v^lv) is
independent of the Lengthy and of the Direction, of the Line thus ope-
rated on ;" or, in other words, that this Product has one common Va-
lue., for all possible Lines (a) in Space: which common or constant
value may be regarded as a kind of new Geometrical Unit, and is equal
to what we have lately denoted, in 300, III., and VII., by the tem-
porary symbol u; because, in the last cited formula, the line op is
the index of the right quotient oq,: ol'. Retaining, then, for the
moment, this symbol, u, we have, for every line a in space, considered
as the index of a right quaternion, v, the four equations :
VI. . . v'^a = u ; VII. . . a = Vlt ; VIII. . . V- a:u;
IX. . . V"' = w: «;
in which it is understood that a = Jv, and the three last are here re-
garded as being merely transfoi^mations o{ the fivst, which is deduced
and interpreted as above. And hence it is easy to infer, that for
any given system of three rectangular lines a, /3, 7, we have the general
expression :
X. . . (/3 : a) . 7 = XU, if aJ-^,^JL^,<^_i,a\
where the scalar co-efficient, x, of the new unit, u, is determined by
the equation,
XI. . .a; = ±(Ty3:Ta).T7, according as XII. . . U7 = + Ax. (a: /3).
This coefficient x is therefore always equal, in magnitude (or absolute
quantity), to the fourth proportional to the lengths of the three given
lines 0^7 ; but it is positively or negatively taken, according as the
rotation round the third line 7, from the second line /3, to the first line
a, is itseM positive or negative: or in other words, according as the
rotation round the first line, from the second to the third, is on the
contrary negative ox positive (compare 294, (3.) ).
(I.) In illustration of the constancy of that fourth proportional whicli has been,
for the present, denoted by u, while the system of the three rectangular unit-lines
2 z
354 ELEMENTS OF QUATERNIONS. [bOOK III.
from \vhi<'h it is conceived to be derived is in any manner turned about, we may ob-
serve that the three equations, or proportions,
XIII. . . u : y =j3: a ; y:a = a:-y; i8:-y = y:/3,
conduct immediately to this fourth equation of the same kmd,
XIV. . . M:a = y:/3, or* « = (y:j3).a;
if we admit that this new quantity, or symbol, u, is to be operated on at all, or com-
bined with other symbols, according to the general rules of vectors and quaternions.
(2.) It is, then, permitted to change the three letters a, /3, y, by a cyclical per-
mutation, to the three other letters /3, y, a (considered again as representing unit-
lines), without altering the value of the fourth proportional, w, or in other woi'ds, it
is allowed to make the system of the three rectangular lines revolve, through the third
part of four right angles^ round the interior and co-initial diagonal of the unit-cube,
of which they are three co-initial edges.
(3.) And it is still more evident, that no such change of value will take place, if
we merely cause the system of the two first lines to revolve, through any angle, in
its own plane, round the third line as an axis ; since thus we shall merely substitute,
for the factor (i : «, another factor equal thereto. But by combining these two last
modes of rotation, we can represent ang rotation whatever, round an origin supposed
to be fixed.
('{.) And as regards the scalar ratio of any one fourth proportional, such as
(3' : a' . y', to any other, of the kind here considered, such as j3 : a . y, or «, it is suffi-
cient to suggest that, mthout any real change in the former, we are allowed to sup-
pose it to be so prepared, that we shall have
XV. ..a' = a; /3' = /3; y' = xy;
X being some scalar coefficient, and representing the ratio required.
304. In the more general case, when the three given lines are
not rectangular, nor unit-lines, we may on similar principles de-
termine their fourth proportional, without referring to Fig. 68, as
follows. Without any real loss of generality, we may suppose that
the planes of a, /3 and a, 7 are perpendicular to each other; since
this comes merely to substituting, if necessary, for the quotient
)3 : a, another quotient equal thereto. Having thus
I. . . Ax.(/3:a) JL Ax.(7:a), let II. . . /3 = /3' + )3'^ ry = y + y',
where /3' and 7' are parallel to a, but ^" and 7'' are perpendicular
to it, and to each other; so that, by 203, I. and II., we shall have
the expressions,
III. ..^' = S^.a, y=S^.a,
a tt
* In equations of this form, the parentheses may be omitted, though for greater
clearness they are here retained.
CHAP. I.] SPHEUICAL PARALLELOGRAM. 355
and W... ^" = Y^.a, y/ = V^.o.
a a
We may then deduce, by the distributive principle (300, (2.) ), the
t ran s formations,
a a a a
where
VI..
. ^ = )3S^+7''S^=7S- + /3''S^, and VII. . . o^w = ^' 7'
a a a a a
The latter part, xu, is what we have called (300) the (geometrically)
scalar part, of the sought fourth proportional ; while the former part
B may (still) be called its vector part : and we see that this part is
represented by a line^ which is at once m thetwo planes^ of /3, 7'', and
of 7, ^" ; or in two planes which may be generally constructed as fol-
lows, without now assuming that the planes ajS and ar^ are rectangu-
lar, as in I. Let 7' be the projection of the line 7 on the plane of
a, j3, and operate on this projection by the quotient yS: a as a multi-
plier ; the plane which is drawn through the line /3 : a . 7' so obtained,
at right angles to the plane a^, is one locus for the sought line d :
and the plane through 7, which is perpendicular to the plane 77^
is another locus for that line. And as regards the length of this line,
or vector part ^, and the magnitude (or quantity) of the scalar part
xu, it is easy to prove that
VIII. . . T^ = / cos 5, and IX. . . a; = + ^sin 5,
where
X... . ^ = T/3:Ta.T7, and XI. . . sin 5 = sine sin p,
if c denote the angle between the two given lines a, )3, and jo the
inclination of the third given line 7 to their plane: the sign of the
scalar coefficient, x, being positive or negative, according as the rota-
tion round a from yS to 7 is negative or positive.
(L) Comparing the recent construction with Fig. 68, we see that when the con-
dition L is satisfied, the four unit-lines Uy, Ua, U/3, Vd take the directions of the
four radii oc, oq, or, od, which terminate at the four comers of what may be called
a tri -rectangular quadrilateral CQRD on the sphere.
(2.) It may be remarked that the area of this quadrilateral is exactly equal to
h(dfthe area 2 of the triangle def ; which may be inferred, either from the circum-
356 ELEMENTS OF QUATERNIONS. [boOK 111.
stance that its spherical excess (over four right angles) is constructed by the angle
MDN ; or from the triangles dbr and eas being together equal to the triangle abf,
60 that the area of desk is 2, and therefore that of cqrd is ^S, as before.
(3.) The two sides CQ, qr of this quadrilateral, which are remote from the obtuse
angle at d, being still called p and c, and the side cd which is opposite to c being
still denoted by c', let the side dr which is opposite to p be now called p' ; also let
the diagonals CR, qd be denoted by d and d' ; and let s denote the spherical excess
(ODR - ^tt), or the area of the quadrilateral. "We shall then have the relations,
!cos d = cosp cos c ; cos d' = cosp cos c' ;
tanc'= cosp tan c ; tan p' = cos c tan j» ;
cos s = cos p sec/>' = cos c sec c' = cos d sec d' ;
of which some have virtually occurred before, and all are easily proved by right-an-
gled triangles, arcs being when necessary prolonged.
(4.) If we take now two points, A and b, on the side qr, which satisfy the arcual
equation (comp. 297, XL., and Fig. 68),
XIII. . . (^ AB= nQB;
and if we then join AC, and let fall on this new arc the perpendiculars bb', dd' ; it
is easy to prove that the projection b'd' of the side bd on the arc AC is equal to that
arc, and that the angle dbb' is right : so that we have the two new equations,
XIV. . . n b'd' = o AC ; XV. . . dbb' = |7r ;
and the neiv quadrilateral bb'd'd is also tri-rectangular.
(5.) Hence the point d may be derived from the three points A'BC, by any two of
the four following conditions: 1st, the equality XIII. of the arcs ab, qr ; Ilud, the
cori'espondiug equality XIV. of the arcs AC, b'd'; Ilird, the tri-rectangular charac-
ter of the quadrilateral CQRD ; IVth, the corresponding character of bb'd'd.
(6.) In other words, this derived point D is the common intersection of the four
perpendiculars, to the four arcs ab, ac, cq, bb', erected at the four points R, d', C, b ;
CQ, bb' being still the perpendiculars from c and b, on ab and AC; and r and d'
bohig deduced from Q and b', by equal arcs, as above.
305. These consequences of the construction employed in 297,
&c., are here mentioned merely in connexion with that theory of
fourth proportionals to vectors, which they have thus served to illus-
trate; but they are perhaps numerous and interesting enough, to
justify us in suggesting the name^ ''^ Sp>herical Parallelogram,''^* for
the quadrilateral cabd, or bacd, in Fig. 68 (or 67) ; and in proposing
to say that d is the Fourth Point, which completes such d^ parallelogram,
when the three points c, A, B, or B, a, c, are given upon the sphere,
{kS first, second, and third. It must however be carefully observed,
that the analogy to the plane is here thus far imperfect, that in the
* By the same analogy, the quadrilateral cqrd, in Fig. 68, may be called a
Sjiherical Rectangle.
