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Full text of "Elements of quaternions"






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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. 


[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. 




ANGLES, OR TO ROTATIONS, . • . . 1-102 


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. 





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. 



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 


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. 


TIONS, • 103-300 


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 


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, 


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. 



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 



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. 




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 




TIONS OF QUATERNIONS, 301 to the end. 



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 


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. 


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- 

Section 11 — On Powers and Logarithms of Diplanar Qua- 
ternions ; with some Additional Formulae, .... 384-390 


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.* 




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. 


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- 

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, 

unless q and d^ be complanar ; and therefore that we have not generally, 

dlp = ^, 

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 

Section 5. — On Successive Differentials and Developments, 

of Functions of Quaternions, 420-435 


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). 


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. 



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 


being followed by several subarticles, which form with it a sort of 

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 

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. 


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) 


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 ) 


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) 


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') 


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. 


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 

(/). 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. 


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. 


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- 


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) 


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 


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 


CONTENTS. xxiii 

(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. 


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 


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. 



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 

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. 


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- 

(<?). 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. 


{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 

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 

p-2^-2 = Ti.2/Udp = /i = const; (Si) 

vfhexe F is the perpendicular from the centre o on the tangent plane, 


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 

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. 



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 

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 

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- 

(/). 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'. 




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') 


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), 


(<?). 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 

(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 


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- 

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 


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 

CONTENTS. xxxvii 

(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. 



(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. 


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. 


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- 

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, 



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- 

(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. 


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 



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- 

(h). It may happen (p. 701) that the differential equation, 

S»'dp = 0, (Y2) 

CONTENTS. xliii 

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- 

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 

(/). 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. 


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 

(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. 


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. 




(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. 


. CONTENTS. xlvii 

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- 

(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. 


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 

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. 


(/). 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 

(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.) 



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. 


(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 


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\ 


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. 


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 



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. 

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- 

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 


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 

^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. 


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- 



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. 


Table* of Initial Pages of Aeticles. 






Page, t 


































































































187 ; 












190 1 




















„ ; 




192 : 











193 i 























202 I 























204 ; 











207 ! 






















































































































































































































































































































































j 142 
































* 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. 

Table of Initial Taq-es— continued. 












Art. Page. 










\ 409 664 











410 667 











i 411 674 











! 412 679 






423 ' 





413 ( 












414 694 










415 698 





! 344 





416 ' 












1 417 709 1 











418 ' 












i 419 ' 












420 ' 












421 ' 












422 ' 












423 ' 























































































1 407 


. , 









1 408 



• • 

Table of Pages foe the Figuees. 














21 i 









i 22 

25 1 








1 23 




66 bis 




! 24 










41 bis 



















42 bis 





































! 31 


46 bis 







1 32 




63 bis 






i 33 











47 bis 




























i 85bis 










; 36 










1 B6bis 







. . 



i " 







• • 

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. 






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 







[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 

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- 

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. 


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, 


c - D =A-B, 


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, 


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 


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 


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. 


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 


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.) 


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; 


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- 

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, 

then y is a fraction of j3j which is denoted as follows, 7 = — jS ; 



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. 


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^ = ^; 


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 


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 


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 


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. 



Section 1. — On Linear Equations connecting two Co-uiitial 


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 


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 


« + 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 

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 



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 


«a -f ^>/3 + C7 = ; 

and if we make 

, - aa 

oa = , 


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. 




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. 




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 


case (comp. Art. 17, Fig. 13), the quotient — must be equal 


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 = ; 


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 

« + i + c = 0. 

This condition may also be thus written, 

Fig. 10. 

-a -b 
c c ' 


or — + — = 1 ; 


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; 


= 6.AB 4 C.AC = C.BC + «.BA = a.CA + 6.CB. 

Hence follows this proportion, between coefficients and seg- 

« :6:c = Bc : CA : ab. 


We might also have 

! observed that the 

proposed equations 


bf3 + Cy 


cy + aa 

aa + bf3 
a + b ' 



y -a 

= ^=-*<S 


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, 


then, in 




shall have this 





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, 


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 


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 

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.) 


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 


whence the sought vectors are expressed in either of the two 
following ways : 


J , -aa 
1. , . a =7 , 

b + c 

^ c + a 

'^~a + b' 


, 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 


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 

ABC + BAC = 0, 

exactly as we had (in Art. 5) for any two points, the equa- 

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, 

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. 



