macfarlane a papers on space analysis- imaginary algebra

Astudent of physics finds a difficulty in the principle of Quaternions whichmakes thesquare of a vector negative.. 1,A denote avector of length a and direction a, and B another vector of

Papers on Space Analysis.

BY

A MACFARLANE, M.A., D.SC, L.L.D.

Fellow of the Royal Society of Edinburgh,

Lately Professor of Physics in the University of Texas.

CONTENTS

i. —Principles of the Algebra of Physics 1891 56 pages

2.—(roiitimiation-: The Imaginary offtg'Algebra 1892, 26 pages

(Reprinted from the Proceedings of the American Association for

the Advancement of Science.)

3.— The Fundamental Theorems of Analysis generalized for Space

B WESTERMANN & CO.,

Fellnw of theRoyal Society of Edinburgh, Professor of Physics

in the University of Texas.

PMNTBDBT

mentofscience,Vol.xl, 1891.]

Principles op the algebra of physics By Prof A Mactarlane,

University ofTexas,Austin,Texas ;

[Thispaperwasread before a joint session of SectionsAand Bon August21.]

Laseulemanierede bien traiter leselemensd'une science exacte et rigoureuse, c'est

d'y mettre toute la rigueur et l'exactitude possible.

D'Alehbert

Thequestion as to the possibility of representing areasandsolids bymeansof the

apparentmultiplication of thesymbolsfor lines hasalways appeared tometo beone

of great difficulty in the application of algebra togeometry;nor has the difficulty, I

think,beenproperlymetinworkson the subject.

D F.Gregory

Tant quel'algebre et la geometrie ont et^ separees, leur progres ont ete lents et leurs

usagesbornes,mais lorsque cesdeuxsciences se sont reunies, elles se sont prfitees

des forces mntuelles, et ontmarche ensembled'un pas rapide vers la perfection.

Lagrange

Intheprefacetotheneweditionof theTreatiseonQuaternionsProfessor

beenmadewiththedevelopmentof Quaternions Onecause,whichhasbeen

intenton modifyingthe notation, or themodeof presentationof the

fun-damentalprinciples, thanon extending the applications of the Calculus."

Attheendof the prefacehe quotesafewwords fromaletterwhichhe

re-ceivedlongago from Hamilton—" Could anything besimpler ormore

sat-isfactory? Don't youfeel, aswell as think, thatweareonthe right track,

devotion to exact science Ihave alwaysfeltthatQuaternionsisonthe

righttrack,andthatHamilton andTaitdeserveandwillreceivemoreand

moreastimegoes on thanksof the highest order But atthesametime

I amconvincedthatthe notationcan be improved;thatthe principles

re-quiretobecorrectedand extended; thatthereisamorecompletealgebra

which unifies Quaternions,Grassmann's method and Determinants, and

appliestophysical quantitiesinspace Theguidingideain this paperis

generalization Whatissoughtfor isan algebrawhichwillapplydirectly

tophysicalquantities, will includeand unify theseveralbranchesof

isopportuneforadiscussion of thisproblem isshown bythe recent

66 SECTION A.

cussionbetween Professors Taltand Gibbs inthecolumns of Natureon

the merits of Quaternions, Vector Analysis, and Grassmnnn's method;

andalsobythediscussioninthesameJournal of themeaningof algebraic

symbolsinapplied mathematics

Astudent of physics finds a difficulty in the principle of Quaternions

whichmakes thesquare of a vector negative Himilton says, Lectures,

page53,"Everyline intri-dimensional space hasitssquareequaltoa

the whole quaternion theory." Now, a physicist isaccustomed to

the expression Jimi2

. In that expression |m is positive, and as the

wholeis positive, v"mustbe positive; butvdenotes thevelocity,which

is adirectedquantity If this isamatterofconvention merely, then the

convention in quaternionsoughtto conform withtheestablished

conven-tionof analysis;if it is amatteroftruth,whichistrue?

The question ispartof the wider question—Is it necessaryto take, as

isdoneinquaternions, thescalarpartof the productof twovectors

be worked out without the minus,but that the expressions soobtained

are moreconsistent withthose of algebra Let, for example (fig 1),A

denote avector of length a and direction a, and B another vector of

length6 anddirectionft,theirsumisA +B,andthesquare oftheirsum

I take to be a* + 2o6 cosaft+ V, where cos aft denotes the cosine of

the anglebetweenthe directionsa andft.

SupposeBtochangeuntil its direction is

the same as that of A, the above ex- A-t"f3

pression becomes as+lab+ 62, which

agrees with the expression in algebra.

But the quaternion method makes it

— (a2+2ab+62). The sumofA and

the opposite of B is A — B; its square a_n^

is a2 —2a6 cosaft + 6swhich becomes

os—2ab +6 s

, when A and B have the

nionsit is—a?+2a6—b".

In ordinary algebra thereare twokinds of quantity, the arithmetical or

(fig 2), can be adequately representedonastraightline produced

jY > > — Allthe additive quantities arelaidoffend

"~ ~" to end, and from thefinal point the

sub-Fig 2. tractive quantities arelaidoffendto end,

under the supposition that the quantity is signless. But the algebraic

quantity requires for its representation (fig 3),a straightlineproduced

ASTRONOMY 67

quan-tity, which mayhave one or other of two directions. But thoughthis

quantityhas a sign, itssquareis signless, oressentially positive. Hence

only a positivequantity hasasquareroot,andthatrootisambiguous,on

accountof thetwodirectionswhichthe algebraic quantitymayhave The

generalization ofthis forspaceis that thesquare ofanydirectedquantity

Thereis a wantofharmonybetween the notation of Quaternionsand

thatofDeterminants Let, asusual,

B= xi+yj +zk, p=x'i+y'j+ziJc, y=x"i+y"j +z"k,

then

Sa/3r = — X

68 SECTION A.

mentalrulesof quaternions Thesewefind intherules forthe

combina-tionofthesymbolsi, j, andk,namely

jk=i ki=j ij=k

Intheprefaceto hisLecturesHamilton narrateshow,in his searchfor

the extensiontospaceof theimaginaryalgebraof the plane,he arrivedat

theserules,andhowhaving

that thenewinstrument for

apply-ing calculationtogeometry had been

physics? Writers on quaternions -J

things—the summing of angles in

space, and the rotation of a line

about an axis. Let(fig 4) i, j, k,

denotethreemutually perpendicular

axes which areusually designated

as the axes ofx, y and s. In

or-dertodistinguishclearlybetweenanaxisandaquadrantof rotationabout

J J J

direc-tion abouttherespectiveaxes. Thedirectionsof positive rotation are

in-2 2"

notexpressed explicitly) a quadrantof the greatcircleroundjfollowed

'byaquadrantof the greatcircleround k;thesumof these isthequadrant

a

fromkto j,whichisthe negative of aquadrant roundiori ; oritmay

beconsidered as a quadrant round —i, and therefore denoted by—i

Hence, supposing the order of the summing to be from right to left,

Again(see fig 4)byj k ismeantaquadrantof thegreatcircleround

kfollowedbyaquadrantof the greatcircleroundj;this isequivalent to

thequadrantfrom—jto—k,whichis aquadrantof the greatcircleround

an angleJanda multiplier

l/(«»»'—nm'y +(nV—In')1+ Qm1—ml')1

Thesetwotermstogetherdenote the arc of a greatcirclewhich isthe

sumofthe two givenarcs, its axis being theaxis specifiedand its angle

suchthat— (W +»»»»'+ran') is its cosine.

Wehave nexttoconsider the other meaning whichisgiventothe

fun-damentalrules :that theyexpress the effectof arotationon aline. Let

TT

s

i j denote the turning bya quadrant roundiof a line initiallyalongj ;

andhere Iintroduce theZ to denoteexplicitlywhatismeant bythefirst

symbol Hamilton obtains thesameelementaryrules asbefore,namely,

or, tospeakmorecorrectly,thefirst sixare obtained, while the

remain-ing three areassumed Aquadrant rotationroundj(seeflg 4) changes

IT

a

equa-tionsof thefirst set. Aquadrantrotationinthe positive direction round

kturns aline originallyalongjto aline in the direction opposite to i;

TT

hence k j= —i. Similarlyfortheothertwoequations of thesecondset.