CHAP. 1.] SERIES OF SniERICAL PARALLELOGRAMS. 357
gefieral case, when the three given points are not co-arcucd, but on the
contrary are corners of a spherical triangle abc, then if we take c, d, b,
or B, D, c, for the three first points of a new spherical parallelogram^ of
the kind here considered, the new fourth pointy say a„ will not coin-
cide with the old second point a; although it will very nearly do so,
if the sides of the triangle abc be small: the deviation aAj being in
fact found to be small of the third order, if those sides of the given
triangle be supposed to be small of the first order; and being always
directed towards the foot of the perpendicular, let fall from a on bc.
(L) To investigate the Zaw of this deviation, let /3, y be still any two given
unit-vectors, ob, oc, making with each other an angle equal to a, of which the co-
sine is I ; and let p or op be any third vector. Then, if we write,
I. . . pi = ^(p) = ANp. -y+-/3 , OQ=Up, OQi = Upi,
\9 P I
the new or derived vector, <pp or pi, or OPi, will be the common vector pai't of the
two fourth proportionals, to p, /3, y, and to p, y, (3, multiplied hy the square of the
length of Q ; and BQCQi will be what we have lately called a spherical parallelogram.
We shall also have the transformation (compare 297, (2.)),
IL..pi = 0p=^S^+yS|-pS|;
and the distributive symbol of operation <p will be such that
III...^p|||Ay, and >V = P, if Plll/^,y;
but IV. ..^p = -Zp, if p II Ax. (y : /3).
(2.) This being understood, let
V. ..p = p' + p"; ^p' = p'i; p'lli/3, y, p"|| Ax.(y:/3);
so that p', or op', is the projection of p on the plane of (3y ; and p", or op", is the
part (or component) of p, which is perpendicular to that plane. Then we shall have
an indefinite series of derived vectors, pi, pg) P3» • • or rather two such series, suc-
ceeding each other alternately, as follows :
VI. . . fP^'^'^P"^ f'^ ~ '^ " ' P2 = <P^9 = p' + l^p" ;
lp3 = <p^p = p'l - i^p"'-, p4 = 0V = p' + i^p" ; &c- ;
the two series of derived points, Pi, P2, P3, P4, . . . being thus ranged, alternately,
on the two perpendicular SfW' and PiP'i, which are let fall from the points p and Pi,
on the given plane BOO ; and the intervals, PP2, P1P3, P2P4, • • . forming a geometri-
cal progression, in which each is equal to the one before it, multiplied by the con-
stant factor - I, or by the negative of the cosine o£ the given angle boc.
(3.) If then this angle be still supposed to be distinct from and tt, and also
in general from the intermediate value ^tt, we shall have the two limiting values,
VII. . . p2n = p', p2rt+l = p'l, if n = 00 ;
or in words, the derived points r2, P4, . . of even orders, tend to the point p', and the
other derived points, Pi, 1% . . oi odd orders, tend to the other point p'l, as limiting
358 ELEMENTS OF QUATERNIONS. [bOOK III.
positions: these two limit points being the feet of the two (rectilinear) perpendicu-
lars, let fall (as above) from p and p' on the plane boc.
(4.) But even \h.Q first deviation ppg, is small of the third order, if the length Tp
of the line op be considered as neither large nor small, and if the sides of the spheri-
cal triangle BQC be small otth.Q first order. For we have by VI. the following ex-
pressions for that deviation,
VIII. . . pp2 = p2-p = (^^ -l)p"=-sina2.sinpa.Tp .Up";
where pa denotes the inclination of the line p to the plane (3y ; or the arcual perpen-
dicular from the point Q on the side bc, or a, of the triangle. The statements lately
made (305) are therefore proved to have been correct.
(5.) And if we now resume and extend the spherical construction, and conceive
that Di is deduced from baiC, as Ai was from bdc, or d from bac ; while A2 may
be supposed to be deduced by the same rule from bd^c, and D2 from BA2C, &c.,
through an indefinite series of spherical parallelograms, in which t\i.Q fourth point
of any one is treated as the second point of the next, while the first and third points
remain constant : we see that the points Ai, A2, . . are all situated on the arcual
perpendicular let fall from A on bc ; and that in like manner the points Dj, D2, . .
are all situated on that other arcual perpendicular, which is let fall from d on bc.
We see also that the ultimate positions, a<x. and Dw, coincide precisely with the feet
of those two perpendiculars : a remarkable theorem, which it would perhaps be diffi-
cult to prove, by any other method than that of the Quaternions, at least with calcu-
lations so simple as those wliich have been employed above.
(6.) It may be remarked that the construction of Fig. 68 might have been other-
wise suggested (comp. 223, IV.), by the principles of the Second Book, if we had
sought to assign i]ie fourth proportional (297) to three right quaternions; for ex-
ample, to three right versors, v, v', v", whereof the unit lines a, (5, y should be sup-
posed to be the axes. For the result would be in general a quaternion v'v^v", with
e for its scalar part, and with d for the itidex of its right part : e and d denoting
the same scalar, and the same vector, as in the sub-articles to 297.
306. Quaternions may also be employed to furnish a new con-
struction, which shall complete (comp. 305, (5.)) the ^mj^/izW deter-
mination of the two series of derived points,
I. . . D, Ai, D„ A2, D2, &C.,
when the three points a, b, c are given upon the unit-sphere ; and
thus shall render visible (so to speak), with the help of anew Figure,
the tendencies of those derived points to approach, alternately and
indefinitely, to the/ee^, say D'and a', of the two arcual perpendiculars
let fall from the two opposite corners, d and a, of the first spherical
parallelogram, baod, on its given diagonal bc ; which diagonal (as we
have seen) is common to all the successive parallelograms.
(1.) The given triangle abc being supposed for simplicity to have its sides ahc
less than quadrants, as in 297, so that their cosines Imn are positive, let a', b', c' be
CHAP.
••]
CONSTRUCTION OF THK SERIES.
359
the feet of the perpendiculars let fall on these three sides from the points A, b, c ;
also let M and n be two auxiliary points, determined by the equations,
II. . . r> BM = r> MC, ^ AM = r\ mn ;
so that the arcs an and bc bisect each other in m. Let fall from n a perpendicular
nd' on BC, so that
III. . . «-> bd'= n a'c ;
and let b", o" be two other auxiliary points, on the sides b and c, or on those sides
prolonged, which satisfy these two other equations,
IV. . . o b'b" = r^ AC, f^ C'C" = n AB,
(2.) Then the perpendiculars to these last sides, CA and AB, erected at these last
points, b" and c", will intersect each other in the point D, which completes (ZQb^ the
spherical parallelogram bacd ; and the foot of the perpendicular from this point d,
on the third side bc of the given triangle, will coincide (comp. 305, (2.) ) with the
foot d' of the perpendicular on the same side from n ; so that this last perpendicular
nd' is one locus of the point D.
(3.) To obtain another locus for that point, adapted to our present purpose, let
E denote now* that new point in which the two diagonals, ad and bc, intersect each
other ; then because (comp. 297, (2.) ) we have the expression,
V. . . OD = u(mj3 + ny - ?a),
we may write (comp. 297, (25.), and (30.)),
VI. . . OE = u (m/3 + ny), whence VII. . . sin be : sin ec = w : m = cos ba' : cos a'c ;
the diagonal ad thus dividing the arc bc into segments, of which the sines are pro-
portional to the cosines of the adjacent sides of the given triangle, or to the cosines
of their projections ba' and a'c on bc ; so that the greater segment is adjacent to the
lesser side, and the middle point M of bc (1.) lies between the points a' and E.
(4.) The intersection e is therefore a known point, and the great circle through
A and e is a second known locus for
D ; which point may therefore be
found, as the intersection of the arc
AE prolonged, with the perpendicular
nd' from N (1.). And because e lies
(3.) beyond the middle point m of bc,
with respect to the foot a' of the per-
pendicular on bc from a, but (as it
is easy to prove) not so far beyond
M as the point d', or in other words
falls between M and d' (when the arc
BC is, as above supposed, less than a
quadrant), the prolonged arc ae cuts
nd' between N and d'; or in other
words, the perpendicular distance of
the sought fourth point D, from the
given diagonal BC of the parallelo-
gram, is less than the distance of the
given second point A, from the same given diagonal, (Compare the annexed Fig. 73.)
Fig. 73.
It will be observed that m, n, e have not here the same significations as in
360 ELKMENTS OF QUATERNIONS. [bOOK III.
(6.) Proceeding next (305) to derive a new point Ai from b, i>, c, as d has been
derived from b, a, c, we see that we have only to determine a new* auxiliary point
F, by the equation,
VIII. . . --> EM = r. MF ;
and then to draw df, and prolong it till it meets a a' in the required point Ai, which
will thus complete the second parallelogram, bdcai, with bc (as before) for a given
diagonal.