[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 


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; 



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), 




The first expressions in Art. 27 for |3', 7', give the equa- 


Fig. 21. 

(c -f «) j3' + ^>i3 = 0, (a + &)y + C7 = ; 


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 


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, 


{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' 


„, _ 2aa + bj5 + Cy ^,„ _ 2^/3 + cy + aa 

^ 2a + b^c ' ^ ~ 26 + c + « ' 
,„ _ 2cy -{ aa^bf5 
^ 2c + « + ft 


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 

„, (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). 


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 



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 

«a + 6/3 + cy = 0, 

shows us at once that these two 
vectors are. 


1 - Pl ; 

Fig. 22. 

001 = ^1 = 


{z-y)b + za * 
whence we derive (bj 25) these two other anharmonics, 

(cb'aBi) = 


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 




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) = - ; 


(bac'Ci) = (ba'cAi) =-; 

and we see that (as was to be expected from known princi- 


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, 

aXa-p) + b'(f5-p) + c(y-p) = 0, 
which gives 

a a + b'Q + cy 

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), 


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 


zcy + xaa 


zc\ xa ' 

xaa + yhfi ^ 

^ xa^yh Fig. 23 

which give at once the following anharmonics of pencils, or of 

(a . BOCP) = (ba CPi) = - ; 


(B . COAP) = (cb'aPz) = - ; 

(C . AOBP) = (ac'bPs) = - ; 


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). 


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 


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 

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 


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, 

[-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] ; 


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, 

— = (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, 


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 


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- 


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^. 


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 

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. 


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. 



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. 


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 

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 ; 

-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) 


under the form, ^— < —7—, we see that the conic is an ellipse, 

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 


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 

= 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- 

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. 


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. 


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) 




[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 


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. 


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- 

(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. 


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 

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 


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. 


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-\ 


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 

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. 


we may therefore assume for its tangential co-ordinates the 

/ = 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- 

= 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 


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. 


(/+ 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. 



Section 1. — On Linear Equations between Vectors not Com^ 


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'^. 




[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 

aa + b(5 + cy + dd = ; 
which will give 

g = 



where oa', ob', oc'. 



or od = oa'+ ob'+ oc' 


-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 


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 


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 

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 



[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 + 


cy + aa 

&/3 + 


cv a 

- b + 


aa + b^ 

Cy + 


a +b 

c + 


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 = ; 


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 


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 

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 

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. 





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, 


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. 


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 = ; 

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. 


Section 2. — On Quinary Symbols far Points and Planes in 


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), 



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 

* 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. 


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 


(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 


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- 

/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 

^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. 


X=:y = z-u', w^tz+u', V = «4 + ?/; 


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 


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 

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 


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, 

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 


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- 

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 


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 


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 : 

(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, 


(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: 


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," 



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, 


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 


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, 


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. 


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). 


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, 


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, 


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 

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. 


[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. 


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 

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. 



(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 


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, 


(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 


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. 


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. 





+ 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. 



of Intersections A 















































































(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 


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- 

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. 




[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 


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. 


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- 

(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 

(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 : 


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, 

^ 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 

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- 

= (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 


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), 


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- 

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 

(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) = &. 


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. 

/(3)-l, 0(3) = 3, x(3) = 4: 

2x (s) = 3»-i - 1, 

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 ; 

F(s) = 22»-3+2«-2-3«-i; 
so that 

P(» + 1) - 4f(») = 3*-» - 2«-i, 

^(* + l)-4,^(*) = 3/(.); 

which last equation in finite differences admits of an independent geometrical inter- 

(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, 


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. 


B = — — or 65=20.4, if 6 = a. 

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 


(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. 



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 

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- 

f As regards the uninterpreted character of such imaginary contacts in geometry, 
the preceding Note to the present Article, resptcting imaginary intersections, may be 



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, 

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) ; 


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. 


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- 

= 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 ; 

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. 


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- 

(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 

the centre K is on the plane [a, /», r, d] ; or in other words, it is infinitely distant : so 


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- 

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. 


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, 

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.) ), 


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 

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. 


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 

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. 


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. 

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 

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. 


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. 


(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 

As is well illustrated by Atwood's machine. 


(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- 

(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 

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. 