7T

2

If wekeep tothesame meaning of the symbols as before,i i ought to

meantheeffectofaquadrantrotationroundiupona line inthe direction

TT

ofi;andasthatproducesnochange,weoughttohavei'i=i. Similarly

j j=jand k k=k Itfollows that the truemeaningof therules lies in

the summing of versors or arcs of great

attemptto formtheproductofa

quadran-tal rotation round any axis and any line.

Let li-\-mj+nk denotetheaxisa (fig 5),

round which there is aquadrant of

turned Ifthedistributive rule applies,we

get the result by decomposing the

quad-rant rotationroundthegiven axisintothe

sumof threecomponentrotations

MATHEMATICS AND ASTRONOMY 71and finding their several effects on the several components of the line

xi-\-yj-\-zk. Accordingtothe quaternionrulesweobtain—(Ix+ my +nz)

+(mz—ny)i-{- (nx—lz)j+(ly—mx)k. Now this expression is not

the expressionforthe resultingline,orforanyline,unlesslx-\-my-\-nz=0.

Whatisthetrueexpression? It is (Ix+ my-f-nz) (li+mj+nk) which

isthecomponentalongtheaxis,and (mz—ny)i -\-(nx—lz)j+(ly—mx)k

istheexpressionforthe other component,which is perpendiculartothe

axisandtheinitial line. The argument here is, of course,not somuch

aboutthe proper expression for the resultof the rotation, asabout the

meaningof thefundamentalrules.

necessary to identify a vector of unit length with a quadrantal versor

havingthe same axis. In the newedition ofhis Elements, p 46, Prof.

Tait makes the transition fromversors tovectors thus "One most

im-portant stepremainstobemade "We havetreatedi, j,k simplyas

quad-rantalversors,andi, j,kasunit-vectorsatrightanglestoeachother,and

coincidingwiththe axesof rotation of these versors Butifwecollate

and compare the equations justproved,wehavei = —1, i2= —1, § 9

ij=kandi j=k;j i= —k andj i= —k Nowthemeaningswehave

assigned to i, j,k are quite independent of, and not inconsistent with,

thoseassignedto i, j,k Andit issuperfluoustousetwosetsof

charac-terswhenonewill suffice. Hence itappears thati, j,kmaybe substituted

fori, j,k;in otherwords,a unit-vectorwhenemployedasafactormaybe

considered asaquadrantalversorwhoseplane isperpendicularto the vector.

Ofcourse,itfollows thateveryvectorcan betreatedas theproductof a

numberanda quadrantal versor Thisisoneof themainelements ofthe

singular simplicityof thequaternioncalculus."

TT 'Z

Byi isheremeantwhatwehave designatedbyi andbyi aunit-vector

alongthe axis of i. We have alreadyseen one difficulty opposing the

7T

2

that insuperable objection, there stillremains forconsideration the case

ofthecombinationoftwovectors This kindof product,inwhichboth

factorsare vectors, has in recent times been generally neglected This

isevident from what is said by Clifford (Mathematical papers, p. 386)

"In every equation we must regard the last symbol in every term as

eithera vector or an operation; but all the others must be regardedas

operations." Thisviewdoesnot explain the productof physical

quanti-ties.

then accordingtothe principles of quaternions

(yj) (yj) = —y2 (zk) (zk)= —z*.

Asthe distributive principleis tobeapplied,themeaning of these

par-tialproductsmustbesuchthat theproductofanytwovectorsisobtained

Fig 6.

bytaking the products of the severalcomponents of theone with the

sev-eralcomponentsof theother.

Letyjzkdenoteor be represented (fig 6)bythe rectangle included

be-tweenyjand zk;its magnitudeis yzand its orientation is defined byjk.

Butinspace of threedimensions the aspect or orientation jkmaybe

rep-resented so far as direction is

con-cerned by the complementary axis i.

Hence we maywriteyjzk=yzjk—yzi.

xiyj=xyij=xyk. Theexpressionzylcj

denotes the same area in magnitude

and plane as yzjk,butistaken the

op-posite way round; the

complement-ary axis is —i. In the same sense

quaternion rules appear to hold good

closer consideration We havetaken the vectorsintheorderofwriting

and obtain jk=i; if, aswas pointedout,we take the versorsalso inthe

TT IT -n

order of writingweobtainj k = —i

Thequestion remains: What consistent meaning must beattached to

xixiandyjyjandzkzkin order that whenthey are taken along withthe

other partial products we may obtain a

completedistributiveproduct? Theviewwhich I have arrived at is that the ex-pression xixi=xH' i meansthearea of thesquare which isformed bythe projection

of xi on itsowndirection; andthatit is

termsaretobe combinedbyarithmetical

Hence I take the rules to be ii = -f-,

jj=+, and kk=+

Let B,=xi+yj+zk and R'=k'i+y'j+z'k be any two line-vectors.

Byapplying theaboverules distributivelyweobtain:

RR' =(xi+yj+zk) (x'i+y'j+z'k)

=xx'-fyy>+zz'+(yz'—zy') i-f(zx'—xz>) j+(xy'—yx<) k.

LetOPand OP' bethe projections of RandR'onthe planeofiandj.

OPP'=xy'—kxy—b x'y'—J (a;—x') (y<— y)=h(xy'—yx')

Thus (xy'—yx>)kdenotes themagnitude andorientation of the

parallel-ogramformed bythe projections ofRandR'onthe plane ofiandj.

Sim-ilarly (yz'—zy') idenotes the oriented areaformed by theprojections of

AND ASTRONOMT 73

BandR'ontheplane ofjandk,and (zx'-xz')jthat for the plane ofk

andi. The geometrical sum of these areas is equal in magnitude and

orientationtothe area of the parallelogramformedinspacebyRandR',

or rather the areaformed byRandthecomponentofR' whichis

perpen-dicular toR

The expression xx'+yy'+zz' isthe areaformed byRand the

projec-tion ofR' upon R For (flg 8) the projection ofR' is equal to ON,

whichisequalto OL + LM+MN,thesumof the projectionsonRofx'i,

y'jandz'krespectively HencetheproductofRandthe projection ofR'

is

f x'— +y'^-+ z'-\ =xx'+yy>+zz>

Fig 8.

Hencebythecomplete productRR'we meantheproductofRandthe

com-ponent ofR' which is parallel toR,

to-gether with the product of R and the

component ofR' whichis perpendicular

to R This product is distributive,that

take theproductdirectly,or takethe

sev-eralproductsof thecomponentsofRand

R' and add them together, the

non-di-rected products byordinary addition, the

directedproducts by geometricaladdition.

The expression xx1 +yy'+zz' isoneof

the fundamental expressions of the

Car-tesian analysis; the other term is

ex-pressed by the square rootof thesumof the squares of itscomponents,

namely,

-[/ (yz>~ zy'f+(zx<—xz>) *+(xy 1—yx') %

becausethat analysisdoes notprovideanexplicitnotationfor direction.

What reasondo writerson quaternions give for takingxx1

+yy1+zz'

negatively in the case of the product of two vectors? In the passage

quotedaboveProfessor Taitrefers tosection9ofhis Treatise fortheproof

that the square ofaunitvectoris—1. Therewefind :"Itmaybe

interest-ing, at this stage, toanticipate sofarastoremarkthatinthetheory of

qua-ternions the analogue ofcosd-\- -\/— 1 sin is cos8 + m sin 0, where

di-rectedunit-linewhateverinspace."

