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Full text of "Vector analysis, a text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs"

STAT 



05tcentennial 

VECTOR ANALYSIS 



Bicentennial publications 



With the approval of the President and Fellows 
of Yale University, a series of volumes has been 
prepared by a number of the Professors and In 
structors, to be issued in connection with the 
Bicentennial Anniversary, as a partial indica 
tion of the character of the studies in which the 
University teachers are engaged. 

This series of volumes is respectfully dedicated to 

raDuate$ of tfc 



/ 



VECTOR ANALYSIS^ 



A TEXT-BOOK FOR THE USE OF STUDENTS 
OF MATHEMATICS AND PHYSICS 



FOUNDED UPON THE LECTURES OF 

J. WILLARD GIBBS, PH.D., LL.D. 

Formerly Professor of Mathematical Physics in Yale University 



BY 

EDWIN BIDWELL WILSON, PH.D. 

Professor of Vital Statistics in 
Harvard School of Public Health 



NEW HAVEN 
YALE UNIVERSITY PRESS 



Copyright, 1901 and 1929 
BY YALE UNIVERSITY 



Published, December, 1901 

Second Printing, January , 19/3 

Third Printing, July, 1916 

fourth Printing^ April, 1922 

Fifth Printing, October, 1925 

Sixth Printing, April, 1020 

Seventh Printing, October, 1951 

Eighth Printing, April, 1943 

Ninth Printing, April, 1947 

All rights reserved. This book may not be re 
produced, in whole or in part, in any form, ex 
cept by written permission from the publishers. 



PRINTED IN THE UNITED STATES OF AMERICA 



PKEFACB BY PROFESSOR GIBBS 

SINCE the printing of a short pamphlet on the Elements of 
Vector Analysis in the years 1881-84, never published, but 
somewhat widely circulated among those who were known to 
be interested in the subject, the desire has been expressed 
in more than one quarter, that the substance of that trea 
tise, perhaps in fuller form, should be made accessible to 
the public. 

As, however, the years passed without my finding the 
leisure to meet this want, which seemed a real one, I was 
very glad to have one of the hearers of my course on Vector 
Analysis in the year 1899-1900 undertake the preparation of 
a text-book on the subject. 

I have not desired that Dr. Wilson should aim simply 
at the reproduction of my lectures, but rather that he should 
use his own judgment in all respects for the production of a 
text-book in which the subject should be so illustrated by an 
adequate number of examples as to meet the wants of stu 
dents of geometry and physics. 

J. WILLARD GIBBS. 
YALE UNIVERSITY, September, 1901. 



GENERAL PREFACE 

WHEN I undertook to adapt the lectures of Professor Gibbs 
on VECTOR ANALYSIS for publication in the Yale Bicenten 
nial Series, Professor Gibbs himself was already so fully 
engaged upon his work to appear in the same series, Elementary 
Principles in Statistical Mechanics, that it was understood no 
material assistance in the composition of this book could be 
expected from him. For this reason he wished me to feel 
entirely free to use my own discretion alike in the selection 
of the topics to be treated and in the mode of treatment. 
It has been my endeavor to use the freedom thus granted 
only in so far as was necessary for presenting his method in 
text-book form. 

By far the greater part of the material used in the follow 
ing pages has been taken from the course of lectures on 
Vector Analysis delivered annually at the University by 
Professor Gibbs. Some use, however, has been made of the 
chapters on Vector Analysis in Mr. Oliver Heaviside s Elec 
tromagnetic Theory (Electrician Series, 1893) and in Professor 
Foppl s lectures on Die Maxwell sche Theorie der Electricitdt 
(Teubner, 1894). My previous study of Quaternions has 
also been of great assistance. 

The material thus obtained has been arranged in the way 
which seems best suited to easy mastery of the subject. 
Those Arts, which it seemed best to incorporate in the 
text but which for various reasons may well be omitted at 
the first reading have been marked with an asterisk (*). Nu 
merous illustrative examples have been drawn from geometry, 
mechanics, and physics. Indeed, a large part of the text has 
to do with applications of the method. These applications 
have not been set apart in chapters by themselves, but have 



x GENERAL PREFACE 

been distributed throughout the body of the book as fast as 
the analysis has been developed sufficiently for their adequate 
treatment. It is hoped that by this means the reader may be 
better enabled to make practical use of the book. Great care 
has been taken in avoiding the introduction of unnecessary 
ideas, and in so illustrating each idea that is introduced as 
to make its necessity evident and its meaning easy to grasp. 
Thus the book is not intended as a complete exposition of 
the theory of Vector Analysis, but as a text-book from which 
so much of the subject as may be required for practical appli 
cations may be learned. Hence a summary, including a list 
of the more important formulae, and a number of exercises, 
have been placed at the end of each chapter, and many less 
essential points in the text have been indicated rather than 
fully worked out, in the hope that the reader will supply the 
details. The summary may be found useful in reviews and 
for reference. 

The subject of Vector Analysis naturally divides itself into 
three distinct parts. First, that which concerns addition and 
the scalar and vector products of vectors. Second, that which 
concerns the differential and integral calculus in its relations 
to scalar and vector functions. Third, that which contains 
the theory of the linear vector function. The first part is 
a necessary introduction to both other parts. The second 
and third are mutually independent. Either may be taken 
up first. For practical purposes in mathematical physics the 
second must be regarded as more elementary than the third. 
But a student not primarily interested in physics would nat 
urally pass from the first part to the third, which he would 
probably find more attractive and easy than the second. 

Following this division of the subject, the main body of 
the book is divided into six chapters of which two deal with 
each of the three parts in the order named. Chapters I. and 
II. treat of addition, subtraction, scalar multiplication, and 
the scalar and vector products of vectors. The exposition 
has been made quite elementary. It can readily be under 
stood by and is especially suited for such readers as have a 
knowledge of only the elements of Trigonometry and Ana- 



GENERAL PREFACE xi 

lytic Geometry. Those who are well versed in Quaternions 
or allied subjects may perhaps need to read only the sum 
maries. Chapters III. and IV. contain the treatment of 
those topics in Vector Analysis which, though of less value 
to the students of pure mathematics, are of the utmost impor 
tance to students of physics. Chapters V. and VI. deal with 
the linear vector function. To students of physics the linear 
vector function is of particular importance in the mathemati 
cal treatment of phenomena connected with non-isotropic 
media ; and to the student of pure mathematics this part of 
the book will probably be the most interesting of all, owing 
to the fact that it leads to Multiple Algebra or the Theory 
of Matrices. A concluding chapter, VII., which contains the 
development of certain higher parts of the theory, a number 
of applications, and a short sketch of imaginary or complex 
vectors, has been added. 

In the treatment of the integral calculus, Chapter IV., 
questions of mathematical rigor arise. Although modern 
theorists are devoting much time and thought to rigor, and 
although they will doubtless criticise this portion of the book 
adversely, it has been deemed best to give but little attention 
to the discussion of this subject. And the more so for the 
reason that whatever system of notation be employed ques 
tions of rigor are indissolubly associated with the calculus 
and occasion no new difficulty to the student of Vector 
Analysis, who must first learn what the facts are and may 
postpone until later the detailed consideration of the restric 
tions that are put upon those facts. 

Notwithstanding the efforts which have been made during 
more than half a century to introduce Quaternions into 
physics the fact remains that they have not found wide favor. 
On the other hand there has been a growing tendency espe 
cially in the last decade toward the adoption of some form of 
Vector Analysis. The works of Heaviside and Foppl re 
ferred to before may be cited in evidence. As yet however 
no system of Vector Analysis which makes any claim to 
completeness has been published. In fact Heaviside says : 
"I am in hopes that the chapter which I now finish may 



x ii GENERAL PREFACE 

serve as a stopgap till regular vectorial treatises come to be 
written suitable for physicists, based upon the vectorial treat 
ment of vectors" (Electromagnetic Theory, Vol. I., p. 305). 
Elsewhere in the same chapter Heaviside has set forth the 
claims of vector analysis as against Quaternions, and others 
have expressed similar views. 

The keynote, then, to any system of vector analysis must 
be its practical utility. This, I feel confident, was Professor 
Gibbs s point of view in building up his system. He uses it 
entirely in his courses on Electricity and Magnetism and on 
Electromagnetic Theory of Light. In writing this book I 
have tried to present the subject from this practical stand 
point, and keep clearly before the reader s mind the ques 
tions: What combinations or functions of vectors occur in 
physics and geometry ? And how may these be represented 
symbolically in the way best suited to facile analytic manip 
ulation ? The treatment of these questions in modern books 
on physics has been too much confined to the addition and 
subtraction of vectors. This is scarcely enough. It has 
been the aim here to give also an exposition of scalar and 
vector products, of the operator y, of divergence and curl 
which have gained such universal recognition since the ap 
pearance of Maxwell s Treatise on Electricity and Magnetism, 
of slope, potential, linear vector function, etc., such as shall 
be adequate for the needs of students of physics at the 
present day and adapted to them. 

It has been asserted by some that Quaternions, Vector 
Analysis, and all such algebras are of little value for investi 
gating questions in mathematical physics. Whether this 
assertion shall prove true or not, one may still maintain that 
vectors are to mathematical physics what invariants are to 
geometry. As every geometer must be thoroughly conver 
sant with the ideas of invariants, so every student of physics 
should be able to think in terms of vectors. And there is 
no way in which he, especially at the beginning of his sci 
entific studies, can come to so true an appreciation of the 
importance of vectors and of the ideas connected with them 
as by working in Vector Analysis and dealing directly with 



GENERAL PREFACE xiii 

the vectors themselves. To those that hold these views the 
success of Professor Foppl s Vorlesungen uber Technische 
Mechanik (four volumes, Teubner, 1897-1900, already in a 
second edition), in which the theory of mechanics is devel 
oped by means of a vector analysis, can be but an encour 
aging sign. 

I take pleasure in thanking my colleagues, Dr. M. B. Porter 
and Prof. H. A. Bumstead, for assisting me with the manu 
script. The good services of the latter have been particularly 
valuable in arranging Chapters III. and IV* in their present 
form and in suggesting many of the illustrations used in the 
work. I am also under obligations to my father, Mr. Edwin 
H. Wilson, for help in connection both with the proofs and 
the manuscript. Finally, I wish to express my deep indebt 
edness to Professor Gibbs. For although he has been so 
preoccupied as to be unable to read either manuscript or 
proof, he has always been ready to talk matters over with 
me, and it is he who has furnished me with inspiration suf 
ficient to carry through the work. 

EDWIN BIDWELL WILSON. 
YALE UNIVERSITY, October, 1901. 



PREFACE TO THE SECOND EDITION 

THE only changes which have been made in this edition are 
a few corrections which my readers have been kind enough to 
point out to me. 

E. B. W. 



TABLE OF CONTENTS 



PAGE 
PREFACE BY PROFESSOR GIBBS vii 

GENERAL PREFACE ix 



CHAPTER I 

ADDITION AND SCALAR MULTIPLICATION 
ARTS. 

1-3 SCALARS AND VECTORS 1 

4 EQUAL AND NULL VECTORS 4 

5 THE POINT OF VIEW OF THIS CHAPTER 6 

6-7 SCALAR MULTIPLICATION. THE NEGATIVE SIGN .... 7 

8-10 ADDITION. THE PARALLELOGRAM LAW 8 

11 SUBTRACTION 11 

12 LAWS GOVERNING THE FOREGOING OPERATIONS .... 12 
13-16 COMPONENTS OF VECTORS. VECTOR EQUATIONS .... 14 

17 THE THREE UNIT VECTORS 1, j, k 18 

18-19 APPLICATIONS TO SUNDRY PROBLEMS IN GEOMETRY. . . 21 

20-22 VECTOR RELATIONS INDEPENDENT OF THE ORIGIN ... 27 

23-24 CENTERS OF GRAVITY. BARYCENTRIC COORDINATES . . 39 

25 THE USE OF VECTORS TO DENOTE AREAS 46 

SUMMARY OF CHAPTER i 51 

EXERCISES ON CHAPTER i . . 52 



CHAPTER II 
DIRECT AND SKEW PRODUCTS OF VECTORS 

27-28 THE DIRECT, SCALAR, OR DOT PRODUCT OF TWO VECTORS 55 

29-30 THE DISTRIBUTIVE LAW AND APPLICATIONS 58 

31-33 THE SKEW, VECTOR, OR CROSS PRODUCT OF TWO VECTORS 60 

34-35 THE DISTRIBUTIVE LAW AND APPLICATIONS 63 

36 THE TRIPLE PRODUCT A* B C 67 



XVI 



CONTENTS 



ARTS. PAGE 

37-38 THE SCALAR TRIPLE PRODUCT A* B X C OR [ABC] . . 68 

39-40 THE VECTOR TRIPLE PRODUCT A X (B X C) 71 

41-42 PRODUCTS OF MORE THAN THREE VECTORS WITH APPLI 
CATIONS TO TRIGONOMETRY 75 

43-45 RECIPROCAL SYSTEMS OF THREE VECTORS 81 

46-47 SOLUTION OF SCALAR AND VECTOR EQUATIONS LINEAR IN 

AN UNKNOWN VECTOR 87 

48-50 SYSTEMS OF FORCES ACTING ON A RIGID BODY .... 92 

51 KINEMATICS OF A RIGID BODY 97 

52 CONDITIONS FOR EQUILIBRIUM OF A RIGID BODY ... 101 

53 RELATIONS BETWEEN TWO RIGHT-HANDED SYSTEMS OF 

THREE PERPENDICULAR UNIT VECTORS 104 

54 PROBLEMS IN GEOMETRY. PLANAR COORDINATES . . . 106 

SUMMARY OF CHAPTER n 109 

EXERCISES ON CHAPTER n 113 



CHAPTER III 
THE DIFFERENTIAL CALCULUS OF VECTORS 

55-56 DERIVATIVES AND DIFFERENTIALS OF VECTOR FUNCTIONS 

WITH RESPECT TO A SCALAR VARIABLE 115 

57 CURVATURE AND TORSION OF GAUCHE CURVES .... 120 

58-59 KINEMATICS OF A PARTICLE. THE HODOGRAPH . . . 125 

60 THE INSTANTANEOUS AXIS OF ROTATION 131 

61 INTEGFATION WITH APPLICATIONS TO KINEMATICS . . . 133 

62 SCALAR FUNCTIONS OF POSITION IN SPACE 136 

63-67 THE VECTOR DIFFERENTIATING OPERATOR V 138 

68 THE SCALAR OPERATOR A V 147 

69 VECTOR FUNCTIONS OF POSITION IN SPACE 149 

70 THE DIVERGENCE V* AND THE CURL VX 150 

71 INTERPRETATION OF THE DIVERGENCE V 152 

72 INTERPRETATION OF THE CURL V X 155 

73 LAWS OF OPERATION OF V> V * > V X 157 

74-76 THE PARTIAL APPLICATION OF V- EXPANSION OF A VEC 
TOR FUNCTION ANALOGOUS TO TAYLOR S THEOREM. 

APPLICATION TO HYDROMECHANICS 159 

77 THE DIFFERENTIATING OPERATORS OF THE SECOND ORDER 166 

78 GEOMETRIC INTERPRETATION OF LAPLACE S OPERATOR 

V* V AS THE DISPERSION 170 

SUMMARY OF CHAPTER in 172 

EXERCISES ON CHAPTER in 177 



CONTENTS 



xvii 



CHAPTER IV 

THE INTEGRAL CALCULUS OF VECTORS 

ARTS. PAGE 
79-80 LINE INTEGRALS OF VECTOR FUNCTIONS WITH APPLICA 
TIONS 179 

81 GAUSS S THEOREM 184 

82 STOKES S THEOREM 187 

83 CONVERSE OF STOKES S THEOREM WITH APPLICATIONS . 193 

84 TRANSFORMATIONS OF LINE, SURFACE, AND VOLUME IN 

TEGRALS. GREEN S THEOREM 197 

85 REMARKS ON MULTIPLE-VALUED FUNCTIONS 200 

86-87 POTENTIAL. THE INTEGRATING OPERATOR " POT " . . 205 

88 COMMUTATIVE PROPERTY OF POT AND V 211 

89 REMARKS UPON THE FOREGOING 215 

90 THE INTEGRATING OPERATORS "NEW," "LAP," " MAX " 222 

91 RELATIONS BETWEEN THE INTEGRATING AND DIFFER 

ENTIATING OPERATORS 228 

92 THE POTENTIAL " POT " is A SOLUTION OF POISSON S 

EQUATION 230 

93-94 SOLENOIDAL AND IRROTATIONAL PARTS OF A VECTOR 

FUNCTION. CERTAIN OPERATORS AND THEIR INVERSE . 234 

95 MUTUAL POTENTIALS, NEWTONIANS, LAPLACIANS, AND 

MAXWELLIANS 240 

96 CERTAIN BOUNDARY VALUE THEOREMS 243 

SUMMARY OF CHAPTER iv 249 

EXERCISES ON CHAPTER iv 255 

CHAFIER V 

LINEAR VECTOR FUNCTIONS 

97-98 LINEAR VECTOR FUNCTIONS DEFINED 260 

99 DYADICS DEFINED 264 

100 ANY LINEAR VECTOR FUNCTION MAY BE REPRESENTED 

BY A DYADIC. PROPERTIES OF DYADICS .... 266 

101 THE NONION FORM OF A DYADIC 269 

102 THE DYAD OR INDETERMINATE PRODUCT OF TWO VEC 

TORS IS THE MOST GENERAL. FUNCTIONAL PROPERTY 

OF THE SCALAR AND VECTOR PRODUCTS 271 

108-104 PRODUCTS OF DYADICS 276 

105-107 DEGREES OF NULLITY OF DYADICS 282 

108 THE IDEMFACTOR 288 



XV111 



CONTENTS 



ARTS. PAGE 
109-110 RECIPROCAL DYADICS. POWERS AND ROOTS OF DYADICS 290 
111 CONJUGATE DYADICS. SELF-CONJUGATE AND ANTI- 
SELF-CONJUGATE PARTS OF A DYADIC 294 

112-114 ANTI-SELF-CONJUGATE DYADICS. THE VECTOR PROD 
UCT. QUADRANTAL VER8ORS 297 

115-116 REDUCTION OF DYADICS TO NORMAL FORM .... 302 

117 DOUBLE MULTIPLICATION OF DYADICS 306 

118-119 THE SECOND AND THIRD OF A DYADIC . . ... 310 

120 CONDITIONS FOR DIFFERENT DEGREES OF NULLITY . 313 

121 NONION FORM. DETERMINANTS 315 

122 INVARIANTS OF A DYADIC. THE HAMILTON-CAYLEY 

EQUATION .319 

SUMMARY OF CHAPTER v 321 

EXERCISES ON CHAPTER v 329 



CHAPTER VI 

ROTATIONS AND STRAINS 

123-124 HOMOGENEOUS STRAIN REPRESENTED BY A DYADIC . 332 

125-126 ROTATIONS ABOUT A FIXED POINT. VERSORS . . . 334 

127 THE VECTOR SEMI-TANGENT OF VERSION 339 

128 BlQUADRANTAL VERSORS AND THEIR PRODUCTS . . . 343 

129 CYCLIC DYADICS 347 

130 RIGHT TENSORS 351 

131 TONICS AND CYCLOTONICS 353 

132 REDUCTION OF DYADICS TO CANONICAL FORMS, TONICS, 

CYCLOTONICS, SIMPLE AND COMPLEX SHEARERS . . 356 

SUMMARY OF CHAPTER vi 368 

CHAPTER VII 

MISCELLANEOUS APPLICATIONS 

136-142 QUADRIC SURFACES 372 

143-146 THE PROPAGATION OF LIGHT IN CRYSTALS .... 392 

147-148 VARIABLE DYADICS 403 

149-157 CURVATURE OF SURFACES 411 

158-162 HARMONIC VIBRATIONS AND BIVECTORS .... 426 



VECTOR ANALYSIS 



VECTOR ANALYSIS 



CHAPTER I 

ADDITION AND SCALAR MULTIPLICATION 

1.] IN mathematics and especially in physics two very 
different kinds of quantity present themselves. Consider, for 
example, mass, time, density, temperature, force, displacement 
of a point, velocity, and acceleration. Of these quantities 
some can be represented adequately by a single number 
temperature, by degrees on a thermometric scale ; time, by 
years, days, or seconds ; mass and density, by numerical val- . 
ues which are wholly determined when the unit of the scale 
is fixed. On the other hand the remaining quantities are not 
capable of such representation. Force to be sure is said to be 
of so many pounds or grams weight; velocity, of so many 
feet or centimeters per second. But in addition to this each 
of them must be considered as having direction as well as 
magnitude. A force points North, South, East, West, up, 
down, or in some intermediate direction. The same is true 
of displacement, velocity, and acceleration. No scale of num 
bers can represent them adequately. It can represent only 
their magnitude, not their direction. 

2.] Definition : A vector is a quantity which is considered 
as possessing direction as well as magnitude. 

Definition : A scalar is a quantity which is considered as pos 
sessing magnitude but no direction. 



2 VECTOR ANALYSIS 

The positive and negative numbers of ordinary algebra are the 
typical scalars. For this reason the ordinary algebra is called 
scalar algebra when necessary to distinguish it from the vector 
algebra or analysis which is the subject of this book. 

The typical vector is the displacement of translation in space. 
Consider first a point P (Fig. 1). Let P be displaced in a 
straight line and take a new position P f . 
This change of position is represented by the 
line PP. The magnitude of the displace 
ment is the length of PP 1 ; the direction of 
it is the direction of the line PP 1 from P to 
P 1 . Next consider a displacement not of one, 
but of all the points in space. Let all the 
points move in straight lines in the same direction and for the 
same distance D. This is equivalent to shifting space as a 
rigid body in that direction through the distance D without 
rotation. Such a displacement is called a translation. It 
possesses direction and magnitude. When space undergoes 
a translation T, each point of space undergoes a displacement 
equal to T in magnitude and direction; and conversely if 
the displacement PP which any one particular point P suf 
fers in the translation T is known, then that of any other 
point Q is also known : for Q Q must be equal and parallel 
to PP. 

The translation T is represented geometrically or graphically 
by an arrow T (Fig. 1) of which the magnitude and direction 
are equal to those of the translation. The absolute position 
of this arrow in space is entirely immaterial. Technically the 
arrow is called a stroke. Its tail or initial point is its origin; 
and its head or final point, its terminus. In the figure the 
origin is designated by and the terminus by T. This geo 
metric quantity, a stroke, is used as the mathematical symbol 
for all vectors, just as the ordinary positive and negative num 
bers are used as the symbols for all scalars. 



ADDITION AND SCALAR MULTIPLICATION 3 

* 3.] As examples of scalar quantities mass, time, den 
sity, and temperature have been mentioned. Others are dis 
tance, volume, moment of inertia, work, etc. Magnitude, 
however, is by no means the sole property of these quantities. 
Each implies something besides magnitude. Each has its 
own distinguishing characteristics, as an example of which 
its dimensions in the sense well known to physicists may 
be cited. A distance 3, a time 3, a work 3, etc., are very 
different. The magnitude 3 is, however, a property common 
to them all perhaps the only one. Of all scalar quanti- 
tities pure number is the simplest. It implies nothing but 
magnitude. It is the scalar par excellence and consequently 
it is used as the mathematical symbol for all scalars. 

As examples of vector quantities force, displacement, velo 
city, and acceleration have been given. Each of these has 
other characteristics than those which belong to a vector pure 
and simple. The concept of vector involves two ideas and 
two alone magnitude of the vector and direction of the 
vector. But force is more complicated. When it is applied 
to a rigid body the line in which it acts must be taken into 
consideration; magnitude and direction alone do not suf 
fice. And in case it is applied to a non-rigid body the point 
of application of the force is as important as the magnitude or 
direction. Such is frequently true for vector quantities other 
than force. Moreover the question of dimensions is present 
as in the case of scalar quantities. The mathematical vector, 
the stroke, which is the primary object of consideration in 
this book, abstracts from all directed quantities their magni 
tude and direction and nothing but these ; just as the mathe 
matical scalar, pure number, abstracts the magnitude and 
that alone. Hence one must be on his guard lest from 
analogy he attribute some properties to the mathematical 
vector which do not belong to it ; and he must be even more 
careful lest he obtain erroneous results by considering the 



4 VECTOR ANALYSIS 

vector quantities of physics as possessing no properties other 
than those of the mathematical vector. For example it would 
never do to consider force and its effects as unaltered by 
shifting it parallel to itself. This warning may not be 
necessary, yet it may possibly save some confusion. 

4.] Inasmuch as, taken in its entirety, a vector or stroke 
is but a single concept, it may appropriately be designated by 
one letter. Owing however to the fundamental difference 
between scalars and vectors, it is necessary to distinguish 
carefully the one from the other. Sometimes, as in mathe 
matical physics, the distinction is furnished by the physical 
interpretation. Thus if n be the index of refraction it 
must be scalar ; m, the mass, and , the time, are also 
scalars ; but /, the force, and a, the acceleration, are 
vectors. When, however, the letters are regarded merely 
as symbols with no particular physical significance some 
typographical difference must be relied upon to distinguish 
vectors from scalars. Hence in this book Clarendon type is 
used for setting up vectors and ordinary type for scalars. 
This permits the use of the same letter differently printed 
to represent the vector and its scalar magnitude. 1 Thus if 
C be the electric current in magnitude and direction, C may 
be used to represent the magnitude of that current ; if g be 
the vector acceleration due to gravity, g may be the scalar 
value of that acceleration ; if v be the velocity of a moving 
mass, v may be the magnitude of that velocity. The use of 
Clarendons to denote vectors makes it possible to pass from 
directed quantities to their scalar magnitudes by a mere 
change in the appearance of a letter without any confusing 
change in the letter itself. 

Definition : Two vectors are said to be equal when they have 
the same magnitude and the same direction. 

1 This convention, however, is by no means invariably followed. In some 
instances it would prove just as undesirable as it is convenient in others. It is 
chiefly valuable in the application of vectors to physics. 



ADDITION AND SCALAR MULTIPLICATION 5 

The equality of two vectors A and B is denoted by the 
usual sign =. Thus A = B 

Evidently a vector or stroke is not altered by shifting it 
about parallel to itself in space. Hence any vector A = PP r 
(Fig. 1) may be drawn from any assigned point as origin ; 
for the segment PP f may be moved parallel to itself until 
the point P falls upon the point and P upon some point T. 



In this way all vectors in space may be replaced by directed 
segments radiating from one fixed point 0. Equal vectors 
in space will of course coincide, when placed with their ter 
mini at the same point 0. Thus (Fig. 1) A = PP\ and B = Q~Q f , 
both fall upon T = ~OT. 

For the numerical determination of a vector three scalars 
are necessary. These may be chosen in a variety of ways. 
If r, </>, be polar coordinates in space any vector r drawn 
with its origin at the origin of coordinates may be represented 
by the three scalars r, </>, 6 which determine the terminus of 
the vector. r~(r,*,0). 

Or if #, y 9 z be Cartesian coordinates in space a vector r may 
be considered as given by the differences of the coordinates a/, 
y i z f of its terminus and those #, y, z of its origin. 

r~ (x r x,y r y,z r z). 

If in particular the origin of the vector coincide with the 
origin of coordinates, the vector will be represented by the 
three coordinates of its terminus 

r -(* ,*, , * ) 

When two vectors are equal the three scalars which repre 
sent them must be equal respectively each to each. Hence 
one vector equality implies three scalar equalities. 



6 VECTOR ANALYSIS 

Definition : A vector A is said to be equal to zero when its 
magnitude A is zero. 

Such a vector A is called a null or zero vector and is written 
equal to naught in the usual manner. Thus 

A = if A = 0. 

All null vectors are regarded as equal to each other without 
any considerations of direction. 

In fact a null vector from a geometrical standpoint would 
be represented by a linear segment of length zero that is to 
say, by a point. It consequently would have a wholly inde 
terminate direction or, what amounts to the same thing, none at 
all. If, however, it be regarded as the limit approached by a 
vector of finite length, it might be considered to have that 
direction which is the limit approached by the direction of the 
finite vector, when the length decreases indefinitely and ap 
proaches zero as a limit. The justification for disregarding 
this direction and looking upon all null vectors as equal is 
that when they are added (Art. 8) to other vectors no change 
occurs and when multiplied (Arts. 27, 31) by other vectors 
the product is zero. 

5.] In extending to vectors the fundamental operations 
of algebra and arithmetic, namely, addition, subtraction, and 
multiplication, care must be exercised riot only to avoid self- 
contradictory definitions but also to lay down useful ones. 
Both these ends may be accomplished most naturally and 
easily by looking to physics (for in that science vectors con 
tinually present themselves) and by observing how such 
quantities are treated there. If then A be a given displace 
ment, force, or velocity, what is two, three, or in general x 
times A? What, the negative of A? And if B be another, 
what is the sum of A and B ? That is to say, what is the 
equivalent of A and B taken together ? The obvious answers 
to these questions suggest immediately the desired definitions. 



ADDITION AND SCALAR MULTIPLICATION 1 

Scalar Multiplication 

6.] Definition: A vector is said to be multiplied by a 
positive scalar when its magnitude is multiplied by that scalar 
and its direction is left unaltered 

Thus if v be a velocity of nine knots East by North, 2 times 
v is a velocity of twenty-one knots with the direction still 
East by North. Or if f be the force exerted upon the scale- 
pan by a gram weight, 1000 times f is the force exerted by a 
kilogram. The direction in both cases is vertically down 
ward. 

If A be the vector and x the scalar the product of x and A is 

denoted as usual by 

x A or A x. 

It is, however, more customary to place the scalar multiplier 
before the multiplicand A. This multiplication by a scalar 
is called scalar multiplication, and it follows the associative law 

x (y A) = (x y) A = y (x A) 

as in ordinary algebra and arithmetic. This statement is im 
mediately obvious when the fact is taken into consideration 
that scalar multiplication does not alter direction but merely 
multiplies the length. 

Definition : A unit vector is one whose magnitude is unity. 

Any vector A may be looked upon as the product of a unit 
vector a in its direction by the positive scalar A, its magni 
tude. 

A = A a = a A. 

The unit vector a may similarly be written as the product of 
A by I/A or as the quotient of A and A. 

1 A 

a = ^ A = -I 

A A 



8 VECTOR ANALYSIS 

7.] Definition : The negative sign, prefixed to a vector 
reverses its direction but leaves its magnitude unchanged. 

For example if A be a displacement for two feet to the right, 
A is a displacement for two feet to the left. Again if the 
stroke A~B be A, the stroke B A, which is of the same length 
as A but which is in the direction from B to A instead of 
from A to 5, will be A. Another illustration of the use 
of the negative sign may be taken from Newton s third law 
of motion. If A denote an "action," A will denote the 
" reaction." The positive sign, + , may be prefixed to a vec 
tor to call particular attention to the fact that the direction 
has not been reversed. The two signs + and when used 
in connection with scalar multiplication of vectors follow the 
same laws of operation as in ordinary algebra. These are 
symbolically 

+ + = + ; +- = -; - + = -; = +; 

(ra A) = m ( A). 

The interpretation is obvious. 

Addition and Subtraction 

8.] The addition of two vectors or strokes may be treated 
most simply by regarding them as defining translations in 
space (Art. 2), Let S be one vector and T the other. Let P 
be a point of space (Fig. 2). The trans 
lation S carries P into P 1 such that the 
line PP 1 is equal to S in magnitude and 
direction. The transformation T will then 
carry P 1 into P 11 the line P P" being 
parallel to T and equal to it in magnitude. 
FIG. 2. Consequently the result of S followed by 

T is to carry the point P into the point 
P". If now Q be any other point in space, S will carry Q 
into Q such that Q~Q r = S and T will then carry Q f into Q" 



ADDITION AND SCALAR MULTIPLICATION 



such that Q Q" = T. Thus S followed by T carries Q into Q". 
Moreover, the triangle Q Q f Q" is equal to PP P". For 
the two sides Q Q f and Q Q", being equal and parallel to S 
and T respectively, must be likewise parallel to P P 1 and 
P P" respectively which are also parallel to S and T. Hence 
the third sides of the triangles must be equal and parallel 

That is 

Q Q" is equal and parallel to PP". 

As Q is any point in space this is equivalent to saying that 
by means of S followed by T all points of space are displaced 
the same amount and in the same direction. This displace 
ment is therefore a translation. Consequently the two 
translations S and T are equivalent to a single translation R. 
Moreover 



if S = PP and T = P P", then R = PP". 

The stroke R is called the resultant or sum of the two 
strokes S and T to which it is equivalent. This sum is de 
noted in the usual manner by 

R = S + T. 

From analogy with the sum or resultant of two translations 
the following definition for the addition of any two vectors is 
laid down. 

Definition : The sum or resultant of two vectors is found 
by placing the origin of the second upon the terminus of the 
first and drawing the vector from the origin of the first to the 
terminus of the second. 

9.] Theorem. The order in which two vectors S and T are 
added does not affect the sum. 

S followed by T gives precisely the same result as T followed 
by S. For let S carry P into P (Fig. 3) ; and T, P into P". 
S + T then carries P into P". Suppose now that T carries P 
into P ". The line PP " is equal and parallel to PP". Con- 



10 VECTOR ANALYSIS 

sequently the points P, P 9 P ff , and P m lie at the vertices of 

a parallelogram. Hence 
pm pn j s e q ua l an( J par- 

allel to PP. Hence S 
carries P" f into P". T fol 
lowed by S therefore car 
ries P into P" through P\ 
whereas S followed by T 
carries P into P" through 
P m . The final result is in 
either case the same. This may be designated symbolically 

by writing 

R = S + T = T + S. 

It is to be noticed that S = PP 1 and T = PP m are the two sides 
of the parallelogram pprpp" which 1 have the point P as 
common origin ; and that JL=PP" is the diagonal drawn 
through P. This leads to another very common way of 
stating the definition of the sum of two vectors. 

If two vectors be drawn from the same origin and a parallelo 
gram be constructed upon them as sides, their sum will be that 
diagonal which passes through their common origin. 

This is the well-known " parallelogram law " according to 
which the physical vector quantities force, acceleration, veloc 
ity, and angular velocity are compounded. It is important to 
note that in case the vectors lie along the same line vector 
addition becomes equivalent to algebraic scalar addition. The 
lengths of the two vectors to be added are added if the vectors 
have the same direction ; but subtracted if they have oppo 
site directions. In either case the sum has the same direction 
as that of the greater vector. 