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- 


tion — in Art. 14; 

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 


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 


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. 


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. 



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 ' 


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- 


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- 

going Section, stated in Art. 104), abstain fi-om writing also 

^ =^x, under the same conditions : x still denoting a scalar. 

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. 


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. 


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. 



[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- 

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. 


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. 



[book II. 

rence to an angle on a desk, that the Four Elements which it involves are the follow- 

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. 


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. 


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 


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. 



and therefore also the tioo equations between quotients (117, 



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, 

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. 


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, 

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» 


124. With the same notation for complanarity, we may 
write generally, 

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^- 


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. 


(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 


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 



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. 


(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 


defined in 130) remains unchanged^ but that the axis (127, 
1 28) is reversed in direction : so that we may write gene- 

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. 




[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, 


= <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- 


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. 


zero, or to two right angles. We may therefore write, gene- 

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 

? = 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. 


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 

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 


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). 



[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, 


P-kP = 


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 


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; 

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 


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. 


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 


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 

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, 


Compare the Note to sub -art, 3. 




[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- 


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- 

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. 


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, 


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, 


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. 


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. 


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 

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. 


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. 


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. 


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- 


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 

* 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. 



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, 


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. 


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, 



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, 


^^ ^ \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 


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 


^ -. 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 



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. 


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 



then U- must be (by 154) a right versor ; and reciprocally, every right versor, with 

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 

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. 


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, 


A AOB OC COD, — = — ; 


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- 


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. 


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. 



(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. 


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, 


(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 


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. 


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., 

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, 

if ^^ = - 1, 2''^ = - 1, and Ax. q x Ax. q, 

{qqy=-{qqy^-\, q'q = -qq\ 


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, 



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 


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 


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). 



[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. 


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). 



[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, 


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 


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. 


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 

(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 





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 

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 


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, 

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'. 




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 


(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 

(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. 


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. 


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), 



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, 


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.). 


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, 


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. 


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, 

(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. 


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, 


which expresses that the scalar part* of the quaternion - is ze/o, and therefore that 

this quaternion is a right quotient (132). 
(7.) In like manner, the equation 

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 


(11.) If U/3 = - Ua, then T(/3 + a) = + (T/3 - Ta), according as T/3 > or < Ta ; 

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 

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. 


(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^ ^ ,„^ ,_^ 


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 


I. . q = 0b: o\ = (ob : oa') . (oa' : oa) = T^' . Uq ; 

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^ 


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. 


(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. 


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. 


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^'. 

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. 


we shall have, by III., 

^=K-, or ADOCa'AOE; 

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 


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, 


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.) 


Section 12. — Oii the Sum or Difference of any tico Quater- 
nions ; and on the Scalar (or Scalar Part) of a Quater- 

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. 


(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, 


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 

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 


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 


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 

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, 

(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. 


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.). 
(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- 







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, 


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. 


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, 

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. 


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, 
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. 


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, 

II. . . SS^ = 2S^, or briefly, SS = 2S ; 
where 2 is used as a sign of Summation : and may say that 

* Comp. 146, (10.), &c. 


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 


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 

(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- 

(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.). 


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 


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. 


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', 

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 


(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.); 

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.). 

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., 


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 


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, 


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 

* Compare 145, (10.) ; and several subsequent sub-articles. 


(4.) Conversely if we take the equation, , ""^ ' ^ /Vl^ 

N^ = a2, of 145, (12.), ^ 


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. 

(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- 

(10.) By changing 7 to 7 + 1 in (8), we find that 

if Tq=\, then S - — - = 0, and reciprocally. 


(11.) Hence if T|0=Trt, then S^ — ^ = 0, and reciprocally ; because (by 106) 
|0 + a 

-a p—ap+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. 


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 


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, 


OA) = 

b'b : OA, 


cause, by construction, b'b = ob". 


C2.) In like manner, for Fig. 36, we have the equation, 

P !>^ 


OA) = 

a'b : OA. 

(3.) Under the recent conditions, 



S(/3":a) = 0. 

(4.) In general, it is evident that 

V. ..g=0, if S^=0, 


V5'=0; and] 


(5.) More generally, 

VI. ..9' =9, if ^q=Sq, 


Yq = Yq ; with the converse. 

(6.) Also VII. ..Y9 = 0, 


lq = 0, 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. 