Inthe aboveexpression <o reallymeans theversoro> . The algebraic

imaginaryi/—1means, asiswellknown,aturningofJ; what is

constant,ifthe rulesaboutthemanipulation ofj/—1aretoholdgood

The true reason for taking the expression negativelyisto satisfythe

thatiftheproduct

(xi+yj+zk) (x'i+y'j+z'k) (x"i+y"j+z"k)

number,thenwemusthave

xix'i= —xx' yj y'j= —yy' zhz'h= —zz'

"Onthis planeveryline intridimensional spacehasitssquare equalto

a negative number."

But what quantityinspace possessessuchassociativeand distributive

properties? It isprovedtobe true of thesummingof versors, thatis,of

arcs of great circles on a sphere,when the

portion of the arc designated bythe versor

may betaken anywhere on the great circle

(fig 9). As any two great circles have a

commonlineof intersection,the arcsmay be

movedalonguntilthesecondstartsfromthe

ABandBC, denoted by {AB) (BO) isequal

toAC,the arc of the greatcirclewhichjoins

Fig 9.

general passthroughAor C,butit willmeet thegreatcircle ACinsome

pointasD Shift ACback toFD; then the versorFEis thesumof FD

andDE,and therefore the sum of AB, BC,DE. Theassociative

prop-ertymeans,that if.BOand DEare first summed and then ABwith the

and on the same circle as the arcobtained by theformer modeof

pro-cedure The proofof thetheorem is not simple; in Tait's Elementary

thecurves knownas Spherical Conies,discovered onlyinrecent timesby

MagnusandChasles Doubtless manyaonehasbeen discouraged from

the study of quaternionsbythe abstrusenature of thefundamental

prin-ciples.

ade-quately represented byaversor rotating aline atright anglesto its axis.

TheversorABfollowedbytheversor.BCmayrotate aline non-conically

thesamewayupon thelineat C. Todoso,thegreatcircleofDEmust

intersectthe greatcircleofBCinthepoint C.

T =pi +<m

v

+rkandS=ui+vj-f-wk,is

Hence thatthere is ageneralized product whichincludes the product of

AND 75the propositions of the second book of Euclid, the products of Grass-

tnann's Ausdehnungslehre, determinants, and generally the products of

physical magnitudes Bya physical magnitude I mean asymbol which

representsnot onlyratioanddirectionbutthesecombinedwiththe

comple-mentof the algebra of physics

Second,Thattheproductoftwoquadrantalversors orgeometricratios

•a

rr'= —{xx1+yy'+zz')—J {yz'—zy')i+(zx'—xz<)j+(xy'—yx>)k \

Hencethat thereIsageneralizedproductwhichincludes the productof

analyticaltrigonometry, spherical trigonometryand the method of

(Ix+ my +nz) (li+mj+nk)+(mz—ny)i+(nx—lz) j-\- (ly—mx)k.

Thesubjectof rotation andtheeffectof rotation on alinemaybe

con-sideredasbelongingtotheversor part of the algebra of space Theeffect

ofa rotation of anyangleuponaline is still morecomplex, and doesnot

answertothedefinitionof a productasadistributivefunction

Beforethe timeofDesCartes, analgebraicquantitywasrepresentedby

aline,the productoftwoquantitiesbytherectangleformed bythelines,

theproduct of aquantitybyitself asthe square formed by the line, the

productof three quantitiesbythe rightsolid formed bythe lines,which

equa-tion was interpreted as denoting asolid, and the equation was actually

solvedbycuttingupa cube Inordertoexplain higher powers thanthe

cube,spaceof four orany adequatenumberof dimensions wasimagined

Thisconcreteviewofa product correspondstothevector partof

gener-alizedalgebra

Thedoctrine ofDesCarteswas thatthe algebraicsymboldidnot

repre-sent a concrete magnitude, but a mere number or ratio,expressing the

relationof themagnitude tosome unit. Hencethat theproduct of two

quantitiesistheproductofratios, andinsteadof beingrepresented by a

rectanglemayberepresented inthesame wayas either factor; that the

powersofa quantity areratios likethe quantityitself,andthereforethere

is no needofimaginingspace ofmorethanonedimension Thisview of

a product corresponds totheversor part of the generalized algebra

Thetheoryhereadvancedwillbeelaboratedand developed inthepages

which follow; but before proceeding to that development, I propose to

consider several other objectionswhichhave beenormaybemadeagainst

the variousmethodsofextending algebratoquantitiesin space, with the

viewof discussingtheir validity ;and,if theyappear tobevalid,whether

they areremoved bythetheory advanced

Some mathematicians have objected to the negative character of the

col-umns of Nature (Vol xliii, p 511), Professor Gibbssays, "When we

cometofunctionshavingan analogyto multiplication, theproductof the

lengths oftwo vectors and the cosine of the angle whichtheyinclude,

from anypoint ofview exceptthat of the quaternionist,seemsmore

sim-ple thanthesamequantitytakennegatively Thereforewewanta

This agrees withthetheory here advanced But Ido notlookuponthe

productoftwo vectors asmerely having an analogyto muliplication,but

asmultiplicationitselfgeneralized

It has beenobjected that while thescalarproduct andthevector

prod-uct areeachofprimaryImportance, the quaternionproperwhichis their

sum,isofverysecondaryimportance Thus,ProfessorHyde, inapaper

on the "Calculus of Direction and Position" (Amer Journ of Math.,

vector renders the productoftwo vectorswhichare neitherparallelnor

perpendiculartoeachother necessarily acomplexquantity,havinga

sca-larandavector partcorrespondingtotherealand imaginarypartsof the

ordinarycomplex a+1\/—1, thus makingathingwhichshould be

sim-ple justthe opposite Itseemstomethat quaternions proper,i e.,these

complex quantities, are practicallyoflittle use. Innearlyallthe

appli-cations to geometry and mechanics, scalars and vectors areused

directedquantityisnot needed."

In reply it may be said that the works of Hamilton andTaitmakeit

abundantly evident that the quaternion idea is essential tothe algebraic

treatment of spherical trigonometry and of rotations. As regards the

use of the complex o+&|/—1, it is indefinite, unless restricted to a

isintroduced,manyof theknowntheoremsintrigonometry canbegreatly

extended,andthattheentiremeaningof the formula?becomesevidentas

truths in geometry, not mere consequencesfromtheconventional use of

symbols

In the letter to Nature quoted above,Professor Gibbs urges thesame

claim a prominent and fundamentalplacein asystem ofvectoranalysis.

vec-torproduct The geometrical sum a-\-ft represents the third sideof a

mag-nitude the area of the parallelogramdetermined bythesidesaand/5,and

in direction the normaltotheplane of the parallelogram S)-VajS

repre-sents thevolumeof the parallelopipeddetermined bytheedgeso, /Jandy.

Theseconceptions are thevery foundationsofgeometry We mayarrive

at the same conclusionfrom a somewhat narrower but very practical

MATHEMATICS AND ASTRONOMY 77point of view It will hardlybe denied that sines and cosines play the

leading parts in trigonometry Now, the notations Vo.fl and Soft

rep-resent the sine and cosine of the angle included between aand/9

com-bined in each case with certain other simple notions Butthesineand

cosinecombinedwiththese auxiliary notions areincomparablymore

trig-onometry, exactly asnumericalquantitiescombined (as inalgebra) with

the notionof positive or negative quality areincomparablymoreamenable

arith-metic I do not know of anything which can be urgedinfavor of the

quaternionicproductoftwovectorsas a,fundamentalnotioninvector

above considerations The sameistrueof the quaternionic quotientand

of thequaternioningeneral."

It maybe observedthatProfessorGibbs doesnot give the geometrical

meaningof Soft but that of SoVfty. The geometricalmeaninggivento

meaningwhena, /S,ydenotequadrantal versors, but thecommon meaning

isnotso evidentwhena,/?,y denotevectors The meaningwhich Iattach

benothingtodeterminethe positive sign; itratheristhe areaformed by

a and the component offtwhichisperpendiculartoa; and asa

comple-ment wehavetheareaformed by a and thecomponentof/Swhich is

par-allel to a If a and /? are both of unit lengthor, rather, ifweconsider

their direction apart from their physical magnitude, Vaft expressesthe

sine and Saft the cosine of the angle between the directions a and /9

it is of the greatest importance that the angle should be treated as a

whole, notmerelythe sinepartseparately andthecosine part separately

Thus, the argument from trigonometryleads to theoppositeconclusion

to thatatwhichProfessorGibbs arrives.