10.] After the definition of the sum of two vectors has 
been laid down, the sum of several may be found by adding 
together the first two, to this sum the third, to this the fourth, 
and so on until all the vectors have been combined into a sin- 



ADDITION AND SCALAR MULTIPLICATION 11 

gle one. The final result is the same as that obtained by placing 
the origin of each succeeding vector upon the terminus of the 
preceding one and then drawing at once the vector from 
the origin of the first to the terminus of the last. In case 
these two points coincide the vectors form a closed polygon 
and their sum is zero. Interpreted geometrically this states 
that if a number of displacements R, S, T are such that the 
strokes R, S, T form the sides of a closed polygon taken in 
order, then the effect of carrying out the displacements is nil. 
Each point of space is brought back to its starting point. In 
terpreted in mechanics it states that if any number of forces 
act at a point and if they form the sides of a closed polygon 
taken in order, then the resultant force is zero and the point 
is in equilibrium under the action of the forces. 

The order of sequence of the vectors in a sum is of no con 
sequence. This may be shown by proving that any two adja 
cent vectors may be interchanged without affecting the result. 

To show 



Let A = A, B = A B, C = B C, D = D, E = D E. 
Then 



_ 

Let now B C 1 = D. Then C ! B C D is a parallelogram and 
consequently C f D = C. Hence 

OJ = A + B + D + C + E, 

which proves the statement. Since any two adjacent vectors 
may be interchanged, and since the sum may be arranged in 
any order by successive interchanges of adjacent vectors, the 
order in which the vectors occur in the sum is immaterial. 

11.] Definition : A vector is said to be subtracted when it 
is added after reversal of direction. Symbolically, 

A - B = A + (- B). 
By this means subtraction is reduced to addition and needs 



12 



VECTOR ANALYSIS 



no special consideration. There is however an interesting and 
important way of representing the difference of two vectors 
geometrically. Let A = OA, B = 0IT(Fig. 4). Complete 

the parallelogram of which A and B 
are the sides. Then the diagonal 
~OG = C is the sum A + B of the 
two vectors. Next complete the 
parallelogram of which A and B 
= OB are the sides. Then the di 



agonal 02) = !) will be the sum of 
A and the negative of B. But the 
segment OD is parallel and equal 

to BA. Hence BA may be taken as the difference to the two 
vectors A and B. This leads to the following rule : The differ 
ence of two vectors which are drawn from the same origin is 
the vector drawn from the terminus of the vector to be sub 
tracted to the terminus of the vector from which it is sub 
tracted. Thus the two diagonals of the parallelogram, which 
is constructed upon A and B as sides, give the sum and dif 
ference of A and B. 

12.] In the foregoing paragraphs addition, subtraction, and 
scalar multiplication of vectors have been defined and inter 
preted. To make the development of vector algebra mathe 
matically exact and systematic it would now become necessary 
to demonstrate that these three fundamental operations follow 
the same formal laws as in the ordinary scalar algebra, al 
though from the standpoint of the physical and geometrical 
interpretation _of vectors this may seem superfluous. These 
laws are 

m (n A) = n (m A) = (m n} A, 

(A + B) + C = A+ (B + C), 
II A + B r, B + A, 

III a (m + n) A = m A + n A, 

m (A + B) = m A + m B, 
III, - (A + B) = - A - B. 



ADDITION AND SCALAR MULTIPLICATION 13 

1 is the so-called law of association and commutation of 
the scalar factors in scalar multiplication. 

I 6 is the law of association for vectors in vector addition. It 
states that in adding vectors parentheses may be inserted at 
any points without altering the result. 

11 is the commutative law of vector addition. 

III a is the distributive law for scalars in scalar multipli 
cation. 

III 6 is the distributive law for vectors in scalar multipli 
cation. 

Ill, is the distributive law for the negative sign. 

The proofs of these laws of operation depend upon those 
propositions in elementary geometry which have to deal with 
the first properties of the parallelogram and similar triangles. 
They will not be given here; but it is suggested that the 
reader work them out for the sake of fixing the fundamental 
ideas of addition, subtraction, and scalar multiplication more 
clearly in mind. The result of the laws may be summed up 
in the statement : 

The laws which govern addition, subtraction, and scalar 
multiplication of vectors are identical with those governing these 
operations in ordinary scalar algebra. 

It is precisely this identity of formal laws which justifies 
the extension of the use of the familiar signs =, +, and 
of arithmetic to the algebra of vectors and it is also this 
which ensures the correctness of results obtained by operat 
ing with those signs in the usual manner. One caution only 
need be mentioned. Scalars and vectors are entirely different 
sorts of quantity. For this reason they can never be equated 
to each other except perhaps in the trivial case where each is 
zero. For the same reason they are not to be added together. 
So long as this is borne in mind no difficulty need be antici 
pated from dealing with vectors much as if they were scalars. 

Thus from equations in which the vectors enter linearly with 



14 VECTOR ANALYSIS 

scalar coefficients unknown vectors may be eliminated or 
found by solution in the same way and with the same limita 
tions as in ordinary algebra; for the eliminations and solu 
tions depend solely on the scalar coefficients of the equations 
and not at all on what the variables represent. If for 

instance 

aA + &B + cC + dD = 0, 

then A, B, C, or D may be expressed in terms of the other 
three 

as D = --:OA + &B + cC). 

a 

And two vector equations such as 

3 A+ 4B=E 

and 2 A + 3 B = F 

yield by the usual processes the solutions 

A=3E-4F 
and B = 3 F - 2 E. 

Components of Vectors 

13.] Definition : Vectors are said to be collinear when 
they are parallel to the same line; coplanar, when parallel 
to the same plane. Two or more vectors to which no line 
can be drawn parallel are said to be non-collinear. Three or 
more vectors to which no plane can be drawn parallel are 
said to be non-coplanar. Obviously any two vectors are 
coplanar. 

Any vector b collinear with a may be expressed as the 
product of a and a positive or negative scalar which is the 
ratio of the magnitude of b to that of a. The sign is positive 
when b and a have the same direction ; negative, when they 
have opposite directions. If then OA = a, the vector r drawn 



ADDITION AND SCALAR MULTIPLICATION 15 

from the origin to any point of the line A produced in 

either direction is 

r = x a. (1) 

If x be a variable scalar parameter this equation may there 
fore be regarded as the (vector) equation of all points in the 
line OA. Let now B be any point not 
upon the line OA or that line produced 
in either direction (Fig. 5). 

Let OB = b. The vector b is surely 
not of the form x a. Draw through B Flo 5 " 

a line parallel to OA and Let R be any 
point upon it. The vector BE is collinear with a and is 
consequently expressible as #a. Hence the vector drawn 
from to R is 

0~E=0~B + ITR 

or r = b + #a. (2) 

This equation may be regarded as the (vector) equation of 
all the points in the line which is parallel to a and of which 
B is one point. 

14.] Any vector r coplanar with two non-collinear vectors 
a and b may be resolved into two components parallel to a 
and b respectively. This resolution may 
be accomplished by constructing the par 
allelogram (Fig. 6) of which the sides are 
parallel to a and b and of which the di 
agonal is r. Of these components one is 
x a ; the other, y b. x and y are respec 
tively the scalar ratios (taken with the 
proper sign) of the lengths of these components to the lengths 

of a and b, Hence 

r = x a + y b (2) 

is a typical form for any vector coplanar with a and b. If 
several vectors r x , r 2 , r 3 may be expressed in this form as 



16 



VECTOR ANALYSIS 



their sum r is then 



r l = x l a + y l b, 
r 2 = # 2 a + 2/ 2 b, 
r 3 = x z a + 2/3 b. 



+ (ft + ft + ft + ) 

This is the well-known theorem that the components of a 
sum of vectors are the sums of the components of those 
vectors. If the vector r is zero each of its components must 
be zero. Consequently the one vector equation r = is 
equivalent to the two scalar equations 



y\ + ft + ft + = 



(3) 



15.] Any vector r in space may be resolved into three 
components parallel to any three given non-coplanar vectors. 

Let the vectors be a, b, 
and c. The resolution 
may then be accom 
plished by constructing 
the parallelepiped (Fig. 
7) of which the edges 
are parallel to a, b, and 
c and of which the di 
agonal is r. This par- 
allelopiped may be 
drawn easily by passing 
three planes parallel re 
spectively to a and b, b and c, c and a through the origin 
of the vector r ; and a similar set of three planes through its 
terminus It. These six planes will then be parallel in pairs 



FIG. 7. 



ADDITION AND SCALAR MULTIPLICATION 17 

and hence form a parallelepiped. That the intersections of 
the planes are lines which are parallel to a, or b, or c is 
obvious. The three components of r are x a, y b, and zc; 
where x, y, and z are respectively the scalar ratios (taken with 
the proper sign) of the lengths of these components to the 
length of a, b, and c. Hence 

r = # a + 7/b + zc (4) 

is a typical form for any vector whatsoever in space. Several 
vectors r lf r 2 , r 3 . . . may be expressed in this form as 

r x = x l a + y l b + z l c, 
r 2 = # 2 a + y 2 b + *2 c 

1*3 = X Z a + 2/3 b 



Their sum r is then 

1 = r l + r 2 + F 3 + * = 0*1 + *2 + X Z + a 

+ (2/i + 2/2+ 3/3 + )!> 

+ Ol +^2 + ^3+ "O - 

If the vector r is zero each of its three components is zero. 
Consequently the one vector equation r = is equivalent to 
the three scalar equations 

x l + # 2 + # 3 + - = v 

2/i + 2/2 + 2/3 + = y r = 0. (5) 

*i + * 2 + % + = / 

Should the vectors all be coplanar with a and b, all the com 
ponents parallel to c vanish. In this case therefore the above 
equations reduce to those given before. 

16.] If two equal vectors are expressed in terms of the 
same three non-coplanar vectors, the corresponding scalar co 
efficients are equal. 



18 VECTOR ANALYSIS 

Let r = r , 

r = x 9 a + y 1 b + z c, 







Then x = x , y = y 

For r - r = = (x - x f ) a + (y - y ) b + (* - z 1 ) c. 

Hence x - * = 0, y - y = 0, z - * = 0. 

But this would not be true if a, b, and c were coplanar. In 
that case one of the three vectors could be expressed in terms 

of the other two as 

c = m a + n b. 

Then r = #a + y b + s c = (a + m z) a + (y + TI z) b, 
r = x ! a + y 1 b + z ; c = (x 1 + m z ) a + (y + n z ) b, 
r r = [(x + m z ) (x + m z )] a, 



Hence the individual components of r r in the directions 
a and b (supposed different) are zero. 

Hence x + mz = x r + mz r 

y -f n z = y f + n z 1 . 

But this by no means necessitates x, y, z to be equal respec 
tively to x\ y\ z 1 . In a similar manner if a and b were col- 
linear it is impossible to infer that their coefficients vanish 
individually. The theorem may perhaps be stated as follows : 
. In case two equal vectors are expressed in terms of one vector, 
or two non-collinear vectors, or three non-coplanar vectors, the 
corresponding scalar coefficients are equal. But this is not ne 
cessarily true if the two vectors be collinear ; or the three vectors, 
coplanar. This principle will be used in the applications 
(Arts. 18 et seq.). 

The Three Unit Vectors i, j, k. 

17.] In the foregoing paragraphs the method of express 
ing vectors in terms of three given non-coplanar ones has been 
explained. The simplest set of three such vectors is the rect- 



ADDITION AND SCALAR MULTIPLICATION 19 

angular system familiar in Solid Cartesian Geometry. This 
rectangular system may however be either of two very distinct 
types. In one case (Fig. 8, first part) the Z-axis l lies upon 
that side of the X Y- plane on which rotation through a right 
angle from the X-axis to the F-axis appears counterclockwise 
or positive according to the convention adopted in Trigonome 
try. This relation may be stated in another form. If the X 
axis be directed to the right and the F-axis vertically, the 
^-axis will be directed toward the observer. Or if the X- 
axis point toward the observer and the F-axis to the right, 
the ^-axis will point upward. Still another method of state- 

Z 



,,k 



Right-handed 



FIG. 8. 



Left-handed 



ment is common in mathematical physics and engineering. If 
a right-handed screw be turned from the Xaxis to the F- 
axis it will advance along the (positive) Z-axis. Such a sys 
tem of axes is called right-handed, positive, or counterclock 
wise. 2 It is easy to see that the F-axis lies upon that side of 
the ^X-plane on which rotation from the ^-axis to the X- 
axis is counterclockwise ; and the X-axis, upon that side of 

1 By the X-, Y-, or Z-axis the positive half of that axis is meant. The X Y- 
plane means the plane which contains the X- and Y-axis, i. e., the plane z = 0. 

2 A convenient right-handed system and one which is always available consists 
of the thumb, first finger, and second finger of the right hand. If the thumb and 
first finger be stretched out from the palm perpendicular to each other, and if the 
second finger be bent over toward the palm at right angles to first finger, a right- 
handed system is formed by the fingers taken in the order thumb, first finger, 
second finger. 



20 VECTOR ANALYSTS 

the F^-plane on which rotation from the F-axis to the Z- 
axis is counterclockwise. Thus it appears that the relation 
between the three axes is perfectly symmetrical so long as the 
same cyclic order XYZXY is observed. If a right-handed 
screw is turned from one axis toward the next it advances 
along the third. 

In the other case (Fig. 8, second part) the ^-axis lies upon 
that side of the X F-plane on which rotation through a right 
angle from the JT-axis to the F-axis appears clockwise or neg 
ative. The F-axis then lies upon that side of the ^X-plane 
on which rotation from the ^-axis to the X-axis appears 
clockwise and a similar statement may be made concerning 
the X-axis in its relation to the F^-plane. In this case, too, 
the relation between the three axes is S3 r mmetrical so long 
as the same cyclic order X YZX Y is preserved but it is just 
the opposite of that in the former case. If a fe/Mianded screw 
is turned from one axis toward the next it advances along 
the third. Hence this system is called left-handed, negative, 
or clockwise. 1 

The two systems are not superposable. They are sym 
metric. One is the image of the other as seen in a 
mirror. If the JT- and F-axes of the two different systems be 
superimposed, the ^-axes will point in opposite directions. 
Thus one system may be obtained from the other by reversing 
the direction of one of the axes. A little thought will show 
that if two of the axes be reversed in direction the system will 
not be altered, but if all three be so reversed it will be. 

Which of the two systems be used, matters little. But in 
asmuch as the formulae of geometry and mechanics differ 
slightly in the matter of sign, it is advisable to settle once for 
all which shall be adopted. In this book the right-handed or 
counterclockwise system will be invariably employed. 

1 A left-handed system may be formed by the left hand just as a right-handed 
one was formed by the right. 



ADDITION AND SCALAR MULTIPLICATION 21 

Definition : The three letters i, j, k will be reserved to de 
note three vectors of unit length drawn respectively in the 
directions of the JT-, T-, and Z- axes of a right-handed rectan 
gular system. 

In terms of these vectors, any vector may be expressed as 

r = xi + y] + zk. (6) 

The coefficients x y y, z are the ordinary Cartesian coordinates 
of the terminus of r if its origin be situated at the origin of 
coordinates. The components of r parallel to the X-, F-, and 
^f-axes are respectively 

x i, y j, z k. 

The rotations about i from j to k, about j from k to i, and 
about k from i to j are all positive. 

By means of these vectors i, j, k such a correspondence is 
established between vector analysis and the analysis in Car 
tesian coordinates that it becomes possible to pass at will 
from either one to the other. There is nothing contradic 
tory between them. On the contrary it is often desirable 
or even necessary to translate the formulae obtained by 
vector methods into Cartesian coordinates for the sake of 
comparing them with results already known and it is 
still more frequently convenient to pass from Cartesian 
analysis to vectors both on account of the brevity thereby 
obtained and because the vector expressions show forth the 
intrinsic meaning of the formulae. 

Applications 

*18.J Problems in plane geometry may frequently be solved 
easily by vector methods. Any two non-collinear vectors in 
the plane may be taken as the fundamental ones in terms of 
which all others in that plane may be expressed. The origin 
may also be selected at pleasure. Often it is possible to 



22 VECTOR ANALYSIS 

make such an advantageous choice of the origin and funda 
mental vectors that the analytic work of solution is materially 
simplified. The adaptability of the vector method is about 
the same as that of oblique Cartesian coordinates with differ 
ent scales upon the two axes. 

Example 1 : The line which joins one vertex of a parallelo 
gram to the middle point of an opposite side trisects the diag 
onal (Fig. 9). 

Let A BCD be the parallelogram, BE the line joining the 
vertex B to the middle point E of the side 
AD, R the point in which this line cuts the 
diagonal A C. To show A R is one third of 
FlG 9 AC. Choose A as origin, A B and AD as the 

two fundamental vectors S and T. Then 
A C is the sum of S and T. Let AR = R. To show 

R = 1 (S + T). 



- 



where x is the ratio of ER to EB an unknown scalar. 

And R = y ( S + T), 

where y is the scalar ratio of A R to A C to be shown equal 



to. 



Hence \ T + x (S -i T) = y (S + T) 

or * S + 1 (1 - X ) T = y S + y T. 

Hence, equating corresponding coefficients (Art. 16), 



2 (1 - x) = y. 



ADDITION AND SCALAR MULTIPLICATION 23 
From which y = . 

Inasmuch as x is also - the line j&2? must be trisected as 

o 

well as the diagonal A C. 

Example 2 : If through any point within a triangle lines 
be drawn parallel to the sides the sum of the ratios of these 
lines to their corresponding sides is 2. 

Let ABC be the triangle, R the point within it. Choose 
A as origin, A B and A C as the two fundamental vectors S 
and T. Let 

AR = R = w S + 7i T. (a) 

m S is the fraction of A B which is cut off by the line through 
R parallel to A C. The remainder of A B must be the frac 
tion (1 m) S. Consequently by similar triangles the ratio of 
the line parallel to A C to the line A C itself is (1 ra). 
Similarly the ratio of the line parallel to A B to the line A B 
itself is (1 n ). Next express R in terms of S and T S the 
third side of the triangle. Evidently from (a) 

R = (m + ri) S + n (T - S). 

Hence (m + ri) S is the fraction of A B which is cut off by the 
line through R parallel to B C. Consequently by similar tri 
angles the ratio of this line to BC itself is (m + n). Adding 
the three ratios 

(1 - m) + (1 - n) + (m + ri) = 2, 

and the theorem is proved. 

Example 3 : If from any point within a parallelogram lines 
be drawn parallel to the sides, the diagonals of the parallelo 
grams thus formed intersect upon the diagonal of the given 
parallelogram. 

Let A B CD be a parallelogram, R a point within it, KM 
and LN two lines through R parallel respectively to AB and 



24 VECTOR ANALYSIS 

AD, the points K, Z, M, N lying upon the sides DA, AS, 
B C, CD respectively. To show that the diagonals KN and 
LM of the two parallelograms KRND and LBME meet 
on A C. Choose A as origin, A B and A D as the two funda 
mental vectors S and T. Let 

R = AB = m S 4- ft T, 
and let P be the point of intersection of KN with LM. 



Then KN=KR + BN = m S + (1 - rc) T, 



=(1 -m) S + 7i T, 



Hence P = n T + x [m S + (1 n) T], 

and P = m S + y [(1 - m) S + n T]. 

Equating coefficients, 

x m = m + y (1 m) 



By solution, ; 



m + n 1 

m 
~ m + n 1 

Substituting either of these solutions in the expression for P, 
the result is 

P^-^-^S + T), 

which shows that P is collinear with A C. 

* 19.] Problems in three dimensional geometry may be 
solved in essentially the same manner as those in two dimen 
sions. In this case there are three fundamental vectors in 
terms of which all others can be expressed. The method of 
solution is analogous to that in the simpler case. Two 



ADDITION AND SCALAR MULTIPLICATION 25 

expressions for the same vector are usually found. The co 
efficients of the corresponding terms are equated. In this way 
the equations between three unknown scalars are obtained 
from which those scalars may be determined by solution and 
then substituted in either of the expressions for the required 
vector. The vector method has the same degree of adapta 
bility as the Cartesian method in which oblique axes with 
different scales are employed. The following examples like 
those in the foregoing section are worked out not so much for 
their intrinsic value as for gaining a familiarity with vectors. 

Example 1 : Let A B CD be a tetrahedron and P any 
point within it. Join the vertices to P and produce the lines 
until they intersect the opposite faces in A\ B , C 1 , D f . To 

show 

PA PB PC 1 PD 



A~A f TTB ~C~O f 



" 



Choose A as origin, and the edges A J?, A C, AD as the 
three fundamental vectors B, C, D. Let the vector A P be 



P = A P=IE + raC + 7i D, 



Also A = A A = A B + BA . 



The vector BA 1 is coplanar with WC = C B and BD 
D B. Hence it may be expressed in terms of them. 

A = B + ^ 1 (C-B)+y 1 (D~B). 
Equating coefficients Jc l m = x v 



Hence &., = 

PA _ V 

ZZ 7 ~~& 



1 I + m + n 

PA* JL-1 
and " ^ 7 






26 VECTOR ANALYSIS 



In like manner A B = # 2 C + y 2 D 



and A B = ^t + B B = B + & 2 (P - B). 

Hence o; 2 C + y 2 D = B + A: 2 (ZB + mC + ^D-B 
and = 1 + *, (J - 1), 



Hence 



2 -i __ 
- 

and 



In the same way it may be shown that 
PC .PL 1 



CC* 3D 

Adding the four ratios the result is 

i d JL vn -4- <w ^ _L 7 J_ w -I- 77 1 

Example % : To find a line which passes through a given 
point and cuts two given lines in space. 

Let the two lines be fixed respectively by two points A 
and B, C and D on each. Let be the given point. Choose 
it as origin and let 

C = ~OC, D=d~D. 



Any point P of A B may be expressed as 

P= OP= 0~A + xA = A + x (B- A). 
Any point Q of CD may likewise be written 



If the points P and Q lie in the same line through 0, P and 
are collinear That is 



ADDITION AND SCALAR MULTIPLICATION 



27 



Before it is possible to equate coefficients one of the four 
vectors must be expressed in terms of the other three. 

Then P = A + x (B - A) 

& Tf _1_ ( 1 A _J_ m Tl _1_ >w P I^^T 

Hence 1 x = z y /, 

x = zy m, 
= z [1 + y (n - 1)J. 



Hence 



m 



x = 



y = 



2 = 



i 
i- 

_________ 

I + m 



Substituting in P and ft 



I A+ m B 



+ m 



ft = 



Either of these may be taken as defining a line drawn from 
and cutting A B and CD. 

Vector Relations independent of the Origin 

20.] Example 1 : To divide a line A B in a given ratio 
m : n (Fig. 10). 

Choose any arbitrary point as 
origin. Let OA = A and OB = B. 
To find the vector P = ~OP of which 
the terminus P divides AB in the 
ratio m : n. 

m 



B 



FIG. 10. 



That is, 



P = 



B = A 

-f- 7i 

n A + m B 



n 



(B - A). 
(7) 



28 VECTOR ANALYSIS 

The components of P parallel to A and B are in inverse ratio 
to the segments A P and PB into which the line A B is 
divided by the point P. If it should so happen that P divided 
the line AB externally, the ratio A P / PE would be nega 
tive, and the signs of m and n would be opposite, but the 
formula would hold without change if this difference of sign 
in m and n be taken into account. 

Example 2 : To find the point of intersection of the medians 
of a triangle. 

Choose the origin at random. Let A BC be the given 
triangle. Let 0~A = A, ()B = B, and "00 = C. Let A f , ,C 
be respectively the middle points of the sides opposite the 
vertices A, B, (7. Let M be the point of intersection of the 
medians and M = M the vector drawn to it. Then 



and 

~< = B 



Assuming that has been chosen outside of the plane of the 
triangle so that A, B, C are non-coplanar, corresponding coeffi 
cients may be equated. 



Hence x = y - 

9 3 

Hence M =4 (A + B + C). 



ADDITION AND SCALAR MULTIPLICATION 29 

The vector drawn to the median point of a triangle is equal 
to one third of the sum of the vectors drawn to the vertices. 

In the problems of which the solution has just been given 
the origin could be chosen arbitrarily and the result is in 
dependent of that choice. Hence it is even possible to disre 
gard the origin entirely and replace the vectors A, B, C, etc., 
by their termini A, B, C, etc. Thus the points themselves 
become the subjects of analysis and the formulae read 

n A + m B 
m + n 

and M=~(A + B + C). 

This is typical of a whole class of problems soluble by vector 
methods. In fact any purely geometric relation between the 
different parts of a figure must necessarily be independent 
of the origin assumed for the analytic demonstration. In 
some cases, such as those in Arts. 18, 19, the position of the 
origin may be specialized with regard to some crucial point 
of the figure so as to facilitate the computation ; but in many 
other cases the generality obtained by leaving the origin un- 
specialized and undetermined leads to a symmetry which 
renders the results just as easy to compute and more easy 
to remember. 

Theorem : The necessary and sufficient condition that a 
vector equation represent a relation independent of the origin 
is that the sum of the scalar coefficients of the vectors on 
one side of the sign of equality is equal to the sum of the 
coefficients of the vectors upon the other side. Or if all the 
terms of a vector equation be transposed to one side leaving 
zero on the other, the sum of the scalar coefficients must 
be zero. 

Let the equation written in the latter form be 



30 VECTOR ANALYSIS 

Change the origin from to by adding a constant vector 
B = OO 1 to each of the vectors A, B, C, D ---- The equation 
then becomes 

a (A 4- B) + 6 (B + B) + c (C + B) + d (D + R) + - = 



If this is to be independent of the origin the coefficient of B 
must vanish. Hence 



That this condition is fulfilled in the two examples cited 
is obvious. 



if 



m + n 



If M = \ (A f B + C), 



m + n m + n 

l 

3 



* 21.] The necessary and sufficient condition that two 
vectors satisfy an equation, in which the sum of the scalar 
coefficients is zero, is that the vectors be equal in magnitude 
and in direction. 

First let a A + 6 B = 

and a + 6 = 0. 

It is of course assumed that not both the coefficients a and b 
vanish. If they did the equation would mean nothing. Sub 
stitute the value of a obtained from the second equation into 
the first. 

-&A + 6B = 0. 

Hence A = B. 



ADDITION AND SCALAR MULTIPLICATION 31 

Secondly if A and B are equal in magnitude and direction 

the equation 

A-B = 

subsists between them. The sum of the coefficients is zero. 

The necessaiy and sufficient condition that three vectors 
satisfy an equation, in which the sum of the scalar coefficients 
is zero, is that when drawn from a common origin they termi 
nate in the same straight line. 1 

First let aA + 6B + cC = 

and a + b + c = 0. 

Not all the coefficients a, J, c, vanish or the equations 
would be meaningless. Let c be a non-vanishing coefficient. 
Substitute the value of a obtained from the second equation 
into the first. 



or 

Hence the vector which joins the extremities of C and A is 
collinear with that which joins the extremities of A and B. 
Hence those three points -4, -B, C lie on a line. Secondly 
suppose three vectors A= OA, B = OB,G= 00 drawn from 
the same origin terminate in a straight line. Then the 
vectors 

AB = B - A and A~C = C - A 

are collinear. Hence the equation 



subsists. The sum of the coefficients on the two sides is 
the same. 

The necessary and sufficient condition that an equation, 
in which the sum of the scalar coefficients is zero, subsist 

1 Vectors which have a common origin and terminate in one line are called by 
Hamilton " termino-collinear: 



82 VECTOR ANALYSIS 

between four vectors, is that if drawn from a common origin 
they terminate in one plane. 1 

First let a A + 6B + cC + dV = 

and a + b + c + d = Q. 

Let d be a non-vanishing coefficient. Substitute the value 
of a obtained from the last equation into the first. 



or d (D - A) = 6 (A - B) + c (A - C). 

The line A D is coplanar with A B and A C. Hence all four 
termini A, B, (7, D of A, B, C, D lie in one plane. Secondly 
suppose that the termini of A, B, C, D do lie in one plane. 
Then AZ) = D - A, ~AC = C - A, and ~AB = B - A are co 
planar vectors. One of them may be expressed in terms of 
the other two. This leads to the equation 

/ (B - A) + m (C - A) + n (D - A) = 0, 

where /, m, and n are certain scalars. The sum of the coeffi 
cients in this equation is zero. 

Between any five vectors there exists one equation the sum 
of whose coefficients is zero. 

Let A, B, C,D,E be the five given vectors. Form the 
differences 

E-A, E--B, E-C, E-D. 

One of these may be expressed in terms of the other three 
- or what amounts to the same thing there must exist an 
equation between them. 

ft (E - A) + / (E - B) + m (E - C) + n (E - D) = 0. 
The sum of the coefficients of this equation is zero. 

1 Vectors which have a common origin and terminate in one plane are called 
by Hamilton " termino-complanar." 



ADDITION AND SCALAR MULTIPLICATION 33 

*22.] The results of the foregoing section afford simple 
solutions of many problems connected solely with the geo 
metric properties of figures. Special theorems, the vector 
equations of lines and planes, and geometric nets in two and 
three dimensions are taken up in order. 

Example 1: If a line be drawn parallel to the base of a 
triangle, the line which joins the opposite vertex to the inter 
section of the diagonals of the 
trapezoid thus formed bisects the 
base (Fig. 11). 

Let ABC be the triangle, ED 
the line parallel to the base CB, 
G the point of intersection of the 
diagonals EB and DC of the tra 
pezoid CBDE, and Fthe intersec 
tion of A G with CB. To show FI(J n 
that F bisects CB. Choose the 

origin at random. Let the vectors drawn from it to the 
various points of the figure be denoted by the corresponding 
Clarendons as usual. Then since ED is by hypothesis paral 
lel to CB, the equation 

E - D = n (C - B) 

holds true. The sum of the coefficients is evidently zero as 
it should be. Rearrange the terms so that the equation 

takes on the form 

E nC = "D 7i B. 

The vector E n C is coplanar with E and C. It must cut 
the line EC. The equal vector D 7&B is coplanar with D 
and B. It must cut the line DB. Consequently the vector 
represented by either side of this equation must pass through 
the point A. Hence 

E 7iC = D ?iB = #A. 



34 VECTOR ANALYSIS 

However the points E, 0, and A lie upon the same straight 

line. Hence the equation which connects the vectors E,C, 

and A must be such that the sum of its coefficients is zero. 

This determines x as 1 n. 

Hence B - C = D - B = (1 - w) A. 

By another rearrangement and similar reasoning 

E + 7i B =D + 7iC= (1 + n)Qt. 
Subtract the first equation from the second : 

n (B + C) = (1 + n) G - (1 - n) A. 

This vector cuts EC and AQ. It must therefore be a 
multiple of F and such a multiple that the sum of the coeffi 
cients of the equations which connect B, C, and F or 0, A, 
and F shall be zero. 

Hence n (B + C) = (1 + )G - (1 - ) A = 2 nf. 





Hence F = 



and the theorem has been proved. The proof has covered 
considerable space because each detail of the reasoning has 
been given. In reality, however, the actual analysis has con 
sisted of just four equations obtained simply from the first. 

Example % : To determine the equations of the line and 
plane. 

Let the line be fixed by two points A and B upon it. Let 
P be any point of the line. Choose an arbitrary origin. 
The vectors A, B, and P terminate in the same line. Hence 



aA + 6B 
and a + I + p = 0. 



, 

Therefore P = 

a + b 



ADDITION AND SCALAR MULTIPLICATION 35 

For different points P the scalars a and b have different 
values. They may be replaced by x and y, which are used 
more generally to represent variables. Then 



x + y 

Let a plane be determined by three points -4, B, and C. 
Let P be any point of the plane. Choose an arbitrary origin. 
The vectors A, B, C, and P terminate in one plane. Hence 



6B + cC 

and a + b + c+p = Q. 

aA + 6B + cC 



Therefore P = 



-f c 



As a, 6, c, vary for different points of the plane, it is more 
customary to write in their stead x, y t z. 






+ y + z 

Example 3 ; The line which joins one vertex of a com 
plete quadrilateral to the intersection of two diagonals 
divides the opposite sides har 
monically (Fig. 12). 

Let A, B, C, D be four vertices 
of a quadrilateral. Let A B meet 
CD in a fifth vertex E, and AD 
meet BC in the sixth vertex F. 
Let the two diagonals AC and p 12 

BD intersect in G. To show 

that FG intersects A B in a point i" and CD in a point E 1 
such that the lines AB and (7I> are divided internally at 
E 1 and 2?" in the same ratio as they are divided externally 
by E. That is to show that the cross ratios 



86 VECTOR ANALYSIS 

Choose the origin at random. The four vectors A, B, C, D 
drawn from it to the points A, B, C, D terminate in one 
plane. Hence 



and a + b + e + d = 0. 

Separate the equations by transposing two terms : 



Divide : = 



a + c = (b + d). 
cC 6B + 



a + c b + d 

aA + d D __ 6B + cC 

a + d b + c 

(a + C )G (a + d)F cC di 



In like manner F = 
Form: 



(a + c) - (a + d) " (a + c) (a + d) 
(a + c)Q (a + rf)F cC 



or 

c a c a 

Separate the equations again and divide : 

aA + EB _ cC + 
a -f b c + d 



(6) 



Hence 2? divides A B in the ratio a : b and CD in the ratio 
c / d. But equation (a) shows that JE ff divides C D in the 
ratio c:d. Hence E and E" divide CD internally and 
externally in the same ratio. Which of the two divisions is 
internal and which external depends upon the relative signs 
of c and d. If they have the same sign the internal point 
of division is E; if opposite signs, it is E 1 . In a similar way 
E 1 and E may be shown to divide A B harmonically. 

Example 4 - To discuss geometric nets. 

By a geometric net in a plane is meant a figure composed 
of points and straight lines obtained in the following manner. 
Start with a certain number of points all of which lie in one 



ADDITION AND SCALAR MULTIPLICATION 37 

plane. Draw all the lines joining these points in pairs. 
These lines will intersect each other in a number of points. 
Next draw all the lines which connect these points in pairs. 
This second set of lines will determine a still greater number 
of points which may in turn be joined in pairs and so on. 
The construction may be kept up indefinitely. At each step 
the number of points and lines in the figure increases. 
Probably the most interesting case of a plane geometric net is 
that in which four points are given to commence with. 
Joining these there are six lines which intersect in three 
points different from the given four. Three new lines may 
now be drawn in the figure. These cut out six new points. 
From these more lines may be obtained and so on. 