(6.) In like manner, to complete (comp. 305, (5.) ), the third parallelogram,
BAiCDi, with the same given diagonal bc, we have only to draw the arc AiE, and
prolong it till it cuts nd' in Di ; after which we should find the point A2 of a fourth
successive parallelogram BD1CA2, by drawing DiF, and so on for ever.
(7.) The constant and indefinite tendency, of the derived points d, Di, . . to the
limit-point d', and of the other (or alternate^ derived points Ai, Ag, . • to the other
limit-point a', becomes therefore evident from this new construction ; the final (or
limiting') results of which, we may express by these two equations (comp. again
305,(5.)),
IX. . . Dd) = d' ; A<p = a'.
(8.) But the smallness (305) of the first deviation AAi, when the sides of the
given triangle abc are small, becomes at the same time evident, by means of the
same construction, with the help of the formula VII. ; which shows that the intervalf
EM, or the equal interval mf (5.), is small of the third order, when the sides of the
given triangle are supposed to be small of ihe first order: agreeing thus with the
equation 305, VIII.
(9.) The theory of such spherical parallelograms admits of some interesting ap-
plications, especially in connexion with spherical conies ; on which however we can-
not enter here, beyond the mere enunciation of a Theorem, % of which (comp. 271)
the proof by quaternions is easy : —
Fig. 68 ; and that the present letters c' and c" correspond to q and r in that Fi-
gure.
* This new point, and the intersection of the perpendiculars of the given trian-
gle, are evidently not the same in the new Figure 73, as the points denoted by the
same letters, f and p, in the former Figure 68 ; although the four points A, b, c, d
are conceived to bear to each other the same relations in the two Figures, and indeed
in Fig. 67 also ; bacd being, in that Figure also, what we have proposed to call a
spherical parallelogram. Compare the Note to (3.).
t The formula VII. gives easily the relation,
VII'. . . tan EM = tan ma' ( tan - T ;
hence the interval em is small of the third order, in the case (8.) here supposed ; and
generally, if o < -, as in (1,), while 6 and c are unequal, the formula shows that this
interval em is less than ma', or than d'm, so that e falls between m and d', as in (4.),
X This Theorem was communicated to the Royal Irish Academy in June, 1845,
as a consequence of the principles of Quaternions. See the Proceedings of that date
(Vol. III., page 109).
CHAP. I.J THIRD INTERPRETATION OF A PRODUCT. 3G I
" T/'klmn be any spherical quadrilateral, and.l any point on the sphere ; if also
we complete the spherical parallelograms,
X. . . KILA, LIMB, MINC, NIKD,
and determine the poles E and F of the diagonals km and ln of the quadrilateral :
then these two poles are the foci* of a spherical conic, inscribed in the derived quadri-
lateral ABCD, or touching its four sides."
(10.) Hence, in a notationf elsewhere proposed, we shall have, under these con«
ditions of construction, the formula :
XL . . EF (. .) ABCD ; or XI'. . . EF (. .) BCDA ; &C.
(11.) Before closing this Article and Section, it seems not irrelevant to remark,
that the projection y' of the unit-vector y, on the plane of a and /3, is given by the
formula,
_,__ , a sin a cos B + /3 sin i cos A
XII. . . y = . ;
smc
and that therefore the point p, in which (see again Fig. 73) the three arcual perpen-
diculars of the triangle abc intersect, is on the vector,
XIII. . . p = a tan a + /3 tan B + y tan c.
(12.) It may be added, as regards the construction in 305, (2.), that the right
lines,
XIV. . . PPi, P1P2, P2P3, P3P4, . . •
however far their series may be continued, intersect the given plane boc, alternately,
in two points s and T, of which the vectors are,
VTT 9 1 + Ip' P'+Ip'l
XV...03=-j^, OT=-^-;
and which thus become two fixed points in the plane, when the position of the point
p in space is given, or assumed.
Section 9. — On a Third Method of interpreting a Product or
Function of Vectors as a Quaternion ; and on the Consis-
tency of the Results of the Interpretation so obtained^ with
those which have been deduced from the two preceding Me-
thods of the present Book.
307. The Conception of the Fourth Proportional to Three
Rectangular Unit-Lines^ as being itselfa species of i^6W?*^^ Uyiit
in Geometry^ is eminently characteristic of the present Calcu-
lus ; and offers a Third Method of interpreting a Product of
two Vectors as a Quaternion : which is however found to be
* In the language of modem geometry, the conic in question may be said to
touch eight given arcs ; four real, namely the sides ab, bc, CD, da ; and/owr ima-
ginary, namely two from each of the focal points, B and F.
t Compare the Second Note to page 295.
3 A
362 ELEMENTS OF QUATERNIONS. [bOOK III.
consistent^ in all its results^ with the two former methods (278,
284) of the present Book ; and admits of being easily extended
to products of three or more lines in space ^ and generally to
Functions of Vectors (289). In fact we have only to conceive*
* It was in a somewhat aaalogous way that Des Cartes showed, in his Geome-
<na (Schooten's Edition, Amsterdam, 1659), that all products and powers of lines,
considered relatively to their lengths alone, and without any reference to their direc-
tions, could be interpreted as lines, by the suitable introduction of a line taken for
unity, however high the dimension of the product or power might be. Thus (at
page 3 of the cited work) the following remark occurs: —
" Ubi notandum est, quod per a2 vel 6^, similesve, communiter, non nisi lineas
omnino simplices concipiam, licet illas, ut nominibus in Algebra usitatis utar, Qua-
drata aut Cubos, &c. appellem."
But it was much more difficult to accomplish the corresponding multiplication of
directed lines in space ; on account of the non-existence of any such line, which is
symmetrically related to all other lines, or common to all possible planes (comp. the
Note to page 248). The Unit of Vector -Multiplication cannot properly be itself a.
Vector, if the conception of the Symmetry of Space is to be retained, and duly com-
bined with the other elements of the question. This difficulty however disappears,
at least in theory, when we come to consider that new Unit, of a scalar kind (300),
which has been above denoted by the temporary symbol u, and has been obtained,
in the foregoing Section, as a certain Fourth Proportional to Three Rectangular
Unit-Lines, such as the three co-initial edges, AB, AC, ad of what we have called an
Unit- Cube : for this fourth proportional, by the proposed conception of it, undergoes
no change, when the cube abcd is in any manner moved, or turned ; and therefore
may be considered to be symmetrically related to all directions of lines in space, or to
all possible vections (or translations) of a pointy or body. In fact, we conceive its de-
termination, and the distinction of it (as + u) from the opposite unit of the same kind
(— «), to depend only on the tisual assumption of an uiiit of length, combined with
the selection of a hand (as, for example, the right hand), rotation towards which
hand shall be considered to he positive, and contrasted (^as such) with rotation to-
wards the other hand, round the same arbitrary axis. Now in whatever manner the
supposed cube may be thrown about in space, the conceived rotation round the edge
AB, from AC to AD, will have the same character, as right-handed or left-handed, at
the end as at the beginnhig of the motion. If then the fourth proportional to these
three edges, taken in this order, be denoted by + «, or simply by + 1, at one stage of
that arbitrary motion, it may (on the plan here considered) be denoted by the same
symbol, at er^ery other stage: while the opposite character of the (conceived) rota-
tion, round the same edge ab, from AD to AC, leads us to regard the fourth propor-
tional to AB, AD, AC as being on the contrary equal to — «, or to — 1, It is true that
this conception of a new unit for space, symmetrically related (as above) to all linear
directions therein, may appear somewhat abstract and metaphysical ; but readers
who think it such can of course confine their attention to the rules of calculation ,
which have been above derived from it, and from other connected considerations : and
which have (it is hoped) been stated and exemplified, in this and in a fonner Vo-
lume, with sufficient clearness and fullness.
CHAP. I.] CONCEPTION OF THE FOURTH UNIT. 363
that each proposed vector, a, is divided by the neio or fourth
unit, u, above alluded to ; and that the quotient so obtained,
which is always (by 303, VIII.) the ripht quaternion T^a,
whereof the vector a is the index^ is substituted for that vec-
tor ; the resulting quaternion being finally, if we think it con-
venient, multiplied into the same fourth unit. F6r in this way
we shall merely reproduce the process of 284, or 289, although
now as a consequence of a different train of thought ^ or of a dis-
tinct but Consistent Interpretation : which thus conducts, by a
new Method, to the same Rules of Calculation as before.
(1.) The equation of the unit-sphere, p2 + i = Q (282, XIV.), may thus be con-
ceived to be an abridgment of the following fuller equation :
i...(ey=-i;
\uj
the quotient p : u being considered as equal (by 303) to the rigfit quaternion, I'/o,
which must here be a right versor (154), because its square is negative unity.