It seems to methat the essence of aproductisthatit is a distributive

function of thefactors. Thus'in ordinary algebra(a+6+c) (a'+b'+c')

con-sisting of three parts, isnot complete, unless itcontains theninepartial

products; otherwise, theproductisnotageneralizationof theproductof

ordinary algebra As a consequence ofnot treating together the two

complementary parts of the product of two vectors,Grassmannandhis

followershaverestrictedtheirattention toassociative productsand treat

of these onlyin a detached manner Intreatingof theproductof a

num-ber ofvectors, that isa veryarbitrary principle whichholds thatallthe

terms into which two similar directions enter must vanish; but thatis

aprincipleof theAusdehnungslehre andof determinants

78 SECTION A.

the-ory ofdimensions which has played so important a part in

mathemati-cal physics since the time ofFourier? Do theyremoveGregory's

systemisfounded on andabsolutelyconsistent with the idea ofgeometric

dimensions, while Hamilton'sis not. Wefind this objection amplifiedin

thepaperreferredto,Am.Jour.Math.,"Vol vi, p 3. "Fromthis

assump-tion it follows as above, that ij=k andalsothati/j= —ij= —h, i e.,

ideas ofgeometricquantitieswithout any correspondingadvantage If,

in the equation 1/1= 1X1, 1 be taken as the unitof length, then the

membersof the equationhaveevidentlynot thesamemeaning, 1/1being

merelyanumericalquantity, while1 X 1 is a unitof area, itbeinga

fun-damental geometricalconception that theproductof a lengthby alength

num-ber of theorderzero. In quaternions,however,wehave theremarkable

by, but actually equal to alength perpendiculartothe plane of the two."

This objectionisnotvalidagainst themethodof quaternions asthe

al-gebraof versors or directed quotients, thatis, geometricratios; butit is

product, oftwo directed lines. "From the purely geometrical point of

view, a quaternionmayberegardedasthequotient of two directedlines

inspace, orwhat comestothesame thiugasthe factoror operatorwhich

changes one directed line into another," Ency Brit., Art. Quaternions

the former; the formeris the primaryandtruedefinition. Theproduct

oftwo vectors is derivedanalytically fromthe quotient of two vectors

no geometric meaning is attached to it as awhole, butit isinterpreted

asa quaternion Thus, Hamilton,Elements,p. 303: "Weproceedto

acertainquotient or quaternion."

Ifthe product oftwo vectors is a quaternion, then thedefinitionof a

quaternionasthequotient oftwolines isnotcorrect. Butthis confusion

vanisheswhentheproductoftwovectorsisperceivedtobedistinctfrom

and independentof thatof twoversors Thedirected part of a versor,

or ofany number of versors is not a line in thesense of involving the

unit of length; it is of zero dimensions like the ordinary sine of

trigo-nometry Adirectedtermin theproductof vectorsmaybeof one,two,

three or anynumber ofdimensions A

dimensions in length is not necessarily a scalar,noris ittrue thata

di-rected quantityisnecessarilyofone dimensioninlength Theidea ofan

uponthesymbolsi, j,kasdenotingnota unit-vector,butdirectionsimply,

the idea contained inthewordaxis. In writingij=k,wedonot equate

a product oflines to a line, but the axisdenoted byij tothe axisk. In

spaceoffour dimensions thisequationisnottrue; itdependsfor itstruth

on the tridimensional character of space In suchanexpressionas xi it

is morephilosophicaland correcttoconsider thexasembodyingtheunit,

while i denotes simply the axis I lookuponthemagnitudeas

contain-ing the physical unit, to be arithmetical ratio and unit combined; and

haslengthlorunit; a linearvelocityinvolveslengthdirectlyand time

in-versely;momentum involvesmassandlengthdirectlyand timeinversely.

An axis is not a physical quantity, but merely adirection Itfollows

from the theory of vector-algebrahereadvancedthatthe reciprocal of a

vectorhasthesameaxisasthe vectorbut the reciprocalmagnitude As

the dimensions depend on the magnitude notontheaxis, itfollows that

thatis,the axisof thetermwhichinvolvesiandj,or of thetermwhich

involves one directly and onereciprocally, or of thetermwhichinvolves

bothreciprocally is k.

Itappearstome thatthissame principleofdimensions isnotobserved

strictly inGrassmann'smethodorin the"Directional Calculus." "Wemeet

such an equation asp2—pl +£wherep, andjo2 denote pointsande

de-notes a vector Notwithstandingthata point isof zerodimensions and

e isusedtodenotealine-vactor,we haveapointequatedtothesumof a

point and aline. That£ isof one dimension inlengthisevident,forthe

expression £x£2 denotes the area of a parallelogram, and c^s denotes

the volume of a solid, while £jfdenotes the momentof aforce It

ap-pears that either the equationisheterogeneous, orelsep! and p2mustbe

understood as denoting vectors from some common point; ifthelatter

view is correct, the point-analysis reduces to a vector-analysis From

the physical point ofviewit ismorecorrectto treatofa mass-vector than

of apoint having weight; for the differential coefficientwithrespectto

time of a mass-vector is the momentum, which is itselfa mass-vector

prod-uct of a pointandamassisnota physicalidea.

ProfessorHydeindicatesanother elementinwhichGrassmann'smethod

appears superior to Hamilton's "Now quaternions deal onlywith the

vector or line directionand the scalar—fora quaternion isonly thesum

of these two; it knows nothing of a vector having adefinite position,

which is the complete representation of the space qualitiesof aforce."

This is the distinctionwhichCliffordemphasized betweena vectorwhich

may be anywhere and one which is restricted to a_deflnite line ; to

dis-tinguish the latter from the former he introduced the wordrotor, short

contrast between vector and rotoris of great importance, and it is

con-venient to have a notationwhichspecifiesa rotorcompletelyas

depend-ingontwovectors In theworksof Hamilton andTait a force is

directionof the force,thelatterthevectorfroman origintothe pointof

Grassmann'smethodbyp, anditappears thatpisequivalenttothe vector

The methodofGrassmannis applicable, sofarasitgoes, tospace of n

dimensions, while the method ofHamilton appears to be restricted to

space of three dimensions How is it possible to unify the two and

developan algebra not only of three dimensional space but of four

generality ofGrassmann's processes—all results being obtained for

n-dimensional space—has been one of the main hindrances tothegeneral

discus-siontospace of twoor threedimensions Itseemsscarcely possible

that any method can ever be devised, comparable withthis, for

investi-gating n-dimensional space."

On this subject ProfessorGibbssays, Nature,Vol xliv,p 82,"Such

a comparison (ofHamilton'sandofGrassmann's systems) Ihave

endeav-oredtomake,or ratherto indicatethebasisonwhichitmaybemade,so

far as systems of geometrical algebra are concerned Asa contribution

to analysis in general, I suppose that there is no question that

Grass-mann's system is ofindefinitelygreater extension,having nolimitation

toanyparticularnumberof dimensions." Alsoin Nature, Vol xliii, p.

512, "Howmuch moredeeply rootedinthe natureof thingsarethe

func-tions So.f3 and Vaftthananywhicli depend onthedefinition of a

space of four or more dimensions It will not be claimed that the

no-tions of quaternions will applyto such a space, exceptindeedinsucha

limited and artificialmanneras to robthemof theirvaluein asystemof

geometricalalgebra. Butvectors existinsuchaspace,andthere mustbe

a vector analysis forsuch aspace." InreplyProfessor Taitsaid,"It is

singular thatone of Professor Gibbs' objections to quaternionsshouldbe

preciselywhatIhave alwaysconsidered(afterperfectinartificiality) their

chiefmerit, viz., that they are uniquely adapted to Euclidian space,and

therefore specially useful in some of the most important branches of

physical science. What have students of physics, as such, to do with

space ofmorethan threedimensions?"