To treat this net analytically write down the equations 



= (c) 

and a + b + c + d = Q 

which subsist between the four vectors drawn from an unde 
termined origin to the four given points. From these it is 
possible to obtain 

a A + 6B cC + dD 



Tjl 



a + b c + d 

A + cC Z>B + dD 



a + c b + d 

A + dJ) &B + cC 



a + d b + c 

by splitting the equations into two parts and dividing. Next 
four vectors such as A, D, E, F may be chosen and the equa 
tion the sum of whose coefficients is zero may be determined. 
This would be 

aA + dV + (a + b) E-f (a + c) P = 0. 

By treating this equation as (c) was treated new points may 
be obtained* 



38 VECTOR ANALYSIS 

a A + dD (a + 6)E + (a + c)F 



H = 
1 = 



a + d 2a + b + c 

aA + (a + ft)E __ <?D+ (a + c)F 



a 4- c + d 
(a + 6) E 



c a + b + d 

Equations between other sets of four vectors selected from 
A, B, C, D, E, F, may be found ; and from these more points 
obtained. The process of finding more points goes forward 
indefinitely. A fuller account of geometric nets may be 
found in Hamilton s " Elements of Quaternions," Book I. 

As regards geometric nets in space just a word may be 
said. Five points are given. From these new points may be 
obtained by finding the intersections of planes passed through 
sets of three of the given points with lines connecting the 
remaining pairs. The construction may then be carried for 
ward with the points thus obtained. The analytic treatment 
is similar to that in the case of plane nets. There are 
five vectors drawn from an undetermined origin to the given 
five points. Between these vectors there exists an equation 
the sum of whose coefficients is zero. This equation may be 
separated into parts as before and the new points may thus 
be obtained. 



+ 6B cC + dD + 



then F = 



a + b c + d + e 

A + cC 6B + dV + e 



a + b b + d + c 

are two of the points and others may be found in the same 
way. Nets in space are also discussed by Hamilton, loc. cit. 



ADDITION AND SCALAR MULTIPLICATION 39 

Centers of Gravity 

* 23.] The center of gravity of a system of particles may 
be found very easily by vector methods. The two laws of 
physics which will be assumed are the following: 

1. The center of gravity of two masses (considered as 
situated at points) lies on the line connecting the two masses 
and divides it into two segments which are inversely pro 
portional to the masses at the extremities. 

2. In finding the center of gravity of two systems of 
masses each system may be replaced by a single mass equal 
in magnitude to the sum of the masses in the system and 
situated at the center of gravity of the system. 

Given two masses a and b situated at two points A and B. 
Their center of gravity G is given by 



where the vectors are referred to any origin whatsoever. 
This follows immediately from law 1 and the formula (7) 
for division of a line in a given ratio. 

The center of gravity of three masses a, J, c situated at the 
three points -4, B, C may be found by means of law 2. The 
masses a and b may be considered as equivalent to a single 
mass a + b situated at the point 

a A + &B 

a + b 

Then G = (a + 6) " A + 6B + c C 

a -f- b 



TT aA-h&B-f-cC 

Hence G = 

a -f b + c 



40 VECTOR ANALYSIS 

Evidently the center of gravity of any number of masses 
a, &, c, d, ... situated at the points A, B, C, D, ... may 
be found in a similar manner. The result is 

aA + ftB + cO + rfD + ... ^ 
a + b + c + d + ... 

Theorem 1 : The lines which join the center of gravity of a 
triangle to the vertices divide it into three triangles which 
are proportional to the masses at the op 
posite vertices (Fig. 13). Let A, B, C 
be the vertices of a triangle weighted 
with masses a, &, c. Let G be the cen 
ter of gravity. Join A, B, C to G and 
produce the lines until they intersect 
the opposite sides in A f , B\ C 1 respectively. To show that 
the areas 

G B C : G C A : G A B : A B C = a : b : c : a + b + c . 

The last proportion between ABC and a + b + c comes 
from compounding the first three. It is, however, useful in 
the demonstration. 

ABC AA A G . GA b + c 

+ 1. 



Hence 



GBC~ GA! CTA G~A f 

ABC a + b + c 



In a similar manner 
and 



GBC a 

BCA a + I + c 



GCA~ b 

CAB _ a + b + c 
GAB ~ ~~c 



Hence the proportion is proved. 

Theorem 2 : The lines which join the center of gravity of 
a tetrahedron to the vertices divide the tetrahedron into four 



ADDITION AND SCALAR MULTIPLICATION 41 

tetrahedra which are proportional to the masses at the oppo 
site vertices. 

Let -4, B, C, D be the vertices of the tetrahedron weighted 
respectively with weights a, &, c, d. Let be the center of 
gravity. Join A, B, C, D to G and produce the lines until 
they meet the opposite faces in A , B\ G\ D . To show that 
the volumes 

BCDG:CDAG:DABG:ABCG:ABCD 



BCDA 



BCDG 

In like manner 

and 

and 



ABCD d 



which proves the proportion. 

* 24.] By a suitable choice of the three masses, a, J, c lo 
cated at the vertices A, B, (7, the center of gravity G may 
be made to coincide with any given point P of the triangle. 
If this be not obvious from physical considerations it cer 
tainly becomes so in the light of the foregoing theorems. 
For in order that the center of gravity fall at P, it is only 
necessary to choose the masses a, 6, c proportional to the 
areas of the triangles PEG, PCA^ and PAB respectively. 
Thus not merely one set of masses a, &, c may be found, but 
an infinite number of sets which differ from each other only 
by a common factor of proportionality. These quantities 



42 VECTOR ANALYSIS 

a, 6, c may therefore be looked upon as coordinates of the 
points P inside of the triangle ABC. To each set there 
corresponds a definite point P, and to each point P there 
corresponds an infinite number of sets of quantities, which 
however do not differ from one another except for a factor 
of proportionality. 

To obtain the points P of the plane ABC which lie outside 
of the triangle ABC one may resort to the conception of 
negative weights or masses. The center of gravity of the 
masses 2 and 1 situated at the points A and B respectively 
would be a point G dividing the line A B externally in the 
ratio 1 : 2. That is 



Any point of the line A B produced may be represented by 
a suitable set of masses a, b which differ in sign. Similarly 
any point P of the plane ABC may be represented by a 
suitable set of masses a, 6, c of which one will differ in sign 
from the other two if the point P lies outside of the triangle 
ABC. Inasmuch as only the ratios of a, 6, and c are im 
portant two of the quantities may always be taken positive. 

The idea of employing the masses situated at the vertices 
as coordinates of the center of gravity is due to Mobius and 
was published by him in his book entitled " Der barycentrische 
Calcul" in 1827. This may be fairly regarded as the starting 
point of modern analytic geometry. 

The conception of negative masses which have no existence 
in nature may be avoided by replacing the masses at the 
vertices by the areas of the triangles GBC, GO A, and 
GAB to which they are proportional. The coordinates of 
a point P would then be three numbers proportional to the 
areas of the three triangles of which P is the common vertex ; 
and the sides of a given triangle ABC, the bases. The sign 
of these areas is determined by the following definition. 



ADDITION AND SCALAR MULTIPLICATION 43 

Definition: The area ABC of a triangle is said to be 
positive when the vertices A, B, C follow each other in the 
positive or counterclockwise direction upon the circle de 
scribed through them. The area is said to be negative when 
the points follow in the negative or clockwise direction. 

Cyclic permutation of the letters therefore does not alter 
the sign of the area. 



Interchange of two letters which amounts to a reversal of 
the cyclic order changes the sign. 

A CB = BA = CBA = -A B C. 
If P be any point within the triangle the equation 
PAB+PBC+PCA=ABC 

must hold. The same will also hold if P be outside of the 
triangle provided the signs of the areas be taken into con 
sideration. The areas or three quantities proportional to 
them may be regarded as coordinates of the point P. 

The extension of the idea of " barycentric " coordinates to 
space is immediate. The four points A, B, C, D situated at 
the vertices of a tetrahedron are weighted with mass a, J, c, d 
respectively. The center of gravity G is represented by 
these quantities or four others proportional to them. To 
obtain points outside of the tetrahedron negative masses 
may be employed. Or in the light of theorem 2, page 40, 
the masses may be replaced by the four tetrahedra which 
are proportional to them. Then the idea of negative vol 
umes takes the place of that of negative weights. As this 
idea is of considerable importance later, a brief treatment of 
it here may not be out of place. 

Definition : The volume A B CD of a tetrahedron is said 
to be positive when the triangle ABC appears positive to 



44 VECTOR ANALYSIS 

the eye situated at the point D. The volume is negative 
if the area of the triangle appear negative. 

To make the discussion of the signs of the various 
tetrahedra perfectly clear it is almost necessary to have a 
solid modeL A plane drawing is scarcely sufficient. It is 
difficult to see from it which triangles appear positive and 
which negative. The following relations will be seen to 
hold if a model be examined. 

The interchange of two letters in the tetrahedron A BCD 
changes the sign. 

ACBD = CBAD=BACD=DBCA 



The sign of the tetrahedron for any given one of the pos 
sible twenty-four arrangements of the letters may be obtained 
by reducing that arrangement to the order A B C D by 
means of a number of successive interchanges of two letters. 
If the number of interchanges is even the sign is the same 
as that of A B CD ; if odd, opposite. Thus 



If P is any point inside of the tetrahedron A B CD the 
equation 

ABCP-BCDP+ CDAP-DABP=ABCD 

holds good. It still is true if P be without the tetrahedron 
provided the signs of the volumes be taken into considera 
tion. The equation may be put into a form more symmetri 
cal and more easily remembered by transposing all the terms 
to one number. Then 



The proportion in theorem 2, page 40, does not hold true 
if the signs of the tetrahedra be regarded. It should read 

BCDG:CDGA:DGAB:GABC:ABCD 



ADDITION AND SCALAR MULTIPLICATION 45 

If the point G- lies inside the tetrahedron a, J, c, d repre 
sent quantities proportional to the masses which must be 
located at the vertices A,B,C,D respectively if G is to be the 
center of gravity. If G lies outside of the tetrahedron they may 
still be regarded as masses some of which are negative or 
perhaps better merely as four numbers whose ratios determine 
the position of the point Gr. In this manner a set of "bary- 
centric " coordinates is established for space. 

The vector P drawn from an indeterminate origin to any 
point of the plane A B C is (page 35) 

aA + yB + zC 

x + y + z 

Comparing this with the expression 

aA + &B + cC 
a + b + c 

it will be seen that the quantities x, y, z are in reality nothing 
more nor less than the barycentric coordinates of the point P 
with respect to the triangle ABO. In like manner from 

equation 

__#A + yB + 2C + wD 

x + y + z + w 

which expresses any vector P drawn from an indeterminate 
origin in terms of four given vectors A, B, C, D drawn from 
the same origin, it may be seen by comparison with 

+ &B + c C + rfD 



= 



a + b + c + d 



that the four quantities x, y, 2, w are precisely the bary 
centric coordinates of P, the terminus of P, with respect to 
the tetrahedron A B CD. Thus the vector methods in which 
the origin is undetermined and the methods of the " Bary 
centric Calculus " are practically co-extensive. 

It was mentioned before and it may be well to repeat here 



46 



VECTOR ANALYSIS 



that the origin may be left wholly out of consideration and 
the vectors replaced by their termini. The vector equations 
then become point equations 

x A + y B 4- z 



and 



x + y + z 

xA + yB + zC + wD 

w. 



At 



x + y + z 

This step brings in the points themselves as the objects of 
analysis and leads still nearer to the " Barycentrische Calcul " 
of Mobius and the "Ausdehnungslehre " of Grassmann. 

The Use of Vectors to denote Areas 

25.] Definition: An area lying in one plane MN and 
bounded by a continuous curve PQR which nowhere cuts 
itself is said to appear positive from the point when the 

letters PQR follow each 
other in the counterclockwise 
or positive order; negative, 
when they follow in the 
negative or clockwise order 
(Fig. 14). 

It is evident that an area 
can have no determined sign 
per se, but only in reference 
to that direction in which its 

boundary is supposed to be traced and to some point out 
side of its plane. For the area P R Q is negative relative to 
PQR; and an area viewed from is negative relative to the 
same area viewed from a point O f upon the side of the plane 
opposite to 0. A circle lying in the X F-plane and described 
in the positive trigonometric order appears positive from every 
point on that side of the plane on which the positive axis 
lies, but negative from all points on the side upon which 



ADDITION AND SCALAR MULTIPLICATION 47 

the negative ^-axis lies. For this reason the point of view 
and the direction of description of the boundary must be kept 
clearly in mind. 

Another method of stating the definition is as follows : If 
a person walking upon a plane traces out a closed curve, the 
area enclosed is said to be positive if it lies upon his left- 
hand side, negative if upon his right. It is clear that if two 
persons be considered to trace out together the same curve by 
walking upon opposite sides of the plane the area enclosed 
will lie upon the right hand of one and the left hand of the 
other. To one it will consequently appear positive ; to the 
other, negative. That side of the plane upon which the area 
seems positive is called the positive side ; the side upon 
which it appears negative, the negative side. This idea is 
familiar to students of electricity and magnetism. If an 
electric current flow around a closed plane curve the lines of 
magnetic force through the circuit pass from the negative to 
the positive side of the plane. A positive magnetic pole 
placed upon the positive side of the plane will be repelled by 
the circuit. 

A plane area may be looked upon as possessing more than 
positive or negative magnitude. It may be considered to 
possess direction, namely, the direction of the normal to the 
positive side of the plane in which it lies. Hence a plane 
area is a vector quantity. The following theorems concerning 
areas when looked upon as vectors are important. 

Theorem 1 : If a plane area be denoted by a vector whose 
magnitude is the numerical value of that area and whose 
direction is the normal upon the positive side of the plane, 
then the orthogonal projection of that area upon a plane 
will be represented by the component of that vector in the 
direction normal to the plane of projection (Fig. 15). 

Let the area A lie in the plane MN. Let it be projected 
orthogonally upon the plane M N . Let M N&nd M* N r inter- 



48 



VECTOR ANALYSIS 



sect in the line I and let the diedral angle between these 
two planes be x. Consider first a rectangle PQJRS in MN 
whose sides, PQ, RS and QR, SP are respectively parallel 
and perpendicular to the line /. This will project into a 
rectangle P Q R S 1 in M N . The sides P Q f and JR S 
will be equal to PQ and US; but the sides Q 1 R and S P 
will be equal to QR and SP multiplied by the cosine of #, 
the angle between the planes. Consequently the rectangle 



At 



FIG. 15. 



Hence rectangles, of which the sides are respectively 
parallel and perpendicular to I, the line of intersection of the 
two planes, project into rectangles whose sides are likewise 
respectively parallel and perpendicular to I and whose area is 
equal to the area of the original rectangles multiplied by the 
cosine of the angle between the planes. 

From this it follows that any area A is projected into an 
area which is equal to the given area multiplied by the cosine 
of the angle between the planes. For any area A may be di 
vided up into a large number of small rectangles by drawing a 
series of lines in MN parallel and perpendicular to the line I. 



ADDITION AND SCALAR MULTIPLICATION 49 

Each of these rectangles when projected is multiplied by the 
cosine of the angle between the planes and hence the total 
area is also multiplied by the cosine of that angle. On the 
other hand the component A of the vector A, which repre 
sents the given area, in the direction normal to the plane 
M f N f of projection is equal to the total vector A multiplied 
by the cosine of the angle between its direction which is 
the normal to the plane M ^and the normal to M N r . This 
angle is x ; for the angle between the normals to two planes 
is the same as the angle between the planes. The relation 
between the magnitudes of A and A is therefore 

A 1 = A cos x, 

which proves the theorem. 

26.] Definition : Two plane areas regarded as vectors are 
said to be added when the vectors which represent them are 
added. 

A vector area is consequently the sum of its three com 
ponents obtainable by orthogonal projection upon three 
mutually perpendicular planes. Moreover in adding two 
areas each may be resolved into its three components, the 
corresponding components added as scalar quantities, and 
these sums compounded as vectors into the resultant area. 
A generalization of this statement to the case where the three 
planes are not mutually orthogonal and where the projection 
is oblique exists. 

A surface made up of several plane areas may be repre 
sented by the vector which is the sum of all the vectors 
representing those areas. In case the surface be looked upon 
as forming the boundary or a portion of the boundary of a 
solid, those sides of the bounding planes which lie outside of 
the body are conventionally taken to be positive. The vec 
tors which represent the faces of solids are always directed 
out from the solid, not into it 

4 



50 VECTOR ANALYSIS 

Theorem 2 : The vector which represents a closed polyhedral 
surface is zero. 

This may be proved by means of certain considerations of 
hydrostatics. Suppose the polyhedron drawn in a body of 
fluid assumed to be free from all external forces, gravity in 
cluded. 1 The fluid is in equilibrium under its own internal 
pressures. The portion of the fluid bounded by the closed 
surface moves neither one way nor the other. Upon each face 
of the surface the fluid exerts a definite force proportional 
to the area of the face and normal to it. The resultant of all 
these forces must be zero, as the fluid is in equilibrium. Hence 
the sum of all the vector areas in the closed surface is zero. 

The proof may be given in a purely geometric manner. 
Consider the orthogonal projection of the closed surface upon 
any plane. This consists of a double area. The part of the 
surface farthest from the plane projects into positive area ; 
the part nearest the plane, into negative area. Thus the 
surface projects into a certain portion of the plane which is 
covered twice, once with positive area and once with negative. 
These cancel each other. Hence the total projection of a 
closed surface upon a plane (if taken with regard to sign) is 
zero. But by theorem 1 the projection of an area upon a 
plane is equal to the component of the vector representing 
that area in the direction perpendicular to that plane. Hence 
the vector which represents a closed surf ace has no component 
along the line perpendicular to the plane of projection. This, 
however, was any plane whatsoever. Hence the vector is 
zero. 

The theorem has been proved for the case in which the 
closed surface consists of planes. In case that surface be 



1 Such a state of affairs is realized to all practical purposes in the case of a 
polyhedron suspended in the atmosphere and consequently subjected to atmos 
pheric pressure. The force of gravity acts but is counterbalanced by the tension 
in the suspending string. 



ADDITION AND SCALAR MULTIPLICATION 51 

curved it may be regarded as the limit of a polyhedral surface 
whose number of faces increases without limit. Hence the 
vector which represents any closed surface polyhedral or 
curved is zero. If the surface be not closed but be curved it 
may be represented by a vector just as if it were polyhedral. 
That vector is the limit l approached by the vector which 
represents that polyhedral surface of which the curved surface 
is the limit when the number of faces becomes indefinitely 
great. 

SUMMARY OF CHAPTER I 

A vector is a quantity considered as possessing magnitude 
and direction. Equal vectors possess the same magnitude 
and the same direction. A vector is not altered by shifting it 
parallel to itself. A null or zero vector is one whose mag 
nitude is zero. To multiply a vector by a positive scalar 
multiply its length by that scalar and leave its direction 
unchanged. To multiply a vector by a negative scalar mul 
tiply its length by that scalar and reverse its direction. 

Vectors add according to the parallelogram law. To subtract 
a vector reverse its direction and add. Addition, subtrac 
tion, and multiplication of vectors by a scalar follow the same 
laws as addition, subtraction, and multiplication in ordinary 
algebra. A vector may be resolved into three components 
parallel to any three non-coplanar vectors. This resolution 
can be accomplished in only one way. 

r = x* + yb + zc. (4) 

The components of equal vectors, parallel to three given 
non-coplanar vectors, are equal, and conversely if the com 
ponents are equal the vectors are equal. The three unit 
vectors i, j, k form a right-handed rectangular system. In 

1 This limit exists and is unique. It is independent of the method in which 
the polyhedral surface approaches the curved surface. 



52 VECTOR ANALYSIS 

terms of them any vector may be expressed by means of the 
Cartesian coordinates #, y, z. 

r = xi + yj+zk. (6) 

Applications. The point which divides a line in a given 
ratio m : n is given by the formula 

(7) 



m + n 

The necessary and sufficient condition that a vector equation 
represent a relation independent of the origin is that the sum 
of the scalar coefficients in the equation be zero. Between 
any four vectors there exists an equation with scalar coeffi 
cients. If the sum of the coefficients is zero the vectors are 
termino-coplanar. If an equation the sum of whose scalar 
coefficients is zero exists between three vectors they are 
termino-collinear. The center of gravity of a number of 
masses a, &, c situated at the termini of the vectors 
A, B, C supposed to be drawn from a common origin is 
given by the formula 



A vector may be used to denote an area. If the area is 
plane the magnitude of the vector is equal to the magnitude 
of the area, and the direction of the vector is the direction of 
the normal upon the positive side of the plane. The vector 
representing a closed surface is zero. 

EXERCISES ON CHAPTER I 

1. Demonstrate the laws stated in Art. 12. 

2. A triangle may be constructed whose sides are parallel 
and equal to the medians of any given triangle. 



ADDITION AND SCALAR MULTIPLICATION 53 

3. The six points in which the three diagonals of a com* 
plete quadrangle l meet the pairs of opposite sides lie three 
by three upon four straight lines. 

4. If two triangles are so situated in space that the three 
points of intersection of corresponding sides lie on a line, then 
the lines joining the corresponding vertices pass through a 
common point and conversely. 

5. Given a quadrilateral in space. Find the middle point 
of the line which joins the middle points of the diagonals. 
Find the middle point of the line which joins the middle 
points of two opposite sides. Show that these two points are 
the same and coincide with the center of gravity of a system 
of equal masses placed at the vertices of the quadrilateral. 

6. If two opposite sides of a quadrilateral in space be 
divided proportionally and if two quadrilaterals be formed by 
joining the two points of division, then the centers of gravity 
of these two quadrilaterals lie on a line with the center of 
gravity of the original quadrilateral. By the center of gravity 
is meant the center of gravity of four equal masses placed at 
the vertices. Can this theorem be generalized to the case 
where the masses are not equal ? 

7. The bisectors of the angles of a triangle meet in a 
point. 

8. If the edges of a hexahedron meet four by four in three 
points, the four diagonals of the hexahedron meet in a point. 
In the special case in which the hexahedron is a parallelepiped 
the three points are at an infinite distance 

9. Prove that the three straight lines through the middle 
points of the sides of any face of a tetrahedron, each parallel 
to the straight line connecting a fixed point P with the mid 
dle point of the opposite edge of the tetrahedron, meet in a 

1 A complete quadrangle consists of the six straight lines which may he passed 
through four points no three of which are collinear. The diagonals are the lines 
which join the points of intersection of pairs of sides 



54 VECTOR ANALYSIS 

point E and that this point is such that PE passes through 
and is bisected by the center of gravity of the tetrahedron. 

10. Show that without exception there exists one vector 
equation with scalar coefficients between any four given 
vectors A, B, C, D. 

11. Discuss the conditions imposed upon three, four, or 
five vectors if they satisfy two equations the sum of the co 
efficients in each of which is zero. 



CHAPTER II 

DIRECT AND SKEW PRODUCTS OF VECTORS 

Products of Two Vectors 

27.] THE operations of addition, subtraction, and scalar 
multiplication have been defined for vectors in the way 
suggested by physics and have been employed in a few 
applications. It now becomes necessary to introduce two 
new combinations of vectors. These will be called products 
because they obey the fundamental law of products ; i. e., the 
distributive law which states that the product of A into the 
sum of B and C is equal to the sum of the products of A into 
B and A into C. 

Definition : The direct product of two vectors A and B is 
the scalar quantity obtained by multiplying the product of 
the magnitudes of the vectors by the cosine of the angle be 
tween them. 

The direct product is denoted by writing the two vectors 
with a dot between them as 

A-B. 

This is read A dot B and therefore may often be called the 
dot product instead of the direct product. It is also called 
the scalar product owing to the fact that its value is sca 
lar. If A be the magnitude of A and B that of B, then by 
definition 

A-B = ^cos (A,B). (1) 

Obviously the direct product follows the commutative law 

A-B = B A. (2) 



56 VECTOR ANALYSIS 

If either vector be multiplied by a scalar the product is 
multiplied by that scalar. That is 

(x A) B = A (x B) = x (A B). 

In case the two vectors A and B are collinear the angle be 
tween them becomes zero or one hundred and eighty degrees 
and its cosine is therefore equal to unity with the positive or 
negative sign. Hence the scalar product of two parallel 
vectors is numerically equal to the product of their lengths. 
The sign of the product is positive when the directions of the 
vectors are the same, negative when they are opposite. The 
product of a vector by itself is therefore equal to the square 

of its length 

A.A=^4 2 . (3) 

Consequently if the product of a vector by itself vanish the 
vector is a null vector. 

In case the two vectors A and B are perpendicular the 
angle between them becomes plus or minus ninety degrees 
and the cosine vanishes. Hence the product A B vanishes. 
Conversely if the scalar product A B vanishes, then 

A B cos (A, B) = 0. 

Hence either A or B or cos (A, B) is zero, and either the 
vectors are perpendicular or one of them is null. Thus the 
condition for the perpendicularity of two vectors, neither of 
which vanishes, is A B = 0. 

28.] The scalar products of the three fundamental unit 
vectors i, j, k are evidently 

ii = jj = kk = l, (4) 

i . j = j . k = k . i = 0. 

If more generally a and b are any two unit vectors the 
product 

a b = cos (a, b). 



DIRECT AND SKEW PRODUCTS OF VECTORS 57 

Thus the scalar product determines the cosine of the angle 
between two vectors and is in a certain sense equivalent to 
it. For this reason it might be better to give a purely 
geometric definition of the product rather than one which 
depends upon trigonometry. This is easily accomplished as 
follows : If a and b are two unit vectors, a b is the length 
of the projection of either upon the other. If more generally 
A and B are any two vectors A B is the product of the length 
of either by the length of projection of the other upon it. 
From these definitions the facts that the product of a vector 
by itself is the square of its length and the product of two 
perpendicular vectors is zero follow immediately. The trigo 
nometric definition can also readily be deduced. 

The scalar product of two vectors will appear whenever the 
cosine of the included angle is of importance. The following 
examples may be cited. The projection of a vector B upon a 
vector A is 

AB AB 

A = A a cos (A, B) = B cos (A, B) a, (5) 



A A 

where a is a unit vector in the direction of A. If A is itself a 
unit vector the formula reduces to 

(A-B) A = cos (A,B) A. 

If A be a constant force and B a displacement the work done 

by the force A during the displacement is A B. If A repre 

sent a plane area (Art. 25), and if B be a 

vector inclined to that plane, the scalar prod 

uct A B will be the volume of the cylinder 

of which the area A is the base and of 

which B is the directed slant height. For 

the volume (Fig. 16) is equal to the base FlG 

A multiplied by the altitude h. This is 

the projection of B upon A or B cos (A, B). Hence 

v = A h = A B cos (A, B) = A B. 



58 VECTOR ANALYSIS 

29.] The scalar or direct product follows the distributive 
law of multiplication. That is 

(A + B) .C = A-C + B.C. (6) 

This may be proved by means of projections. Let C be equal 
to its magnitude C multiplied by a unit vector c in its direc 
tion. To show 

(A + B) (<7c) = A (0o) + B. (0o) 
or (A + B) c = A c + B c. 

A c is the projection of A upon c ; B c, that of B upon c ; 
(A + B) c, that of A + B upon c. But the projection of the 
sum A + B is equal to the sum of the projections. Hence 
the relation (6) is proved. By an immediate generalization 

(A + B + ...) (P + a+-") = A-P + A.Q+... 

+ B.P + B.Q + ... <ey 



The scalar product may be used just as the product in ordi 
nary algebra. It has no peculiar difficulties. 

If two vectors A and B are expressed in terms of the 
three unit vectors i, j, k as 

A = ^[ 1 i + ^ 2 j + ^ 8 k, 
and B = ^ i + JB 2 j + B k, 

then A- B = (A l i + A z j + A B k) . (^i + 2 j + ^k) 
= A l B l i . i + A l 2 i j + A l B% i k 



By means of (4) this reduces to 

A-* = A 1 l + A 2 E, + A B JB,. (7) 

If in particular A and B are unit vectors, their components 
A l ,A^,A 3 and B 19 S 29 S B are the direction cosines of the 
lines A and B referred to X, Y, Z. 



DIRECT AND SKEW PRODUCTS OF VECTORS 59 

A l = cos (A, JT), A<i = cos (A, F), A z = cos (A, ^f), 
^! = cos (B, JT), .# 2 = cos (B, T), 3 = cos (B, ). 

Moreover A B is the cosine of the included angle. Hence 
the equation becomes 

cos (A, B) = cos (A, X) cos (B, X) + cos (A, T) cos (B, T) 

+ cos (A,) cos (B,Z). 

In case A and B are perpendicular this reduces to the well- 
known relation 

= cos (A, JT) cos (B, X) + cos (A, Y) cos (B, F) 

+ cos (A,^) cos (B,) 

between the direction cosines of the 
line A and the line B. 

30.] If A and B are two sides A 



and OB of a triangle OAB, the third o 

side AisG = -B-JL (Fig. 17). PlG 17 * 



C*C = (B-A). (B-A)=B-B 

or (7 2 = A 2 + J5 2 -2 A^cos(AB). 



That is, the square of one side of a triangle is equal to the 
sum of the squares of the other two sides diminished by twice 
their product times the cosine of the angle between them. 
Or, the square of one side of a triangle is equal to the sum of 
the squares of the other two sides diminished by twice the 
product of either of those sides by the projection of the other 
upon it the generalized Pythagorean theorem. 

If A and B are two sides of a parallelogram, C = A + B 
and D = A B are the diagonals. Then 

C.C = (A + B).(A + B)=A.A + 2A.B + B.B, 
D.D=(A-B).(A-B)=A-A-2A.B + B.B, 

C-C + D.D = 2(A-A + BB), 
or a 2 + 7) 2 = 2 (A* 



60 VECTOR ANALYSIS 

That is, the sum of the squares of the diagonals of a parallelo 
gram is equal to twice the sum of the squares of two sides. 
In like manner also 



or C*-D 2 = 4A cos (A, B). 

That is, the difference of the squares of the diagonals of a 
parallelogram is equal to four times the product of one of the 
sides by the projection of the other upon it. 

If A is any vector expressed in terms of i, j, k as 

A = A l i + A 2 j + A B k, 
then A A = A* = A* + A* + A*. (8) 

But if A be expressed in terms of any three non-coplanar unit 
vectors a, b, c as 



+ 2 J c bc + 2 c a e a 

A 2 = a? + 6 2 + c 2 + 2 a b cos (a, b) + 2 b c cos (b, c) 

+ 2 ca cos (c, a). 

This formula is analogous to the one in Cartesian geometry 
which gives the distance between two points referred to 
oblique axes. If the points be x v y v z v and # 2 , y v z% the 
distance squared is 

D 2 = (* 2 - x^ + (y a - yi ) 2 + (z 2 - zj* 
+ 2 (a, - xj (y a - 2/0 cos (X, Y) 
+ 2 (y t - ft) (,-*!> cos (F.S) 
+ 2 (z 2 -24) (x 2 - xj cos (^,-T). 

31.] Definition: The skew product of the vector A into 
the vector B is the vector quantity C whose direction is the 
normal upon that side of the plane of A and B on which 



DIRECT AND SKEW PRODUCTS OF VECTORS 61 

rotation from A to B through an angle of less than one 
hundred and eighty degrees appears positive or counter 
clockwise ; and whose magnitude is obtained by multiplying 
the product of the magnitudes of A and B by the sine of the 
angle from A to B. 

The direction of A x B may also be defined as that in 
which an ordinary right-handed 
screw advances as it turns so as c= AXB 
to carry A toward B (Fig. 18). 

The skew product is denoted by 
a cross as the direct product was ^ 

by a dot. It is written FIG. 18. 

C = A x B 

and read A cross B. For this reason it is often called the cross 
product. More frequently, however, it is called the vector prod 
uct, owing to the fact that it is a vector quantity and in con 
trast with the direct or scalar product whose value is scalar. 
The vector product is by definition 

C = A x B = ^J5sin (A,B)c, (9) 

when A and B are the magnitudes of A and B respectively and 
where c is a unit vector in the direction of C. In case A and 
B are unit vectors the skew product A X B reduces to the 
unit vector c multiplied by the sine of the angle from A to B. 
Obviously also if either vector A or B is multiplied by a scalar 
x their product is multiplied by that scalar. 

A) X B = A X (zB) = xC. 

If A and B are parallel the angle between them is either zero 
or one hundred and eighty degrees. In either case the sine 
vanishes and consequently the vector product A X B is a null 
vector. And conyersely if A X B is zero 

A B sin (A, B) = 0. 



62 VECTOR ANALYSIS 

Hence A or B or sin (A, B) is zero. Thus the condition for 
parallelism of two vectors neither of which vanishes is A X B 
= 0. As a corollary the vector product of any vector into 
itself vanishes. 

32.] The vector product of two vectors will appear wher 
ever the sine of the included angle is of importance, just as 
the scalar product did in the case of the cosine. The two prod 
ucts are in a certain sense complementary. They have been 
denoted by the two common signs of multiplication, the dot 
and the cross. In vector analysis they occupy the place held 
by the trigonometric functions of scalar analysis. They are 
at the same time amenable to algebraic treatment, as will be 
seen later. At present a few uses of the vector product may 
be cited. 

If A and B (Fig. 18) are the two adjacent sides of a parallel 
ogram the vector product 

C = A x B = A B sin (A, B) c 

represents the area of that parallelogram in magnitude and 
direction (Art. 25). This geometric representation of A X B 
is of such common occurrence and importance that it might 
well be taken as the definition of the product. From it the 
trigonometric definition follows at once. The vector product 
appears in mechanics in connection with couples. If A and 
A are two forces forming a couple, the moment of the 
couple is A X B provided only that B is a vector drawn from 
any point of A to any point of A. The product makes its 
appearance again in considering the velocities of the individ 
ual particles of a body which is rotating with an angular ve 
locity given in magnitude and direction by A. If R be the 
radius vector drawn from any point of the axis of rotation A 
the product A X & will give the velocity of the extremity of 
B (Art. 51). This velocity is perpendicular alike to the axis 
of rotation and to the radius vector B. 



DIRECT AND SKEW PRODUCTS OF VECTORS 63 

33.] The vector products A X B and B x A are not the 

same. They are in fact the negatives of each other. For if 
rotation from A to B appear positive on one side of the plane 
of A and B, rotation from B to A will appear positive on the 
other. Hence A X B is the normal to the plane of A and B 
upon that side opposite to the one upon which B x A is the 
normal. The magnitudes of A X B and B X A are the same. 

Hence 

AxB = -BxA. (10) 

The factors in a vector product can be interchanged if and only 
if the sign of the product be reversed. 