(2.) The equation of the ellipsoid,
T(tp + pfc) = fc2 - t2 (282, XIX.),
may be supposed, in like manner, to be abridged from this other equation :
\u u uu j \u I \tt/
and similarly in other cases.
(3.) We might also write these equations, of the sphere and ellipsoid, under these
other, but connected forms :
III...^p = -«; IV...Tf-p+-
u
with intepretations which easily offer themselves, on the principles of the foregoing
Section.
(4.) It is, however, to be distinctly understood, that we do not propose to adopt
this Form of Notation, in the practice of the present Calculus : and that we merely
suggest it, in passing, as one which may serve to throw some additional light on the
Conception, introduced in this Third Book, of a Product of two Vectors as a Qua-
ternion.
(5.) In general, the Notation of Products, which has been employed throughout
the greater part of the present Book and Chapter, appears to be much more conve-
nient, for actual use in calculation, than any Notation of Quotients : either such as
has been just now suggested for the sake of illustration, or such as was employed in
the Second Book, in connexion with that First Conception of a Quaternion (112),
to which that Book mainly related, as the Quotient of two Vectors (or of two di-
rected lines in space). The notations of the two Books are, however, intimately con-
nected, and the former was judged to be an useful preparation for the latter, even as
3G4 ELEMENTS OF QUATERNIONS. [bOOK III.
regarded the quotient-forms of many of the expressions used : while the Characteris-
tics of Operation, such as
S, V, T, U, K, N,
are employed according to exactly the same laws in both. In short, a reader of the
Second Book has nothing to unlearn in the Third; although he may be supposed to
have become prepared for the use of somewhat shorter and more convenient pro-
cesses, than those before employed.
Section 10 — On the Interpretation of a Power of a Vector
as a Quaternion.
308. The only symbols, of the kinds mentioned in 277,
which we have not yet interpreted, are the cube a% and the
general power a\ of an arbitrary vector base^ a, with an arbi-
trary scalar exponent, t ; for we have already assigned inter-
pretations (282, (1.), (14.), and 299, (8.)) for th^ particular
symbols a^, a'S a"'^, which are included in this last^rm. And
we shall preserve those particular interpretations if we now
define, in fall consistency with the principles of the present and
preceding Books, that this Power a* is generally a Quaternion,
which may be decomposed into two factors, of the tensor and
vers or kinds, as follows :
I. ..a^=Ta^Ua';
IV denoting the arithmetical value of the t^^"- power of the po-
sitive number Ta, which represents (as usual) the length of the
base-line a ; and Ua^ denoting a versor, which causes any line
p, perpendicular to that line a, to revolve round it as an axis,
through t right angles^ or quadrants, and in a positive or nega-
tive direction, according as the scalar exponent, t, is itself a
positive or negative number (comp. 234, (5.) ).
(1.) As regards the omission of parentheses in the formula I., we may observe
that the receut definition, or interpretation, of the symbol a*, enables us to write
(comp. 237, 11. III.),
II. . . T(aO = (Tay = Ta«; III. . . U (a*) = (Ua)< = Ua*.
(2.) The ascis and angle of the power a*, considered as a quaternion, are generally
determined by the two following formulae :
IV. . . Ax. a< = ± Ua ; V. . . ^ . a' = 2n7r ± ^tn ;
the signs acctmipanying each other, and the (positive or negative or null) integer, «,
being so chosen as to bring the angle within the usual limits, and it.
CHAP. 1.] POWER OF A VECTOR A QUATERNION. 365
(3.) In general (comp. 235), we may speak of the (positive or negative) product
i<7r, as being the amplitude of the same power, with reference to the line a as an
axis of rotation ; and may write accordingly,
VI. , . am. a* = ^tir.
(4.) We may write also (comp. 234, VII. VIII.),
VII. . . Ua< = cos Y + Ua . sin — ; or briefly, VIII. . . Ua« = cas —.
(5.) In particular,
IX. . . Ua^"^ cas «7r = ± 1 ; IX'. . . Ua2«+i= ± Ua ;
upper or lower signs being taken, according as the number n (supposed to be whole)
is even or odd. For example, we have thus the cubes,
X. . . Ua3 = -Ua; X'. . . a3 = -aNa.
(6.) The coiijugate and norm of the power a' may be thus expressed (it being
remembered that to turn a line -^ a through - |f7r round + a, is equivalent to turn-
ing that line through + Itir round -a):
XI. . . Ka< = Ta« . Ua' = (- a)« ; XII. . . Na* = Ta*« ;
parentheses being unnecessary, because (by 295, VIII.) Ka = — a.
(7.) The scalar, vector, and reciprocal of the same power are given by the for-
mulae :
XIII. . . S.a« = Ta<.cos~; XIV. . . V. a« = Ta^.Ua. sin ^;
2 A
XV. . . 1 : a^= Ta-«.Ua-' = a-«= Ka« : Na< (comp. 190, (3.)).
(8.) If we decompose any vector p into parts p' and p", which are respectively
parallel and perpendicular to a, we have the general transformation :*
XVI. . . atpa-t=^at{p + p") a-«= p' + Va^K p",
= the new vector obtained by causing p to revolve conically through an angular quan-
tity expressed by tir, round the line a as an axis (comp. 297, (15.)),
(9.) More generally (comp. 191, (5.) ), if q be any quaternion, and if
XVII. ..a*qa-*=q,
the new quaternion q is formed from q by such a conical rotation of its own axis
Ax. 5, through tir, round a, without any change of its angle L q, or of its tensor Tq.
(10.) Treating ijk as three rectangular unit-lines (295), the symbol, or expres-
sion,
XVIII. . .p = rktjskj-^kt, or XIX. . . p = r¥j^^k^^,
in which
XX. ..r>0, s>0, s^l, t^O, *<2,
may represent any vector ; the length or tensor of this line p being r ; its inclina-
tion\ to k being sir ; and the angle through which the variable />Zane kp may be
* Compare the shortly following sub-article (11.).
t If we conceive (compare tlie first Note to page 322) that the two hnes i andy
are directed respectively towards the south and west points of the horizon, while the
third line k is directed towards the zenith^ then sir is the zenith-distance of p; and
tTT is the azimuth of the same line, measured /rom south to west, and thence (if ne-
cessary) through north and east, to south again.
366 ELEMENTS OF QUATERNIONS. [bOOK III.
conceived to have revolved, frem the initial position ki, with an initial direction to-
wards the position kj, being t-jr.
(11.) In accomplishing the transformation XVI., and in passing from the ex-
pression XVIII. to the less symmetric but equivalent expression XIX., we employ
the principle that
XXI. . . */-* = S-i = - K (kj-o) =j^k ;
which easily admits of extension, and may be confirmed by such transformations as
VII. or VIII.
(12.) It is scarcely necessary to remark, that the definition or interpretation I.,
of the power a* of an?/ vector a, gives (as in algebra) the exponential property,
XXII. ..a*a< = a«+«,
whatever scalars may be denoted by s and t ; and similarly when there are more than
two factors of this form.
(13.) As verifications of the expression XVIII., considered as representing a vec-
tor, we may observe that it gives,
XXIII... p = -Kp; and XXI V. . . p2 = _ r2.
(14.) More generally, it will be found that if m* be any scalar, we have the
eminently simple transformation :
XXV. . . |0« = (rk^j^kj-^k-^y = r^'ktfktfj-«k-*.
In fact, the two last expressions denote generally two equal quaternions, because
they have, 1st, equal tensors, each = r" ; Ilnd, equal angles, each = L (^'0 ; and
Ilird, equal (or coincident) axes, each formed from + A by one common system of
two successive rotations, one through stt round j, and the other through tn round k.
309. Ani/ quaternion, q, which is not simply a scalar^ may
be brought to i\\Qform a\ by a suitable choice of the base, a,
and of the exponent, t ; which latter may moreover be supposed
to fall between the limits and 2 ; since for this purpose we
have only to write,
1...^=^^; II. . .Ta = T^^ III. . .Ua = Ax.^;
TT
and thus the general dependence of a Quaternion, on a Scalar
and a Vector Element, presents itself in a new ivay (comp. 17,
207, 292). When the proposed quaternion is a versor, T^- = 1,
* The emplojonent of this letter u, to denote what we called, in the two preced-
ing Sections, a. fourth unit, &c., was stated to be a merely temporary' one. In gene-
ral, we shall henceforth simply equate that scalar unit to the number one ; and die-
note it (when necessary to be denoted at all) by the usual symbol, 1, for that num-
ber.