The view which I have arrived at,unifyingHamilton and Grassmann

anddeveloping amorecomprehensivealgebra' is : Thati2 = + f = +

k*= + do not involve the condition of three dimensions,-beingtruefor

space ofany number ofdimensions, while ij=k jk=i ki=j do

in-volve and indeed express the condition of three dimensions Therules

= = — = —

dimensions we require fourmutually rectangularaxes; letthe fourthbe

denoted by«. ThenIt isnottruethatij=k;butit istrue thatijk=u,

jku= —i,kui=j, uij= —k.

sca-larand vector terms Hamiltonwasneverquitesatisfied, andspeculated

onan extraspatialunit. Now, the heterogeneityisnotin dimensions,for

allthe terms have the same numberofdimensions withrespecttoeach

unitinvolvedinthe units of thefactor-vectors. Thetheoryof axeshere

advanced and theextension of algebra tospace of fourdimensionsshow

thatallthetermsarehomogeneousinthesense ofhavinganaxis,but that

for some terms itmaybeanyaxis; forothers, the fourth axisin aspace

offourdimensions

DEFINITIONSAND NOTATION

I proposetouse a notationwhichshallconformasfaraspossible with

the notationofalgebra, the Cartesiananalysis,quaternions,etc.,butshall

atthesametimeembodywhatIconceive tobethe'principles of the

alge-bra ofphysics The most logical procedure is to generalize as far as

possible the notation of algebra

quantity; ithasnodirection or anydirection. For example, themassof

a body,oritskineticenergy

Byascalarismeant aquantitywhich has magnitude, and maybe

posi-tiveor negative,butisdestitute of adefiniteaxis; orit isthe elementof

aphysicalquantitywhichisindependentof the axis It isequivalent to

theordinary algebraic quantity,andisdenoted,as nsual,by anItalic letter

asa, 6, x, X,etc. The workdone byor against aforce,and thevolume

of ageometricfigureareexamples Thesequantities,though bothscalar,

requires threenumbersto specifyitcompletely Thesimplestexampleis

thedisplacementof apoint,representedbya straightlinedrawn fromits

originalto its final position. Other examples are alinear velocity, an

dimensionsand inthenature of the physical unit; and there are vectors

which have the same dimensions in length, yet have different kinds of

axes Whattheyhaveincommonisawantofsymmetryinspace

A vector is denoted by a black capital letter as A,its magnitude by

a and its axis by a. Thus A =aa,B =6/9,R=rp Sometimes it is

necessaryto introducea dottoseparate the expression for themagnitude

from the expression for the direction; but when the two symbols are

thata denotesthealgebraicmagnitude and a merely its axis,notanother

algebraicmagnitude InClerk-Maxwell'sElectricityand Magnetism,

andplainblacklettershavealreaaybeen usedforthe purpose,asby

Flem-ing in hisbook on Alternate Current Transformers The simple aand o

are more commodious than Taand Uaasusedinworkson quaternions,

82 SECTION A.

and the notation is also more in harmony with the Cartesinn analysis.

What is doneismerelytointroduce atospecifythe axis in space,

leav-ing the expressionforthescalarpart of themagnitudethesameas before

In the case ofmutually rectangular components, i, jand k are used to

denotethe axes

Vector quantities may be classified according to the nature of the

—

a vector in the primarymeaningof thewordasusedby Hamilton It is

ofone dimensioninlength

By the poleoftwoaxesismeantthe axiswhichisperpendicular to both

The pole of a and/S isdenoted by aft; the pole ofa/3and yis denoted

byajSy; that of a and fiy byaj3yandso on Anaxiswhich is

nota-tionenables us to expressexplicitlythreemutually rectangular axes Let

a and/Jbe anytwoindependentaxes;then, a and a/?and apadenote three

mutuallyrectangular axes. In theworksonquaternions, thereisno

sys-tematic notationfor direction; consequentlytospecifythe axis whichis

perpendiculartotwogiven axes,it isnecessarytouseaspecial

non-syste-matic symbol

By&tensor ismeant an arithmetical ratio or quantity destitute of

di-mensions and ofaxis. Thisistheprimary meaningof theword asused

byHamilton; it isprimarily used todenote themagnitudeof the

quater-nion quotient defined as a ratiooftwolines inspace Toconceivea, 6,

x, X,etc., as tensors, is tosuppose theunitthrownintothesymbolsi,j, Jc.

mathematical analysis to regard them as axes, and a, b, x, X, etc., as

magnitudes, notmeretensors

;

forexample,thephysical quantitywhichClerkMaxwellcallsa mass-vector;

it isproportionaltothemassandto the vectorfrom anorigin tothemass

Sucha quantity maybedenoted byA•m ,where theItalic letterdenotes

to theposition of thequantity. This idea corresponds to the weighted

pointof the Ausdehnungslehre

magni-tude,direction andposition;forexample,aforceor a rotationalvelocity.

Itmaybedenoted by suchasymbol asA•F whereAdenotesthe vector

from an origin to the point of application, and F denotes the vector

quantity.

it

has anaxisandan amountofangle. Aversor, asa whole,maybe denoted'by

a smallblackletter as a, andanalyticallybyaA,wherea denotes its axis,

and Atheamountofitsanglein circularmeasure Thus J?isthe

ima<*-MATHEMATICS 83

inary\/— 1 for theaxisa; while a nisequivalent tothe

trigonometri-cal+,provided that in thiscase adenote anyaxis Iconsider thatit is

moreconvenient, andmorein harmony withtrigonometryandthelawof

indicestoconsiderJ, not1, astheindexofaquadrantal versor

Bya quaternionismeantageometricratio ; it isanordinary arithmetical

denotesthe ratioanda the versor. Theratioandaxismaybe expressed

syntheticallyas avector-ratioA,giving the expressionAA

for the ternion

qua-Byadyadismeant aphysicalratio, or therate connecting twovector

thedependentvector,Rthe

independ-ent; if the former is directly

expressed by the rate R—1 S.

Pro-fessor Gibbs in his Vector Analysis

bases thetreatment of vectorslargely

on the conception of adyad; andthe

word, I believe, is due to him The

dyadJ is in a certain sense alocalized

. .

, Fig 10.

quaternion; ithas an axis andan

an-gle, but the angle is localized, that is, it must start from aspecific

dimen-sions in its magnitude,while the quaternion quotient has not.

Byamatrixismeant thesumoftheratesconnectinga vector quantity

withthe three independent components ofanother vector quantity In

itssimplest^ormit isequivalentto ahomogeneous strainor linear-vector

operator Asit isasumof dyads,ProfessorGibbscalls it a dyadic. The

syntheticsymbolusedtodenoteamatrixisaGreekcapital letter as0,

ADDITION AND SUBTRACTION OF VKCTORS

Addition.— Byaddingtwoquantitiesof thesamekindof vector quantity

is meantfindingtheirgeometricresultant, or whatis called inmechanics

compounding them This processis called addition, because when the

vectors have acommon axis,the process reduces to ordinary algebraic

addition Supposetwoquantitiesof a vectorA andB tohaveacommon

point of application (fig 10), theirresultant or sum is thediagonal of

the parallelogram ofwhich AandBarethesides. The principle of the

parallelogram of forces isthus one of thefundamental principles of

the-algebraof physics

Subtraction.— By subtracting one quantity of a vector from another

quantityismeantfinding the quantitywhichaddedtotheformerproduces

the latter. Let A(fig 11) be the quantityto be subtracted, and B the

quantitytobesubtractedfrom; theremainderisthe vector from theend

ofAtotheend ofB,the cross-diagonal of the parallelogram formed by

Aand B, and takeninthe directionfromAtoB

84 SECTTON A.

Tosubtract a quantity of a vectorisequivalent toreversing the axisand

then adding In thefigure (fig 11)—Aisthe opposite ofAindirection;

andthe diagonalfromthe corner of the parallelogramformed by— Aand

Bisequaltothe cross-diagonal of the

parallelogram formedbyAandB To

reversal seems to me less accurate \ # \ _

than to recognize thetwoprocesses of

composition and resolution of vector

the A denote reversal ofaxis,while -* _

a large minus denotes subtraction,

Fig 11thenwe havethe theoremorprinciple " '