This is the first instance in which the laws of operation in 
vector analysis differ essentially from those of scalar analy 
sis. It may be that at first this change of sign which must 
accompany the interchange of factors in a vector product will 
give rise to some difficulty and confusion. Changes similar to 
this are, however, very familiar. No one would think of inter 
changing the order of x and y in the expression sin (x y) 
without prefixing the negative sign to the result. Thus 

sin (y x) = sin (x y), 

although the sign is not required for the case of the cosine, 
cos (y x) = cos ( x y). 

Again if the cyclic order of the letters ABC in the area of a 
triangle be changed, the area will be changed in sign (Art. 

25). 

AB C = -ACB. 

In the same manner this reversal of sign, which occurs 
when the order of the factors in a vector product is reversed, 
will appear after a little practice and acquaintance just as 
natural and convenient as it is necessary. 

34.] The distributive law of multiplication holds in the 
case of vector products just as in ordinary algebra except 



64 



VECTOR ANALYSTS 






that the order of the factors must be carefully maintained 
when expanding. 

A very simple proof may be given by making use of the ideas 
developed in Art. 26. Suppose that C 
is not coplanar with A and B. Let A 
and B be two sides of a triangle taken 
in order. Then (A + B) will be the 
third side (Fig. 19). Form the prism 
of which this triangle is the base and 
of which C is the slant height or edge. 
The areas of the lateral faces of this 
prism are 

A x C, B x C, (A -f B) x C. 
The areas of the bases are 

5 (A x B) and - - (A x B). 

But the sum of all the faces of the prism is zero; for the 
prism is a closed surface. Hence 



4 

FIG. 19. 



AxC + BxC-(A + B)xC = 0, 
or A X C + B X C = (A + B) X C. (11) 

The relation is therefore proved in case C is non-coplanar 
with A and B. Should C be coplanar with A and B, choose D, 
any vector out of that plane. Then C + D also will lie out of 
that plane. Hence by (11) 

A X (C + D) + B X (C + D) = (A + B) x (C + D). 

Since the three vectors in each set A, C, D, and B, C, D, and 
A + B, C, D will be non-coplanar if D is properly chosen, the 
products may be expanded. 



DIRECT AND SKEW PRODUCTS OF VECTORS 65 

AxC + AxD-fBxC + BxD 

= (A + B) x C + (A + B) x D. 
But by (11) AxD + BxD = (A + B)xD. 
Hence AxC + BxC = (A + B)xC. 

This completes the demonstration. The distributive law holds 
for a vector product. The generalization is immediate. 

(A + B+---)x(P + a + ---) = AxP + Axa + --- (11) 

+ B x P + B x <J + 



35.] The vector products of the three unit vectors i, j, k are 
easily seen by means of Art. 17 to be 

ixi = jxj = kxk = 0, 

ixj=-j xi = k, (12) 

jxk = k x j = i, 
kxi = ixk=j. 

The skew product of two equal l vectors of the system i, j, k 
is zero. The product of two unequal vectors is the third taken 
with the positive sign if the vectors follow in the cyclic order 
i j k but with the negative sign if they do not. 

If two vectors A and B are expressed in terms of i, j, k, 
their vector product may be found by expanding according 
to the distributive law and substituting. 



A x B = (A l i + -4 2 j + ^ 3 k) x (^i + 2 j + 3*) 

= A l l ixi + A l B 2 ixj + A l BzixTt 

+ A 2 l j x i + AI 2 j x j + AZ B B j x k, 

+ A z S 1 k x i + A B BZ k x j + A z B z k x k. 

Hence A x B = (A^B^ - A B 2 ) i + (A Z B 1 -A,B B )j 

4- (A, z - A 2 BJ k. 

1 This follows also from the fact that the sign is changed when the order of 
factors is reversed. Hence j X j = j Xj=0. 

5 



66 VECTOR ANALYSIS 

This may be written in the form of a determinant as 

Ax B = 

The formulae for the sine and cosine of the sum or dif 
ference of two angles follow immediately from the dot and 
cross products. Let a and b be two unit vectors lying in the 
i j-plane. If x be the angle that a makes with i, and y the 
angle b makes with i, then 



a = 



Hence 
If 

Hence 



Hence 



Hence 



a b : 
a b : 

cos (y x) : 
V 

a.V: 

cos (y + x) : 

a x b : 

a x b = 

sin (y x) - 

axb = 

ax b = 

sin (y + x) - 



cos x i + sin x j, 

cos y i -f sin y j, 

cos (a, b) = cos (y x), 

cos x cos y + sin x sin y. 

cos y cos x + sin y sin x. 

cos y i sin y j, 

cos (a, b ) = cos (y + x). 

cos y cos x sin y sin x. 

k sin (a, b) = k sin (y x), 

k (sin y cos x sin x cos y). 

sin y cos x sin x cos y. 

k sin (a, b ) = k sin (y + x) 9 

k (sin y cos x + sin x cos y). 

sin y cos x + sin # cos y. 



If /, m, 7i and Z , w , TI are the direction cosines of two 
unit vectors a and a referred to JT, F, , then 

a = li + m j + 7i k, 



m j 



a a = cos (a, a ) = IV + m m r + n n f , 

as has already been shown in Art. 29. The familiar formula 
for the square of the sine of the angle between a and a may 
be found. 



DIRECT AND SKEW PRODUCTS OF VECTORS 67 

a x a = sin (a, a ) e = (mn f m ri) i + (n V n ? I) j 
+ (Jm -f m) k, 

where e is a unit vector perpendicular to a and a . 

(a x a ) (a x a ) = sin 2 (a, a f ) e e = sin 2 (a, a ). 
sin 2 (a, a ) = (mn m r n)*+ (nl n f /) 2 +(lm I m)*. 

This leads to an easy way of establishing the useful identity 



= (72 + w 2 + 7i 2 ) (V* + m 2 + n *) (ll + mm + n n ) 2 . 

Products of More than Two Vectors 

36.] Up to this point nothing has been said concerning 
products in which the number of vectors is greater than 
two. If three vectors are combined into a product the result 
is called a triple product. Next to the simple products 
A-B and AxB the triple products are the most important. 
All higher products may be reduced to them. 

The simplest triple product is formed by multiplying the 
scalar product of two vectors A and B into a third C as 

(A-B) C. 

This in reality does not differ essentially from scalar multi 
plication (Art. 6). The scalar in this case merely happens to 
be the scalar product of the two vectors A and B. Moreover 
inasmuch as two vectors cannot stand side by side in the 
form of a product as BC without either a dot or a cross to 
unite them, the parenthesis in (AB) C is superfluous. The 
expression ^ n 

cannot be interpreted in any other way * than as the product 
of the vector C by the scalar AB. 

i Later (Chap. V.) the product BC, where no sign either dot or cross occurs, 
will be defined. But it will be seen there that (A.B) C and A-(B C) are identical 
and consequently no ambiguity can arise from the omission of the parenthesis. 



68 VECTOR ANALYSIS 

37.] The second triple product is the scalar product of 
two vectors, of which one is itself a vector product, as 
A-(BxC) or (AxB>C. 

This sort of product has a scalar value and consequently is 

often called the scalar triple prod 
uct. Its properties are perhaps most 
easily deduced from its commonest 
geometrical interpretation. Let A, B, 
and C be any three vectors drawn 
from the same origin (Fig. 20). 
Then BxC is the area of the par 

allelogram of which B and C are two adjacent sides. The 

scalar . * (14) 



will therefore be the volume of the parallelepiped of which 
BxC is the base and A the slant height or edge. See Art. 28. 
This volume v is positive if A and BxC lie upon the same 
side of the B C-plane ; but negative if they lie on opposite 
sides. In other words if A, B, C form a right-handed or 
positive system of three vectors the scalar A* (BxC) is posi 
tive; but if they form a left-handed or negative system, it 
is negative. 

In case A, B, and C are coplanar this volume will be 
neither positive nor negative but zero. And conversely if 
the volume is zero^the three edges A, B, C of the parallelo- 
piped must lie in one plane. Hence the necessary and suffi 
cient condition for the coplanarity of three vectors A, B, C none 
of which vanishes is A-(BxC) = 0. As a corollary the scalar 
triple product of three vectors of which two are equal or 
collinear must vanish ; for any two vectors are coplanar. 

The two products A(BxC) and (AxB)-C are equal to the 
same volume v of the parallelepiped whose concurrent edges 
are A, B, C. The sign of the volume is the same in both 
cases. Hence (AxB) . c = A . (BxC) = ,. (14) 



DIRECT AND SKEW PRODUCTS OF VECTORS 69 

This equality may be stated as a rule of operation. The dot 
and the cross in a scalar triple product may be interchanged 
without altering the value of the product. 

It may also be seen that the vectors A, B, C may be per 
muted cyclicly without altering the product 

A-(BxC) = B-(CxA) = C-(AxB). (15) 

For each of the expressions gives the volume of the same 
parallelepiped and that volume will have in each case the 
same sign, because if A is upon the positive side of the B C- 
plane, B will be on the positive side of the C A-plane and C 
upon the positive side of the A B-plane. The triple product 
may therefore have any one of six equivalent forms 

A<BxC) = B-(CxA) =: C.(AxB) (35) 

= (AxB)-C = (BxC)-A = (CxA)-B 

If however the cyclic order of the letters is changed the 
product will change sign. 

A-(BxC) = - A<CxBV (16) 

This may be seen from the figure or from the fact that 
BxC = CxB. 

Hence : A scalar triple product is not altered by interchanging 
the dot or the cross or by permuting cyclicly the order of the 
vectors, but it is reversed in sign if the cyclic order be changed. 

38.] A word is necessary upon the subject of parentheses 
in this triple product. Can they be omitted without am 
biguity ? They can. The expression 

A-BxC 

can have only the one interpretation 

A<BxC). 

For the expression (A-B)xC is meaningless. It is impos 
sible to form the skew product of a scalar AB and a vector 



70 VECTOR ANALYSIS 

C. Hence as there is only one way in which ABxC may 
be interpreted, no confusion can arise from omitting the 
parentheses. Furthermore owing to the fact that there are 
six scalar triple products of A, B, and C which have the same 
value and are consequently generally not worth distinguish 
ing the one from another, it is often convenient to use the 
symbol 

[ABC] 

to denote any one of the six equal products. 

[A B C] = A.BxC = B*CxA = C AxB 
= AxB.C = BxC-A = CxA-B 
then [A B C] = - [A B]. (16) 

The scalar triple products of the three unit vectors i, j, k 
all vanish except the two which contain the three different 
vectors. 

[ijk] = _[ikj] = l. (17) 

Hence if three vectors A, B, C be expressed in terms of i, j, k 
as 



B = ^ i + A, j + 8 k, 

C = C 1 i+C 2 j + C 3 k, 
then [ABC] =A 1 Z C 3 + , C 2 A.+ 






This may be obtained by actually performing the multiplica 
tions which are indicated in the triple product. The result 
may be written in the form of a determinant. 1 



[A B C] = 



-4i 



\ 



B 



1 This is the formula given in solid analytic geometry for the volume of a 
tetrahedron one of whose vertices is at the origin. For a more general formula 
see exercises. 



DIRECT AND SKEW PRODUCTS OF VECTORS 71 



If more generally A, B, C are expressed in terms of any three 
non-coplanar vectors a, b, c which are not necessarily unit 

vectors, 

A = a x a + a 2 b + 8 c 

B = &! a + & 2 b + J 8 c 
C = c l a 



c 2 b 



where a^ # 2 , #3," ftp & 2 , 
stants, then 

[A B 0] = (a l & 2 C B + 



are certain con 



[a b c]. 



or 



[A B C] = 



[a be] 







(19) 



39.] The third type of triple product is the vector product 
of two vectors of which one is itself a vector product. Such 

are 

Ax(BxC) and (AxB)xC. 

The vector Ax(BxC) is perpendicular to A and to (BxC). 
But (BxC) is perpendicular to the plane of B and C. Hence 
Ax (BxC), being perpendicular to (BxC) must lie in the 
plane of B and C and thus take the form 

Ax(BxC) = x B + y C, 

where x and y are two scalars. In like manner also the 
vector (AxB)xC, being perpendicular to (AxB) must lie 
in the plane of A and B. Hence it will be of the form 

(AxB)xC = ra.A + n B 

where m and n are two scalars. From this it is evident that 
in general 

(AxB)xC is not equal to Ax(BxC). 

The parentheses therefore cannot be removed or inter 
changed. It is essential to know which cross product is 



72 VECTOR ANALYSIS 

formed first and which second. This product is termed the 
vector triple product in contrast to the scalar triple product. 

The vector triple product may be used to express that com 
ponent of a vector B which is perpendicular to a given vector 
A. This geometric use of the product is valuable not only in 

itself but for the light it sheds 



AXB 
AXB* 



B 



upon the properties of the product. 
Let A (Fig. 21) be a given vector 
and B another vector whose com 
ponents parallel and perpendicular 
to A are to be found. Let the 
components of B parallel and per- 
A X (AXB) pendicular to A be B and B" re- 

2i spectively. Draw A and B from a 

common origin. The product AxB 

is perpendicular to the plane of A and B. The product 
Ax (AxB) lies in the plane of A and B. It is furthermore 
perpendicular to A. Hence it is collinear with B". An 
examination of the figure will show that the direction of 
Ax (AxB) is opposite to that of B". Hence 

Ax(AxB) = cB", 
where c is some scalar constant. 

Now Ax (AxB) = - A* B sin (A, B) V 

but - c B"-^= - c B sin (A, B) b", 

if b" be a unit vector in the direction of B". 
Hence c A 2 A* A. 

Hence B" = - Ax(AxB) . (20) 

The component of B perpendicular to A has been expressed 
in terms of the vector triple product of A, A, and B. The 
component B parallel to A was found in Art 28 to be 



DIRECT AND SKEW PRODUCTS OF VECTORS 73 
B =?A (21) 

B = B + B = ?A-^>. (22) 

AA AA 

40.] The vector triple product Ax (BxC) may be expressed 
as the sum of two terms as 

Ax(BxC)=A-C B-A-B C 

In the first place consider the product when two of the 
vectors are the same. By equation (22) 

A-A B = A-B A - Ax(AxB) (22) 

or Ax(AxB) = A*B A - A- A B (23) 

This proves the formula in case two vectors are the same. 
To prove it in general express A in terms of the three 
non-coplanar vectors B, C, and BxC. 

A = bE + cC + a (BxC), (I) 

where #, &, c are scalar constants. Then 

Ax(BxC) = SBx(BxC) + cCx(BxC) (II) 

+ a (BxC)x(BxC). 

The vector product of any vector by itself is zero. Hence 

(BxC)x(BxC) = 

Ax(BxC) = 6Bx(BxC) + c Cx(BxC). (II) 
By (23) Bx(BxC) = B-C B - B-B C 

Cx(BxC) = - Cx(CxB) = - C-B C + C-C B. 
Hence Ax(BxC) = [(&B-C + cC-C)B- (6B-B + cCB)C]. (II)" 
But from (I) A-B = JBB + cC-B + a (BxC>B 
and A-C = b B-C + c C*C + a (BxC)-O. 

By Art. 37 (BxC)-B = and (BxC)-C = 0. 
Hence A-B = JB-B + cC-B, 

A-C = 5B-C + cC-C. 



74 VECTOR ANALYSIS 

Substituting these values in (II)", 

Ax(BxC) = A.C B - A.B C. (24) 

The relation is therefore proved for any three vectors A, B, C. 

Another method of giving the demonstration is as follows. 

It was shown that the vector triple product Ax(BxC) was 

of the form 

Ax(BxC) = #B + yC. 

Since Ax(AxC) is perpendicular to A, the direct product of 
it by A is zero. Hence 

A-[Ax(BxC)] = a; A*B + yAC = 
and x : y = A*C : AB. 

Hence Ax(BxC) = n (A-0 B - A-B C), 

where n is a scalar constant. It remains to show n = 1. 
Multiply by B. 

Ax(BxC>B = n (A-C B.B-A-B C-B). 

The scalar triple product allows an interchange of dot and 
cross. Hence 

Ax(BxC>B = A<BxC)xB = - A-[Bx(BxC)], 
if the order of the factors (BxC) and B be inverted. 

-A-[Bx(BxC)] = -A-[B.C B-B.BC] 
= B-C AB + B-B AC. 

Hence n = 1 and Ax(BxC) = A.C B A-B C. (24) 

From the three letters A, B, C by different arrangements, 
four allied products in each of which B and C are included in 
parentheses may be formed. These are 

Ax(BxC), Ax(CxB), (CxB)xA, (BxC)xA. 

As a vector product changes its sign whenever the order of 
two factors is interchanged, the above products evidently 
satisfy the equations 

Ax (BxC) = - Ax(CxB) = (CxB)xA = - (BxC)xA. 



DIRECT AND SKEW PRODUCTS OF VECTORS 75 

The expansion for a vector triple product in which the 
parenthesis comes first may therefore be obtained directly 
from that already found when the parenthesis comes last. 

(AxB)xC = - Cx(AxB) = - C-B A + CA B. 

The formulae then become 

Ax(BxC) = A-C B - A.B C (24) 

and (AxB)xC = A*C B - C-B A. (24) 

These reduction formulae are of such constant occurrence and 
great importance that they should be committed, to memory. 
Their content may be stated in the following rule. To expand 
a vector triple product first multiply the exterior factor into the 
remoter term in the parenthesis to form a scalar coefficient for 
the nearer one, then multiply the exterior factor into the nearer 
term in the parenthesis to form a scalar coefficient for the 
remoter one, and subtract this result from the first. 

41.] As far as the practical applications of vector analysis 
are concerned, one can generally get along without any 
formulae more complicated than that for the vector triple 
product. But it is frequently more convenient to have at 
hand other reduction formulae of which all may be derived 
simply by making use of the expansion for the triple product 
Ax(BxC) and of the rules of operation with the triple pro 
duct ABxC. 

To reduce a scalar product of two vectors each of which 
is itself a vector product of two vectors, as 

(AxB>(CxD). 

Let this be regarded as a scalar triple product of the three 
vectors A, B, and CxD thus 

AxB-(CxD). 

Interchange the dot and the cross. 



76 VECTOR ANALYSIS 

AxB.(CxD) = A-Bx(CxD) 
Bx(CxD) = B-D C - B-C D. 

Hence (AxB>(CxD) = A-C B-D - A-D B.C. (25) 

This may be written in determinantal form. 

(25) 



If A and D be called the extremes ; B and C the means ; A 
and C the antecedents: B and D the consequents in this 
product according to the familiar usage in proportions, then 
the expansion may be stated in words. The scalar product 
of two vector products is equal to the (scalar) product of the 
antecedents times the (scalar) product of the consequents 
diminished by the (scalar) product of the means times the 
(scalar) product of the extremes. 

To reduce a vector product of two vectors each of which 
is itself a vector product of two vectors, as 

(AxB)x(CxD). 
Let CxD = E. The product becomes 

(AxB)xE = A-E B - B-E A. 
Substituting the value of E back into the equation : 

(AxB)x(CxD) = (A-CxD)B - (B-CxD) A. (26) 
Let F = AxB. The product then becomes 

Fx(CxD) = FD C F-C D 
(AxB)x(CxD) = (AxB-D)C - (AxB-C) D. (26) 

By equating these two equivalent results and transposing 
all the terms to one side of the equation, 

[B C D] A - [C D A] B + [D A B] C - [A B C] D - 0. (27) 

This is an equation with scalar coefficients between the four 
vectors A, B, C, D. There is in general only one such equa- 



DIRECT AND SKEW PRODUCTS OF VECTORS 11 

tion, because any one of the vectors can be expressed in only 
one way in terms of the other three : thus the scalar coeffi 
cients of that equation which exists between four vectors are 
found to be nothing but the four scalar triple products of 
those vectors taken three at a time. The equation may also 
be written in the form 

[A B C] D = [B C D] A + [C A D] B + [A B D] C. (27) 

More examples of reduction formulae, of which some are 
important, are given among the exercises at the end of the 
chapter. In view of these it becomes fairly obvious that 
the combination of any number of vectors connected in 
any legitimate way by dots and crosses or the product of any 
number of such combinations can be ultimately reduced to 
a sum of terms each of which contains only one cross at most. 
The proof of this theorem depends solely upon analyzing the 
possible combinations of vectors and showing that they all 
fall under the reduction formulae in such a way that the 
crosses may be removed two at a time until not more than 
one remains. 

* 42.] The formulae developed in the foregoing article have 
interesting geometric interpretations. They also afford a 
simple means of deducing the formulae of Spherical Trigo 
nometry. These do not occur in the vector analysis proper. 
Their place is taken by the two quadruple products, 

(AxB>(CxD) = A-C B-D - B-C A-D (25) 
and (AxB)x(CxD) = [ACD] B - [BCD] A 

= [ABD] C - [ABC] D, (26) 

which are now to be interpreted. 

Let a unit sphere (Fig. 22) be given. Let the vectors 
A, B, C, D be unit vectors drawn from a common origin, the 
centre of the sphere, and terminating in the surface of the 
sphere at the points A,B, (7, D. The great circular arcs 



78 



VECTOR ANALYSIS 



FIG. 22. 



AB, A C) etc., give the angles between the vectors A and B, 
A and C, etc. The points A, B, C, D determine a quadrilateral 
upon the sphere. A C and BD are one 
pair of opposite sides ; A D and B C> the 
other. A B and CD are the diagonals. 

(AxB).(CxD) = A-C B-D - A-D B-C 
AxB = sin (A, B), CxD = sin (C, D). 

The angle between AxB and CxD is the 
angle between the normals to the AB- 
and CD-planes. This is the same as 

the angle between the planes themselves. Let it be denoted 

by x. Then 

(AxB). (CxD) = sin (A,B) sin (C,D) cos a:. 

The angles (A, B), (C, D) may be replaced by the great 
circular arcs AB, CD which measure them. Then 

(AxB).(CxD) = sin A B sin CD cos#, 
A-C B-D- A.D B*C = cos AC cosBD - cos AD cos BC. 

Hence 

sin A B sin CD cos x = cos A C cos B D cos AD cos B C. 

In words : The product of the cosines of two opposite sides 
of a spherical quadrilateral less the product of the cosines of 
the other two opposite sides is equal to the product of the 
sines of the diagonals multiplied by the 
cosine of the angle between them. This 
theorem is credited to Gauss. 

Let A, B, C (Fig. 23) be a spherical tri 
angle, the sides of which are arcs of great 
circles. Let the sides be denoted by a, 6, c 
respectively. Let A, B, C be the unit vectors 
drawn from the center of the sphere to the points -A, B, C. 
Furthermore let p a , p b , p e be the great circular arcs dropped 



FIG. 23. 



DIRECT AND SKEW PRODUCTS OF VECTORS 79 

perpendicularly from the vertices -4, J9, C to the sides a, 6, . 
Interpret the formula 

(AxB)-(CxA) = A-C B-A - B.C A-A. 

(AxB) = sin (A, B) = sin c, (CxA) = sin (C, A) = sin 6. 
Then (AxB) (CxA) = sin c sin b cos #, 

where x is the angle between AxB and CxA. This 
angle is equal to the angle between the plane of A, B and the 
plane of C, A. It is, however, not the interior angle A which 
is one of the angles of the triangle : but it is the exterior 
angle 180 A, as an examination of the figure will show. 

Hence 

(AxB). (CxA) = sin c sin b cos (180 A) 

= sin c sin 6 cos A 
AC BA BC A- A = cos & cos c cos a 1. 

By equating the results and transposing, 

cos a = cos 6 cos c sin 6 sin c cos A 
cos 6 = cos c cos a sin c sin a cos B 
cos c = cos a cos 6 sin a sin 6 cos C. 

The last two may be obtained by cyclic permutation of the 
letters or from the identities 

(BxC).(AxB) = B-A C B - C-A, 
(CxAHBxC) = C-B A.C - B-C. 

Next interpret the identity (AxB)x(CxD) in the special 
cases in which one of the vectors is repeated. 

(AxB)x(AxC) = [A B C] A. 

Let the three vectors a, b, c be unit vectors in the direction of 
BxC, CxA, AxB respectively. Then 

AxB = c sin c, AxC = b sin 6 

(AxB)x(AxC) = cxb sin c sin & = A sin c sin 6 sin A 
[A B C] = (AxB)-C = cC sin c = cos (90 p c ) sin c 
[ABC] A = sin c sin p c A. 



80 VECTOR ANALYSIS 

By equating the results and cancelling the common factor, 

sin^ c = sin b sin A 
sin^? a = sin c sin B 
sin p b = sin a sin C. 

The last two may be obtained by cyclic permutation of the 
letters. The formulae give the sines of the altitudes of the 
triangle in terms of the sines of the angle and sides. Again 

write 

(AxB)x(AxC) = [ABC]A 

(BxC)x(BxA) = [BCA]B 
(CxA)x(CxB) = [CAB]C. 

Hence sin c sin b sin A = [A B C] 

sin a sin c sin B = [B C A] 
sin b sin a sin C = [C A B]. 

The expressions [ABC], [BCA], [CAB] are equal. Equate 
the results in pairs and the formulae 

sin b sin A = sin a sin B 
sin c sin B = sin b sin C 
sin a sin C = sin c sin A 

are obtained. These may be written in a single line. 

sin A sin B sin C 
sin a sin b sin c 

The formulae of Plane Trigonometry are even more easy to 
obtain. If A B C be a triangle, the sum of the sides taken 
as vectors is zero for the triangle is a closed polygon. 
From this equation 

a + b-f c = 

almost all the elementary formulae follow immediately. It 
is to be noticed that the angles from a to b, from b to c, from 



DIRECT AND SKEW PRODUCTS OF VECTORS 81 

o to a are not the interior angles A, B, (7, but the exterior 
angles 180 -A, 180 - B, 180 - C. 

a = b + c 
aa = (b + c)*(b + c) = b-b + c-c + 2 bc. 

If a, J, c be the length of the sides a, b, c, this becomes 



c 2 = a 2 + 6 2 - 2 a 6 cos C. 

The last two are obtained in a manner similar to the first 
one or by cyclic permutation of the letters. 
The area of the triangle is 

^axb = ^bxc = 2 cxa = 
2 a b sin C = % b c sin A = ^ c a sin B. 

If each of the last three equalities be divided by the product 
a b c, the fundamental relation 

sin A sin B sin 



is obtained. Another formula for the area may be found from 

the product 

(bxc)(bxc) = (cxa)-(axb) 

2 Area (6 c sin A} = (c a sin B) (a b sin (7) 
a 2 sin -Z?sin C 



2 Area = 



sin A 



Reciprocal Systems of Three Vectors. Solution of Equations 

43.] The problem of expressing any vector r in terms of 
three non-coplanar vectors a, b, c may be solved as follows. 
Let 



82 VECTOR ANALYSIS 

where a, J, c are three scalar constants to be determined 
Multiply by b x c. 

r.bxc = a abxc + 6 bbxc + cc-bxo 
or [rbc] = a [a be]. 

In like manner by multiplying the equation by c x a and 
. a X b the coefficients b and c may be found. 

[r c a] = I [b c a] 
[r a b] = c [c a b] 



Hence r = a+ b + , (28) 

[be a] [c a b] 



The denominators are all equal. Hence this gives the 
equation 

[a b c] r [b c r] a + [c r a] b - [r a b] c = 

which must exist between the four vectors r, a, b, c. 
The equation may also be written 

rb x c ro x a, ra x b 
r = - r . - a + r v .. b + e 

[abe] [abc] [a be] 

bxc cxa, axb 

or r = r r _ a + r r b + r o. 

[abc] [abc] [abc] 

The three vectors which appear here multiplied by !, namely 

bxc cxa axb 

_ * _ - > _ 

[a be] [a b c] [a b c] 

are very important. They are perpendicular respectively to 
the planes of b and c, c and a, a and b. They occur over and 
over again in a large number of important relations. For 
this reason they merit a distinctive name and notation. 
Definition : The system of three vectors 

b x c ^ cxa axb 
[abc] [abc] [abc] 



DIRECT AND SKEW PRODUCTS OF VECTORS 83 

which are found by dividing the three vector products bxc, 
c x a, a x b of three non-coplanar vectors a, b, c by the scalar 
product [abc] is called the reciprocal system to a, b, c. 

The word non-coplanar is important. If a, b, c were co 
planar the scalar triple product [a b c] would vanish and 
consequently the fractions 

bxc cxa axb 

j j ________ 

[a be] [a b c] [a b c] 

would all become meaningless. Three coplanar vectors have 
no reciprocal system. This must be carefully remembered. 
Hereafter when the term reciprocal system is used, it will be 
understood that the three vectors a, b, c are not coplanar. 
The system of three vectors reciprocal to system a, b, c 
will be denoted by primes as a , b , c , 

,_bxc ? h , _ c x a , , __ a x b (29) 
""[abc] [abc] ~[ac] 

The expression for r reduces then to the very simple form 
r = r-a a + r-b b + r.c c. (30) 

The vector r may be expressed in terms of the reciprocal 
system a , b , c instead of in terms of a, b, c. In the first 
place it is necessary to note that if a, b, c are non-coplanar, 
a , b , c which are the normals to the planes of b and c, 
c and a, a and b must also be non-coplanar. Hence r may 
be expressed in terms of them by means of proper scalar 
coefficients #, y, z. 

r x a + ?/b + z c 

Multiply successively by -a, -b, -c. This gives 
[a b c] r-a = x [b c a], x = r-a 
[abc]r-b = y [cab], y = r-b 
[a b c] r-c = z [a b c], z = r-c 

Hence r = r-a a + r-b b + r-c c . (31) 



84 VECTOR ANALYSIS 

44.] If a , V, c be the system reciprocal to a, b, c the 
scalar product of any vector of the reciprocal system into the 
corresponding vector of the given system is unity ; but 
the product of two non-corresponding vectors is zero. That is 

a .a = bM>=:c .c = l (32) 

a .b = a .c = b -a = b *c = c -a = c -b = 0. 

This may be seen most easily by expressing a , V, c in 
terms of themselves according to the formula (31) 

r = raa + r*bb + rcc . 
Hence a = a -aa + a b V + a cc 

b = b .aa + b -bb + bW 
c = c aa + c -bb + c .cc . 

Since a , b , c are non-coplanar the corresponding coeffi 
cients on the two sides of each of these three equations must 
be equal. Hence from the first 

1 = a *a = a -b = a c. 
From the second = b a l=b b = b e. 
From the third = c a = c b l = c o. 

This proves the relations. They may also be proved 
directly from the definitions of a , b , c . 

bxc bxca [be a] 

a a = a = = = 1 

[abc] [abc] [abc] 

bxc bxc-b 

a . b = b = = =0 

[abc] [abc] [abc] 

and so forth. 

Conversely if two sets of three vectors each, say A, B, C, 
and a, b, c, satisfy the relations 

Aa = Bb = Cc = 1 
A-b = Ac = Ba = B-c = Ca = Cb = 



DIRECT AND SKEW PRODUCTS OF VECTORS 85 

then the set A, B, C is the system reciprocal to a, b, c. 
By reasoning similar to that before 

A = A-a a + Ab b + A-c c 
B = B-a a + B-b b + B-c c 
C = Caa + C-bb + C-c c . 

Substituting in these equations the given relations the re 
sult is 

A = a , B = b , C = c . 
Hence 

Theorem : The necessary and sufficient conditions that the 
set of vectors a , b , c be the reciprocals of a, b, c is that 
they satisfy the equations 

a .a = b .b = c .c = l (32) 

a -b = a -c = b a = b .c = c -a = c .b = 0. 

As these equations are perfectly symmetrical with respect 
to a , b , c and a, b, c it is evident that the system a, b, c may 
be looked upon as the reciprocal of the system a , b , c just 
as the system a , b , c may be regarded as the reciprocal of 
a, b, c. That is to say, 

Theorem: If a , b , c be the reciprocal system of a, b, c, 
then a, b, c will be the reciprocal system of a , V, c . 

V x c c x a a x b (29V 

- b=- - - . v / 



[a b e ] [a b c ] [a b c ] 

These relations may be demonstrated directly from the 
definitions of a , b , c . The demonstration is straightfor 
ward, but rather long and tedious as it depends on compli 
cated reduction formulae. The proof given above is as short 
as could be desired. The relations between a , b ,c and 
a, b, c are symmetrical and hence if a , b , c is the reciprocal 
system of a, b, c, then a, b, c must be the reciprocal system of 



86 



VECTOR ANALYSIS 



45.] Theorem : If a , V, c and a, b, c be reciprocal systems 
the scalar triple products [a b c ] and [a b c] are numerical 
reciprocals. That is 



t. b . ]=[i 



[a bV] [abc]=l 

xc cxa axb"| 
[a "be] [abc] [abc] J 

[bxc cxa axb]. 



(33) 



But 
Hence 

Hence 



~[abc] 3 
[bxc cxa axb] = (bxc)x(cxa>(axb). 

(bxc) x (cxa) = [abc]c. 
[bxc cxa axb] = [abc] c-axb = [abc] 2 . 
1 1 



[a bV] = 



[abc] : 



[abc] 2 = 



[abc] 



(33) 



By means of this relation between [a b c ] and [a b c] it 
is possible to prove an important reduction formula, 



(P.axE)(ABxC) = 



P-A P.B p.c 
Q.A Q.B a-c 

B*A *B *C 



(34) 



which replaces the two scalar triple products by a sum of 
nine terms each of which is the product of three direct pro 
ducts. Thus the two crosses which occur in the two scalar 
products are removed. To give the proof let P, ft, B be 
expressed as 

P = P-A A + P.B B + P.C C 



Then 
But 



B = B-A A + B.B B + BC C . 

P.A P.B P.C 



[POB] = 



a-A Q.B a-c 

R-A R.B R.C 

1 
[ABC] 



[A B C ]. 



[A B C J = 



DIRECT AND SKEW PRODUCTS OF VECTORS 87 



Hence [PQE] [ABC] = 



P.A P-B P-C 

Q.A a-B Q.C 

R.A R.B B*C 



The system of three unit vectors i, j, k is its own reciprocal 
system. 

jxki kxi ,, i x j 

J k==k - (35) 



For this reason the primes i , j , k are not needed to denote 
a system of vectors reciprocal to i, j, k. The primes will 
therefore be used in the future to denote another set of rect 
angular axes i, j, k , just as X* , F , Z* are used to denote a 
set of axes different from X, F, Z. 

The only systems of three vectors which are their own reciprocals 
are the right-handed and left-handed systems of three unit 
vectors. That is the system i, j, k and the system i, j, k. 