CHAP. I.] EXPRESSIONS FOR VERSORS AS POWERS. 367
we have thus Ta = 1 ; or in other words, the base a, of the
equivalent jooi^er a', is an unit-line. Conversely, every versor
may be considered as a power of an unit-line^ with a scalar ex-
ponent^ t^ which may be supposed to be m. general positive, ^-rA
less than two ; so that we may write generally^
lY...Vq^a\ with V. . .a = Ax.y = T-U,
and VI. . . ^ > 0, t<2\
although if this versor degenerate into 1 or - 1, the exponent
t becomes or 2, and the base a has an indeterminate or ar-
bitrary direction. And from such transformations ofversors
new methods may be deduced, for treating questions of sphe-
rical trigonometry, and generally of spherical geometry.
(1.) Conceive that p, q, k, in Fig. 46, are replaced by a, b, c, with unit-vec-
tors a, j3, y as usual ; and let a;, y, z be three scalars between and 2, determined
by the three equations,
VII. . . x7r = 2A, ^7r = 2B, 27r = 2c',
where a, b, c denote the angles of the spherical triangle. The three versors, indi-
cated by the three arrows in the upper part of the Figure, come then to be thus de-
noted :
VIII. . . 9 = a^ ; 9' = /32/ ; q'q = y2-z .
so that we have the equation,
IX. . . /3J/a*= 72-a ; or X. . . y^^va^^- 1 ;
from which last, by easy divisions and multiplications, these two others immediately
follow :
X'. . . a^y^i^v = - 1 ; X", . . ^va^'y^ = - 1 ;
the rotation round a from /3 to y being again supposed to be negative.
(2.) In X. we may write (by 308, VIII.),
XI. . . a»^ = casA ; /3J' = c/3sb; y« = cySC;
and then the formula becomes, for any spherical triangle, in which the order of ro-
tation is as above :
XII. . . cysc . c/3sB . caSA = — 1;
or (com p. IX.),
XIII. . . - COS c + y sin c = (cos b + jS sin b) (cos a + a sin a).
(3.) Taking the scalars on both sides of this last equation, and remembering that
S/3a= - cos c, we thus immediately derive one form of ihQ fundamental equation of
spherical trigonometry ; namely, the equation,
XIV. . . cos c + cos a cos b = cos c sin a sin b,
(4.) Taking the vectors, we have this other formula :
XV. . . y sin c = a sin a cos B + jS sin b cos a + V/3« sin a sin E ;
which is easily seen to agree with 306, XII., and may also be usefully compared
with the equation 210, XXXVII.
368 ELEMENTS OF QUATERNIONS. [bOOK III.
(5.) The result XV. may be euunciated in the form of a Theorem^ as follows : —
" If there be any spherical triangle abc, and three lines he drawn from the
centre O of the sphere, one towards the point a, with a length = sin A cos B ; another
towards the point b, with a length = sin b cos A ; and the third perpendicular to the
plane aob, and towards the same side of it as the point C, with a length = sin c sin A
sin B ; and if with these three lines as edges, we construct a parallelepiped : the
intermediate diagonal from o will he directed towards c, and will have a length
= sinc."
(6.) Dividing both members of the same equation XV. by p, and taking scalars,
we find that if p be any fourth point on the sphere, and q ih.^ foot of the perpendi-
cular let fall from this point on the arc ab, this perpendicular pq being considered as
positive when c and p are situated at one common side of that arc (or in one common
hemisphere, of the two into which the great circle through a and b divides the sphe-
ric surface), we have then,
XVI. . . sin c cos pc = sin a cos b cos pa + sin b cos a cos pb + sin a sin b sin c sin pq ;
a formula which might have been derived from the equation 210, XXXVIIL, by first
cyclically changing aftcABC to 6caBCA, and then passing from the former triangle to
its polar, or supplementary : and from which many less general equations may be
deduced, by assigning particular positions to p.
(7.) For example, if we conceive the point p to be the centre of the circumscribed
small circle abc, and denote by R the arcual radius of that circle, and by s the
se7nisum of the three angles, so that 2s=A + B4-c=7r + <T, if<7 again denote, as in
297, (47.), the area^ of the triangle abc, whence
XVII. . . PA = PB = po = iE, and sin pq = sin R sin (s — c),
the formula XVI. gives easily,
XVIII. . . 2 cot ^ sin — = sin a sin b sin c :
2
a relation between radius and area, which agrees with kno\^n results, and from which
we may, by 297, LXX., &c., deduce the known equation :
abc
XIX. . . e tan i? = 4 sin - sin - sin - ;
2 2 2'
in which we have still, as in 297, (47.), &c.,
XX. . . e = (Sa/3y =) sin a sin & sin c = &c.
(8.) In like manner we might have supposed, in the corresponding general equa-
tion 210, XXXVIII., that p was placed at the centre of the inscribed small circle,
and that the arcual radius of that circle was r, the semisum of the sides being s ;
and thus should have with ease deduced this other known relation, which is a sort
of polar reciprocal of XVI II.,
XXI . . . 2 tan r . sin s = e.
But these results are mentioned here, only to exemplify the fertility of the formulae,
to which the present calculus conducts, and from which the theorem in (5.) was
early seen to be a consequence.
Compare the Note to the cited sub-article.
CHAP. I.] EXPONENTIAL FORMULA FOR THE SPHERE. 369
(9.) We might devefope the ternary product in the equation XII., as we deve-
loped the 6^nary/Jrorfttc< XIII. ; compare scalar and vector parts; and operate on
the latter, by the symbol S . p-\ New general theorems, or at least new general
forms, wonld thus arise, of which it may be sufficient in this place to have merely
suggested the investigation.
(10.) As regards the order of rotation (1.) (2.), it is clear, from a mere inspec-
tion of the formula XV., that the rotation round y from /3 to a, or that round c from
B to A, must be positive, when that equation XV. liolds good; at least if the angle*
A, B, c, of the triangle ABC, be (as usual) treated as positive : because the rotation
round the line Yj3a from /3 to a is always positive (by 281, (3.) ).
(11.) If, then, for any given spherical triangle, ABC, with angles still supposed
to be positive, the rotation round c from b to a should happen to be (on the con-
trary) negative, we should be obhged to modify the formula XV. ; which could be
done, for example, so as to restore its correctness, by interchanging a with j8, and at
the same time A with b.
(12.) There is, however, a sense in which the formula might be considered as
still remaining true, without any change in the mode of writing it ; namely, if we were
to interpret the symbols A, b, c as denoting negative angles, for the case last sup-
posed (11.)- Accordingh', if we take the reciprocal of the equation X., we get this
other equation,
XXII. . . a-^/3-yy-^=-l;
where x, y, z are positive, as before, and therefore the new exponents, —x, —y, — z,
are negative, if the rotation round a from j8 to y be iV^e//" negative, as in (1).
(13.) On the whole, then, if a, j3, y be any given system of three co-initial and
diplanar unit-lines, OA, OB, oc, we can always assign a system of three scalars,
X, y, z, which shall satisfy the exponential equation X., and shall have relations of
the form VII. to the spherical angles A, b, c; but these three scalars, if determined
so as to fall between the limits + 2, will be all positive, or all negative, according as
the rotation round a from /3 to y is negative, as in (1.), or positive, as in (11.).
(14.) As regards the limits just mentioned, or the inequalities,
XXIII. .. a; < 2, y<2, z<2; x>-2, y>-2, z>-2,
they are introduced with a view to render the problem of finding the exponents xyz
in the formula X. determinate ; for since we have, by 308,
XXIV. . .a4 = ^4 = y4=+l, if Ta = T/3 = Ty = l,
we might otherwise add any multiple (positive or negative) of the number four, to
the value of the exponent of any unit-line, and the value of the resulting /jower would
not be altered.
(15.) If we admitted exponents = + 2, we might render the problem of satisfy-
ing the equation X. indeterminate in another way ; for it would then be sufficient to
suppose that any one of the three exponents was thus equal to + 2, or —2, and that
the two others were each = ; or else that all three were of the form + 2.
(16) When it was lately said (13.), that the exponents, x, y, z, in the formula
X., if limited as above, would have one common sign, the case was tacitly excluded,
for which those exponents, or some of them, when multiplied each by a quadrant,
give angles not equal to those of the spherical triangle abc, whether positively or
3 B
370 ELEMENTS OF QUATERKIONS. [bOOK III.
negatively taken ; but equal to the supplements of those angles, or to the negatives
of those supplements.
(17.) In fact, it is evident (because a^ = /32 = 72 = _ 1), that the equation X., or
the reciprocal equation XXII., if it be satisfied by any one system of values of xj/z,
will still be satisfied, when we divide or multiply any two of the three exponential
factors, by the squares of the two unit-vectors, of which those factors are supposed to
he powers: or in other words, if we subtract or add the number two, in each of two
exponents,
(18.) We may, for example, derive from XXII. this other equation :
XXV. . . a2 ^/3^-3/y-s' = - 1 ; or XXVI. . . a^-^(3^-y= y'-^ ;
which, when the rotation is as supposed in (1.), so that xyz are positive, maybe in-
terpreted as follows.