B — A = B +_A. Hence we have the rules '— A = + _ A and

+ A = —~A, which mean respectively: tosubtract a quantityis

equiv-alenttoaddingthe opposite quantity; andtoadd aquantityisequivalent

to subtracting the opposite quantity

CommutativeRule.— Whenthe pointofapplicationof a vectoris

are applied simultaneously, orAfirstandthenB, or Bfirst and thenA

Hencethecommutativerulein^addingandsubtracting quantities of avector

A + B = B +A

vfchirdquantityCis tobecompounded,it isimmaterialwhetherthesumof

.Aand Bbeadded toC,or Abeadded tothesum ofBand C Hence

the associativerule in addingandsubtracting quantities of a vector

It follows thattherules forthe transformationof equationsbetween

quan-titiesof a vectorby addingor subtracting equal terms onthe two sides

are thesameasthoseinordinary algebra,wherethe axis ofalltheterms

Sisconstant

Given themagnitudeandaxisof each ofthecomponents; tofindthe

mag-nitudeandaxisofthesum

GivenA =aa, and B =6/J;

tana+bcosafi

=l/o8+6s+2a&cos a/9 •a/S a

Hereya2+&2+2a&cos a/Sgives the magnitude of the sum, while the

rest of the expression denotesits axis in terms of thegiven quantities.

t 6 sin a8

In that expression a/9denotes theaxis,andtan.-1

a ,

b cota a the angle

of the versorwhich changes aintothe direction of thesum

Forthe generalizedaddition whichapplies toquantitiesofa scalar

sit-uatedat different points or to quantities of a vector applied at different

points,seethe>endof the paper

PRODUCT OF TWO VECTORS.

and

B =b^i + b2j -f b3k

be any two vector quantities, not necessarily of thesame kind. Their

product, accordingtothe rules (p 72), is

AB = C«ii+a2 j+a3 fc) (M +&ai+&a*)

=aibiii+atb2jj+a3 b3kk+ a^bzjk+a3& 2 &j+a3&iM +a-fi 3 ik

Here the vector part is written intheformof a determinant In the

Cartesian analysis this vectordeterminant is imperfectly expressed by

meansof thecomposite determinant

B, a2a3

&, 62 63

Let AandBbe givenintheform aa and6/Srespectively; then it is

ev-vident(fromp 72) that

Gi&i ~f"a2&2+8363 =abcosa/3;

and ce 1 a2a3 =absinaft • a/3

61 62 63

i j k

where a/3 is used to denote the axis which isperpendicular toa and /3.

Hence

AB abcos a/3 •aa+a&sire a/3 a?

=ab(cos a/3+sin a/3 • a/3).

Notation forthetwo parts ofthe product.—Inquaternions the negativeof

«!&! + o26 2 + a3b3 is called the scalar ofABandis denoted bySAB,

while theothertermis calledthevector of ABandisdenoted by FAB.

Theobjectionto thisnotationisthe association of the negative signwith

of tiie vector part. As they arenotlinkedtoanythinginordinary

alge-bra,theymake theconnectionobscure andthetransition difficultfrom

or-dinaryalgebra to the algebra of space

I have found it convenient to use for this purpose the functional

ex-pressionscosandSin. Theypossessalltheadvantageofalogical

they then have their trigonometrical meaning Theymakethe formulae

AB =cosAB + SinAB,

Sin with a capital denoting the complete vector quantity, while sin

Hence, it is commutative only ifSin AB =0, that is if §=a. This

condition is satisfied by the quantities of ordinary algebra, but notby

quantities ina plane

Squareofavector.—LetB =A,

thenAa=a,a+a2 + a3 = a*.

The square of a vector has noaxis, or, whatisprobablymorecorrect

in-determinate problem, when the vectoris in space Ifthevectoris

backwards Hencethedoable signforthesquareroot. Again, sincethe

square of anyvector is positive, a negative scalarcannotbe the square

sca-lar isnot only imaginary,it isimpossible

Beciprocal ofa vector.— Bythe reciprocal of a vectorismeantthe

vec-tor which combined as a factor with the original vector produces the

product+1 Since

AB =ab(cos a/?-f-sin aj3 • a/3)

in order that the product may be 1, 6 mustequala-1and/?beidentical

witha. Thus,A-1=a-1a Itfollows that

»_i= ^L = A _ai'+ g2J+a3 fe

b

andthatA_1B = —(cosa/5+sinaj3 •a/?).

The expression inside the parenthesisdepending ontheaxesisthesame

for AB,A-^B,AB-i,A~iB-i

In quaternions the reciprocal of a vector has the opposite axisto that

ofthevector,butthis arisesfromtreatingavector asaquadrantal versor

geometry, whenthe constant quantityis 1. Curvature,denoted bv—

isa directed quantity;its reciprocal, denotedby (^)

_1

, isthe radius ofcurvature; theyhavethesame axis,butreciprocalmagnitudes

It explains whytheruleof signsforaquotientisthesame astheruleof

signs for a product Forexample,— —b= —b ,which meansthatit is

im-material to the resultwhethertheminussignoccursinthe numeratoror

the denominator This view of the generalized reciprocalalsoexplains

the change of signs of the trigonometrical functionsinthe several

quad-rants.

Generalized trigonometricalfunctions.—The other trigonometrical

func-tions maybe definedintermsof the generalized cosineandsine. Thus,

"While TanAB denotesboththemagnitude andtheaxis,tanAB maybe

usedtodenotethemagnitudeapart fromthe

ofB, Tan AB has its simple

trigonometri-calmeaning, only it has an axis in space

For

abcosa§

Complementaryvector.— Bythe

complemen-taryvector (flg 12)ofAwithrespect to B,

Grassmann means the vectorwhich has the

samemagnitude asA and is drawn perpendicularto A in the plane of

wheresinAB =j/(a2& a3&2)2+(a3&i—a1b3y+{a1 b2

and CosAB =(0,6!+a2 &2+a3b3 ) aft •

-Mi)2

PRODUCT OF THREE VECTORS

not necessarily of the same kind; bytheir product ismeanttheproduct

SECTION A.

of the product ofA and B with C, accordingto the rules forvectors

Thus

+ {(a2 &3 —a3 &2)*+(«3&i — a^s^'+Ca^a — 2 6 1)ft|(c1 i+C2J+c 3 J)

=(o 1 6 1+a2 6 2+o3 &3) fcli+c2j+c3 7c)+ a2a3 1

MATHEMATICS AND ASTRONOMY 89

of fourdimensions If the vectorsA, B, Careeach ofonedimensionin

length,eachof thetermsof theproductIsof threedimensionsinlength

Thethirdterminvolves the threeaxesof spacesymmetrically,hence has

no axes It isascalar,but not of thesamekindas cosAB. This view

ofthe term becomes clearer, when theproduct of three line-vectorsin

spaceof fourdimensionsisconsidered

Toexpressthesecond term asthe differenceof two terms similarto thefirst.

—Thesecond term Sin (SinAB) Cexpressedinterms ofi, j,kis

j—(& 2 c 2+63C3) a: +(fi^at +c3a3) Ojji

+ {—(63C3 +&i<a)02 + (fi 3a3+ c^O&2}j

By adding the null term (ftjCiOj — Cia^o^ito the i term, we get

— cosBCaji +cosCA&,i

Bytreating similarly theothertwo componentsand addingtheresults,

weobtain

Sin(SinAB) C = —cosBC •A +cosCA•B

Hence,

ABC =cosAB C — cosBC •A +cosCA B +cos(SinAB) C

The vector which is the sum ofallthe vectortermsmaybecalledthe

totalvector

Theproductofthree vectors isnotindifferentasregardsassociation.—The

expression ABC,without anyparenthesis, meansthat the association of

the factors begins at theleft, whileA(BC) denotesthatthe association

beginsatthe right. Byapplying therulesof multiplicationweget

On comparing these terms with those ofABC, it will be seen,bya

well-known propertyof the determinant, that the third termsareequal.