Let A, B, C be a set of vectors which is its own reciprocal. 

Then by (32) 

AA = B-B = CNC = 1. 

Hence the vectors are all unit vectors. 

A-B = A-C = 0. 
Hence A is perpendicular to B and C. 

B-A = B-C = 0. 
Hence B is perpendicular to A and C. 

C-A =C.B = O. 

Hence C is perpendicular to A and B. 

Hence A, B, C must be a system like i, j, k or like i, j, k. 

* 46.] A scalar equation of the first degree in a vector r is 
an equation in each term of which r occurs not more than 
once. The value of each term must be scalar. As an exam 
ple of such an equation the following may be given. 

a a-bxr + 6(oxd)(exr) + c fr + d = 0, 



88 VECTOR ANALYSIS 

where a, b, c, d, e, f are known vectors ; and a, &, c, d, known 
scalars. Obviously any scalar equation of the first degree in 
an unknown vector r may be reduced to the form 

r-A = a 

where A is a known vector ; and a, a known scalar. To ac 
complish this result in the case of the given equation proceed 

as follows. 

a axbor + "b (cxd)xe-r + c fr + d = 

{a axb + b (cxd)xe + c f}r = d. 

In more complicated forms it may be necessary to make use 
of various reduction formulae before the equation can be made 

to take the desired form, 

]>A = a. 

As a vector has three degrees of freedom it is clear that one 
scalar equation is insufficient to determine a vector. Three 
scalar equations are necessary. 

The geometric interpretation of the equa 
tion 

r.A => a (36) 



is interesting. Let r be a variable vector 
(Fig. 24) drawn from a fixed origin. Let 
A be a fixed vector drawn from the same 
origin. The equation then becomes 

r A cos (r,A) = a, 

a 
or T cos (r,A) = , 

if r be the magnitude of r ; and A that of A. The expression 

r cos (r, A) 

is the projection of r upon A. The equation therefore states 
that the projection of r upon a certain fixed vector A must 



DIRECT AND SKEW PRODUCTS OF VECTORS 89 

always be constant and equal to a/ A. Consequently the ter 
minus of r must trace out a plane perpendicular to the vector 
A at a distance equal to a/ A from the origin. The projec 
tion upon A of any radius vector drawn from the origin to a 
point of this plane is constant and equal to a/ A. This gives 
the following theorem. 

Theorem : A scalar equation in an unknown vector may be 
regarded as the equation of a plane, which is the locus of the 
terminus of the unknown vector if its origin be fixed. 

It is easy to see why three scalar equations in an unknown 
vector determine the vector completely. Each equation de 
termines a plane in which the terminus of r must lie. The 
three planes intersect in one common point. Hence one vec 
tor r is determined. The analytic solution of three scalar 
equations is extremely easy. If the equations are 

rA = a 

r-B = b (37) 

r-C = c 9 

it is only necessary to call to mind the formula 

r = r.A A + r-BB + r-C C . 

Hence r = a A + 6 B + c C . (38) 

The solution is therefore accomplished. It is expressed in 
terms A , B , C which is the reciprocal system to A, B, C. One 
caution must however be observed. The vectors A, B, C will 
have no reciprocal system if they are coplanar. Hence the 
solution will fail. In this case, however, the three planes de 
termined by the three equations will be parallel to a line. 
They will therefore either not intersect (as in the case of the 
lateral faces of a triangular prism) or they will intersect in a 
common line. Hence there will be either no solution for r or 
there will be an infinite number. 



90 VECTOR ANALYSIS 

From four scalar equations 

r-A = a 

r.B = 6 (39) 

rC = c 

rD =d 

the vector r may be entirely eliminated. To accomplish this 
solve three of the equations and substitute the value in the 

fourth. 

r = aA + 6B + cC 

a A D + &B .D + cC -D = d 

or a [BCD] + b [CAD] + c [ABD] = d [ABC]. (40) 

* 47.] A vector equation of the first degree in an unknown 
vector is an equation each term of which is a vector quantity 
containing the unknown vector not more than once. Such 
an equation is 

(AxB)x(Cxr) + D ET + n r + F =0, 

where A, B, C, D, E, F are known vectors, n a known scalar, 
and r the unknown vector. One such equation may in gen 
eral be solved for r. That is to say, one vector equation is in 
general sufficient to determine the unknown vector which is 
contained in it to the first degree. 

The method of solving a vector equation is to multiply it 
with a dot successively by three arbitrary known non-coplanar 
vectors. Thus three scalar equations are obtained. These 
may be solved by the methods of the foregoing article. In the 
first place let the equation be 

A ar + B br + C c-r = D, 

where A, B, C, D, a, b, c are known vectors. No scalar coeffi 
cients are written in the terms, for they may be incorporated in 
the vectors. Multiply the equation successively by A , B , C . 
It is understood of course that A, B, C are non-coplanar. 



DIRECT AND SKEW PRODUCTS OF VECTORS 91 

a-r = D-A r 

b-r = D-B 

c-r = D-C . 

But r = a a-r + b b-r + c c-r. 

Hence r = D-A a + D-B b + D-C c . 

The solution is therefore accomplished in case A, B, C are non- 
coplanar and a, b, c also non-coplanar. The special cases in 
which either of these sets of three vectors is coplanar will not 
be discussed here. 

The most general vector equation of the first degree in an 
unknown vector r contains terms of the types 

A a-r, n r, Exr, D. 

That is it will contain terms which consist of a known 
vector multiplied by the scalar product of another known vec 
tor and the unknown vector ; terms which are scalar multi 
ples of the unknown vector; terms which are the vector 
product of a known and the unknown vector ; and constant 
terms. The terms of the type A a-r may always be reduced 
to three in number. For the vectors a, b, c, which are 
multiplied into r may all be expressed in terms of three non- 
coplanar vectors. Hence all the products a-r, b-r, or, 
may be expressed in terms of three. The sum of all terms of 
the type A a-r therefore reduces to an expression of three 
terms, as 

A a-r + B b-r + C c-r. 

The terms of the types n r and Exr may also be expressed 
in this form. 

n r = 7i a a-r + n b b-r + n c c-r 
Exr = Exa a-r + Exb b-r+Exc c-r. 

Adding all these terms together the whole equation reduces 
to the form 

L a-r + M b-r + N c-r = K. 



92 VECTOR ANALYSIS 

This has already been solved as 

r = K.L a + K-M b + XJT c . 

The solution is in terms of three non-coplanar vectors a , V, c f . 
These form the system reciprocal to a, b, c in terms of which 
the products containing the unknown vector r were expressed. 

* SUNDRY APPLICATIONS OF PKODUCTS 
Applications to Mechanics 

48.] In the mechanics of a rigid body a force is not a 
vector in the sense understood in this book. See Art. 3. 
A force has magnitude and direction ; but it has also a line 
of application. Two forces which are alike in magnitude 
and direction, but which lie upon different lines in the body 
do not produce the same effect. Nevertheless vectors are 
sufficiently like forces to be useful in treating them. 

If a number of forces f x , f 2 , f 3 , ---act on a body at the 
same point 0, the sum of the forces added as vectors is called 

the resultant R. 

E = f 1 + f 2 + f 8 + ... 

In the same way if f x , f 2 , f 8 do not act at the same point 
the term resultant is still applied to the sum of these forces 
added just as if they were vectors. 

B = f 1 + f a + f 8 + ... (41) 

The idea of the resultant therefore does not introduce the 
line of action of a force. As far as the resultant is concerned 
a force does not differ from a vector. 

Definition: The moment of a force f about the point is 
equal to the product of the force by the perpendicular dis 
tance from to the line of action of the force. The moment 
however is best looked upon as a vector quantity. Its mag 
nitude is as defined above. Its direction is usually taken to 



DIRECT AND SKEW PRODUCTS OF VECTORS 93 

be the normal on that side of the plane passed through the 
point and the line f upon which the force appears to pro 
duce a tendency to rotation about the point in the positive 
trigonometric direction. Another method of defining the 
moment of a force t = PQ about the point is as follows : 
The moment of the force f = PQ about the point is equal 
to twice the area of the triangle PQ. This includes at once 
both the magnitude and direction of the moment (Art. 25). 
The point P is supposed to be the origin ; and the point Q, 
the terminus of the arrow which represents the force f. The 
letter M will be used to denote the moment. A subscript will 
be attached to designate the point about which the moment is 
taken. 



The moment of a number of forces f x , f 2 , is the (vector) 
sum of the moments of the individual forces. 



If 



This is known as the total or resultant moment of the forces 

* v *& 

49.] If f be a force acting on a body and if d be the vector 
drawn from the point to any point in the line of action of 
the force, the moment of the force about the point is the 
vector product of d into f . 

Mo W = dxf (42) 

For dxf = d f sin (d, f) e, 

if e be a unit vector in the direction of dxf. 



dxf = dsm (d, f)/e. 

Now d sin (d, f) is the perpendicular distance from to f. 
The magnitude of dxf is accordingly equal to this perpen 
dicular distance multiplied by/, the magnitude of the force. 



94 VECTOR ANALYSIS 

This is the magnitude of the moment MO {f} . The direction 
of dxf is the same as the direction of the moment. Hence 
the relation is proved. 

Mo {f} = dxf. 

The sum of the moments about of a number of forces 
f p f 2 , acting at the same point P is equal to the moment 
of the resultant B of the forces acting at that point. For let 
d be the vector from to P. Then 

Mo {f x> = dxf l 
Mo {f a | = dxf a 



+ ... (43) 

= dx( 1 + f a + -..)=dxB 

The total moment about f of any number of forces f x , f 2 , 
acting on a rigid body is equal to the total moment of those 
forces about increased by the moment about of the 
resultant BO considered as acting at 0. 

M<x { f i> f 2 >} = Mo {f r f 2 , } + Mo< {Bo \. (44) 

Let dj, d 2 , be vectors drawn from to any point in 
f r f 2 , respectively. Let d/, d 2 , be the vectors drawn 
from O f to the same points in f x , f 2 , respectively. Let o 
be the vector from to_0 . Then 

d^d/H-c, d 2 = d 2 + c, 

Mo {f i, f 2 , } = d x xf j + d 2 xf 2 + 
Mo {f!,f 2 , J=d 1 xf 1 + d a xf a + ... 

^ = (d x - c)xf ! + (d 2 - c)xf a + - . 
= d x xf x +d 2 xf 2 + ---- cx(f j + f a + . . .) 

But c is the vector drawn from to 0. Hence c x f, 
is the moment about of a force equal in magnitude and 
parallel in direction to f 1 but situated at 0. Hence 



DIRECT AND SKEW PRODUCTS OF VECTORS 95 



f a + ...) = - cxBo = Mo {Bo}. 
Hence MO/ {f x , f 2 , } = M {f r f a ;,} + MCX {Bo |. (44) 

The theorem is therefore proved. 

The resultant is of course the same at all points. The 
subscript is attached merely to show at what point it is 
supposed to act when the moment about O f is taken. For 
the point of application of E affects the value of that moment. 

The scalar product of the total moment and the resultant 
is the same no matter about what point the moment be taken. 
In other words the product of the total moment, the result 
ant, and the cosine of the angle between them is invariant 
for all points of space. 

E MO {f i, f 2 }= B MO {f ! , f 2 9 } 

where O f and are any two points in space. This -important 
relation follows immediately from the equation 

Mo {*i, f a , } = Mo {fj , f 2 , } + Mo {Eo}. 
For E.Mo if!,f 2 , }=* M {f^, } + E- M {B }. 

But the moment of E is perpendicular to E no matter what 
the point of application be. Hence 

E-MO* IE O } = o 

and the relation is proved. The variation in the total 
moment due to a variation of the point about which the 
moment is taken is always perpendicular to the resultant. 

50.] A point O r may be found such that the total moment 
about it is parallel to the resultant. The condition for 
parallelism is 

{f x , f a , -}=<) 



=0 



96 VECTOR ANALYSIS 

where is any point chosen at random. Replace Mo {Eo} 
by its value and for brevity omit to write the f v f 2 , in the 
braces { }. Then 

RxMcy = ExMo - Ex(cxE) = 0. 
The problem is to solve this equation for c. 

ExMo EE c + R.c E = 0. 

Now R is a known quantity. M o is also supposed to be 
known. Let c be chosen in the plane through perpen 
dicular to E. Then Ec = and the equation reduces to 

ExM = EE c 
ExMo 
E-E 

If c be chosen equal to this vector the total moment about 
the point O r , which is at a vector distance from equal to c, 
will be parallel to E. Moreover, since the scalar product of 
the total moment and the resultant is constant and since the 
resultant itself is constant it is clear that in the case where 
they are parallel the numerical value of the total moment 
will be a minimum. 

The total moment is unchanged by displacing the point 
about which it is taken in the direction of the resultant. 

For Mo jf !, f 2 , } = Mo {f ! , f 2 , } - cxE. 

If c = O f is parallel to E, cxE vanishes and the moment 
about O f is equal to that about 0. Hence it is possible to 
find not merely one point O r about which the total moment 
is parallel to the resultant ; but the total moment about any 
point in the line drawn through parallel to E is parallel 
to E. Furthermore the solution found in equation for c is 
the only one which exists in the plane perpendicular to E 
unless the resultant E vanishes. The results that have been 
obtained may be summed up as follows : 



DIRECT AND SKEW PRODUCTS OF VECTORS 97 

If any system of forces f 19 f 2 , whose resultant is not 
zero act upon a rigid body, then there exists in space one 
and only one line such that the total moment about any 
point of it is parallel to the resultant. This line is itself 
parallel to the resultant. The total moment about all points 
of it is the same and is numerically less than that about any 
other point in space. 

This theorem is equivalent to the one which states that 
any system of forces acting upon a rigid body is equivalent 
to a single force (the resultant) acting in a definite line and 
a couple of which the plane is perpendicular to the resultant 
and of which the moment is a minimum. A system of forces 
may be reduced to a single force (the resultant) acting at any 
desired point of space and a couple the moment of which 
(regarded as a vector quantity) is equal to the total moment 
about of the forces acting on the body. But in general the 
plane of this couple will not be perpendicular to the result 
ant, nor will its moment be a minimum. 

Those who would pursue the study of systems of forces 
acting on a rigid body further and more thoroughly may 
consult the Traite de Mecanique Rationnelle l by P. APPELL. 
The first chapter of the first volume is entirely devoted to 
the discussion of systems of forces. Appell defines a vector 
as a quantity possessing magnitude, direction, and point of 
application. His vectors are consequently not the same as 
those used in this book. The treatment of his vectors is 
carried through in the Cartesian coordinates. Each step 
however may be easily converted into the notation of vector 
analysis. A number of exercises is given at the close of 
the chapter. 

51.] Suppose a body be rotating about an axis with a con 
stant angular velocity a. The points in the body describe 
circles concentric with the axis in planes perpendicular to 

1 Paris, Gauthier-Villars et Fils, 1893. 

7 



98 



VECTOR ANALYSIS 



FIG. 25. 



the axis. The velocity of any point in its circle is equal 
to the product of the angular velocity and the radius of the 
circle. It is therefore equal to the product of the angular 

velocity and the perpendicular dis 
tance from the point to the axis. 
The direction of the velocity is 
perpendicular to the axis and to 
the radius of the circle described 
by the point. 

Let a (Fig. 25) be a vector drawn 
along the axis of rotation in that 
direction in which a right-handed 
screw would advance if turned in 
the direction in which the body is 
rotating. Let the magnitude of a 

be a, the angular velocity. The vector a may be taken to 
represent the rotation of the body. Let r be a radius vector 
drawn from any point of the axis of rotation to a point in the 
body. The vector product 

axr = a rsin(a,r) 

is equal in magnitude and direction to the velocity v of the 
terminus of r. For its direction is perpendicular to a and r 
and its magnitude is the product of a and the perpendicular 
distance r sin (a, r) from the point to the line a. That is 

v = axr. (45) 

If the body be rotating simultaneously about several axes 
a i* a 2> a a which pass through the same point as in the 
case of the gyroscope, the velocities due to the various 

rotations are 

v i -=a 1 xr 1 

v 8 = a 8 xr 8 



DIRECT AND SKEW PRODUCTS OF VECTORS 99 

where r x , r 2 , r 3 , are the radii vec tores drawn from points 
on the axis a 19 a 2 , a 3 , to the same point of the body. Let 
the vectors r x , r 2 , r 8 , be drawn from the common point of 
intersection of the axes. Then 

TJ = r a = r 8 = = r 
and 

v = v t + v 2 + v 3 + == a x xr + a 2 xr + a 8 xr + 



This shows that the body moves as if rotating with the 
angular velocity which is the vector sum of the angular 
velocities a 19 a 2 , a 8 , This theorem is sometimes known 
as the parallelogram law of angular velocities. 

It will be shown later (Art.) 60 that the motion of any 
rigid body one point of which is fixed is at each instant of 
time a rotation about some axis drawn through that point. 
This axis is called the instantaneous axis of rotation. The 
axis is not the same for all time, but constantly changes its 
position. The motion of a rigid body one point of which is 
fixed is therefore represented by 

v = axr (45) 

where a is the instantaneous angular velocity; and r, the 
radius vector drawn from the fixed point to any point of the 
body. 

The most general motion of a rigid body no point of which 
is fixed may be treated as follows. Choose an arbitrary 
point 0. At any instant this point will have a velocity v . 
Relative to the point the body will have a motion of rotation 
about some axis drawn through 0. Hence the velocity v of 
any point of the body may be represented by the sum of 
V the velocity of and axr the velocity of that point 

relative to 0. 

v = v + axr. (46) 



100 VECTOR ANALYSIS 

In case v is parallel to a, the body moves around a and 
along a simultaneously. This is precisely the motion of a 
screw advancing along a. In case v is perpendicular to a, it 
is possible to find a point, given by the vector r, such that 
its velocity is zero. That is 



This may be done as follows. Multiply by xa. 

(axr)xa = v xa 
or aa r a-r a = v xa. 

Let r be chosen perpendicular to a. Then ar is zero and 

aa r = v x a 
f = - v x a 
aa 

The point r, thus determined, has the property that its veloc 
ity is zero. If a line be drawn through this point parallel to 
a, the motion of the body is one of instantaneous rotation 
about this new axis. 

In case v is neither parallel nor perpendicular to a it may 
be resolved into two components 

v v 4- v " 

n v n r n 



which are respectively parallel and perpendicular to a. 

v = v + v " + axr 
A point may now be found such that 

v " = axr. 

Let the different points of the body referred to this point be 
denoted by r . Then the equation becomes 

v = v + axr . (46) 

The motion here expressed consists of rotation about an axis 
a and translation along that axis. It is therefore seen that 
the most general motion of a rigid body is at any instant 



DIRECT AND SKEW PRODUCTS OF VECTORS 101 

the motion of a screw advancing at a certain rate along a 
definite axis a in space. The axis of the screw and its rate 
of advancing per unit of rotation (i. e. its pitch) change from 
instant to instant. 

52.] The conditions for equilibrium as obtained by the 
principle of virtual velocities may be treated by vector 
methods. Suppose any system of forces f x , f 2 , act on a 
rigid body. If the body be displaced through a vector dis 
tance D whether this distance be finite or infinitesimal the 
work done by the forces is 



The total work done is therefore 

W^^i l + D.f 2 + ... 

If the body be in equilibrium under the action of the forces 
the work done must be zero. 



W= D-fj + D-f 2 + = D-Cfj + f 2 + = D.E = 0. 

The work done by the forces is equal to the work done by 
their resultant. This must be zero for every displacement 

D. The equation 

D-E = 

holds for all vectors D. Hence 

E = 0. 

The total resultant must be zero if the body be in equilibrium. 

The work done by a force f when the rigid body is dis 

placed by a rotation of angular velocity a for an infinitesimal 

time t is approximately 

a-dxf t, 

where d is a vector drawn from any point of the axis of rota 
tion a to any point of f. To prove this break up f into two 
components f , f " parallel and perpendicular respectively to a. 

a-dxf = a-dxf + a-dxf ". 



102 VECTOR ANALYSIS 

As f is parallel to a the scalar product [a d f ] vanishes. 
a-dxf = a-dxf ". 

On the other hand the work done by t" is equal to the work 
done by f during the displacement. For f being parallel to 
a is perpendicular to its line of action. If h be the common 
vector perpendicular from the line a to the force f ", the work 
done by f " during a rotation of angular velocity a for time 
t is approximately 



The vector d drawn from any point of a to any point of f may 
be broken up into three components of which one is h, another 
is parallel to a, and the third is parallel to f ". In the scalar 
triple product [adf] only that component of d which is 
perpendicular alike to a and f " has any effect. Hence 

W= a-hxf " t = a-dxf t f = a-dxf t. 

If a rigid body upon which the forces f v f 2 , act be dis 
placed by an angular velocity a for an infinitesimal time t 
and if d x , d 2 , be the vectors drawn from any point of 
a to any points of f v f 2 , - respectively, then the work done 
by the forces f v f 2 , - will be approximately 

W= (a-djXfj + ad 2 xf 2 + ) t 

= a.(d 1 xf 1 + d 2 xf 2 + .-.)* 
= a.M {f 1 ,f 2 ,...} t. 

If the body be in equilibrium this work must be zero. 
Hence a*M \t l9 f 2 , } t = 0. 

The scalar product of the angular velocity a and the total 
moment of the forces t v f 2 , about any point must be 
zero. As a may be any vector whatsoever the moment itself 
must vanish. 

Mo {f r f r - } = 0. 



DIRECT AND SKEW PRODUCTS OF VECTORS 103 

The necessary conditions that a rigid body be in equilib 
rium under the action of a system of forces is that the result 
ant of those forces and the total moment about any point in 
space shall vanish. 

Conversely if the resultant of a system of forces and the 
moment of those forces about any one particular point in space 
vanish simultaneously, the body will be in equilibrium. 

If E = 0, then for any displacement of translation D 

DE = o. 

JF=D-f 1 + D.f 2 + ... = 

and the total work done is zero, when the body suffers any 
displacement of translation. 

Let Mo {fp f 2 > } be zero for a given point 0. Then for 
any other point O 1 

Mo< {f x , f 2 , -\ = Mo 1 f lf f 2 , - } + M {Bo}- 
But by hypothesis E is also zero. Hence 



Hence 

where a is any vector whatsoever. But this expression is 
equal to the work done by the forces when the body is rotated 
for a time t with an angular velocity a about the line a 
passing through the point O 1 . This work is zero. 

Any displacement of a rigid body may be regarded as a 
translation through a distance D combined with a rotation 
for a time t with angular velocity a about a suitable line a in 
space. It has been proved that the total work done by the 
forces during this displacement is zero. Hence the forces 
must be in equilibrium. The theorem is proved. 



104 VECTOR ANALYSIS 

Applications to Geometry 

53.] Relations between two right-handed systems of three 
mutually perpendicular unit vectors. Let i, j, k and i , j , k 
be two such systems. They form their own reciprocal systems. 
Hence 

r = / I +r *^,t r ^*v, (47) 

and r = ri i + rj j + rk k . 

From this 
/ i = i -i i + i .j j + i -k k = a 1 i + a 2 j + a 3 k 

I k = k -i i + k -j j + k -k k = c l i + c 2 j + c 3 k. 



The scalarsflj, a 2 , a 3 ; b lt Z> 2 , b 3 ; c v 2 , c 3 are respectively the 
direction cosines of i ; j ; k with respect to i, j, k. 
That is 

<&]_ cos (i , i) a 2 = cos (i , j) a 3 = cos (i , k) 

0j " == - COS (j ) l) O t ^ == - COS (j i j) t> 3 - COS (J , Kj ( 4o) 

c x = cos (k , i) c 2 = cos (k , j) c 3 = cos (k , k). 
In the same manner 

^ i = i-i i + i-j j 7 + i-k k = ^ i + \ y + GI k 

j j - j-i i + j-j j + j-k k = a a i + 6 2 j + c 2 k (47)" 

( k = k.i i + k.j j + k-k k = a a i f + J 8 J + C 3 k/ 

!/!_/ "1 9i 9i 9 

t fcf !? I /t ^ I n & j / 4 

and ) j.j = 1 = 2 2 + J 2 2 + c 2 2 (49) 

( k-k = 1 = a s 2 + 6 3 2 + c 3 2 

and ] j .k = = \ cj + & 2 c 2 + b B c s (50) 

I i_^ ! f\ 

\ K ! = U = Cj ttj -f- C< dy -f- C 3 a% 



DIRECT AND SKEW PRODUCTS OF VECTORS 105 

and \ j-k = = a* a 9 + K I* + c c, (50) 



j.k = = # 2 a 3 + 6 2 6 3 + 



and 



But 
Hence 



k = i xj = (a 2 5 3 - a 3 6 2 ) i 



y l "2 



= (a 2 & 3 - a 3 6 2 ), 



(51) 



(52) 



Or 



Co = 



and similar relations may be found for the other six quantities 
a v a 2 , a 3 ; b v & 2 , & 3 . All these scalar relations between the 
coefficients of a transformation which expresses one set of 
orthogonal axes X 1 , F , Z* in terms of another set JT, F, Z are 
important and well known to students of Cartesian methods. 
The ease with which they are obtained here may be note 
worthy. 

A number of vector relations, which are perhaps not so well 
known, but nevertheless important, may be found by multi 
plying the equations 

i = a l i + a 2 j + a 3 k 



in vector multiplication. 

&! k Cj j = a 3 j a 2 k. 



(53) 



The quantity on either side of this equality is a vector. From 
its form upon the right it is seen to possess no component in 



106 VECTOR ANALYSIS 

the i direction but to lie wholly in the jk-plane ; and from 
its form upon the left it is seen to lie in the j k -plane. 
Hence it must be the line of intersection of those two planes. 
Its magnitude is V af + a or V b^ + c^. This gives the 
scalar relations 

af + a* = V + *! 2 = 1 - a*. 

The magnitude 1 a^ is the square*of the sine of the angle 
between the vectors i and i . Hence the vector 

^k -cj ^sj-aak (53) 

is the line of intersection of the j k - and jk-planes, and 
its magnitude is the sine of the angle between the planes. 
Eight other similar vectors may be found, each of which gives 
one of the nine lines of intersection of the two sets of mu 
tually orthogonal planes. The magnitude of the vector is in 
each case the sine of the angle between the planes. 

54.] Various examples in Plane and Solid Geometry may 
be solved by means of products. 

Example 1 : The perpendiculars from the vertices of a trian 
gle to the opposite sides meet in a point. Let A B be the 
triangle. Let the perpendiculars from A to BC and from B 
to CA meet in the point 0. To show is perpendicular 
to A B. Choose as origin and let OA = A, OB = B, and 
=C. Then 



= C-B, 

By hypothesis 

A.(C - B) = 

and B<A - C) = 0. 

Add; C<B - A) = 0, 

which proves the theorem. 

Example 2 : To find the vector equation of a line drawn 
through the point B parallel to a given vector A. 



DIRECT AND SKEW PRODUCTS OF VECTORS 107 

Let be the origin and B the vector OS. Let be the ra 
dius vector from to any point of the required line. Then 
E B is parallel to A. Hence the vector product vanishes. 

Ax(B-B) = 0. 

This is the desired equation. It is a vector equation in the 
unknown vector B. The equation of a plane was seen (page 
88) to be a scalar equation such as 

BC = c 
in the unknown vector B. 

The point of intersection of a line and a plane may be 
found at once. The equations are 

( Ax(B - B) = 
i B-C = e 
AxB = AxB 



A-C B - C-B A = (AxB)xC 
A-C B - c A = (AxB)xC 

Hence (AxB)xC + c A . 

A-C 

The solution evidently fails when AC = 0. In this case how 
ever the line is parallel to the plane and there is no solution ; 
or, if it lies in the plane, there are an infinite number of solu 
tions. 

Example 3: The introduction of vectors to represent planes. 

Heretofore vectors have been used to denote plane areas of 
definite extent. The direction of the vector was normal to 
the plane and the magnitude was equal to the area to be re 
presented. But it is possible to use vectors to denote not a 
plane area but the entire plane itself, just as a vector represents 
a point. The result is analogous to the plane coordinates of 
analytic geometry. Let be an assumed origin. Let M N be 
a plane in space. The plane MN is to be denoted b^ a vector 



108 VECTOR ANALYSIS 

whose direction is the direction of the perpendicular dropped 
upon the plane from the origin and whose magnitude is the 
reciprocal of the length of that perpendicular. Thus the nearer 
a plane is to the origin the longer will be the vector which 
represents it. 

If r be any radius vector drawn from the origin to a point 
in the plane and if p be the vector which denotes the plane, 

then 

r-p = 1 

is the equation of the plane. For 

rp = r cos (r, p) p. 

Now p, the length of p is the reciprocal of the perpendicular 
distance from to the plane. On the other hand r cos (r, p) 
is that perpendicular distance. Hence rp must be unity. 
If r and p be expressed in terms of i, j, k 

r = #i + yj + zk 
p = ui + vj + wit 
Hence rp = xu + yv + zw = L. 

The quantities u, v, w are the reciprocals of the intercepts of 
the plane p upon the axes. 

The relation between r and p is symmetrical. It is a rela 
tion of duality. If in the equation 

r-p = 1 

r be regarded as variable, the equation represents a plane p 
which is the locus of all points given by r. If however p be 
regarded as variable and r as constant, the equation repre 
sents a point r through which all the planes p pass. The 
development of the idea of duality will not be carried out. 
It is familiar to all students of geometry. The use of vec 
tors to denote planes will scarcely be alluded to again until 
Chapter VII. 



DIRECT AND SKEW PRODUCTS OF VECTORS 109 

SUMMARY OF CHAPTER II 

The scalar product of two vectors is equal to the product 
of their lengths multiplied by the cosine of the angle between 

them. 

A-B = A B cos (A, B) (1) 

A-B = B.A (2) 

A.A = ^. (3) 

The necessary and sufficient condition for the perpendicularity 
of two vectors neither of which vanishes is that their scalar 
product vanishes. The scalar products of the vectors i, j, k 
are 

^=J!Uk!=o (4) 

A.B = A 1 B 1 + A,, B 2 + AS B z (7) 

H = A* = A* + A* + A*. (8) 

If the projection of a vector B upon a vector A is B f , 

-R A B A (*\ 

XA 

The vector product of two vectors is equal in magnitude to 
the product of their lengths multiplied by the sine of the an 
gle between them. The direction of the vector product is the 
normal to the plane of the two vectors on that side on which 
a rotation of less than 180 from the first vector to the second 

appears positive. 

AxB = A B sin (A, B) c. (9) 

The vector product is equal in magnitude and direction to the 
vector which represents the parallelogram of which A and B 
are the two adjafcent sides. The necessary and sufficient con 
dition for the parallelism of two vectors neither of which 



110 



VECTOR ANALYSIS 



vanishes is that their vector product vanishes. The com 
mutative laws do not hold. 



AxB = 



AxB = -BxA 
ixi =jxj = kxk = 
ixj = jxi = k 
jxk = kxj =i 
kxi = ixk = j 
2 ) i + (A a B l - A 1 



(10) 
(12) 



AxB = 



B n Bo 



(13) 
(13) 



The scalar triple product of three vectors [A B C] is equal 
to the volume of the parallelepiped of which A, B, C are three 
edges which meet in a point. 



[AB C] = A-BxC = B.CxA = C-AxB 
= AxB-C = BxCA = Cx A-B 
[ABC] =- [A OB]. 



(15) 

(16) 



The dot and the cross in a scalar triple product may be inter 
changed and the order of the letters may be permuted cyclicly 
without altering the value of the product ; but a change of 
cyclic order changes the sign. 



[ABC] = 



(18) 



[ABC] = 



[a be] 



(19) 



DIRECT AND SKEW PRODUCTS OF VECTORS 111 
If the component of B perpendicular to A be B", 

B ,, = _AX(AXB) 

A*A 

Ax(BxC) = A-C B - A-B C (24) 

(AxB)xC = A-C B - C-B A (24) 

(AxB>(CxD) = A.C B-D - A-D B-C (25) 

(AxB)x(CxD) = [A CD] B- [BCD] A 

= [ABD] C-[ABC] D. (26) 

The equation which subsists between four vectors A, B, C, D 
is 

[BCD] A-[CDA]B + [DAB] C- [ABC] D = 0. (27) 

Application of formulae of vector analysis to obtain the for 
mulae of Plane and Spherical Trigonometry. 

The system of vectors a , V, c is said to be reciprocal to the 
system of three non-coplanar vectors a, b, c 

bxc cxa axb 

when a = _ . ., b = = -=> * = (29) 

[a be] [abc] [abc] 

A vector r may be expressed in terms of a set of vectors and 
its reciprocal in two similar ways 

r = r.a a + r.V b + r-c c (30) 

r = r-aa + r.bb + r.cc . (31) 

The necessary and sufficient conditions that the two systems of 
non-coplanar vectors a, b, c and a , b , c be reciprocals is that 

a .a = Vb = c c = 1 
a .b = a -c = b .c = b .a = c -a = e -b = 0. 

If a , b , c form a system reciprocal to a, b, c ; then a, b, c will 
form a system reciprocal to a , b , c . 



112 VECTOR ANALYSIS 

P.A P.B P.C 

a-A ft-B a-c 

R.A B-B R-C 



[PaK][ABC] = 



(34) 



The system i, j, k is its own reciprocal and if conversely a 
system be its own reciprocal it must be a right or left handed 
system of three mutually perpendicular unit vectors. Appli 
cation of the theory of reciprocal systems to the solution of 
scalar and vector equations of the first degree in an unknown 
vector. The vector equation of a plane is 

r-A = a. (36) 

Applications of the methods developed in Chapter II., to the 
treatment of a system of forces acting on a rigid body and in 
particular to the reduction of any system of forces to a single 
force and a couple of which the plane is perpendicular to that 
force. Application of the methods to the treatment of 
instantaneous motion of a rigid body obtaining 

v = v + a x r (46) 

where v is the velocity of any point, v a translational veloc 
ity in the direction a, and a the vector angular velocity of ro 
tation. Further application of the methods to obtain the 
conditions for equilibrium by making use of the principle of 
virtual velocities. Applications of the method to obtain 
the relations which exist between the nine direction cosines 
of the angles between two systems of mutually orthogonal 
axes. Application to special problems in geometry including 
the form under which plane coordinates make their appear 
ance in vector analysis and the method by which planes (as 
distinguished from finite plane areas) may be represented 
by vectors. 