(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, 


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 


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 




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, 


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 


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. 


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, 

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 ), 


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 

(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, 


[^'^-[^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- 


(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, 



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, 



T^=l, by 190, VI.; 

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- 


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. 


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, 


"' [^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- 

(14.) The result of the elimination of x between the two last equations, namely 
this new equation, 


NS ^ + NV §= 1, by XXV., XXVI. ; 

a p 


Nfs^ + v|Ul, by XXVII.; 
or finally, 

Tfs^ + V|^=l, by 190, VI., 


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.) ), 


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, 


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, 


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- 

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. 


(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 : * 

XXXIV. . . tan z: 9 = — - . 5 ; or XXXIV. . . tan Z 9 = (TV : S) 9 ; 


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 


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. 


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- 

(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 

* 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 


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. 


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 

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). 


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 

(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 

(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. 


t = TVq = Tq. sin Lq, f = TYq' =Tq\ sin /.q\ 

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, 
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; = ; 

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. 



[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: 



XXIV. . . s 

^ V- 


(4.) Also, by XIX. XX. XXI. XXII, 
still positive, 



sin a cosec c cos b : 

= + sin a sm c < 

XXVI. . . L 




+ 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. 


= + ^; 

xxxn. . .ivi v^.v 

V — 1 = — /3 sin a cosec c sin b 
a j 


+ (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. 


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 


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 

(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. 


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, 

^q' = qy + q2+"-\-qm> ^q' = q'i + q2 + ' •+q'ny 


-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 

* 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 


(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. ; 

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 


(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; 


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. 


(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- 

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. 

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) ; 


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 ; 


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. 


b'c' = tt — a, c'a' = tt — b, a'b'= tt — c, 

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: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., 


{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. 


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, 

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-. ^^ 




p = a-2(3S- 

* Compare the Notes to page 90, &c. 


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. 


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 


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.] 



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 


+ «^ 

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 


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. 



[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-, 

« £ 


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.) 


(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 ' 


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, 

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- 

* 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. 


(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) 

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. 


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- 


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; 


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- 


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 


6=- • . a, 

' /3-i-a 


which give (by 125) these other expressions, 

we have 

y V 

VII. ..^ + ^ = 2, 
a /3 




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. 




(6.) Under the same conditions, for any arbitraiy vector p, wo have the trans- 

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 



[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, 


— {U^-^^vl^xW 


if we make 


^^"•••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; 





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.), 


and therefore 

CD = -K-.p, 

P . 


Fig. 53, 

xxiir. . . DB 



- rf • 







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. 


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- 

(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 

(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 

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- 


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)' 


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 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 


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, 


[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 ; 


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 

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 ; 

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. 


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 : 


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 


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 


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^ . 


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. 



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^ 


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 


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- 

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" ; 


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. 


(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. 






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 


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- 

"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- 

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 


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' 

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. 


q^ and denoted by 'sj q. We may therefore establish the for- 

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). 


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. 


(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- 

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 

(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. 


(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- 


tained from j3 by two successive and positive rotations, each 
through the third part of a circumference, 
so that 

fi' 15" 13' 




and therefore 

V... (|)- = (|)-=,,*„v....f =(!)•, l-d 

we shall have 


SO that we are equally entitled, at this stage, to write, instead 
of III. or III'., these other equations : 

vii...&'=f^Y, li'M' 


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- 


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 

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- 

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, 

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. 


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 


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; 


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. 


(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 



(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 — : 

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. 



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— . 

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, 


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 


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 

XI. . . amo q = loq = ±lq', 

XII. . . am„ a = am,, 1 = Zn 1 = 2n7r, if a > ; 

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. 


(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 


IX'. . . cis z' . cis z = cis (z' + 2) ; X'. . . (cis 2)2 = cis 2z ; 

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 


interpreted in 234, (5.)j as a symbol satisfying an equation which may be written 

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) ; 


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 

237. Ifp be still a whole number, we have thus the transforma- 

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. 


will conduct to one common value of this whole power q^. And if, 
for I., we substitute this slightly different formula (comp. 235, 

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 

IX. . . (- iyn = clsp(2n+ l)7r; whence (comp. 232), 

X. ..(-l)io = cis-=+t-, (-l)ii = ci8-2- = -i, (-l)i2 = + t, &c.; 

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. 


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 


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^. 