But

SinA(SinBC) = —Sin(SinBC) A =cosCA B —cosAB C

Hencethetotalvector ofA(BC) is

cosBC •A +cosCA•B —cosAB •C,

which is equal in magnitude to the total vector ofABC, but does not

associationtobeappliedis

cosAB • C =cosBO "A,

and (AC)B =cosAC B —cosCB •A +cosBA• C

90 SECTION A.

It is evident (fig 13) thatBAA-1isthevectorwhichisthereflection

ofBinA

ABC =cosAB •C + {— cosBC •A +cosCA B1+cos(SinAS)C,

BCA = cosBCA-f--f— cosCA -B+cosAB c\ +cos(SinBC)A,

CAB =cosCAB + {—cosAB C + cosBC •a| +cos(SinCA)B

Thelasttermhas thesamevalueinthe threeproducts; itexpresses the

volume ofthe parallelopiped formedbythe threevectorsandmaybe

de-notedbyvolABC. Thesumof the threeproductsis

ABC + BCA + CAB =cosAB •C +cosBC •A +cos CA B

+ 3vo*ABC.

Byabstracting thecommonmagnitudeabc of thetotalvectors, the

fol-lowingratio-vectors areobtained

cos aft •

y—cosfty• a'+cosya•

ft (l) cosfty • a—cosya •

/ 5) but hereweareled tothemdirectlybyvarying the product soas

to get the three modes ofassociation. Let a,ft, y (flg 14) be the

ex-tremities of theaxes onthe unit-sphere Asthe vector (1) has a

nega-tivecomponentalong a, it willbeonthe opposite side of the arcftyfrom

(3).

Since(1) + (2)=2 cos aft •

y and (2)+ (3) =2 cosfty • a and

(3)+ (1)=2cosya•

ft ; theaxes a',

ft', y<aresuchthatthetriangle a'ft'y

Notation.—-Thesquare ofeachof the vectors (1), (2), (3) is

whichis the complement to one ofvol 2

afiy. In sphericaltrigonometry

2< sin s sin (s—a) sin (s— b) sin (s—c) I

andtheneedof aname forthe function hasbeenfelt Ithas beencalled

by some the "sine ofthe trihedralangle"formedbya, /3,y; byothers

the " Staudtian" (Casey, Spherical Trigonometry, p 22). The notation

isdenotedby—Safiy andthetotalvectorby Vafiy

PRODUCT 01? FOURVECTORS

((AB) C) D, (A(BC)) D,A((BC) D), A(B (CD)), (AB) (CD),

ofwhichthefirstandlastarethe mostimportant Whennoparenthesis

isused,thefirstformisunderstood

Thefirstform oftheproduct.—Let A,B andCbe expressedasbefore

intermsof i, j,kandletD =dxi-f-ds j+d3k. Then

ABCD = (aA +a262+a3b3 (Cida +c2<2 2+c3d3 (1)

+

+ («!&! +<z2&2 +a3b3)

a2 a3

These five termsare equalinorderto (1), (2), (3), (4), (5)respectively.

By expandingthesecond andfourth terms,

ABCD =abed( cosaft cosyd—cos fty cosad+cosyacos ftd

+cosaftsinyd•yd—cosfty sin aS'a8+cosY sinP*'Ps

+s in aft cos afty 'S>

The productmaybeexpressedmoresyntheticallyby

ABCD = cosABcosCD + cos(Sin (SinAB) C)D + cosAB SinCV

+ Sin{ Sin(SinAB) C}D +cos(SinAB) C D

The symmetrical product.— By the symmetrical product is meant

SinceAB =ab(cos aft+sin aft •

aft)

(.AB) (CD) =abed {cos aft cosyd+cos aft sinyd•yd+cosydsin aft •

aft+sin aft sinydcos aftyd+sin aft sinydsin aftyd•

aftydjThis differs essentiallyfrom the product oftwo quaternions,for in it

the last two termsare negative Howthen canit satisfythelawof the

norms? Byconsidering thefivetermstobe independentof one another

COMPOUND AXES

Byan axis of thefirst degree ismeantthe direction of aline ; it is

de-notedbyanelementarysymbol suchas a.

Byan axis of the seconddegree ismeanttheproductof twoelementary

axes,denotedingeneralbyaft.

Now,

a/3=cos aft-f-sin a/3 •

aft ;

hence, a2= + and whenftIsperpendicular toa$theaxisreduces toaft.

Also/3a= —aft.

afty =cos a/3 •

y—cosfty'a+cosya•

ft+sin a/3 cos afty •

afty,

whereaftydenotes theaxisof the third term

Lety= a; then theaxisreducesto afta,thatis /3.

Lety—ft ; then the axisreducestoaftft,whichisequalto

2 cosaft 'ft—a.

Hence, ifa and ftareat rightangles, aftftreducesto —a.

afty,whichthereforeisanaxis inaspace of four dimensions Insucha

space,Volumehasanaxis It issuchthat

AND ASTRONOMY 93Theraleof signsfora determinantof the third orderistherule forthe

direction along this axis. In a space of three dimensionswhena,ft,y

aremutuallyrectangularafty istheonlyextraspatialaxis, andmaybe

de-noted in acertainsense by1 ; and aftis equivalenttothecomplementary

axisy. Thus,ij=kintroduces the condition of threedimensions.

By anaxisof the fourth degreeismeanttheproductof fourelementary

axes; it isdenotedin generalbyaftyd,andwehaveshownthat

aftyd =cos aft cosyd—cosftycosad+cosyacos ftd

+cos aft sinyd -yd—cosftycosad ad+cosyasinfid ftd

+sin aft cos afty •

afiy d.

If a, ft and y are mutuallyrectangular, the axis reducesto afty d If

S=a, the axis has thesamedirection asfty,butthe signremainstobe

determined As in space of three dimensionsfty=aand afty= 1,the

sign is + Hence, aftya = fty in general Let d = ft; then since

If, in addition d is at right angles to «, ft andy,wehaveanewaxis

the fourth order, namely, aftdy = — ftyda = ydaft— — dafty, etc.

Thefollowingtablecontains thedifferenttypes ofaxesforthefirstfour

degrees, withtheir reducedequivalents It is supposedthati, j, k, uare

mutuallyrectangular

DEGREE

DEGREE

+ ("464 +ai&i +a2 6i) c3fc+ (a^ +rc 2 & 2+a3 & 3 ) c4?i.

2yi = (a2c2+a3 c3 +a4c4 ftji +(a 3 c3+a4c4+ a^) 6 2 j

+ (O4C4+o&iC! +a2 c2 )& 3 fc+(a^j +a2 c2 +a3 c3 64i«.

b l b2&3

C\ c2 c3

axisbeingthe fourth axisinaspaceof four dimensions

Product offour vectors in space offour dimensions.— By means of the

types, given above, the complete product may be formed In space of

threedimensionsallthetypes existexceptingthelast Ithascommonly

been supposed thattheproductoffourlines isimpossible For instance,

De Morgan (Double Algebra, p 107) says that ABCD is unintelligible,

spacenothavingfourdimensions; andGregory,in hispaper onthe

"Ap-plication of Algebraical Symbols to Geometry," says, "If we combine

moresymbolsthanthree,wefind nogeometrical interpretationforthe

justas\'—1 isan impossible arithmetical one."