DIRECT AND SKEW PRODUCTS OF VECTORS 113 

EXERCISES ON CHAPTER II 
Prove the following reduction formulae 

1. Ax{Bx(CxD)} = [ACD]B-A-BCxD 

= BD AxC B-C AxD. 

2. [AxB CxD ExF] = [ABD] [CEF]- [ABC] [DBF] 

= [ABE] [FCD] - [ABF] [BCD] 
= [CD A] [BEF] - [CDB] [AEF]. 

3. [AxB BxC CxA] = [ABC] 2 . 

P.A P.B P 



4 [PQE] (AxB) = 



Q.B Q 



RA RB R 

5. Ax(BxC) + Bx(CxA) + Cx(AxB) = 0. 

6. [AxP Bxtt CxR] + [Axtt BxR CxP] 

+ [AxR BxP Cxtt] = 0. 

7. Obtain formula (34) in the text by expanding 

[(AxB)xP].[Cx(ttxR)] 
in two different ways and equating the results. 

8. Demonstrate directly by the above formulae that if 
a , V, c form a reciprocal system to a, b, c; then a, b, c form 
a system reciprocal to a , b , c . 

9. Show the connection between reciprocal systems of vec 
tors and polar triangles upon a sphere* Obtain some of the 
geometrical formulae connected with polar triangles by inter 
preting vector formulae such as (3) in the above list. 

10. The perpendicular bisectors of the sides of a triangle 
meet in a point. 

11. Find an expression for the common perpendicular to 
two lines not lying in the same plane. 



114 



VECTOR ANALYSIS 



12. Show by vector methods that the formulae for the vol 
ume of a tetrahedron whose four vertices are 



IS 



13. Making use of formula (34) of the text show that 



[abo] = a be 



N 



1 

n 
m 



n 
1 

I 



m 
I 
1 



where a, &, c are the lengths of a, b, c respectively and where 
I = cos (b, c), m = cos (c, a), n = cos (a, b). 

14. Determine the perpendicular (as a vector quantity) 
which is dropped from the origin upon a plane determined by 
the termini of the vectors a, b, c. Use the method of solution 
given in Art. 46. 

15. Show that the volume of a tetrahedron is equal to one 
sixth of the product of two opposite edges by the perpendicu 
lar distance between them and the sine of the included angle. 

16. If a line is drawn in each face plane of any triedral angle 
through the vertex and perpendicular to the third edge, the 
three lines thus obtained lie in a plane. 



CHAPTER III 

THE DIFFERENTIAL CALCULUS OF VECTORS 

Differentiation of Functions of One Scalar Variable 

55.] IF a vector varies and changes from r to r the incre 
ment of r will be the difference between r and r and will be 
denoted as usual by A r. 

Ar = r -r, (1) 

where A r must be a vector quantity. If the variable r be 
unrestricted the increment A r is of course also unrestricted : 
it may have any magnitude and any direction. If, however, 
the vector r be regarded as a function (a vector function) of 
a single scalar variable t the value of A r will be completely 
determined when the two values t and t f of , which give the 
two values r and r , are known. 

To obtain a clearer conception of the quantities involved 
it will be advantageous to think of the vector r as drawn 
from a fixed origin (Fig. 26). When 
the independent variable t changes its 
value the vector r will change, and as t 
possesses one degree of freedom r will 
vary in such a way that its terminus 
describes a curve in space, r will be 
the radius vector of one point P of 
the curve ; r , of a neighboring point P f . A r will be the 
chord PP 1 of the curve. The ratio 

Ar 
A* 



FIG. 26. 



116 VECTOR ANALYSIS 

will be a vector collinear with the chord P P f but magnified 
in the ratio 1 : A t. When A t approaches zero P f will ap 
proach P, the chord PP 1 will approach the tangent at P, and 

the vector 

Ar ... rfr 

will approach 
i\ t (t t 

which is a vector tangent to the curve at P directed in that 
sense in which the variable t increases along the curve. 
If r be expressed in terms of i, j, k as 

r = r x i + r 2 j + r z k 

the components r v r 2 , r 3 will be functions of the scalar t. 
r = (r 1 + Arj)i+ (^ 2 + Ar 2 )j + (r 3 + Ar 3 )k 
Ar = r r = Ar x i + Ar 2 j + Ar 3 k 

A r _ A ?*! . A r 2 . A r 8 

^ 1 " J+ k 



and 



Hence the components of the first derivative of r with re 
spect to t are the first derivatives with respect to t of the 
components of r. The same is true for the second and higher 
derivatives. 



. . ~ 

i j __ _ f , _ 3 
* 



_ __ _ _ 

dt*~ dt* dt* dt* 

(2) 

d n r d n r, . d n r fl d n r* 
- - l i j __ ? i _i __ ? v 

dt n dt dt n J dt* 

In a similar manner if r be expressed in terms of any three 
non-coplanar vectors a, b, c as 

r = aa + &b + cc 
d n r d n a d n b d n c 



THE DIFFERENTIAL CALCULUS OF VECTORS 117 

Example : Let r = a cos t + b sin t. 

The vector r will then describe an ellipse of which a and b 
are two conjugate diameters. This may be seen by assum 
ing a set of oblique Cartesian axes X, Y coincident with a 
and b. Then 

X = a cos t, Y = 6 sin t, 



which is the equation of an ellipse referred to a pair of con 
jugate diameters of lengths a and b respectively. 

dr 

-3 = a sin t + b cos t. 

a t 

Hence = a cos (t + 90) + b sin (t + 90). 

The tangent to the curve is parallel to the radius vector 

for + 90). 2r 

= (a cos t + b sin t). 



The second derivative is the negative of r. Hence 



is evidently a differential equation satisfied by the ellipse. 
Example : Let r = a cosh t + b sinh t. 

The vector r will then describe an hyperbola of which a and 
b are two conjugate diameters. 

dr 

= a sinh t + b cosh t, 

dt 



and - - = a cosh t + b sinh t. 



Hence = r 

d t* 

is a differential equation satisfied by the hyperbola. 



118 VECTOR ANALYSIS 

56.] A combination of vectors all of which depend on the 
same scalar variable t may be differentiated very much as in 
ordinary calculus. 

d 



For 
(a + Aa) . (b 

A(a-b) = (a + Aa) (b + Ab) - a-b 



Ab Aa Aa-Ab 

= a H -- b + - 1 - - 



- 

A* A* A* 

Hence in the limit when A t = 0, 



d_ 
dt 



_(a.bxc) = a-b 

dt v \d t 



X [b X 



The last three of these formulae may be demonstrated exactly 
as the first was. 

The formal process of differentiation in vector analysis 
differs in no way from that in scalar analysis except in this 
one point in which vector analysis always differs from scalar 
analysis, namely : The order of the factors in a vector product 



THE DIFFERENTIAL CALCULUS OF VECTORS 119 

cannot be changed without changing the sign of the product. 
Hence of the two formulae 



d 
and 



the first is evidently incorrect, but the second correct. In 
other words, scalar differentiation must take place without 
altering the order of the factors of a vector product. The 
factors must be differentiated in situ. This of course was to 
be expected. 

In case the vectors depend upon more than one variable 
the results are practically the same. In place of total deriva 
tives with respect to the scalar variables, partial derivatives 
occur. Suppose a and b are two vectors which depend on 
three scalar variables #, y, z. The scalar product ab will 
depend upon these three variables, and it will have three 
partial derivatives of the first order. 



The second partial derivatives are formed in the same way. 

52 






- 

9y \3x5y 



120 VECTOR ANALYSTS 

Often it is more convenient to use not the derivatives but 
the differentials. This is particularly true when dealing with 
first differentials. The formulas (3), (4) become 

d (a b) = da, b + a db, (3) 

d (a X b) = ds, x b + a x db, (4) 

and so forth. As an illustration consider the following 
example. If r be a unit vector 

rr = 1. 

The locus of the terminus of r is a spherical surface of unit 
radius described about the origin, r depends upon two vari 
ables. Differentiate the equation. 



Hence r d r = 0. 

Hence the increment di of a unit vector is perpendicular to 
the vector. This can be seen geometrically. If r traces a 
sphere the variation d r must be at each point in the tangent 
plane and hence perpendicular to r. 

*57.j Vector methods may be employed advantageously 
in the discussion of curvature and torsion of curves. Let r 
denote the radius vector of a curve 



where f is some vector function of the scalar t. In most appli 
cations in physics and mechanics t represents the time. Let 
s be the length of arc measured from some definite point of 
the curve as origin. The increment A r is the chord of the 
curve. Hence A r / A s is approximately equal in magnitude 
to unity and approaches unity as its limit when A s becomes 
infinitesimal. Hence d r / d s will be a unit vector tangent to 
the curve and will be directed toward that portion of the 



THE DIFFERENTIAL CALCULUS OF VECTORS 121 

curve along which s is increasing (Fig. 27). Let t be the 
unit tangent UAt 



The curvature of the curve is the 
limit of the ratio of the angle through 
which the tangent turns to the length 

of the arc. The tangent changes by the increment At. As t 
is of unit length, the length of A t is approximately the angle 
through which the tangent has turned measured in circular 
measure. Hence the directed curvature C is 



LIM = t = 

As=0 As ds ds* 

The vector C is collinear with A t and hence perpendicular to 
t; for inasmuch as t is a unit vector At is perpendicular 
to t. 

The tortuosity of a curve is the limit of the ratio of the 
angle through which the osculating plane turns to the length 
of the arc. The osculating plane is the plane of the tangent 
vector t and the curvature vector C. The normal to this 

planei8 N = txC. 

If c be a unit vector collinear with C 

n = t x c 

will be a unit normal (Fig. 28) to the osculating plane and 

the three vectors t, c, n form an i, j, k system, 

that is, a right-handed rectangular system. 

Then the angle through which the osculating 

plane turns will be given approximately by 

A n and hence the tortuosity is by definition _ 

d n / d s. 

From the fact that t, c, n form an i, j, k system of unit 
vectors 



122 VECTOR ANALYSIS 

t t = c c = nn = 1 
and tc = cn = nt = 0. 

Differentiating the first set 

t-dt = cdc = ndn = 0, 
and the second 

t* do + rft c =cdn + dcn = ndt + dnt==0. 
But d t is parallel to c and consequently perpendicular to n. 

n- dt = 0. 
Hence d n t = 0. 

The increment of n is perpendicular to t. But the increment 
of n is also perpendicular to n. It is therefore parallel to c. 
As the tortuosity is T = dn/ds, it is parallel to dn and hence 
to c. 

The tortuosity T is 



~ds^ 
d*r d*r 

T v 

* j O * 

d s* 



i \ 

VCC/ 



The first term of this expression vanishes. T moreover has 
been seen to be parallel to C = d 2 r/ds 2 . Consequently the 
magnitude of T is the scalar product of T by the unit vec 
tor c in the direction of C. It is desirable however to have 
the tortuosity positive when the normal n appears to turn in 
the positive or counterclockwise direction if viewed from 
that side of the n c-plane upon which t or the positive part 
of the curve lies. With this convention d n appears to move 
in the direction c when the tortuosity is positive, that is, n 
turns away from c. The scalar value of the tortuosity will 
therefore be given by c T. 



THE DIFFERENTIAL CALCULUS OF VECTORS 123 

1 dr d 2 r d 1 

c T = c x 



But c is parallel to the vector d 2 i/d s 2 . Hence 



dr 

ds ds 2 ~~ 

And c is a unit vector in the direction C. Hence 

C 

~ 



Hence T. -c-T = - . x 



(12) 



Or r = . (13) 



The tortuosity may be obtained by another method which 
is somewhat shorter if not quite so straightforward. 

tc = cn = nt = 0. 
Hence dtc = dct 

dc n = dn c 
dn*t = dtn. 

Now d t is parallel to c ; hence perpendicular to n. Hence 
d t n = 0. Hence dnt = 0. Butdnis perpendicular to n. 
Hence d n must be parallel to c. The tortuosity is the mag 
nitude of dn/ds taken however with the negative sign 
because d n appears clockwise from the positive direction of 
the curve. Hence the scalar tortuosity T may be given by 

dn dc 

r=- .c = n. , (14) 

ds ds 

r = txc-^- C , (14) 

as 



124 



VECTOR ANALYSIS 
C 



c = 



dc 
ds 



V cc 

dC , d / 

!; C ---VC.C 

ds ds 

C-C 



But 



t x c C = 0. 

t x c A/C C 



C-C 



m 

1 = 



_ dC 
T~ 

ITTc 






(13) 



ds* 



In Cartesian coordinates this becomes 



T= 



(13) 



Those who would pursue the study of twisted curves and 
surfaces in space further from the standpoint of vectoi-s will 
find the book " Application de la Methode Vectorielle de Grass- 
maun d la Greometrie Infinitesimale" 1 by FEHB extremely 

1 Paris, Carre et Naud, 1899. 



THE DIFFERENTIAL CALCULUS OF VECTORS 125 

helpful. He works with vectors constantly. The treatment 
is elegant. The notation used is however slightly different 
from that used by the present writer. The fundamental 
points of difference are exhibited in this table 

HI X a 2 ~ Oi 2 ] 

a x a 2 x a 3 = [a x a 2 a 3 ] ~ [a x a 2 aj. 

One used to either method need have no difficulty with the 
other. All the important elementary properties of curves 
and surfaces are there treated. They will not be taken 
up here. 

* Kinematics 

58.] Let r be a radius vector drawn from a fixed origin to 
a moving point or particle. Let t be the time. The equation 
of the path is then 

The velocity of the particle is its rate of change of position. 
This is the limit of the increment A r to the increment A t. 

LIM f A r"| d r 

V = A * . 



This velocity is a vector quantity. Its direction is the 
direction of the tangent of the curve described by the par 
ticle. The term speed is used frequently to denote merely 
the scalar value of the velocity. This convention will be 
followed here. Then 

.-, (16) 

if s be the length of the arc measured from some fixed point 
of the curve. It is found convenient in mechanics to denote 
differentiations with respect to the time by dots placed over 
the quantity differentiated. This is the oldfliixional notation 



126 VECTOR ANALYSIS 

introduced by Newton. It will also be convenient to denote 
the unit tangent to the curve by t. The equations become 



--T. <"> 

v = v t. (17) 

The acceleration is the rate of change of velocity. It 
is a vector quantity. Let it be denoted by A. Then by 
definition 

LIM A v d v 



_ _ 

-At=OA7-rf7 = 

dv d 



and 



Differentiate the expression v = v t. 

dv d(vt) dv dt 

A - __ - v . _ * I nj __ . - 

dt ~ dt dt dt 
dv d z s~ 



dt dt d s 

_ _ _ _ = C t? 
d t ds d t 

where C is the (vector) curvature of the curve and v is the 
speed in the curve. Substituting these values in the equation 
the result is 

A = s t + v* C. 

The acceleration of a particle moving in a curve has there 
fore been broken up into two components of which one is paral 
lei to the tangent t and of which the other is parallel to the 
curvature C, that is, perpendicular to the tangent. That this 
resolution has been accomplished would be unimportant were 



THE DIFFERENTIAL CALCULUS OF VECTORS 127 

it not for the remarkable fact which it brings to light. The 
component of the acceleration parallel to the tangent is equal 
in magnitude to the rate of change of speed. It is entirely 
independent of what sort of curve the particle is describing. 
It would be the same if the particle described a right line 
with the same speed as it describes the curve. On the other 
hand the component of the acceleration normal to the tangent 
is equal in magnitude to the product of the square of the 
speed of the particle and the curvature of the curve. The 
sharper the curve, the greater this component. The greater 
the speed of the particle, the greater the component. But the 
rate of change of speed in path has no effect at all on this 
normal component of the acceleration. 
If r be expressed in terms of i, j, k as 

r = # i + y} + z k, 



v = V ** + y* + * 2 , (16) 

A = v = r = ii + yj + * k, (18) 

x x + i/ i/ + z % 
A = v=s = y * 

V x 2 + y* + z 2 

From these formulae the difference between s t the rate of 
change of speed, and A = r, the rate of change of velocity, 
is apparent. Just when this difference first became clearly 
recognized would be hard to say. But certain it is that 
Newton must have had it in mind when he stated his second 
law of motion. The rate of change of velocity is proportional 
to the impressed force ; but rate of change of speed is not. 

59.] The hodograph was introduced by Hamilton as an 
aid to the study of the curvilinear motion of a particle. 
With any assumed origin the vector velocity r is laid off. 
The locus of its terminus is the hodograph. In other words, 
the radius vector in the hodograph gives the velocity of the 



128 VECTOR ANALYSIS 

particle in magnitude and direction at any instant. It is 
possible to proceed one step further and construct the hodo 
graph of the hodograph. This is done by laying off the 
vector acceleration A = r from an assumed origin. The 
radius vector in the hodograph of the hodograph therefore 
gives the acceleration at each instant. 

Example 1 : Let a particle revolve in a circle (Fig. 29) 

of radius r with a uniform 

fV * ^-r^ angular velocity a. The 

speed of the particle will then 
be equal to 

v = a r. 

Let r be the radius vector 
drawn to the particle. The 
velocity v is perpendicular to r and to a. It is 

f = v = a x r. 

The vector v is always perpendicular and of constant magni 
tude. The hodograph is therefore a circle of radius v = a r. 
The radius vector r in this circle is just ninety degrees in 
advance of the radius vector r in its circle, and it conse 
quently describes the circle with the same angular velocity 
a. The acceleration A which is the rate of change of y is 
always perpendicular to v and equal in magnitude to 

A = a v = a 2 r. 

The acceleration A may be given by the formula 
r = A = axv = ax(axr) = ar a a-a r. 

But as a is perpendicular to the plane in which r lies, a r = 0. 

Hence 9 

r = A = aa r = a 2 r. 

The acceleration due to the uniform motion of a particle in 
a circle is directed toward the centre and is equal in magni 
tude to the square of the angular velocity multiplied by the 
radius of the circle. 



THE DIFFERENTIAL CALCULUS OF VECTORS 129 

Example 2: Consider the motion of a projectile. The 
acceleration in this case is the acceleration g due to gravity. 

r = A = g. 

The hodograph of the hodograph reduces to a constant 
vector. The curve is merely a point. It is easy to find 
the hodograph. Let v be the velocity of the projectile 
in path at any given instant. At a later instant the velocity 

will be 

v = v + t g. 

Thus the hodograph is a straight line parallel to g and pass 
ing through the extremity of v . The hodograph of a 
particle moving under the influence of gravity is hence a 
straight line. The path is well known to be a parabola. 
Example 3 : In case a particle move under any central 

acceleration 

r = A = f(r). 

The tangents to the hodograph of r are the accelerations r! 
But these tangents are approximately collinear with the 
chords between two successive values r and f of the radius 
vector in the hodograph. That is approximately 



A* 
Multiply by rx. r x r = r x . 

Since r and r are parallel 

r x (r - r ) = 0. 
Hence r x r = r x r . 

But J r x f is the rate of description of area. Hence the 
equation states that when a particle moves under an ac 
celeration directed towards the centre, equal areas are swept 
over in equal times by the radius vector. 

9 



130 VECTOR ANALYSIS 

Perhaps it would be well to go a little more carefully into 
this question. If r be the radius vector of the particle in 
its path at one instant, the radius vector at the next instant 
is r + A r. The area of the vector of which r and r + A r are 
the bounding radii is approximately equal to the area of the 
triangle enclosed by r, r + A r, and the chord A r. This 
area is 



The rate of description of area by the radius vector is 
consequently 

LIM irx(r+ Ar) Lm 1 AT 1 
A* = 02 A* ~A*-=02 A*~2 r 

Let r and r be two values of the velocity at two points 
P and P which are near together. The acceleration r at P 
is the limit of 

r r _ A r 

A* " A * * 

A * * " 

Break up the vector ^- = ?^IlI? into two components one 
A t A t 

parallel and the other perpendicular to the acceleration r . 
Ar. 



if n be a normal to the vector if . The quantity x ap 
proaches unity when A t approaches zero. The quantity y 
approaches zero when A t approaches zero. 

Ar = r-r = #A*r + yA*n. 
Hence r x (r - r ) = x A* r x r + y A* r x n. 

r x (f - r ) = r x r - (r + ^ A M x f . 



THE DIFFERENTIAL CALCULUS OF VECTORS 131 

Hence 

Ar 

rxr-r xf = xr A* + zA*rxr + yA* rxn. 

/A 6 

But each of the three terms upon the right-hand side is an 
infinitesimal of the second order. Hence the rates of descrip 
tion of area at P and P d differ by an infinitesimal of the 
second order with respect to the time. This is true for any 
point of the curve. Hence the rates must be exactly equal 
at all points. This proves the theorem. 

60.] The motion of a rigid body one point of which is 
fixed is at any instant a rotation about an instantaneous axis 
passing through the fixed point. 

Let i, j, k be three axes fixed in the body but moving in 
space. Let the radius vector r be drawn from the fixed point 
to any point of the body. Then 



But d r = (d r i) i + (d r j) j 4- (d r k) k. 

Substituting the values of d r i, d r j, d r k obtained from 
the second equation 

dr = (xi di+ yi d j + 2 i d k) i 
j di + yj *dj + zj 



But i j =j k = k i = 0. 

Hence i dj +j di = Q or j-c?i = i dj 
j.dk + k.dj = or k.dj = j-dk 
k.di + i.rfk = or idk = k di. 

Moreover i.i=j .j=kk = l. 

Hence i d i = j d j = k d k = 0. 



132 VECTOR ANALYSIS 

Substituting these values in the expression for d r. 

dr = (zi dk yjdi)i+(jdi s k 

+ ( y k . d j - x i d k) k. 

This is a vector product. 

dr = (Wj i + idkj + jdik)x(>i + yj + 2; k). 
Let d j . d k . d i 

k -r l+| -i J+J ii k - 

Then . d r 

r = ;n =axr - 

This shows that the instantaneous motion of the body is one 
of rotation with the angular velocity a about the line a. 
This angular velocity changes from instant to instant. The 
proof of this theorem fills the lacuna in the work in Art. 51. 

Two infinitesimal rotations may be added like vectors. 
Let a x and a 2 be two angular velocities. The displacements 

due to them are 

d l r = a x x r d t, 

d 2 r = a 2 x r d t. 

If r be displaced by a, it becomes 

T + d 1 T = T + a, 1 xrdt. 

If it then be displaced by a 2 , it becomes 

r 4- d r = r + d l r + % x [r + (a x x r) d t] d t. 
Hence d r = aj x r d t + a 2 x r d t + a 2 x (a x x r) (d ) 2 . 

If the infinitesimals (d t) 2 of order higher than the first be 
neglected, 

d r = a x x r d t + a 2 x r d t = (a x + a 2 ) x r d t, 
which proves the theorem. If both sides be divided by d t 

. dr 

r = = (a 1 + a 2 ) x r. 



THE DIFFERENTIAL CALCULUS OF VECTORS 133 

This is the parallelogram law for angular velocities. It 
was obtained before (Art. 51) in a different way. 

In case the direction of a, the instantaneous axis, is con 
stant, the motion reduces to one of steady rotation about a. 



r = a x r. 
The acceleration r = axr + axr = axr + ax (axr). 

As a does not change its direction a must be collinear with 
a and hence a x r is parallel to a x r. That is, it is perpen 
dicular to r. On the other hand ax (a x r) is parallel to r. 
Inasmuch as all points of the rotating body move in con 
centric circles about a in planes perpendicular to a, it is 
unnecessary to consider more than one such plane. 

The part of the acceleration of a particle toward the centre 
of the circle in which it moves is 

a x (a x r). 

This is equal in magnitude to the square of the angular 
velocity multiplied by the radius of the circle. It does not 
depend upon the angular acceleration a at all. It corresponds 
to what is known as centrifugal force. On the other hand 
the acceleration normal to the radius of the circle is 

axr. 

This is equal in magnitude to the rate of change of angular 
velocity multiplied by the radius of the circle. It does not 
depend in any way upon the angular velocity itself but only 
upon its rate of change. 

61.] The subject of integration of vector equations in which 
the differentials depend upon scalar variables needs but a 
word. It is precisely like integration in ordinary calculus. 

If then d r = d s, 

r = s + C, 



134 VECTOR ANALYSIS 

where C is some constant vector. To accomplish the integra 
tion in any particular case may be a matter of some difficulty 
just as it is in the case of ordinary integration of scalars. 

Example 1 : Integrate the equation of motion of a 
projectile. 

The equation of motion is simply 



which expresses the fact that the acceleration is always ver 
tically downward and due to gravity. 

r = g t + b, 

where b is a constant of integration. It is evidently the 
velocity at the time t = 0. 

r = ig*2 + b* + c. 

c is another constant of integration. It is the position vector 
of the point at time t = 0. The path which is given by this 
last equation is a parabola. That this is so may be seen by 
expressing it in terms of x and y and eliminating t. 

Example % : The rate of description of areas when a par 
ticle moves under a central acceleration is constant. 

r = f(r). 

Since the acceleration is parallel to the radius, 
r x r = 0. 

But r x r = (r x r). 

a L> 

For (r x f ) = r x f + r x r. 

u/ t 

Hence (r x r) = 

CL t 

and r x f = C, 

which proves the statement. 



THE DIFFERENTIAL CALCULUS OF VECTORS 135 

Example 3 : Integrate the equation of motion for a particle 
moving with an acceleration toward the centre and equal to 
a constant multiple of the inverse square of the distance 
from the centre. 

^2 
Given 

Then r x r = 0. 

Hence r x r = C. 

Multiply the equations together with x. 

r xC -1 -1 ( 

^- = rx (rxr)= -jjj- {r.r r - r-r r}. 

r r = r 2 . 
Differentiate. Then r r = r r. 



Hence *2L = _ L r 

o o * 



Each side of this equality is a perfect differential. 



Integrate. Then r x C = + e I, 



c* r 



where e I is the vector constant of integration, e is its magni 
tude and I a unit vector in its direction. Multiply the equa 
tion by r . 

r r x C r r 

+ e r I. 



But 



c* r * 

r r x C r x r C C C 



136 VECTOR ANALYSIS 

T f C C 

p = s" and cos u = cos (r, I). 
c 

Then p = r + e r cos u. 

Or p 

r = 



1 + e cos w 

This is the equation of the ellipse of which e is the eccentri 
city. The vector I is drawn in the direction of the major 
axis. The length of this axis is 



It is possible to cany the integration further and obtain 
the time. So far merely the path has been found. 

Scalar Functions of Position in Space. The Operator V 

62. ] A function V (x, y, z) which takes on a definite scalar 
value for each set of coordinates #, y, z in space is called a 
scalar function of position in space. Such a function, for ex 
ample, is 

V O, y, z) == x 2 + y* + z 2 = r\ 

This function gives the square of the distance of the point 
(x, y, z) from the origin. The function V will be supposed to 
be in general continuous and single-valued. In physics scalar 
functions of position are of constant occurrence. In the 
theory of heat the temperature T at any point of a body is a 
scalar function of the position of that point. In mechanics 
and theories of attraction the potential is the all-important 
function. This, too, is a scalar function of position. 

If a scalar function V be set equal to a constant, the equa 
tion 

V(x,y,z)=c. (20) 

defines a surface in space such that at every point of it the 
function V has the same value c. In case V be the tempera- 



THE DIFFERENTIAL CALCULUS OF VECTORS 137 

ture, this is a surface of constant temperature. It is called an 
isothermal surface. In case V be the potential, this surface of 
constant potential is known as an equipotential surface. As 
the potential is a typical scalar function of position in space, 
and as it is perhaps the most important of all such functions 
owing to its manifold applications, the surface 

V O, y,z)=c 

obtained by setting V equal to a constant is frequently spoken 
of as an equipotential surface even in the case where V has 
no connection with the potential, but is any scalar function 
of positions in space. 

The rate at which the function V increases in the X direc 
tion that is, when x changes to x + A x and y and z remain 
constant is 

LIM [" F" (a? + A a, y, g) - T (x, y, z) "1 
Aa = L A x J* 

This is the partial derivative of Fwith respect to x. Hence 
the rates at which V increases in the directions of the three 
axes X, Y) Z are respectively 

3V 3V 3V 
~Wx Ty* Tz 

Inasmuch as these are rates in a certain direction, they may 
be written appropriately as vectors. Let i, j, k be a system 
of unit vectors coincident with the rectangular system of 
axes X, Y) Z. The rates of increase of V are 

3V 3V 3V 
1 JZ* J 5? ~3~z 

The sum of these three vectors would therefore appear to be 
a vector which represents both in magnitude and direction 
the resultant or most rapid rate of increase of V. That this 
is actually the case will be shown later (Art. 64). 



138 VECTOR ANALYSIS 

63.] The vector sum which is the resultant rate of increase 
of Fis denoted by VF 



V V represents a directed rate of change of V a directed 
or vector derivative of F^ so to speak. For this reason VF 
will be called the derivative of V; and F, the primitive of 
VF. The terms gradient and slope of F are also used for 
V F. It is customary to regard V as an operator which obtains 
a vector V F from a scalar function F of position in space. 



This symbolic operator V was introduced by Sir W. R. 
Hamilton and is now in universal employment. There 
seems, however, to be no universally recognized name l for it, 
although owing to the frequent occurrence of the symbol 
some name is a practical necessity. It has been found by 
experience that the monosyllable del is so short and easy to 
pronounce that even in complicated formulae in which V occurs 
a number of times no inconvenience to the speaker or hearer 
arises from the repetition. V F is read simply as " del F." 

Although this operator V has been defined as 

v=i* + ji- +k * 

dx dy 9z 

1 Some use the term Nabla owing to its fancied resemblance to an Assyrian 
harp. Others have noted its likeness to an inverted A and have consequently 
coined the none too euphonious name Ailed by inverting the order of the letters in 
the word Delta. Foppl in his Einfuhrung in die Maxwell 1 sche Theorie der Elec- 
tricitdt avoids any special designation and refers to the symbol as "die Operation 
V. v How this is to be read is not divulged. Indeed, for printing no particular 
name is necessary, but for lecturing and purposes of instruction something is re 
quiredsomething too that does not confuse the speaker or hearer even when 
often repeated. 



THE DIFFERENTIAL CALCULUS OF VECTORS 139 

so that it appears to depend upon the choice of the axes, it 
is in reality independent of them. This would be surmised 
from the interpretation of V as the magnitude and direction 
of the most rapid increase of V. To demonstrate the inde 
pendence take another set of axes, i , j , k and a new set of 
variables # , y ^ z f referred to them. Then V referred to this 
system is 

v/ = i ?T7 + J o^7 + k ^T7 ( 22 ) 

a x a y d z 

By making use of the formulae (47) and (47)", Art. 53, page 
104, for transformation of axes from i, j, k to i , j , k and by 
actually carrying out the differentiations and finally by 
taking into account the identities (49) and (50), V may 
actually be transformed into V. 



The details of the proof are omitted here, because another 
shorter method of demonstration is to be given. 
64] Consider two surfaces (Fig. 30) 



and 



y,z)=c 
V (x, y, z) = c + d c, 



upon which V is constant and which are moreover infinitely 
near together. Let #, y, z be a given point upon the surface 
V=c. Let r denote the ra 
dius vector drawn to this 
point from any fixed origin. 
Then any point near by in 
the neighboring surface V 
c + d c may be represented 
by the radius vector r + d r. 
The actual increase of Ffrom 
the first surface to the second 
is a fixed quantity dc. The rate of increase is a variable 



FIG. 30. 



140 VECTOR ANALYSIS 

quantity and depends upon the direction dr which is fol 
lowed when passing from one surface to the other. The rate 
of increase will be the quotient of the actual increase d c and 
the distance V d r d r between the surfaces at the point 
x, y, z in the direction d r. Let n be a unit normal to the 
surfaces and d n the segment of that normal intercepted 
between the surfaces, n d n will then be the least value for 
d r. The quotient . 

\/d r d r 

will therefore be a maximum when d r is parallel to n and 
equal in magnitude of d n. The expression 



is therefore a vector of which the direction is the direction of 
most rapid increase of Fand of which the magnitude is the 
rate of that increase. This vector is entirely independent of 
the axes JT, Y, Z. Let d c be replaced by its equal d V which 
is the increment of F^in passing from the first surface to the 
second. Then let V V be defined again as 

Vr=4^n. (24) 

d n 

From this definition, V V is certainly the vector which 
gives the direction of most rapid increase of V and the rate 
in that direction. Moreover VFis independent of the axes. 
It remains to show that this definition is equivalent to the one 
first given. To do this multiply by d r. 

dV 

VF.dr = -n.dr. (25) 

d n 

n is a unit normal. Hence n d r is the projection of d r on 
n and must be equal to the perpendicular distance d n between 
the surfaces. 



THE DIFFERENTIAL CALCULUS OF VECTORS 141 

dV 

dT = -dn = dV (25) 

dn 

5V 5V 5V 



But =7r- -= -z 

dx dy 5z 

where (d x? + (d y) 2 + (d *) 2 = d r d r. 

If dr takes on successively the values i dx, j dy, kdz the 
equation (25) takes on the values 

5V 
ids= ~dx 

d x 

sv 

d y = d y (26 ) 

9V 



If the factors rf a;, rf y, rf be cancelled these equations state 
that the components VF i, VF* j, VF- k of VF in the 
i, j, k directions respectively are equal to 

3V 5V 5_V 

5x 5y* 5z 



VF=(VF. i)i + (Vr-j)j + (VT. k)k. 
Henceby(26) VF= i |^ + j |T+ k | 

The second definition (24) has been reduced to the first 
and consequently is equivalent to it. 

*65.] The equation (25) found above is often taken as a 
definition of V V. According to ordinary calculus the deriv 

ed y 
ative - satisfies the equation 

d x 

, dy 

dx = dy 
dx 



142 VECTOR ANALYSIS 

Moreover this equation defines dy / dx. In a similar manner 
it is possible to lay down the following definition. 

Definition: The derivative V^ of a scalar function of 
position in space shall satisfy the equation 



for all values of d r. 

This definition is certainly the most natural and important 
from theoretical considerations. But for practical purposes 
either of the definitions before given seems to be better. 
They are more tangible. The real significance of this last 
definition cannot be appreciated until the subject of linear 
vector functions has been treated. See Chapter VII. 

The computation of the derivative V of a function is most 
frequently carried on by means of the ordinary partial 
differentiation. 



Example 1 : 



(ix + jy + kz) 



The derivative of r is a unit vector in the direction of r. 
This is evidently the direction of most rapid increase of r 
and the rate of that increase. 