(19.) Conceive a lune cc', with points A and b on its two bounding semicircles,
and with a negative rotation round A from b to c ; or, what comes to the same thing,
with a positive rotation round A from b to c'. Then, on the plan illustrated by Fi-
gures 45 and 46, the supplements tt - A, 7r — B, of the angles A and b in the triangle
ABC, or the angles at the sa7ne points A and b in the co-lunar triangle abc', will
represent two versors, a multiplier, and a multiplicand, which are precisely those
denoted, in XXVI., by the two factors, a^"^ and (S^-v ; and the product of these two
factors, taken in this order, is that third versor, which has its axis directed to o',
and is represented, on the same general plan (177), by the external angle of the lune,
at that point c' ; which, in quantity, is equal to the external angle of the same lune
at c, or to the angle rr-c. This product is therefore equal to that power of the
2
unit-line oc', or - y, which has its exponent = - (tt — c) = 2 — z ; we have there-
fore, by this construction, the equation,
XXVII. . . a2-*/3«-y = (-y)2-«;
which (by 308, (6.) ) agrees with the recent formula XXVI.
310. The equation,
2c 2b 2a
I. . . 7'^P'^d^ = -l,
which results from 309, (1.), and in which a, j3, 7 are the
unit-vectors oa, ob, oc of any three points on the unit-sphere ;
while the three scalars a, b, c, in the exponents of the three
factors, represent generally the angular quantities of rotation,
round those three unit-lines, or radii, a, j3, 7, from the plane
Aoc to the plane aob, from boa to bog, and from cob to coa,
and are positive or negative according as these rotations of
planes are themselves positive or negative : must be regarded
as an important formula, in the applications of the present
Calculus. It includes^ for example, the whole doctrine of
Spherical Triangles; not merely because it conducts, as we
CHAP. I.] SPHERICAL SUM OF ANGLES. 371
have seen (309, (3.) ), to one form of the fundamental scalar
equation of spherical trigonometry^ namely to the equation,
II. . . cos c + cos A cos B = COS c sin A sm b ;
but also because it gives a vector equation (309, (4.) ), which
serves to connect the angles^ or the rotations^ a, b, c, with the
directions* of the radii, a, j3, 7, or OA, ob, go, for any system
of three diverging right lines from one origin. It may, there-
fore, be not improper to make here a few additional remarks,
respecting the nature, evidence and extension of the recent
formula I.
(1.) Multiplying both members of the equation I., by the inverse exponential
20
y "" , vfe have the transformation (comp. 309, (1.) ) :
2b 2a 2c 2(7r — c)
IIL . . j3^ a^ =-y ^ =y '^ .
2a
(2.) Again, multiplying both members of I. intof a t, we obtain this other for-
mula:
2c 2b 2a 2(ff — a)
IV. . . y'' (3^ =-a~^ =a ^ .
2a 20
(3.) Multiplying this last equation IV. by a'^, and the equation III. into y"^,
we derive these other forms :
* This may be considered to be another instance of that habitual reference to
direction, as distinguished from mere quantity (or magnitude), although combined
therewith, which pervades the present Calculus, and is eminently characteristic of
it ; whereas Des Cartes, on the contrary, had aimed to reduce all problems of geo-
metry to the determination of the lengths of right lines : although (as all who use
his co-ordinates are of course well aware) a certain reference to direction is even in
his theory inevitable, in connexion with the interpretation of negative roots (by him
called inverse or false roots) of equations. Thus in the first sentence of Schooten's
recently cited translation (1659) of the Geometry of Des Cartes, we find it said:
" Omian Geometriae Problemata facile ad hujusmoditerminosreduci possunt, ut
deinde ad illorum constructionem, opus tantum sit rectarum quarundam longitudinem
cognoscere."
The very different view of geometry, to which the present writer has been led,
makes it the more proper to express here the profound admiration with which he re-
gards the cited Treatise of Des Cartes : containing as it does the germs of so large a
portion of all that has since been done in mathematical science, even as concerns
imaginary roots of equations, considered as marks of geometrical impossibility.
t For the distinction between multiplying a quaternion into and by a factor, see
the Notes to pages 146, 159.
372 ELEMENTS OF QUATERNIONS. [bOOK III.
2a 2c 2b 2b 2a 2c
V. . . a'T y'r /S" =-1; VI. . . /3^ o*^ y'^ = - 1 ;
so that cyclical permutation of the letters, a, /3, y, and A, B, c, is allowed in the
equation I. ; as indeed was to be expected, from the nature of the theorem which
that equation expresses.
(4.) From either V. or VI. we can deduce the formula:
2a 2c 2b 2 (tt — b)
VII. . . a'T y?r = _^ 7r = ^ n ;
by comparing which with III. and IV , we see that cyclical permutation of letters
is permitted, in these equations also.
(5.) Taking the recijaroca/ (or conjugate) of the equation I., we obtain (com-
pare 309, XXII.) this other equation :
2a 2b 2c
VIII. . . a~»r j3~7r y T=_l;
2 (tt — A) 2(7r — B) 2(7r-c )
or IX. . . a If (3 If y •" = + 1;
in which cyclical permutation of letters is again allowed, and from which (or from
III.) we can at once derive the formula,
2a 2b 2c
X. . . a «• ^" TT = _ y »r.
(6.) The equation X. may also be thus written (comp. 309, XXVII.) :
2(7r — A) 2 (TT — B) 2(7r — c) 2 (tt — c)
XI. . . a '^ TT =.j,~ TT =(-y) T .
(7.) And all the foregoing equations may be interpreted {cqvc\'^. 309, (19.) ), and
at the same time/jrorerf, by a reference to that general construction (177) for the
multiplication ofversors, which the Figures 45 and 46 were designed to illustrate; if
we bear in mind that a power a*, of an unit-line a, with a scalar exponent, t, is (by
308, 309) a versor, which has the effect of turning a line -^ a, through t right an-
gles, round a as an axis of rotation.
(8.) The principle expressed by the equation I , from which all the subsequent
equations have been deduced, may be stated in the following manner, if we adopt the
definition proposed in an earlier part of this work (180, (4.) ), for the spherical sum
of two angles on a spheric surface :
" For any spherical triangle, the Spherical Sum of the three angles, if taken in a
suitable Order, is equal to Two Right Angles."
(9.) In fact, when the rotation round A from B to c is negative, i{ we spherically
add the angle b to the angle a, the spherical sum so obtained is (by the definition
referred to) equal to the external angle at c; if then we add to this sum, or supple-
ment of c, the angle c itself, we get di final or total sum, which is exactly equal to
7r ; addition of spherical angles at one vertex, and therefore in one plane, being ac-
complished in the usual manner; but the spherical summation of angles with diffe-
rent vertices being performed according to those new rules, which were deduced in the
Ninth Section of Book II., Chapter I. ; and were connected (180, (6.) ) with the
conception of angular transvection, or of the composition of angular motions, in dif-
ferent and successive planes.
CHAP. I.] ADDITION OF ARCS ON A SPHERE. 373
(10.) "Without pretending to attach importance to the following notation, we may-
just propose it in passing, as one which may serve to recall and represent the con-
ception here referred to. Using a plus in parentheses, as a symbol or characteristic
of such spherical addition of angles, the formula I. may be abridged as follows:
XII. . . c(+)B(+)A=7r;
the symbol of an added angle being written to the left of the symbol of the angle to
which it is added (comp. 264, (4.) ) ; because such addition corresponds (siS above)
to a multiplication ofversors, and we have agreed to write the symbol of the multi-
plier to the left* of the symbol of the multiplicand, in every multiplication of qua-
ternions.
311. There is, however, another view of the important equation
310, I., according to which it is connected rather with addition of
arcs (180, (3.) ), than with addition of angles (180, (4.) ); and may
be interpreted) and proved anew^ with the help of the supplementary
or polar triangle^ a'b'c', as follows.
(1.) The rotation round a from b to o being still supposed to be negative, let
a', b', c' be (as in 175) the positive poles of the sides bc, ca, ab ; and let a', (5', y'
be their unit- vectors. Then, because the rotation round a from y' to /3' is positive
(by 180, (2.) ), and is in quantity the supplement of the spherical angle a, the pro-
duct y'j3' will be (by 281, (2.), (3.)) a versor, of which a is the axis, and a the
angle; with similar results for the two other products, a'y', (5' a'.
(2.) If then we write (comp. 291),
I. . . a' = UV/3y, /3' = UVya, y' = UVa|3,
supposing that
II. ..Ta = T/3 = Ty = l, and III. . . Sa/3y > 0,
we shall have (comp. again 180, (2.) ),
IV. . . a = UVy'/3', (3 = Way', y = UV)3V,
and V. .. A=z.y'/3', B = z.a'y', c = lfS'a'',
whence (by 308 or 309) we have the following exponential expressions for these
three last products of unit-lines,
2a 2b 2c
VI. . . y '^' = a~' ; a'y ' = j3^ ; (i'a = y^.
(3.) Multiplying these three expressions, in an inverted order, we have, there-
fore, the new product :
2c 2b 2a
VII. . .y-^ ^ a"^ = j3'a'. a'y'. y'jS' = y'2/3'2<i'2 = - 1 ;
and the equation 310, I. is in this way proved anew.