QUATERNIONS

com-binedwithan amount of turning It con- _

an amount of angle. Let a denote a

qua-ternion, a its ratio, a its axis and A the

amountof angle; thena=aa It is called

a quaternion, because a requirestwo

num-bers to specify it, while a and A each

re-quires one; in all, four numbers The

quantity; it may involve a physical ratio,

and theangleis fixed (fig 15) IfAand B

Fig 15.

a a~ =a(cosA'a2 "-\- sinA - a *)

Components of a quaternion — Aquaternionmaybeexpressed as the

sumoftwocomponents, oneofwhichhas an indefinite axis,andthe other

-ls

lessthanaquadrant

aaA=a(cosA•a +sinA'a?)

IfAisbetween one andtwoquadrants

w

aaA=a (cosA • a*+ sinA ' a2

IfAisbetweentwoandthreequadrants

3jr

aaA =a (cosA' a? +sinA' a*)

IfAisbetweenthreeandfourquadrants

zA =i

lessthan a revolution

Whenthe quaternions areall inoneplane, ais constant, and need not

be expressed Thequaternion takes theformof thecomplexratio

a•A =a (cosA -\- sinA•

J)the angleJbeing expressed byj/—1.

only be orn; anda• =a, a•n= —a.

Theabove equationsarehomogeneous; aquaternionis equated to the

sumoftwoquaternions, the onlypeculiaritybeingthattheaxisofoneof

thecomponentsmaybeanyaxis.

SUM OF TWO QUATERNIONS

Leta,= aaA andb =&[iB be thetwoquaternions

K

Since a=a (cosA +sinA•a^)

b =& (cosB+sinB-(3s

) , 7T_ It

then a-fb =a +& + j(a, +0l>'+(a2+&0J+ Os + bjkl*

Thisis theaddition ofcomplex numbersnot confinedtooneplane.

PRODUCT Or TWO QUATERNIONS

Bytheproductoftwoquaternions ismeant theproductof the tensors

combinedwiththesumof the versors Theproduct isaquantity of the

samekind as either factor;it is the generalization forspaceof the

prod-uct ofratios.

Letthetwoquaternionsbe

a=a + (<M+arf+a-jk)?

b =6„ + (6 1 i+6 2j+&3 X;)?,

ab =a b —(fli&i +0S2&21+O363)

f a (&,iH

f / a (6,1+b 2 j+b 3 k)+ &„ (a^+a2j+a3k)

Letcosabdenote thecosineofthe angle of theproduct multiplied by

the tensors of aandb, and Sinab the directed sine of the same angle

multipliedinthesame manner; then

cosab =a b — (o^&i +«2&2 +ci3b 3 )

and Sinab =a (bii+b^j+b3k)+6 (a^i+a2j+a3k)—

Ifthe factors areexpressedmoresyntheticallyby

a=a +A2

IT

Trigonometricalformof theproduct.—Let

a =aa , b =bft ;

a.

Leta=6= 1 ; then(fig 16)

whichisthefundamentalpropositioninspherical trigonometry; it isthe

cosine of thesumof theangles. Also

Sina ft =cosBsinA'a-\-cosAsinB ft—sinAsinBsin aft •

aft

isthe expressionfor thedirected sineof thesamesum

Letftcoincide with a; we get the fundamental propositions of plane

trigonometry,namely,

cosaand

A+B A+B

cosAcosB —sinAsin B,_

Sinalr"= (cosBsinA +cosAsinB) • a.

Whenonlyoneplaneis considered, amayheomitted, and the

expres-sionsbecome

cos(A-^-B)=cosAcosB —sinAsinB

sin (A-\-B) =cosBsinA-\- cosAsinB

Here we have evidence that the consistent order of the factors in a

quaternionisfrom left toright; for, when particularized for a plane, we

gettheestablishedorderinplanetrigonometry

Let A = B = s ;

then aa bft = —ab (cos aft+sin a/3 •

aft )Thisistheproductoftwoquadrantal quaternions,which in works on

quaternionsis identifiedwiththeproductoftwo

vectors, onlythesign of thesecondtermismade

positive.

Secondpowerof a quaternion.— Bythe second

power of aquaternion is meantthe product of

thequaternionbyitself. Eromthegeneral prod

net it follows that aaA a aA = ct?aSA

. The

remains the same, the angle is doubled This

is nota squarein thepropersenseoftheword

Reciprocal of a quaternion.—The quaternionb

axis oppositebutits angleequal. Let itbe

MATHEMATICS AND ASTRONOMY 99

bytaking thesecondpowerof theformer

a +2+ a =4cosM

andbytaking thesecondpowerof thelatter

x 2A—2 +oT'2A= —4ifti'A

PRODUCT OF TFJRKE QUATERNIONS

As the product of twoquaternionsis aquaternion, theproductof that

product witha third quaternionisfound by thesamerules asbefore

Leta=a +A?, b =o +B2\ C =c +C*

7T

Now, ab =a„6 —cosAB + (6 A + «oB — (SireAB)3 ';

and bytaking the severalproductsof theseterms withthose ofe, we

ob-tain

it

+ f c A +c <z B +«o 6 oC —cosAB • C — a SinBC\?

symmetricalform

—a SinBC +o SinCA —c SinAB.

Leta=aaA

, c =c^*7.

theaboveexpressionsbecome

cosabe =a6c [ cos 4 cosBcosG —cosAsinBsinCcosj3y

+sinAsinBsinCsinajS cos aj3y

and

SinBtoo=abe ' cosBcosCsinA-a+cos CcosAsinB •§ \

\-cosAcosBsinC y

|

—sinAsinBsin C(cospy.a—cosya • §+cosafi •

y) j

—cosAsinBsinCsinfty • (fy

+cosBsinCsinAsinya • ya

—cosCsinAsinBsin a/3 • a/3

As all the terms are evidently symmetrical withrespect to /?, withthe

exceptionof thefifth, itfollows that (ab)c= a(bc) provided

sina§cos ajSy isequalto sinflycos afiy;

but this is a known truth. Hence in this species of multiplication the

mode of associationofthefactors is indifferent.

Whena, lband carecoplanar, a=ft=Y'i and

coga

A+B+°= cosAcosBcosC—cosAsinBsinO —cosBsin GsinA

and

sina

A+ £ +C=cosBcosCsinA +cos CcosAsinB +cosAcosBsin C

whichareidenticalwith the formulaeinplane trigonometry

If furtherA = B = C,

aSA—cos zA —3cosAsin"A +j3cos sAsinA —sin3A\a?

LetA — B =C =J; then

a2pXjZ=sinap cos aPr+ I—cosjSya+cosya-(i—cos aft • y\

Finiterotation.—The effectofa finiterotationonaline is ingeneralnot

analgebraic product Letabetheaxisand theamountof therotation,

Ealineof lengthrandaxis p. Then (tig 17)

aa y

Fig 17.

a9R =r 1cosap• a+sinapsin • ap -f-sinapcos 6 • apo

Theeffectof asubsequentrotation/S* isgot by applying the samerule

to each of thecomponentsofRin itsnewposition.

In the expression for the quaternion a.A(iB

, letaA=y—C; itwillbe

foundon makingthereductions that

Thus the effect ofy~°

( ) T° uponthequaternion@Bis torotateits

axis by an angle of 20 round y. Hence theeffectof a rotationaBupon

for the multiplication may be associatedinany manner Nowfi *~a ^

is the reciprocal ofu.~'z

It has been shown that a a = a +

; it follows that n being any

numberwholeorfractional,

(a")"=<*""•

Hence by decomposingintocomponentquaternions,

ir ij_n

cosnO+sinnd • a?= (cos +sin • a")

cos?d-g?-"cos"-Qsin'O +

-1T-r%) cos

n-Z0sin3O+ }<?

The component cosn0 having an indefinite axisis equal tothesumof

thecomponents which have anindefiniteaxisandstoredwhich has a

defi-nite axis is equal to thesumofthecomponentshavingthe samedefinite

(aY)n= (o + A*)">

where

a =cos cos<f—sin sin <p cos aft

and

A =cos <p sin • a+cos sin <p '

ft—sin sin<fsin aft •

aft.

Butit isnottrue that