THE DIFFERENTIAL CALCULUS OF VECTORS 143 
Example % : Let 



T V1F 
1 X 



Hence 



T (x z + y z + 2 )* (x z + y z + 2 2 )* 

-k 2 + g 2 )t 

_1 1 



1 -r -r 

and V ~ == 7 - ^i = ~T 

r (r r) r 3 

The derivative of 1/r is a vector whose direction is that 
of r, and whose magnitude is equal to the reciprocal of the 
square of the length r. 

Example 3: V r n n r n ~ 2 r = n r* 

i>r 

The proof is left to the reader. 

Example 4 Let F(#, y, z) = log y# 2 + y*. 

V log V^Tp = i TT 5 + j 2 f 2 + k 
22 22 



If r denote the vector drawn from the origin to the point 
, y, z) of space, the function V may be written as 



2/1 *) = log Vr.r-(k.r) 2 
and ix + )y = T k kr. 

Hence V log V^ + y "" 



r r 
T - k kr 



(r-kk.r).(r-kk.r) 



144 VECTOR ANALYSIS 

There is another method of computing V which is based 
upon the identity 



Example 1 : Let V = Vrr = r. 

d V = =^ = 



Hence 



v r r 



- 
V i>r r 

Example 2 : Let V = r a, where a is a constant vector. 

d F=dr.a = dr Vr. 
Hence V V = a. 

Example 3: Let F= (rxa) (rxb), where a and b are 

constant vectors. 

V = rr ab r-a rb. 

dV = 2cZr*r a-b dr-a r-b drb r-a = di V Fl 
Hence V F == 2 r a-b a r.b b r^a 

Vr= (ra-b-ar-b) + (ra-b -br.a) 
= bx(rxa) -fax (rxb). 

Which of these two methods for computing V shall be 
applied in a particular case depends entirely upon their 
relative ease of execution in that case. The latter method is 
independent of the coordinate axes and may therefore be 
preferred. It is also shorter in case the function Fcan be 
expressed easily in terms of r. But when V cannot be so 
expressed the former method has to be resorted to. 

*66.] The great importance of the operator V in mathe 
matical physics may be seen from a few illustrations. Sup 
pose T (#, y, z) be the temperature at the point #, y, z of a 



THE DIFFERENTIAL CALCULUS OF VECTORS 145 

heated body. That direction in which the temperature de 
creases most rapidly gives the direction of the flow of heat. 
V T, as has been seen, gives the direction of most rapid 
increase of temperature. Hence the flow of heat f is 

f = _& vr, 

where k is a constant depending upon the material of the 
body. Suppose again that V be the gravitational potential 
due to a fixed body. The force acting upon a unit mass at 
the point (#, y, z) is in the direction of most rapid increase of 
potential and is in magnitude equal to the rate of increase 
per unit length in that direction. Let F be the force per unit 

mass. Then 

F = VF. 

As different writers use different conventions as regards the 
sign of the gravitational potential, it might be well to state 
that the potential Preferred to here has the opposite sign to 
the potential energy. If W denoted the potential energy of 
a mass m situated at #, y, z, the force acting upon that mass 

would be 

F = - VfF. 

In case V represent the electric or magnetic potential due 
to a definite electric charge or to a definite magnetic pole re 
spectively the force F acting upon a unit charge or unit pole 

as the case might be is 

F = - VF. 

The force is in the direction of most rapid decrease of 
potential. In dealing with electricity and magnetism poten 
tial and potential energy have the same sign ; whereas in 
attraction problems they are generally considered to have 
opposite signs. The direction of the force in either case is in 
the direction of most rapid decrease of potential energy. The 
difference between potential and potential energy is this. 

10 



146 VECTOR ANALYSIS 

Potential in electricity or magnetism is the potential energy 
per unit charge or pole ; and potential in attraction problems 
is potential energy per unit mass taken, however, with the 
negative sign. 

*67.] It is often convenient to treat an operator as a 
quantity provided it obeys the same formal laws as that 
quantity. Consider for example the partial differentiators 

!_ A !.. 

9x 3y 3z 

As far as combinations of these are concerned, the formal laws 
are precisely what they would be if instead of differentiators 

three true scalars 

a, 6, c 

were given. For instance 
the commutative law 

99 d 9 

= - *, a = a, 
Sx3y 3ySx 

the associative law 

5 9 3\ 3 3 3 



and the distributive law 



3 f 3 3\ 33 33 
( -+_-)=_--_ + ._ -- a(b + c) = a 

3x\3y 3zJ 3x3y dxdz 

hold for the differentiators just as for scalars. Of course such 

formulae as 

3 3 



where u is a function of x cannot hold on account of the 
properties of differentiators. A scalar function u cannot be 
placed under the influence of the sign of differentiators. 
Such a patent error may be avoided by remembering that an 
operand must be understood upon which 3/3 # is to operate. 



THE DIFFERENTIAL CALCULUS OF VECTORS 147 

In the same way a great advantage may be obtained by 
looking upon 

V-if +jf + kf 

3x dy dz 

as a vector. It is not a true vector, for the coefficients 

.., JL, A 

P# dy dz 

are not true scalars. It is a vector differentiator and of 
course an operand is always implied with it. As far as formal 
operations are concerned it behaves like a vector. For 

instance 

V (u + v) = V u + V v, 

V(ttfl) = (Vtt) v + ^(Vtf), 
c V u = V (c u), 

if w and v are any two scalar functions of the scalar variables 
#, y, 2 and if c be a scalar independent of the variables with 
regard to which the differentiations are performed. 

68.] If A represent any vector the formal combination 
A. Vis 

A.V = A l /- x+ A 2 /- + A s j-, (27) 

provided A = A l i + A^ j + A% k. 

This operator A V is a scalar differentiator. When applied 
to a scalar function V (x, y, z) it gives a scalar. 



<^ r -A+^+^- (28) 
Suppose for convenience that A is a unit vector a. 

(a.V)F=a 1 I + a 2 r +a8 r (29) 



148 VECTOR ANALYSIS 

where a v a^a B are the direction cosines of the line a referred 
to the axes Jf, F, Z. Consequently (a V) V appears as the 
well-known directional derivative of V in the direction a. 
This is often written 

3V 3V , 3F, 3V 

T^^+^-^sT- (29) 

It expresses the magnitude of the rate of increase of V in 
the direction a. In the particular case where this direction is 
the normal n to a surface of constant value of F, this relation 
becomes the normal derivative. 



if n x , n 2 , n 3 be the direction cosines of the normal. 

The operator a V applied to a scalar function of position 
V yields the same result as the direct product of a and the 

vector V V. 

(a.V)F=a.(VF). (30) 

For this reason either operation may be denoted simply by 

a- VF 

without parentheses and no ambiguity can result from the 
omission. The two different forms (a V) Fand a- (V F) 
may however be interpreted in an important theorem. 
(a V) F is the directional derivative of F in the direction 
a. On the other hand a ( V V) is the component of V F in 
the direction a. Hence : The directional derivative of F in 
any direction is equal to the component of the derivative 
VFin that direction. If Fdenote gravitational potential the 
theorem becomes : The directional derivative of the potential 
in any direction gives the component of the force per unit 
mass in that direction. In case Fbe electric or magnetic 
potential a difference of sign must be observed. 



THE DIFFERENTIAL CALCULUS OF VECTORS 149 

Vector Functions of Position in Space 
69.] A vector function of position in space is a function 

V (x, y, z) 

which associates with each point x, y, z in space a definite 
vector. The function may be broken up into its three com 
ponents 

V (x, y, z) = F! (x, y,z)i+ F 2 (x, y, z) j + F 3 (a?, y, z) k. 

Examples of vector functions are very numerous in physics. 
Already the function VF has occurred. At each point of 
space V F has in general a definite vector value. In mechan 
ics of rigid bodies the velocity of each point of the body is a 
vector function of the position of the point. Fluxes of heat, 
electricity, magnetic force, fluids, etc., are all vector functions 
of position in space. 

The scalar operator a V may be applied to a vector func 
tion V to yield another vector function. 

Let V = Fi (x, y,z) i + F 2 (as, y, z) j + F 3 (x, y, z) k 
and a = a 1 i + a 2 j + a 3 k. 

Then a - V = i^ + a 3^+-af 3 

(a.V)V = (a.V) F! i + (a.V) F 2 j + (a.V)F, k 

9V 3V 9V\ 

- 



( 
a 



9V, 9V 2 3V 2 



150 VECTOR ANALYSIS 

This may be written in the form 



Hence (a V) V is the directional derivative of the vector 
function V in the direction a. It is possible to write 

(a V) V = a - V V 

without parentheses. For the meaning of the vector symbol 
V when applied to a vector function V has not yet been 
defined. Hence from the present standpoint the expression 
a V V can have but the one interpretation given to it by 
(a V) V. 

70.] Although the operation V V has not been defined and 
cannot be at present, 1 two formal combinations of the vector 
operator V and a vector function V may be treated. These 
are the (formal) scalar product and the (formal) vector prod 
uct of V into V. They are 



T < 82 > 

and VxV = i + ]- +kxV . (88) 

V V is read del dot V; and V x V, del cross V. 


The differentiators , , , being scalar operators, pass 

by the dot and the cross. That is 





(32) 

Qy 3z 

(88) 



These may be expressed in terms of the components F", PI. V, 
ofV. 

i A definition of V V will be given in Chapter VII. , 



THE DIFFERENTIAL CALCULUS OF VECTORS 151 



Now 



Then 



dx 9x 



Sx 



3y dy 9y ~5y 

8V_9V } 5V, 3F 
3* 3* h 3 Jl ~3 



i.fl- 



3V 



(34) 



Hence V V = 

Moreover i > 



rf + ^7 + 3T- ( 32 )" 



This may be written in the form of a determinant 



VxV= 



i j k 

333 



(33)" 



152 VECTOR ANALYSIS 

It is to be understood that the operators^are to be applied to 
the functions V v F" 2 , F 3 when expanding the determinant. 

From some standpoints objections may be brought forward 
against treating V as a symbolic vector and introducing V V 
and V x V respectively as the symbolic scalar and vector 
products of V into V. These objections may be avoided by 
simply laying down the definition that the symbols V and 
V x, which may be looked upon as entirely new operators 
quite distinct from V, shall be 



and V xV = ix + jx-4-kx-. (33) 

But for practical purposes and for remembering formulae it 
seems by all means advisable to regard 

3 5 3 



as a symbolic vector differentiator. This symbol obeys the 
same laws as a vector just in so far as the differentiators 
333 
^ T~~ T~ ^ )e y ^ e same * aws ^ or( ^ nar y sca l ar quantities. 

71.] That the two functions V V and V x V have very 
important physical meanings in connection with the vector 
function V may be easily recognized. By the straight 
forward proof indicated in Art. 63 it was seen that the 
operator V is independent of the choice of axes. From this 
fact the inference is immediate that V V and V x V represent 
intrinsic properties of V invariant of choice of axes. In order 
to perceive these properties it is convenient to attribute to the 
function V some definite physical meaning such as flux or 
flow of a fluid substance. Let therefore the vector V denote 



THE DIFFERENTIAL CALCULUS OF VECTORS 153 



at each point of space the direction and the magnitude of the 
flow of some fluid. This may be a material fluid as water 
or gas, or a fictitious one as heat or electricity. To obtain as 
great clearness as possible let the fluid be material but not 
necessarily restricted to incompressibility like water. 



Then 



= i~+j. + k *I 

dx 3y dz 



is called the divergence of V and is often written 
V V= div V. 

The reason for this term is that VV gives at each point the 
rate per unit volume per unit time at which fluid is leaving 
that point the rate of diminution of density. To prove 
this consider a small cube of matter (Fig. 31). Let the edges 
of the cube be dx, dy, and dz respectively. Let 

V (x, y, z) = V l (x 9 y,z)i+ V^ (x y y, z) j + F 3 (x, y, z) k. 

Consider the amount of fluid which passes through those 

faces of the cube which are parallel to the F^-plane, i. e. 

perpendicular to the X 

axis. The normal to the 

face whose x coordinate is 

the lesser, that is, the nor 

mal to the left-hand face 

of the cube is i. The flux 

of substance through this 

face is 



xy2 



FIG. 31. 



-i.V (x,y,z) dy dz. 

The normal to the oppo- z 

site face, the face whose 

x coordinate is greater by the amount dx, is + i and the flux 

through it is therefore 



164 VECTOR ANALYSIS 

r 3v i 

i V (x + dx, y, z) dy dz = i V(#, y, z) + dx dy dz 

3V 
= i V (x y y, z) dy dz + i - dx dy dz. 

c) x 

The total flux outward from the cube through these two 
faces is therefore the algebraic sum of these quantities. This 

is simply 

3V , , . 3^ 
i -= dx dy dz = -^ - dx dy dz. 
9 x 9 x 

In like manner the fluxes through the other pairs of faces of 
the cube are 

3V,,, j,,c)V 
i ^ dx dy dz and k - dx dy dz. 

9 y 9 z 

The total flux out from the cube is therefore 

/. 3V t 3V t , 3V\ 

( i + j + k . ) dx dy dz. 

\ 9x dy 9zJ 

This is the net quantity of fluid which leaves the cube per 
unit time. The quotient of this by the volume dx dy dz of 
the cube gives the rate of diminution of density. This is 



V.T.I. + , . 

9x dy 9z 9x dy 9z 

Because V V thus represents the diminution of density 
or the rate at which matter is leaving a point per unit volume 
per unit time, it is called the divergence. Maxwell employed 
the term convergence to denote the rate at which fluid ap 
proaches a point per unit volume per unit time. This is the 
negative of the divergence. In case the fluid is incompressible, 
as much matter must leave the cube as enters it. The total 
change of contents must therefore be zero. For this reason 
the characteristic differential equation which any incompres 
sible fluid must satisfy is 



THE DIFFERENTIAL CALCULUS OF VECTORS 155 

where V is the flux of the fluid. This equation is often 
known as the hydrodynamic equation. It is satisfied by any 
flow of water, since water is practically incompressible. The 
great importance of the equation for work in electricity is due 
to the fact that according to Maxwell s hypothesis electric dis 
placement obeys the same laws as an incompressible fluid. If 
then D be the electric displacement, 

div D = V D = 0. 

72.] To the operator V X Maxwell gave the name curl. 
This nomenclature has become widely accepted. 

V x V = curl V. 

The curl of a vector function V is itself a vector function 
of position in space. As the name indicates, it is closely 
connected with the angular velocity or spin of the flux at 
each point. But the interpretation of the curl is neither so 
easily obtained nor so simple as that of the divergence. 

Consider as before that V represents the flux of a fluid. 
Take at a definite instant an infinitesimal sphere about any 
point (#, y, z). At the next instant what has become of the 
sphere ? In the first place it may have moved off as a whole 
in a certain direction by an amount d r. In other words it 
may have a translational velocity of dr/dt. In addition to 
this it may have undergone such a deformation that it is no 
longer a sphere. It may have been subjected to a strain by 
virtue of which it becomes slightly ellipsoidal in shape. 
Finally it may have been rotated as a whole about some 
axis through an angle dw. That is to say, it may have an 
angular velocity the magnitude of which is dw/dt. An 
infinitesimal sphere therefore may have any one of three 
distinct types of motion or all of them combined. First, a 
translation with definite velocity. Second, a strain with three 
definite rates of elongation along the axes of an ellipsoid. 



156 VECTOR ANALYSIS 

Third, an angular velocity about a definite axis. It is this 
third type of motion which is given by the curl. In fact, 
the curl of the flux V is a vector which has at each point of 
space the direction of the instantaneous axis of rotation at 
that point and a magnitude equal to twice the instantaneous 
angular velocity about that axis. 

The analytic discussion of the motion of a fluid presents 
more difficulties than it is necessary to introduce in treating 
the curl. The motion of a rigid body is sufficiently complex 
to give an adequate idea of the operation. It was seen (Art. 
51) that the velocity of the particles of a rigid body at any 
instant is given by the formula 

v = v + a x r. 

curl v = Vxv = Vxv + Vx(axr). 
Let a = a l i + a% j + a 3 k 

r = r 1 i + r 2 j + r 3 k=:;ri + 2/j-fzk 

expand V X (a x r) formally as if it were the vector triple 
product of V, a, and r. Then 

V x v = V x v + (V - r) a - (V a) r. 
v is a constant vector. Hence the term V x v vanishes. 

V . r = + ^ + = 3. 

3 x 3 y 3 z 

As a is a constant vector it may be placed upon the other side 
of the differential operator, V a = a V. 

/ 3 3 3\ 

a - Vr=( ai^ + a 2j-+ a s^ Jr = a 1 i 

Hence Vxv = 3a a = 2a. 

Therefore in the case of the motion of a rigid body the curl 
of the linear velocity at any point is equal to twice the 
angular velocity in magnitude and in direction. 



THE DIFFERENTIAL CALCULUS OF VECTORS 157 
V x v = curl v = 2 a, 
a = ^Vxv=| curl v. 
v = v + ^ (V x v) x r = v + \ (curl v) x r. (34) 

The expansion of V x (a x r) formally may be avoided by 
multiplying a x r out and then applying the operator V X to 
the result. 

73.] It frequently happens, as in the case of the applica 
tion just cited, that the operators V>V% V X, have to be 
applied to combinations of scalar functions, vector functions, 
or both. The following rules of operation will be found 
useful. Let u, v be scalar functions and u, v vector func 
tions of position in space. Then 

V(t6 + t?) = Vw + Vfl (35) 

V.(u + v) = V.u + V-v (36) 

Vx(u + v) = Vxu + Vxv (37) 

V (u v) = v V u + u V v (38) 

V (w v) = V M v + M V v (39) 

v (40) 

(41) 
+ v x (V x u) + u x (V x v) 1 

V.(uxv)=v.Vxu u-Vxv (42) 
Vx (uxv) = v.Vu~vV-u-u.Vv + uV.v. 1 (43) 

A word is necessary upon the matter of the interpretation 
of such expressions as 

V u v, V u v, V u x v. 

The rule followed in this book is that the operator V applies 
to the nearest term only. That is, 

1 By Art. 69 the expressions v V n an d n V v me to be interpreted as 
(V V) uand ( u * V) v - 



158 VECTOR ANALYSIS 

V uv = (V u) v 

V u v = ( V u) v 

V u x v = ( V u) x v. 

If V is to be applied to more than the one term which follows 
it, the terms to which it is applied are enclosed in a paren 
thesis as upon the left-hand side of the above equations. 

The proofs of the formulae may be given most naturally 
by expanding the expressions in terms of three assumed unit 
vectors i, j, k. The sign 2 of summation will be found con 
venient. By means of it the operators V> V* A x take the 
form 



The summation extends over #, y, z. 
To demonstrate Vx (wv) 



^ 9 x 

Hence Vx (wv) = Vwxv + ^Vxv. 

To demonstrate 

V (u v) = v V u + u V v + v x ( V x u) + n x ( V x v). 



THE DIFFERENTIAL CALCULUS OF VECTORS 159 



^ . 3 u ^ . 3 v 
V(u.v) = 2^-v+^^.- 

Now 



,3u 



^ 3 u . ^ . 9 u 

2 v ._ 1 = vx(V xu) + 2v.i- 

or IE v i = v x ( V x u) + v V u. 

** 9 x 

3 v 
In like manner T u ;r- * = u x ( V x v) + u V v. 

d x 



Hence V(uv)=vVu + 

+ v x (V x n) + u x (V x v). 

The other formulae are demonstrated in a similar manner. 

71] The notation l 

V(u-v) u (44) 

will be used to denote that in applying the operator V to the 
product (u v), the quantity u is to be regarded as constant. 
That is, the operation V is carried out only partially upon 
the product (u v). In general if V is to be carried out 
partially upon any number of functions which occur after 
it in a parenthesis, those functions which are constant for the 
differentiations are written after the parenthesis as subscripts. 

Let M = U i 



1 This idea and notation of a partial V so to speak may be avoided by means 
of the formula 41. But a certain amount of compactness and simplicity is 
lost thereby. The idea of V ( u v )u is surely no more complicated than u V v or 
v X (V X n). 



160 VECTOR ANALYSIS 

then n-v = M 1 1 + u^v 2 + u z 

and V (u V) 



3^0 
But 



and V(u.v) T = 

Hence V(u-v) rrr^j Vw x + 



But V(u-v) n = w 1 Vi? 1 + ^ a Vi? a + w 8 V? 8 (44) 

and V(uv) v = ^j V^ x + i; 2 V^ 2 + ^ 3 V^ 3 . 

Hence V (u- v) = V (u- v) u + V (u- v) v . (45) 

This formula corresponds to the following one in the nota 
tion of differentials 

d (u v) = d (u v) u + d (u v) T 
or d (u v) = u d v + d n v. 

The formulae (35)-(43) given above (Art. 73) may be 
written in the following manner, as is obvious from analogy 
with the corresponding formulae in differentials : 

V (u + v*) = V (u + v\ + V (u + v) 9 (35) 

V. (u + v) = V- (u + v) u + V- (u + v) v (36) 

V x (u + v) = V x (u + v) u + V x (u + v) y (37) 



THE DIFFERENTIAL CALCULUS OF VECTORS 161 

V (u v) = V (u v\ + V (u v\ (38) 

V- (u v) = V- (u v). + V- O v) r (39) 

V x (u v) = V x (i* v) a + V x (u v) v (40) 

V (u- v) = V (u. v) u + V (u. v) v (41) 

V (u x v) = V (u x v) a + V (u x v) y (42) 

V x (u x v) = V x (u x v) a + V x (u x v) v . (43) 

This notation is particularly useful in the case of the 
scalar product u^v and for this reason it was introduced. 
In almost all other cases it can be done away without loss of 
simplicity. Take for instance (43) . Expand V x (u x v) u 

formally. 

V x (u x v) u = (V v) u (V u) v, 

where it must be understood that u is constant for the differ 
entiations which occur in V. Then in the last term the 
factor u may be placed before the sign V. Hence 

V X (uX v) u = u V * v u- Vv. 
In like manner V x (u x v) v = v V u v V u. 
Hence Vx(uxv)=vVu v V u u V v + u V v. 

75.] There are a number of important relations in which 
the partial operation V (u v) u figures. 

u x (V x v) = V (u v) a - u V v, (46) 

or V(u-v) u = u. Vv + u x (V x v), (46) 

or u V v = V (u v) u + (V x v) x u. (46)" 

The proof of this relation may be given by expanding in 
terms of i, j, k. A method of remembering the result easily 
is as follows. Expand the product 

u x (V x v) 
ll 



162 VECTOR ANALYSIS 

formally as if V, u, v were all real vectors. Then 
ux(Vxv)=u.vV u V v. 

The second term is capable of interpretation as it stands. 
The first term, however, is not. The operator V has nothing 
upon which to operate. It therefore must be transposed so 
that it shall have u v as an operand. But u being outside 
of the parenthesis in u x (V x v) is constant for the differen 
tiations. Hence 

u v V = V (u v) u 

and u x (V x v) = V (u v) u u V v. (46) 

If u be a unit vector, say a, the formula 

a-Vv = V(av) a + (V x v) x a (47) 

expresses the fact that the directional derivative a V v of a 
vector function v in the direction a is equal to the derivative 
of the projection of the vector v in that direction plus the 
vector product of the curl of v into the direction a. 
Consider the values of v at two neighboring points. 

v (x, y, z) 
and v (x + dx, y + dy> z + dz) 

d v = v (x + dx, y + dy, z + dz) v (#, y, z). 
Let v = v{i + v 2 j + v 8 k 

dv = dv l i + dv%j + dv 3 k. 
But by (25) dv 1 = dr* 

d v% = dr 
dv% = dr 

Hence d v s= d r (V v l i + V v 2 j + V v z k). 
Hence d v = d r V v 

By (46)" d v = V (rfr v) dr + (V x v) x dr. (48) 



THE DIFFERENTIAL CALCULUS OF VECTORS 163 

Or if V denote the value of v at the point (#, y, z) and v the 
value at a neighboring point 

v = v + V (d r v) dr + (V x v) x dr. (49) 

This expression of v in terms of its value v at a given point, 
the dels, and the displacement d r is analogous to the expan 
sion of a scalar functor of one variable by Taylor s theorem, 

/<*>=/(*>+ .TOO ** 

The derivative of (r v) when v is constant is equal to v. 
That is V (r v) v = v. 

For V (r v) v = v V r - (V x r> x v, 



9 



v Vr = v l i + v%j + 8 k = v, 

V x r = 0. 
Hence V (r v) v = v. 

In like manner if instead of the finite vector r, an infinitesimal 
vector d r be substituted, the result still is 

V (d r v) v = v. 

V/fllO*^ 

By (49) v = V + V (d r v) dr + (V x v) x d r 
V (d r v) = V (d r v) d ; + V (d r vV 

Hence V (d r v) dr = V (d r v) v. 

Substituting : 

v = ^ v o + ^V(dr.v) + ^(Vxv)xdr. (50) 

This gives another form of (49) which is sometimes more 
convenient It is also slightly more symmetrical. 



164 VECTOR ANALYSIS 

* 76.] Consider a moving fluid. Let v (#, T/, 3, t) be the 
velocity of the fluid at the point (#, y, z) at the time t. Sur 
round a point (a; , y , z ) with a small sphere. 

dr dr = c 2 . 

At each point of this sphere the velocity is 
v = v + d r V v. 

In the increment of time B t the points of this sphere will have 
moved the distance 



The point at the center will have moved the distance 



The distance between the center and the points that were 
upon the sphere of radius d r at the commencement of the 
interval $ t has become at the end of that interval S t 



To find the locus of the extremity of dr r it is necessary to 
eliminate d r from the equations 



c 2 = d r d r. 

The first equation may be solved for d r by the method of 
Art. 47, page 90, and the solution substituted into the second. 
The result will show that the infinitesimal sphere 



has been transformed into an ellipsoid by the motion of the 
fluid during the time 8 1. 

A more definite account of the change that has taken place 
may be obtained by making use of equation (50) 



THE DIFFERENTIAL CALCULUS OF VECTORS 165 
v = iv + |v(rfr.v) + 2-(Vx v) xdr, 

v = v +i[V(dr.v)-v ] + |(Vxv)xdr; 

S 

or of the equation (49) 

v = v + V(dr-v) dr +(Vx v)xrfr, 
v = v + [V (dr v) dr + I (V x v) x d r]+ ^ (V x v) x d r. 

The first term v in these equations expresses the fact that 
the infinitesimal sphere is moving as a whole with an instan 
taneous velocity equal to V . This is the translational element 
of the motion. The last term 

^(Vxv)xdr = curl v x d r 

shows that the sphere is undergoing a rotation about an 
instantaneous axis in the direction of curl v and with an angu 
lar velocity equal in magnitude to one half the magnitude of 
curl v. The middle term 



or v(dr.v) dr - (Vx v) x dr 

expresses the fact that the sphere is undergoing a defor 
mation known as homogeneous strain by virtue of which it 
becomes ellipsoidal. For this term is equal to 



dx V^j + dy V# 2 + dz 

if Vj, v 2 , v s be respectively the components of v in the direc 
tions i, j, k. It is fairly obvious that at any given point 
(#o> 2A z o) ^ set of three mutually perpendicular axes i, j, k 
may be chosen such that at that point V^, V# 2 , V# 3 are re- 



166 VECTOR ANALYSIS 

spectively parallel to them. Then the expression above 
becomes simply 

dx *i i+dy ^i + dz s ^. 

dx y 9y 9^ 

The point whose coordinates referred to the center of the 
infinitesimal sphere are 

dx, dy, dz 

is therefore endowed with this velocity. In the time S t it 
will have moved to a new position 



The totality of the points upon the sphere 



goes over into the totality of points upon the ellipsoid of 
which the equation is 

dx 2 dy 2 dz* 



y 

The statements made before (Art. 72) concerning the three 
types of motion which an infinitesimal sphere of fluid may 
possess have therefore now been demonstrated. 

77.] The symbolic operator V may be applied several times 
in succession. This will correspond in a general way to 
forming derivatives of an order higher than the first. The 
expressions found by thus repeating V will all be independ 
ent of the axes because V itself is. There are six of these 
dels of the second order. 

Let V (#, y, z) be a scalar function of position in space. 
The derivative VF is a vector function and hence has a curl 
and a divergence. Therefore 

V-VF, VxVF 



THE DIFFERENTIAL CALCULUS OF VECTORS 167 

are the two derivatives of the second order which may be 

obtained from V. 

V-VF=div VF (51) 

V x VF=curl VF. (52) 

The second expression V x V V vanishes identically. That is, 
the derivative of any scalar function V possesses no curl. This 
may be seen by expanding V x V V in terms of i, j, k. All 
the terms cancel out. Later (Art. 83) it will be shown con 
versely that if a vector function W possesses no curl, i. e. if 

V x W = curl W = 0, then W = VF, 

W is the derivative of some scalar function F. 

The first expression V V F when expanded in terms of 
i, j, k becomes 



Symbolically, V V = ^ + -5 + -r 

y2 O <i/2 O /v 2 

The operator V V is therefore the well-known operator of 
Laplace. Laplace s Equation 



becomes in the notation here employed 

V-VF=0. (53) 

When applied to a scalar function F the operator V V yields 
a scalar function which is, moreover, the divergence of the 
derivative. 

Let T be the temperature in a body. Let c be the con 
ductivity, p the density, and k the specific heat. The 
flow f is 



168 VECTOR ANALYSIS 

The rate at which heat is leaving a point per unit volume per 
unit time is V f. The increment of temperature is 

rfr=-^-V.f dt. 

p K 

d - = -^.VT. 

at p K 

This is Fourier s equation for the rate of change of tempera 
ture. 

Let V be a vector function, and V v V v V z its three com 
ponents. The operator V V of Laplace may be applied to V. 

v.vv = v-vr 1 i + v-vr 2 j + v.vr 3 k (54) 

If a vector function V satisfies Laplace s Equation, each of 
its three scalar components does. Other dels of the second 
order may be obtained by considering the divergence and curl 
of V. The divergence V V has a derivative 

VV-V = VdivV. (55) 

The curl V X V has in turn a divergence and a curl, 

and V V x V, VxVxV. 

V - V x V = div curl V (56) 

and V x V x V = curl curl V. (57) 

Of these expressions V V x V vanishes identically. That is, 
the divergence of the curl of any vector is zero. This may be 
seen by expanding V V x V in terms of i, j, k. Later (Art. 
83) it will be shown conversely that if the divergence of a 
vector function W vanishes identically, i. e. if 

V W = div W = 0, then W = V x V = curl V, 
W is the curl of some vector function V. 



THE DIFFERENTIAL CALCULUS OF VECTORS 169 

If the expression V x (V x V) were expanded formally 
according to the law of the triple vector product, 

Vx(VxV) = V-VV-V.VV. 

The term V V V is meaningless until V be transposed to 
the beginning so that it operates upon V. 

VxVxV = VV.V-V-VV, (58) 

or curl curl V = V div V - V VV. (58) 

This formula is very important. It expresses the curl of the 
curl of a vector in terms of the derivative of the divergence 
and the operator of Laplace. Should the vector function V 
satisfy Laplace s Equation, 

V VV = and 
curl curl V = V div V. 

Should the divergence of V be zero, 

curl curl V = V VV. 
Should the curl of the curl of V vanish, 
V div V = V VV. 
To sum up. There are six of the dels of the second order. 

V-VT, VxVF, 

V-VV, V V V, V V x V, V x V x V. 
Of these, two vanish identically. 

VxVr=0, V-VxV = 0. 
A third may be expressed in terms of two others. 

VxVxV = VV.V-V.VV. (58) 

The operator V V is equivalent to the operator of Laplace. 



170 VECTOR ANALYSIS 

* 78.] The geometric interpretation of V Vw is interesting. 
It depends upon a geometric interpretation of the second 
derivative of a scalar function u of the one scalar variable x. 
Let u i be the value of u at the point x t . Let it be required 
to find the second derivative of u with respect to x at the 
point x . Let x l and x 2 be two points equidistant from # . 
That is, let 

Xn *"" XQ XQ ~~~ */ &t 



* a ^^ ni 

o 

Then - ^ 



is the ratio of the difference between the average of u at the 
points x l and # 2 and the value of u at x to the square of the 
distance of the points x v # 3 from x . That 



d*u 



. LIM. 



is easily proved by Taylor s theorem. 

Let u be a scalar function of position in space. Choose 
three mutually orthogonal lines i, j, k and evaluate the 
expressions 



Let o? 2 and a?! be two points on the line i at a distance a from 
x ; # 4 and # 3 , two points on j at the same distance a from 
s > #e and # 6 , two points on k at the same distance a from x . 



= u~ 

?^_. LIM ._2 : 



THE DIFFERENTIAL CALCULUS OF VECTORS 171 



Add: 



-LIM r 

~a=OL 



As V and V- are independent of the particular axes chosen, 
this expression may be evaluated for a different set of axes, 
then for still a different one, etc. By adding together all 
these results 

u \ + u % + * 6 n terms 

- 



a== a 

Let n become infinite and at the same time let the different 
sets of axes point in every direction issuing from # . The 
fraction 

u \ + U 2 + * ^ n terms 
6 n 

then approaches the average value of u upon the surface of a 
sphere of radius a surrounding the point x . Denote this 
by u a . 



= a 



V V u is equal to six times the limit approached by the ratio 
of the excess of u on the surface of a sphere above the value 
at the center to the square of the radius of the sphere. The 
same reasoning holds in case u is a vector function. 

If u be the temperature of a body V-V u (except for a 
constant factor which depends upon the material of the 



172 VECTOR ANALYSIS 

body) is equal to the rate of increase of temperature (Art. 
77). If VV^is positive the average temperature upon a 
small sphere is greater than the temperature at the center. 
The center of the sphere is growing warmer. In the case 
of a steady flow the temperature at the center must remain 
constant. Evidently therefore the condition for a steady 

flow is 

V V u = 0. 

That is, the temperature is a solution of Laplace s Equation. 

Maxwell gave the name concentration to V V u whether 
u be a scalar or vector function. Consequently V V u may 
be called the dispersion of the function u whether it be scalar 
or vector. The dispersion is proportional to the excess of 
the average value of the function on an infinitesimal surface 
above the value at the center. In case u is a vector function 
the average is a vector average. The additions in it are 
vector additions. 

SUMMARY OF CHAPTER III 

If a vector r is a function of a scalar t the derivative of 
r with respect to t is a vector quantity whose direction is 
that of the tangent to the curve described by the terminus 
of r and whose magnitude is equal to the rate of advance of 
that terminus along the curve per unit change of t. The 
derivatives of the components of a vector are the components 
of the derivatives. 

d n r d n r,. d r . d* r~ 

= i H j H k ( 2V 

dt* dt n dt* J dt* 

A combination of vectors or of vectors and scalars may be 
differentiated just as in ordinary scalar analysis except that 
the differentiations must be performed in situ. 