(4.) And because, instead of VI., we might have written,
Compare the Note to page 146.
374 ELEMENTS OF QUATERNIONS. [bOOK III.
- ' — a' — 3'
VIII. . . a'r=- L; (5^ ■■=--,; y" = --„
P y «
we see that the equation to be proved may be reduced to the form of the identity
« 7 /5'
aud may be interpreted as expressing, what is evident, that if a point be supposed to
move first along the side b'c', of the polar triangle a'b'c', from b' to c' ; then along
the successive side c'a', from c' to a' ; and finally along the remaining side a'b',
from a' to b', it will thus have returned to the position from which it set out, or will
on the whole have not changed place at all.
(5.) In this view, then, we perform what we have elsewhere called an addition of
arcs (instead of angles as in 310) ; and in a notation already used (264, (4.) ), we
may express the result by the formula,
X. . . '^ a'b' + r> c'a' + o b'c' = ;
each of the the two left-hand symbols denoting an arc, which is conceived to be added
(as a successive vector-arc, 180, {d.) ), to the arc whose symbol immediately /o//ow«
it, or is written next it, but towards the right-hand.
(6.) The expressions VI. or VIII., for the exponential factors in 310, I., show
in a new way the necessity of attending to the order of those factors, in that formula :
for if we should invert that order, without altering (as in 310, VIII.) the exponents,
we may now see that we should obtain this new product :
2a 2b ic , ,
XI. . . a^ /S"^ y^ =- ^ -, ^ = + (/i8'a')2 ;
(5 y a
which, on account of the diplanarity of the lines a', (3', y', is not equal to negative
unity, but to a certain other versor ; the properties of which may be inferred from
what was shown in 297, (64.), and in 298, (8.), but upon which we cannot here
delay.
312. In general (comp. 221), an equation^ such as
1...?'=?,
between two quaternions, includes a system o//our* scalar equa-
tions, such as the following :
II. . . Sq = ^q; Saq' = Saq ; Sj3^' = S(5q ; Syq = Syq ;
where a, j3, y may be ani/ three actual and diplanar vectors :
and conversely, if* a, /3, y be any three such vectors, then the
four scalar equations II. reproduce, and are sufficiently re-
* The propriety, which such results as this establish, for the use of the name,
Quaternions, as applied to this whole Calculus, on account of its essential connexion
with the number Four, does not require to be again insisted on.
CHAP. I.] A QUATERNION EQUATION INCLUDES FOUR. 375
placed by, the one quaternion equation I. But an equation
between two vectors is equivalent only to a system of three sca-
lar equations^ such as the three last equations II. ; for exam-
ple, in 294, (12.), the one vector equationXXll. is equivalent
to the three scalar equations XXI., under the immediately
preceding condition of diplanarity XX. In like manner, an
equation between two versors of quaternions,* such as the equa-
tion
III. ..JJq'=\Jq,
includes generally a system of three, but of not more than
three, scalar equations ; because the versor \]q depends gene-
rally (comp. 157) on a system of three scalars, namely the two
which determine its axis Ax. q, and the one which determines
its angle /. q ; or because the versor equation III. requires to
be combined with the tetisor equation,
IV. . . Tq=Tq, compare 187 (13.),
in order to reproduce the quaternion equation I. Now the re-
cent equation, 310, I., is evidently of this versor-form III., if
a, j3, 7 be still supposed to be unit-lines. If then we met that
equation, or if one of its form had occurred to us, without any
knowledge of its geometrical signification, we might propose to
resolve it, with respect to the three scalars a, b, c, treated as
three unknown quantities. The few following remarks, on the
problem thus proposed, may be not out of place, nor unin-
structive, here.
(1.) Wiitiug for abridgment,
V, . . cot A = t, cot B = M, cot c = V,
and VI. . . « = — cosec a cosec b cosec c,
the equation to be resolved becomes (by 308, VII., or 309, XII.),
VII. ..(y + y) («+/S) {t + a) = s;
in which the tensors on both sides are already equal, because
* An equation, Up'= Up, or UV9' = UV9, between two versors of vectors (156),
or between the axes of two quaternions (291), is equivalent only to a system of ^ujo
scalar equations ; because the direction of an axis^ or of a vector^ depends on a sys-
tem of two angular elements (111).
376 ELEMENTS OF QUATERNIONS. [bOOK III.
VIII. . . «2 = (y2 + 1) (a2 + 1) (<« + 1).
(2.) Multiplying the equation VII. by t + a, and into t-a, and dividing the re-
sult by i^ + 1, we have this new equation of the same form, but differing by cyclical
permutation (comp. 310, (3.) ) :
IX. ..(« + a)(«+y)(«+)8) = «;
and in like manner,
X. . . (u + p)(t+a)(:o-\-y) = 8.
(3.) Taking the half difference of the two last equations, and observing that (by
279, IV., and 294, II.)
XI V|(i3ar-ariS)=V./3Vay = ySa/3-aS/3y,
"'\i(l3a-a(i) = Y(3a, K(^y-y^) = Y(3y,
we arrive at this new equation, of vector form :
XII. . . = vYpa + tY(3y + ySa/3 - aS/3y ;
which is equivalent only to a system of two scalar equations, because it gives = 0,
when operated on by S./8 (comp. 294, (9.) ).
(4.) It enables us, however, to determine the twoscalars, t and v ; for if we ope-
rate on it by S.a, we get (comp. 298, XXVI. ),
XIII. . . fSa/3y = a2S/3y-S^aSay = S(V/3a.Vay);
and if we operate on the same equation XII. by S . y, we get in like manner,
XIV. . . rSa/3y = y2Sa/3 - SaySy(3 = S(Vay.Vy/3).
(5.) Processes quite similar give the analogous result,
XV. . . uSal3y = |32Sya - Sy/3 S/3a = S (Vy/3 . V/3a) :
and thus the problem is resolved, in the sense that expressions have been found for
the three sought scalars t, u, v, or for the cotangents V. of the three sought angles
A, B, c : whence the fourth scalar, s, in the quaternion equation VII., can easily be
deduced, as follows.
(6.) Since (by 294, (6.), changing S to a, and afterwards cyclically permuting)
we have, for any three vectors a, j3, y, the general transformations,
XVI. . . aSa/3y = Y(Y(3a . Vay), /3Sa)3y = V(Vy/3 .Yl3a),
ySa/3y = V(ay.Vy/3),
the expressions XIII. XV. XIV. give,
Ut +a)Sai3y = Vi3a.Vay;
XVII. ..)(u + (B) Sa(5y = Vy/3 .Y(3a ;
((» + y)Sa/3y = Vay .Vy/3;
whence, by VII ,
XVIII, . . «(Sa/3y)3 = (Vy/3)2(Vi8a)2 (Vay)2;
and thus the remaining scalar, s, is also entirely determined.
( 7.) And the equation VIII. may be verified, by observing that the expressions
XVII. give,
((«« + 1) (Sa/3y)2 = (V/3a)2 (Vay)2 ;
XIX. . . («2 + 1) (Sa/3y)2 = (Vy;8)^ (V,8a)2 ;
( («2 + 1) (SafSyy = (Vay)2 ( Vy/3)=*.
(8.) The equations XIII. XIV. XV. XVI. give, by elimination of Saf3y, these
new expressions :
CHAP. I.J SOLUTION OF THE EXPONENTIAL EQUATION. 377
XX. . . a<-» = (V : S) (Vi3a . Vay) ; /3«-' = (V : S) (Vy/3 . Y(3a) ;
y«-l=(V:S)(Vay.Vy/3);
by comparing which Avith the formula 281, XXVIII., after suppressing (291) the
characteristic I, we find that the three scalars, t, u, v, are either 1st, the cotangents
of the angles opposite to the sides a, b, c, of the spherical triangle in which the three
given unit-lines a, (3, y terminate^ or Ilnd, the negatives of those cotangents, the
angles themselves of that triangle being as usual supposed to hepositive (309, (10.) ),
according as the rotation round a from /3 to y is negative or positive : that is (294,
(3.) ), according as Sa/3y >or < ; or finally, by XVIIL, according as the fourth
scalar, s, is negative or positive, because the second member of that equation XVIII.
is ahvays negative, as being the product of three squares of vectors (282, 292).
(9.) In the 1st case, which is that of 309, (1.), we see then anew, by V. and VI.,
that we are permitted to interpret the scalars A, B, c, in the exponential formula
310, L, as equal to the angles of the spherical triangle (8.), which are usually de-
noted by the same letters. But we see also, that we may add any even multiples of
TT to those three angles, without disturbing the exponential equation ; or any one
even, and two odd multiples of tt, in any order, so as to preserve o, positive product
of cosecants, because s is, for this case, negative in VI., by (8.).