(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. // 


(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/. 


(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 ; 

XXIV. . . (py = S.i-^Fiy = SPi(y-c)=f(y-c), 

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. 


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, 


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.) ). 


(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, 


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. 


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 

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. 


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 


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, 

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 

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 


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. 


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, 


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\, 


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- 

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, 


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.)). 


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 


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. 


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, 


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- 

(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 ; 


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, 

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/>, 


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 


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; 

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 

(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 



[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 


(T = -— , and 


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- 

p +a /3- a 

A OA'P a PAB ; 


a p — a 

whence it follows by the simplest calculations, 



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 


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. 

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. 


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 

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 

(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. 


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, 

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. 


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 

(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. 


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- 

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; 


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. 


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, 


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 


(4.) The inverse similarity I. gives also, generally, the relation, 


(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. 


(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.) ). 



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. 


(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 

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 

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- 


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; 

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., 



.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. 


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. 

(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.). 




Section 1. — On some Enunciatio7is of the Associative Pro- 
perty, or Principle, of Multiplication of Diplanar Quater- 

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 


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- 

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- 

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 


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. 


(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 



[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. 


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 

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. 


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- 

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. 


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. 


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. 


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. 


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- 

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. 


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. 



(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. 


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- 

(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. 


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- 

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. 





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. 


(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- 

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- 

* Compare the Notes to pages 14G, 159. 


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- 

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- 


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- 

{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 


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, 


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 


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; 

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. 


(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 

(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. 


XXL . . Ra = l:i = a-S 


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. 


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- 

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- 

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- 

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.) ) 


(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. 


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, 



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, 


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 


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. 


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- 

(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. 


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 ; 


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. 


(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. 


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 


XL. ..Sap=a, S/3p = &, Syp=r, S^p = d; 


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, 


and may then write the result (comp. 68) under the more symmetric form (because 

— EBCD = BECD = &C.) : 


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), 


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 


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 


(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 

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. 


(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. 


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, 


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 


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, 

IX'. . . p-^ - a-^ = XT : a^p^, 

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. 

(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. 


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 ; 


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; 


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 


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 

(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- 

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, 

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.).) 


(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 


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 

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, 

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 

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- 

(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. 


Section 7. — On the Fourth Proportional to Three Diplanar 


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, 


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 


£ = 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 

(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. 



[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 


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. 


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- 


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- 


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 


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- 

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 


XLII. . . ^ = ny-X, e = ny+\, Ydt=2ny\, UV^e = UyX, 


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. 


(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 ; 

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 


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 

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. 


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 


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 - 

(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. 


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. 


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. 


(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 


(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- 

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. 


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. 


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. 


(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 

(9.) With the same direction of rotation, we have also the conjugate or recipro- 
cal formula, 

XIV. . . ^^- = -(/3a-»y)2 = cos2-OD.sin2. 


(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' ; 

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- 

2 Y 


(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. 


(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). 


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 


/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, 


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 

* The formula here referred to should have been printed as Ra = 1 : a = a-*. 


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 

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 


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 


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.). 


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'^ ; 

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 


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. 


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 


. ^ = )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, 

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- 


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. 


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 


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 





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 


(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 

(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 


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- 

* 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). 


" T/'klmn be any spherical quadrilateral, and.l any point on the sphere ; if also 
we complete the spherical parallelograms, 


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 

_,__ , a sin a cos B + /3 sin i cos A 

XII. . . y = . ; 


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 

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 


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. 


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 : 


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+- 


with intepretations which easily offer themselves, on the principles of the foregoing 

(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- 

(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 


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. 


(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- 

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. 


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.^; 


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- 


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. 


(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 : 

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 : 

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. 


(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 


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 

(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 

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 


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 
y "" , vfe have the transformation (comp. 309, (1.) ) : 

2b 2a 2c 2(7r — c) 

IIL . . j3^ a^ =-y ^ =y '^ . 


(2.) Again, multiplying both members of I. intof a t, we obtain this other for- 

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 

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. 


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. 


(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- 

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. 


- ' — 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 

312. In general (comp. 221), an equation^ such as 


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. 


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- 

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). 


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 : 


XX. . . a<-» = (V : S) (Vi3a . Vay) ; /3«-' = (V : S) (Vy/3 . Y(3a) ; 

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 


= 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 


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^ 


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, 

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