(10.) In the Ilnd case, which is that of 309, (11.), we may, for similar reasons,
interpret the scalars A, B, c, in the formula 310, 1., as equal to the negatives of the
angles of the triangle; and as thus having, what VI. now requires, because s is now
positive (8.), a negative product of cosecants, while their cotangents have the values
required. But we may also add, as in (9.), any multiples of tc, to the scalars thus
found for the formula, provided that the number of the odd multiples, so added, is
itself even (0 or 2).
(11.) The conclusions of 309, or 310, respecting the interpretation of the expo-
nential formula, are therefere confirmed, and might have been anticipated, by the
present new analysis : in conducting which it is evident that we have been dealing
with real scalars, and with real vectors, only.
(12.) If this last restriction were removed, and imaginary values admitted, iu
the solution of the quaternion equation VII., we might have begun by operating, aa
in II., on that equation, by i\\Q four characteristics,
XXI. . . S, S . a, S . /3, and S . y ;
which would have given, with the significations 297, (1.), (3.), of/, m, n, and e,
and therefore with the following relation between those ybi/r scalar data,
XXII. . . e2 = 1-/2-^2 -n2+2Zm»,
a system of four scalar equations, involving theyb«r sought scalars, s, t, u, v; from
which it might have been required to deduce the (real or imaginary) values of those
four scalars, by the ordinary processes of algebra.
(13.) The four scalar equations, so obtained, are the following:
= e + lt-\- mu 4- nr — tuv + s ;
= c< + tiitu + ntv + «w - Z :
^ . . = - ew + ftw + *w + nuv + m - 2/n ;
= ew + <?f + Uv + muv — n ;
3 c
378 ELEMENTS OF QUATERNIONS. [bOOK III,
= e(<2 + l) (fiu-n+Z7n)i
eliminating uv and u between the three last of which, we find, with the help of XXII.,
the determinant,
1, mt, ntv -\-et — l
XXIV. . . = m, t, Itv + ev-n
n, li — e, tv-'r 171 — 2171
and analogous eliminations give,
XXV. . . = c(<2+l) (eu-m+nl),
and XXVI. . . = (t^ + 1) [e^uv - (m - nl) (»-//») + (1 - /«) (et-l + mn)}.
(14.) Rejecting then the factor i^ + 1 we find, as the only real solution of the
problem (12.), the following system of values :
XXVII. .. c^ = ^ — »ran; eu = m-nl; ev = n—lm\
and XXVIII. .. e^s = -(l- P) (1 - m^) (1 - n^) ;
which correspond precisely to those otherwise found before, in (4.) (5.) (6.), and might
therefore serve to reproduce the interpretation of the exponential formula (310).
(15.) But on the purely algebraic side, it is found, by a similar analysis, that
the four equations XXIII. are satisfied also by a system offour imaginary solutions^
represented by the following formulae :
XXIX. ..f'+ 1 = ^5 t,2+l = 0;
\s = tuv — It — mu — «?j — e = ;
which it may be sufficient to have mentioned in passing, since they do not appear to
have any such geometrical interest, as to deserve to be dwelt on here : though, as
regards the consistency of the different processes employed, it may be remembered
that in passing (2.) from the equation VII. to IX., after certain preliminary multi-
plications, we divided by f^ + 1, as we were entitled to do, when seeking only for real
solutions, because t was supposed to be a scalar.
(16.) This seems to be a natural occasion for remarking that the following gene-
ral transformation exists, whatever three vectors may be denoted by a, /3, y :
XXX. . . S(V/3y .Vya .Va/3) =- (Sa/3y)2 ;
which proves in a new way (comp. 180), that the rotation round the line Y(3y, from
Vya to Va/3, is always positive ; or is directed in the same sense (281, (3.) ), as the
rotation round Vaj3 from a to (3, &c.
(17.) In like manner we have generally,
XXXI. . . S (Va/3 .Vya .V/3y) = + (Sa/3y)2,
and XXXII. . . S (Vy/3 . Vay . V/3a) = + (Sa/3y )2 ;
80 that the rotation round Yy (3 from Vay to V/3a is negative, whatever arrange-
ment the three diplanar vectors a, /3, y may have among themselves.
(18.) If then a", b", c" be the negative poles of the three successive sides, BC, CA,
AB, of any spherical triangle, the rotation round a" from b" to c" is negative: which
is entirely consistent with the opposite result (180), respecting the system of the
three positive poles a', b', c'.
(19.) A quantitative interpretation^ of the equation XXX. may also be easily as-
signed : for we may infer from it (by 281, (4.), and 294, (3.) )'that (/"oabo be any
pyramid, and if normals oa', ob', oo' to the three faces BOC, COA, aob have their
lengths numerically equal to the areas of those faces (as bearing the same ratios to
CHAP. I.] EXTENSION TO SPHERICAL POLYGONS. 379
units^ &c,), then (with a similar reference to units) the volume of the new pyramid^
Oa'b'c', will he three quarters of the square of the volume of the old pyramid^
OABC.
313. But an allusion was made, in 310, to an extension oi
the exponential formula which has lately been under discus-
sion ; and in fact, that formula admits of being easily extended,
from triangles to polygons upon the sphere : for we may write,
generally,
2A„ 2A„_i 2A3 2Ai
I. . . a„~ an.~ ... 02"^ ai~= (- 1)»,
if A1A3 . . . A„.i A„ be any spherical polygon^ and if the scalars
Ai, A2, . . . in the exponents denote the positive or negative
angles of that polygon, considered as the rotations a^AiAj,
A1A2A3, . . . namely those from AiA„ to A1A2, &c. ; while n is any
positive whole number* > 2.
(1.) One mode of proving this extended formula is the following. Letoc = y
be the unit-vector of an arbitrary point c on the spheric surface ; and conceive that
arcs of great circles are drawn from this point c to the n successive corners of the
polygon. We shall thus have a system of « spherical triangles, and each angle of
the polygon will (generally) be decomposed into two (positive or negative) partial
angles, which may be thus denoted :
II. . . CA1A3 = Ai', CA2A3 = A2', . . . ;
III. . . AnAiC = Ai", A1A2C = A2", . . . ;
so that, with attention to signs of angles in the additions,
IV. . . Ai = Ai' + Ai", A2 = A2' + A2", &c.
Also let
v. . . AoCAi = Ci, A3CA2 = C3, &c. ;
and therefore
VI. . . Ci + C2 + . . + C,» = an even multiple of tt,
which reduces itself to 27r in the simple case of a polygon with no re-entrant angles,
and with the point c in its interior.
(2.) Then, for the triangle CA1A2, of which the angles Are Ci, Ai', A2", we have,
by 310, III., the equation,
2A2" 2Ai' 2Ci
VII. . . a2 '^ aj T = — y n- ;
and in like manner, for the triangle CA2A3, we have
* The formula admits of interpretation, even for the case n~2.
380 ELEMENTS OF QUATERNIONS. [bOOK III.
2A:j 2A2' 2C2
VIII. . . 03 '^ a2 '^ - - y "■ » &c.
Bat, when we multiply VII. by VIII., we obtain, by IV., the product,
243" 8A2 2Ai' 2(Ci + C2)
IX. . . as "^ a2 '^ ai T = + y 'f ;
and so proceeding, we have at last, by VI., a product of the form,
2Ai" 2A;j 2A2 2Ai'
X. . . ai T a„ T . . . a2 «■ ai «■ = (- 1)'* ;
2Ai" 2Ai"
which reduces itself to I., when it is multiplied hy a "^ , and into a "^ (comp.
310, (3.) ). The theorem is therefore proved.
(3.) In words (comp. 310, (8.) ), " the spherical sum of the successive angles of
any spherical polygon, if taken in a suitable order, is equal to a multiple of two right
angles, which is odd or even, according as the number of the sides (or corners) of the
polygon is itself odd or even'''' : the definition formerly given (180, (4.) ), of a Sphe-
rical Sum of Angles, being of course retained. And the reasoning may be briefly
stated thus. When an arbitrary point c is taken on the spherical surface, as in (1.),
the spherical sum of the two partial angles, at the ends of any one side, is the supple-
ment of the angle which that side subtends, at the point c ; but the sum of all such
subtended angles is either four right angles, or some whole multiple thereof: there-
fore the sum of their supplements can differ only by some such multiple from nir, if
n be the number of the sides.
(4.) Whatever that number may be, if we denote by p„ the exponential product
in the formula I., we have for every vector p, and for every quaternion q, the equa-
tions :
XI. . . pnppn'^ = p', XII. . . pnqpn'^ = ? ;
whereof the former may (by 308, (8.), be thus interpreted: —
" If any line OP, drawn from the centre O of a sphere, he made to revolve coni'
cally round any n radii, OAi, . . OA^, as n successive axes of rotation, through an-
gles equal respectively to the doubles of the angles of the spherical polygon Ai . .An,
the lin