THE DIFFERENTIAL CALCULUS OF VECTORS 173 

(3) 

(4) 

or d (a b) = d a b + a d b, (3) 

d(axb) = daxb + axdb, (4) 

and so forth. The differential of a unit vector is perpendicu 
lar to that vector. 

The derivative of a vector r with respect to the arc s of 
the curve which the terminus of the vector describes is 
the unit tangent to the curves directed toward that part of the 
curve along which $ is supposed to increase. 

r."- <" 

The derivative of t with respect to the arc * is a vector whose 
direction is normal to the curve on the concave side and 
whose magnitude is equal to the curvature of the curve. 



The tortuosity of a curve in space is the derivative of the 
unit normal n to the osculating plane with respect to the 
arc s. 

^n_^_/rfr <? 2 r _1 _ \ 
~ ds~ ds\ds X ds* VCTC/ 

The magnitude of the tortuosity is 



r= 



rdr d*r cZ 3 r"| 
L^s ^T 2 rf^J 



174 VECTOR ANALYSIS 

If r denote the position of a moving particle, t the time, 
v the velocity, A the acceleration, 



*-*---* 

The acceleration may be broken up into two components of 
which one is parallel to the tangent and depends upon the 
rate of change of the scalar velocity v of the particle in its 
path, and of which the other is perpendicular to the tangent 
and depends upon the velocity of the particle and the curva 

ture of the path. 

A = s t + 2 C. (19) 

Applications to the hodograph, in particular motion in a 
circle, parabola, or under a central acceleration. Application 
to the proof of the theorem that the motion of a rigid body 
one point of which is fixed is an instantaneous rotation about 
an axis through the fixed point. 

Integration with respect to a scalar is merely the inverse 
of differentiation. Application to finding the paths due to 
given accelerations. 

The operator V applied to a scalar function of position in 
space gives a vector whose direction is that of most rapid 
increase of that function and whose magnitude is equal to 
the rate of that increase per unit change of position in that 
direction 



THE DIFFERENTIAL CALCULUS OF VECTORS 175 

The operator V is invariant of the axes i, j, k. It may be 
denned by the equation 

n, (24) 



or W-dT = dV. (25) 

Computation of the derivative V V by two methods depend 
ing upon equations (21) and (25) . Illustration of the oc 
currence of V in mathematical physics. 

V may be looked upon as a fictitious vector, a vector 
differentiator. It obeys the formal laws of vectors just in 
so far as the scalar differentiators of 51 5 x> "9 / d y, 9 1 3 z obey 
the formal laws of scalar quantities 

A - VF =^ + ^i^ l7 < 28 > 

If a be a unit vector a V V is the directional derivative of V 
in the direction a. 

a.VF = (a-V) F=a(VF). (30) 

If V is a vector function a VV is the directional derivative 
of that vector function in the direction a. 



- _ ,-J~ + k.^-, (32) 

3 x 9 y 3 z 

VxV=ix|^ + jx + kx , (33) 

3x 3y 3 z 



V.V= ^ J + ^ + ^ 8 , (32) 

3 x 3 y 3 z 






176 VECTOR ANALYSIS 

Proof that V V is the divergence of V and V x V, the curl 

of V. 

V V = div V, 

V X V = curl V. 

V O + t;) = V u + Vtf, (35) 

V (u + v) = V u + V v, (36) 

Vx(u + v)=Vxn + Vxv, (37) 

V (u v) = v V u + u V v, (38) 

V (u v) = V u v + u V v, (39) 

V x (u v) = V u x v + u V x v, (40) 

V(nv)=vVu + U Vv + vx ( V x n) 

+ n x (V x v), (41) 

V (n x v) = v Vxu-u-Vxv, (42) 

Vx (u x v) =v .Vu v V u u Vv + uV* v. (43) 

Introduction of the partial del, V (u v) u , in which the dif 
ferentiations are performed upon the hypothesis that u is 

constant. 

u x (V x v) = V (u v) u n V v. (46) 

If a be a unit vector the directional derivative 

a V v = V (a v) a + (V x v) x a. (47) 

The expansion of any vector function v in the neighborhood 
of a point (x# y# z ) at which it takes on the value of v is 

v = v + V (d r v) dr + (V x v) x dr, (49) 
or v = \ v + V (d r . v) + \ (V x v) x d r. (50) 

Application to hydrodynamics. 
The dels of the second order are six in number. 



THE DIFFERENTIAL CALCULUS OF VECTORS 177 
V x VF= curl VF= 0, (52) 

x) 2 F" wv &v 
V-VJ^vV^f^+^ + Vp (51) 

d x 2 d y 2 9 z 2 

V V is Laplace s operator. If VVF=0, V satisfies La 
place s Equation. The operator may be applied to a vector. 



VV. V = VdivV, (55) 

V V x V = div curl V = 0, (56) 

Vx VxV=curlcurlV = VV.V- V. VV. (58) 

The geometric interpretation of V V as giving the disper 
sion of a function. 

EXERCISES ON CHAPTER III 

1. Given a particle moving in a plane curve. Let the 
plane be the ij-plane. Obtain the formulae for the compo 
nents of the velocity parallel and perpendicular to the radius 
vector r. These are 

rp kxr, 

where is the angle the radius vector r makes with i, and k 
is the normal to the plane. 

2. Obtain the accelerations of the particle parallel and 
perpendicular to the radius vector. These are 



Express these formulae in the usual manner in terms of x 
and y. 

12 



178 VECTOR ANALYSIS 

3. Obtain the accelerations of a moving particle parallel 
and perpendicular to the tangent to the path and reduce the 
results to the usual form. 

4. If r, </>, be a system of polar coordinates in space, 
where r is the distance of a point from the origin, </> the 
meridianal angle, and 6 the polar angle ; obtain the expressions 
for the components of the velocity and acceleration along the 
radius vector, a meridian, and a parallel of latitude. Reduce 
these expressions to the ordinary form in terms of #, y, z. 

5. Show by the direct method suggested in Art. 63 that 
the operator V is independent of the axes. 

6. By the second method given for computing V find 
the derivative V of a triple product [a be] each term of which 
is a function of #, y, z in case 

a = (r r) r, b = (r a) e, c = r x t, 
where d, e, f are constant vectors. 

7. Compute V V F when Fis r 2 , r, -, or -r 

, r r* 

8. Compute V V V, VV V, and V x V x V when V is 
equal to r and when V is equal to -j> and show that in these 
cases the formula (58) holds. 

9. Expand V x V V and V V x V in terms of i, j, k and 
show that they vanish (Art. 77). 

10. Show by expanding in terms of i, j, k that 

Vx VxV=VV. V-V VV. 



11. Prove A.V(7-W) = VA.VW+ WA- VV, 

and 

(VxV) x W=Vx (Vx 



CHAPTER IV 

THE INTEGRAL CALCULUS OF VECTORS 

79.] Let W (#, y, z) be a vector function of position in 
space. Let C be any curve in space, and r the radius vector 
drawn from some fixed origin to the points of the curve. 
Divide the curve into infinitesimal elements dr. From the 
sum of the scalar product of these elements d r and the value 
of the function W at some point of the element 

thus 2 W d r. 

The limit of this sum when the elements dr become infinite 
in number, each approaching zero, is called the line integral of 
W along the curve C and is written 



.dr. 
and dT = i dx + j dy + k dz, 

r r 

I W dr = i [W+dx -\-W*dy +W%dz\. (1) 

t/ (7 t/ C7 

The definition of the line integral therefore coincides with 
the definition usually given. It is however necessary to 
specify in which direction the radius vector r is supposed to 
describe the curve during the integration. For the elements 
d r have opposite signs when the curve is described in oppo- 



180 VECTOR ANALYSIS 

site directions. If one method of description be denoted by 
C and the other by (7, 



/W d r = -- I W d r. 
-G J c 



In case the curve C is a closed curve bounding a portion of 
surface the curve will always be regarded as described in 
such a direction that the enclosed area appears positive 
(Art. 25). 

If f denote the force which may be supposed to vary from 
point to point along the curve (7, the work done by the force 
when its point of application is moved from the initial point 
r of the curve C to its final point r is the line integral 



ff . dr= f f dr. 

J c J r 



Theorem : The line integral of the derivative V F of a 
scalar function V(x,y, z) along any curve from the point 
r to the point r is equal to the difference between the values 
of the function F (#, y, z) at the point r and at the point r . 
That is, 

Vr.dr = F(r) - F(r ) = V(x,y,z) - V(xy*d. 

o 

By definition d r V F" = d V 

fdV= F(r) - F(r ) = Ffey,^) - V(xyz.). (2) 

Theorem : The line integral of the derivative V F" of a 
single valued scalar function of position V taken around a 
closed curve vanishes. 

The fact that the integral is taken around a closed curve 
is denoted by writing a circle at the foot of the integral sign. 
To show 

(3) 



THE INTEGRAL CALCULUJS OF VECTORS 181 
The initial point r and the final point r coincide. Hence 



Hence by (2) fvF.dr = 0. 

Jo 

Theorem : Conversely if the line integral of W about every 
closed curve vanishes, W is the derivative of some scalar 
function V (x, y, z) of position in space. 

Given 

J o 

To show W = V V. 

Let r be any fixed point in space and r a variable point. 
The line integral 

J 

di 



is independent of the path of integration C. For let any two 
paths C and C f be drawn between r and r. The curve which 
consists of the path C from r to r and the path C f from r 
to r is a closed curve. Hence by hypothesis 

/W*cZr+ fw.dr = 0, 
j / c 

/Wdr = / W*dr. 
-c J c 

Hence / W d r = / W dr. 

J c J c 

Hence the value of the integral is independent of the path 
of integration and depends only upon the final point r. 



182 VECTOR ANALYSIS 

The value of the integral is therefore a scalar function of 
the position of the point r whose coordinates are x, y, z. 







Let the integral be taken between two points infinitely near 

together. 

y,z). 



But by definition V V d r = d V. 

Hence W 



The theorem is therefore demonstrated. 

80.] Let f be the force which acts upon a unit mass near 
the surface of the earth under the influence of gravity. Let 
a system of axes i, j, k be chosen so that k is vertical. Then 



The work done by the force when its point of application 
moves from the position r to the position r is 



w 



= I f*dT = I # k d r = I gdz. 
J r J r J r 



Hence w = g (z z ) = g (z z). 

The force f is said to be derivable from a force-function V 
when there exists a scalar function of position V such that 
the force is equal at each point of the derivative VF. 
Evidently if V is one force-function, another may be obtained 
by adding to V any arbitrary constant. In the above ex 
ample the force-function is 

V=w = g(z Q -z). 
Or more simply V = g z. 

The force is f = VF=-0k. 



THE INTEGRAL CALCULUS OF VECTORS 183 

The necessary and sufficient condition that a force-function 
V (z, y, z) exist, is that the work done by the force when its 
point of application moves around a closed circuit be zero. 

The work done by the force is 

w = I f d r . 



If this integral vanishes when taken around every closed 
contour 



And conversely if f = V V 

the integral vanishes. The force-function and the work done 

differ only by a constant. 

V = w + const 

In case there is friction no force-function can exist. For the 
work done by friction when a particle is moved around in a 
closed circuit is never zero. 

The force of attraction exerted by a fixed mass M upon 
a unit mass is directed toward the fixed mass and is propor 
tional to the inverse square of the distance between the 

masses. 

M 

f = -c-r. 
r 6 

This is the law of universal gravitation as stated by Newton. 
It is easy to see that this force is derivable from a force- 
function V. Choose the origin of coordinates at the center 
of the attracting mass M. Then the work done is 

M 
? r d r. 



But r d r = r d r, 

r dr 



r r dr M 1) 

= -c$r I =-cM j --- } 
J r r 2 I r r 3 



184 VECTOR ANALYSIS 

By a proper choice of units the constant c may be made 
equal to unity. The force-function V may therefore be 
chosen as 



If there had been several attracting bodies 
the force-function would have been 



M < 



where r r r 2 , r 8 , are the distances of the attracted unit 
mass from the attracting masses M v M% y M B 

The law of the conservation of mechanical energy requires 
that the work done by the forces when a point is moved 
around a closed curve shall be zero. This is on the assump 
tion that none of the mechanical energy has been converted 
into other forms of energy during the motion. The law of 
conservation of energy therefore requires the forces to be 
derivable from a force-function. Conversely if a force- 
function exists the work done by the forces when a point is 
carried around a closed curve is zero and consequently there 
is no loss of energy. A mechanical system for which a force- 
function exists is called a conservative system. From the 
example just cited above it is clear that bodies moving under 
the law of universal gravitation form a conservative system 
at least so long as they do not collide. 

81.] Let W (x, y, z) be any vector function of position in 
space. Let S be any surface. Divide this surface into in 
finitesimal elements. These elements may be regarded as 
plane and may be represented by infinitesimal vectors of 
which the direction is at each point the direction of the 
normal to the surface at that point and of which the magni 
tude is equal to the magnitude of the area of the infinitesimal 



THE INTEGRAL CALCULUS OF VECTORS 185 

element. Let this infinitesimal vector which represents the 
element of surface in magnitude and direction be denoted by 
d a. Form the sum 



which is the sum of the scalar products of the value of W 
at each element of surface and the (vector) element of 
surface. The limit of this sum when the elements of sur 
face approach zero is called the surface integral of W over 
the surface $, and is written 

(4) 

The value of the integral is scalar. If W and da be ex 
pressed in terms of their three components parallel to i, j, k 



or d a = dy dz i -f dz dx j + dx dy k, 

(5) 



The surface integral therefore has been defined as is cus 
tomary in ordinary analysis. It is however necessary to 
determine with the greatest care which normal to the surface 
d a is. That is, which side of the surface (so to speak) the 
integral is taken over. For the normals upon the two sides 
are the negatives of each other. Hence the surface integrals 
taken over the two sides will differ in sign. In case the 
surface be looked upon as bounding a portion of space d a 
is always considered to be the exterior normal. 

If f denote the flux of any substance the surface integral 

f.rfa 

s 



186 VECTOR ANALYSIS 

gives the amount of that substance which is passing through 
the surface per unit time. It was seen before (Art. 71) that 
the rate at which matter was leaving a point per unit 
volume per unit time was V f . The total amount of mat 
ter which leaves a closed space bounded by a surface S per 
unit time is the ordinary triple integral 

(6) 

Hence the very important relation connecting a surface in 
tegral of a flux taken over a closed surface and the volume 
integral of the divergence of the flux taken over the space 
enclosed by the surface 

/// 

CO 

Written out in the notation of the ordinary calculus this 
becomes 

I I \Xdy dz + Ydzdx + Zdxdy~\ 



3Y, 



where X, F, Z are the three components of the flux f . The 
theorem is perhaps still more familiar when each of the three 
components is treated separately. 

(8) 

This is known as Gauss s Theorem. It states that the surface 
integral (taken over a closed surface) of the product of a 
function X and the cosine of the angle which the exterior 
normal to that surface makes with the X-axis is equal to 
the volume integral of the partial derivative of that function 



THE INTEGRAL CALCULUS OF VECTORS 187 

with respect to x taken throughout the volume enclosed by 
that surface. 

If the surface S be the surface bounding an infinitesimal 
sphere or cube 

ff f-da = V-f dv 

where d v is the volume of that sphere or cube. Hence 

V.f = ^ fff-da. (9) 

dv J J a 

This equation may be taken as a definition of the divergence 
V f . The divergence of a vector function f is equal to the 
limit approached by the surface integral of f taken over a sur 
face bounding an infinitesimal body divided by that volume 
when the volume approaches zero as its limit. That is 



V.f= , A -- f-da. (10) 

dvQ dvJJs 

From this definition which is evidently independent of the 
axes all the properties of the divergence may be deduced. In 
order to make use of this definition it is necessary to develop 
at least the elements of the integral calculus of vectors before 
the differentiating operators can be treated. This definition 
of V f consequently is interesting more from a theoretical 
than from a practical standpoint. 

82.] Theorem : The surface integral of the curl of a vector 
function is equal to the line integral of that vector function 
taken around the closed curve bounding that surface. 



f f 

J J 



V x W-da= w-dr. (11) 

8 J O 

This is the celebrated theorem of Stokes. On account of its 
great importance in all branches of mathematical physics a 
number of different proofs will be given. 



188 VECTOR ANALYSIS 

First Proof : Consider a small triangle 1 23 upon the surface 
S (Fig. 32). Let the value of W at the vertex 1 be W . 
Then by (50), Chap. III., the value at any neighboring point is 

W = ~{ W + V (W* 8 r) + (V x W) x 8 r j , 



where the symbol 8 r has been introduced for the sake of dis 
tinguishing it from d r which is to be used as the element of 
integration. The integral of W taken around the triangle 



FIG. 32. 



Cw-dr=l fw o -dr + g fv(W-Sr).<Zr 
+ 5 f (V x W) x Sr-dr. 

/ A 



The first term I fw o .dr = iw o . Cdr 
2 JA JA 

vanishes because the integral of d r around a closed figure, in 
this case a small triangle, is zero. The second term 

g fv(W-Sr).dr 

J A 

vanishes by virtue of (3) page 180. Hence 



THE INTEGRAL CALCULUS OF VECTORS 189 

Cw*di = l fvxWxSr-dr. 
JA J A 

Interchange the dot and the cross in this triple product. 

V xW-Sr x dr. 



=| J 



When dr is equal to the side 12 of the triangle, Sr is also 
equal to this side. Hence the product 

Sr x di 

vanishes because 8 r and d r are collinear. In like manner 
when dr is the side 31, 8r is the same side 13, but taken 
in the opposite direction. Hence the vector product vanishes. 
When dr is the side #5, Sr is a line drawn from the vertex 
1 at which W= W to this side S3. Hence the product 8 r x d r 
is twice the area of the triangle. This area, moreover, is the 
positive area 1 % 3. Hence 

|r x dr = 

where d a denotes the positive area of the triangular element 
of surface. For the infinitesimal triangle therefore the 

relation 

= V x W 



holds. 

Let the surface 8 be divided into elementary triangles. 
For convenience let the curve which bounds the surface 
be made up of the sides of these triangles. Perform the 
integration 

fw-dr 

J A 

around each of these triangles and add the results together. 



2/ 1 

a JA 



190 



VECTOR ANALYSIS 



The second member ] V x W d a 

3 

is the surface integral of the curl of W. 

2 V x W-rfa=JJv x W 

In adding together the line integrals which occur in the first 
member it is necessary to notice that all the sides of the ele 
mentary triangles except those which lie along the bounding 
curve of the surface are traced twice in opposite directions. 
Hence all the terms in the sum 



which arise from those sides of the triangles lying within the 
surface S cancel out, leaving in the sum only the terms 
which arise from those sides which make up the bounding 
curve of the surface. Hence the sum reduces to the line in 
tegral of W along the curve which bounds the surface S. 



= fw 

Jo 



Hence 



V x W d a = W d r. 



= f 
Jo 



FIG. 33. 



Second Proof : Let C be any closed 
contour drawn upon the surface S 
(Fig. 33). It will be assumed that C 
is continuous and does not cut itself. 
Let C r be another such contour near 
to C. Consider the variation S which 
takes place in the line integral of W 
in passing from the contour C to the 
contour C". 



THE INTEGRAL CALCULUS OF VECTORS 191 
/V.dr = f 

t/ t/ 

S fwdr = f 

But d(W- 

and 

Hence J*W &dT= Cw*dST=f d(W*Sr) -- CdW *ST. 

The expression d (W 8 r) is by its form a perfect differential. 
The value of the integral of that expression will therefore be 
the difference between the values of W d r at the end and at 
the beginning of the path of integration. In this case the 
integral is taken around the closed contour C. Hence 



/^ 
Jc 



Hence 

and S fw-rfr= fsw.dr- f 



9W J J 

But d W = -K d a? + -7T d y + 



PW 3W 3W 

or d W = -^ i d! r + ^ j d r + -^ k d r, 

& x d y d z 



and 

v x & y 



192 VECTOR ANALYSIS 

Substituting these values 



dT i.Br-~ -8r i-dr 

x ox 



+ similar terms in y and z. [ 
But by (25) page 111 



Hence sfwdr=/ j i x ^ Srxdr 

+ similar terms in y and z | . 

or 8 f W d r = f V x W 8 r x d r. 

In Fig. 33 it will be seen that d r is the element of arc 
along the curve C and 8 r is the distance from the curve C to 
the curve C r . Hence 8 r X d r is equal to the area of an ele 
mentary parallelogram included between C and C f upon the 
surface S. That is 



S fw-dr= fv x W da. 

Let the curve C starting at a point in expand until it 
coincides with the contour bounding S. The line integral 



will vary from the value at the point to the value 



/ 

t/O 



THE INTEGRAL CALCULUS OF VECTORS 193 

taken around the contour which bounds the surface S. This 
total variation of the integral will be equal to the sum of the 
variations 8 



Or f Wdr= ff Vx W-da. (11) 

83.] Stokes s theorem that the surface integral of the curl 
of a vector function is equal to the line integral of the func 
tion taken along the closed curve which bounds the surface 
has been proved. The converse is also true. If the surface 
integral of a vector function U is equal to the line integral of the 
function W taken around the curve bounding the surface and if 
this relation holds for all surfaces in space, then TT is the curl of 
W. That is 

if f fll. da = f Wdr, thenU=Vx W. (12) 

Form the surface integral df the difference between IT and 
V x W. 

// (tf~ Vx W)*da=f W*dr - f W-dr = 0, 
or f f (TI- V x W)-da = 0. 

Let the surface S over which the integration is performed be 
infinitesimal. The integral reduces to merely a single term 



(U_V x 

As this equation holds for any element of surface d a, the 
first factor vanishes. Hence 

IT- V x W = 0. 

Hence IT = V x W. 

The converse is therefore demonstrated. 

13 



194 VECTOR ANALYSIS 

A definition of V x W which is independent of the axes 
i, j, k may be obtained by applying Stokes s theorem to an in 
finitesimal plane area. Consider a point P. Pass a plane 
through P and draw in it, concentric with P, a small circle of 
area d a. 

Vx W.da=f W*dT. (13) 



When d a has the same direction as V X W the value of the 
line integral will be a maximum, for the cosine of the angle 
between V x W and d a will be equal to unity. For this 
value of da, 



=rfa IM F/V f W-rfrl (13) 
rfa=:0 Lda.dajo J 



Hence the curl V x W of a vector function W has at each 
point of space the direction of the normal to that plane in 
which the line integral of W taken about a small circle con 
centric with the point in question is a maximum. The mag 
nitude of the curl at the point is equal to the magnitude of 
that line integral of maximum value divided by the area of 
the circle about which it is taken. This definition like the 
one given in Art. 81 for the divergence is interesting more 
from theoretical than from practical considerations. 

Stokes s theorem or rather its converse may be used to de 
duce Maxwell s equations of the electro-magnetic field in a 
simple manner. Let E be the electric force, B the magnetic 
induction, H the magnetic force, and C the flux of electricity 
per unit area per unit time (i. e. the current density). 

It is a fact learned from experiment that the total electro 
motive force around a closed circuit is equal to the negative 
of the rate of change of total magnetic induction through 
the circuit. The total electromotive force is the line integral 
of the electric force taken around the circuit. That is 

Edr. 



THE INTEGRAL CALCULUS OF VECTORS 195 

The total magnetic induction through the circuit is the sur 
face integral of the magnetic induction B taken over a surface 
bounded by the circuit. That is 



B d*. 

i 

Experiment therefore shows that 



or /E-dr=/l B d a. 

J o J J a 

Hence by the converse of Stokes s theorem 

V x E = - B, curl E = - B. 

It is also a fact of experiment that the work done in carry 
ing a unit positive magnetic pole around a closed circuit is 
equal to 4?r times the total electric flux through the circuit. 
The work done in carrying a unit pole around a circuit is 
the line integral of H around the circuit. That is 



The total flux of electricity through the circuit is the 
surface integral of C taken over a surface bounded by the 
circuit. That is 

///* 

Experiment therefore teaches that 



= 47r C f 

J J s 



196 VECTOR ANALYSIS 

By the converse of Stokes s theorem 
V x H = 4 TT C. 

With a proper interpretation of the current C, as the dis 
placement current in addition to the conduction current, 
an interpretation depending upon one of Maxwell s primary 
hypotheses, this relation and the preceding one are the funda 
mental equations of Maxwell s theory, in the form used by 
Heaviside and Hertz. 

The theorems of Stokes and Gauss may be used to demon 
strate the identities. 

V V x W = 0, div curl W = 0. 
Vx VF=0, curl VF=0. 

According to Gauss s theorem 

VX Wdv= 
According to Stokes s theorem 

f fvxW-da = CW dr. 



Hence 



fffv-VxWdtf= Cw*dr. 



Apply this to an infinitesimal sphere. The surface bounding 
the sphere is closed. Hence its bounding curve reduces to a 
point ; and the integral around it, to zero. 

V-VxWdv = fw-dr = 0, 
J o 

V V x W = 0. 



THE INTEGRAL CALCULUS OF VECTORS 197 
Again according to Stokes s theorem 

ffvxvr.<2a = fvr-dr. 

Apply this to any infinitesimal portion of surface. The curve 
bounding this surface is closed. Hence the line integral of 
the derivative VF" vanishes. 



V x 

As this equation holds for any d a, it follows that 

Vx VF=0. 

In a similar manner the converse theorems may be 
demonstrated. If the divergence V TT of a vector function 
TJ is everywhere zero, then TT is the curl of some vector 

function W. 

TJ = V x W* 

If the curl V x II of a vector function TT is everywhere zero, 
then U is the derivative of some scalar function F", 



84.] By making use of the three fundamental relations 
between the line, surface, and volume integrals, and the 
dels / viz. : 



, (2) 

JYv x W-rfa= f W.rfr, (11) 



(7) 

it is possible to obtain a large number of formulae for the 
transformation of integrals. These formulae correspond to 



198 VECTOR ANALYSIS 

those connected with u integration by parts " in ordinary 
calculus. They are obtained by integrating both sides of the 
formulae, page 161, for differentiating. 

First V (u v) = u V v + v V u. 

C C C 

Jc ~Jc V ( T J G V 

Hence I % V v di = [uv] \ vV u* dx. (14) 

r 
The expression [u v] 

represents the difference between the value of (u v) at r, the 
end of the path, and the value at r , the beginning of the path. 
If the path be closed 

f^Vvdr = - C V u*dr. (14) 

Jo Jo 

Second V x (u v) = u V x v + V u x v. 

f* f* f* (* (* f* 

I I V x ( wv )*^ a= / / ^Vxvrfa+/ I Vwxv-da. 

J J S J J S J J 8 

Hence 

f* f* f* f* f* 

I I V^xvda=l uv dr I I wVxvda, (15) 
J J a Jo J J a 

&Vxvda= / uv dr I I V?txvrfa, (15) 
Jo J J a 

Third Vx (wV / y)^^VxV^ + V^xV 2 ; 
But V x V v = 

Hence V x (u V v) = V u x V v, 



or 



THE INTEGRAL CALCULUS OF VECTORS 199 
f* f* f* f* 

J J S J J 8 

Hence 

f* f* f* ,-y , P ^-J 7 S-4 />V 

IIVO AVt/ It I 1 \ / 

J J s Jo Jo 

Fourth V (u v) = u V v + V w v. 

/// r r r C C C * 

JJJV.(T)^=JJJVT^ + JJJ V-vd 

Hence 



^ v a 



or 



C C C ^7uv dv= I I Mvda rrr^V V^i;, (17) ; 
Fifth v(V^xv) = VXV^*v v^-VXv. 
V (V M x v) = V ^ V x v, 



Hence 



rfv^xvrfa = fffv^vxvdi;. (18) 



In all these formulae which contain a triple integral the 
surface $ is the closed surface bounding the body throughout 
which the integration is performed. 

Examples of integration by parts like those above can be 
multiplied almost without limit. Only one more will be 
given here. It is known as Greens Theorem and is perhaps 
the most important of all. If u and v are any two scalar 
functions of position, 



200 VECTOR ANALYSIS 

V (^ V fl) = V ^ V tf + 24 V V ^ 
V O V u) = V u V v + v V V u> 



J J J ^ u ^ vclv== J J J V- (uvv)dv C f Cu^ 



Hence 

/ / /V^-Vtfdfl=/ /^VvcU / r/^V-Vvdi?, 

= / / ^ V ^ d a f j I v^*V udv. (19) 
By subtracting these equalities the formula (20) 

/ / / (^ V V ^ v V V w ) ^ ^ = / / (^ V t> v V ^) ^ a. 

is obtained. By expanding the expression in terms of i, j, k 
the ordinary form of Green s theorem may be obtained. A 
further generalization due to Thomson (Lord Kelvin) is the 
following : 



/ / lw^/u*Vvdv=l I uwVv*d& I I I u\ 

= / I vwVU"d* I I I v\? [w^ u^ dv, (21) 



where w is a third scalar function of position. 

The element of volume dv has nothing to do with the scalar 
function v in these equations or in those that go before. The 
use of v in these two different senses can hardly give rise to 
any misunderstanding. 

* 85.] In the preceding articles the scalar and vector func 
tions which have been subject to treatment have been sup- 



THE INTEGRAL CALCULUS OF VECTORS 201 

posed to be continuous, single-valued, possessing derivatives 
of the first two orders at every point of space under consider 
ation. When the functions are discontinuous or multiple- 
valued, or fail to possess derivatives of the first two orders 
in certain regions of space, some caution must be exercised in 
applying the results obtained. 
Suppose for instance 



VF- 
The line integral 



y dx 



Introducing polar coordinates 

x = r cos 6, 

y = r sin 0, 



7 V d r = I d 0. 

Form the line integral from the point ( + 1,0) to the point 
(1, 0) along two different paths. Let one path be a semi 
circle lying above the JT-axis ; and the other, a semicircle 
lying below that axis. The value of the integral along the 
first path is 



/-* 
along the second path, I d 6 TT. 

From this it appears that the integral does not depend merely 
upon the limits of integration, but upon the path chosen, 



202 VECTOR ANALYSIS 

the value along one path being the negative of the value 
along the other. The integral around the circle which is a 
closed curve does not vanish, but is equal to 2 TT. 

It might seem therefore the results of Art. 79 were false 
and that consequently the entire bottom of the work which 
follows fell out. This however is not so. The difficulty is 
that the function 



1 V 

F=tan ^- 

x 



is not single-valued. At the point (1,1), for instance, the 
function V takes on not only the value 

-i TT 

F= tan l = -r> 
4 

but a whole series of values 

7T 
-+&7T, 

where k is any positive or negative integer. Furthermore at 
the origin, which was included between the two semicircular 
paths of integration, the function V becomes wholly inde 
terminate and fails to possess a derivative. It will be seen 
therefore that the origin is a peculiar or singular point of the 
function V. If the two paths of integration from (+ 1, 0) to 
(1,0) had not included the origin the values of the integral 
would not have differed. In other words the value of the 
integral around a closed curve which does not include the 
origin vanishes as it should. 

Inasmuch as the origin appears to be the point which 
vitiates the results obtained, let it be considered as marked 
by an impassable barrier. Any closed curve which does 
not contain the origin may be shrunk up or expanded at will ; 
but a closed curve which surrounds the origin cannot be 
so distorted as no longer to enclose that point without break 
ing its continuity. The curve C not surrounding the origin 



THE INTEGRAL CALCULUS OF VECTORS 203 

may shrink up to nothing without a break in its continuity ; 
but C can only shrink down and fit closer and closer about 
the origin. It cannot be shrunk down to nothing. It must 
always remain encircling the origin. The curve C is said to 
be reducible ; (7, irreducible. In case of the function F, then, 
it is true that the integral taken around any reducible circuit 
C vanishes; but the integral around any irreducible circuit C 
does not vanish. 

Suppose next that V is any function whatsoever. Let all 
the points at which V fails to be continuous or to have con 
tinuous first partial derivatives be marked as impassable 
barriers. Then any circuit which contains within it no 
such point may be shrunk up to nothing and is said to be 
reducible; but a circuit which contains one or more such 
points cannot be so shrunk up without breaking its continuity 
and it is said to be irreducible. The theorem may then be 
stated: The line integral of the derivative VF" of any function 
V vanishes around any reducible circuit C. It may or may not 
vanish around an irreducible circuit In case one irreducible 
circuit C may be distorted so as to coincide with another 
irreducible circuit C without passing through any of the 
singular points of V and without breaking its continuity, 
the two circuits are said to be reconcilable and the values of 
the line integral of V F about them are the same. 

A region such that any closed curve C within it may be 
shrunk up to nothing without passing through any singular 
point of V and without breaking its continuity, that is, a 
region every closed curve in which is reducible*, is said to be 
acyclic. All other regions are cyclic. 

By means of a simple device any cyclic region may be ren 
dered acyclic. Consider, for instance, the region (Fig. 34) en 
closed between the surface of a cylinder and the surface of a 
cube which contains the cylinder and whose bases coincide 
with those of the cylinder. Such a region is realized in a room 



204 VECTOR ANALYSIS 

in which a column reaches from the floor to the ceiling. It 
is evident that this region is cyclic. A circuit which passes 
around the column is irreducible. It cannot be contracted to 
nothing without breaking its continuity. If 
~^x / now a diaphragm be inserted reaching from 
the surface of the cylinder or column to the 
surface of the cube the region thus formed 
bounded by the surface of the cylinder, the 
surface of the cube, and the two sides of the 
diaphragm is acyclic. Owing to the inser 
tion of the diaphragm it is no longer possible 
to draw a circuit which shall pass completely around the cyl 
inder the diaphragm prevents it. Hence every closed cir 
cuit which may be drawn in the region is reducible and the 
region is acyclic. 

In like manner any region may be rendered acyclic by 
inserting a sufficient number of diaphragms. The bounding 
surfaces of the new region consist of the bounding surfaces of 
the given cyclic region and the two faces of each diaphragm. 

In acyclic regions or regions rendered acyclic by the fore 
going device all the results contained in Arts. 79 et seq. 
hold true. For cyclic regions they may or may not hold 
true. To enter further into these questions at this point is 
unnecessary. Indeed, even as much discussion as has been 
given them already may be superfluous. For they are ques 
tions which do not concern vector methods any more than the 
corresponding Cartesian ones. They belong properly to the 
subject of integration itself, rather than to the particular 
notation which may be employed in connection with it and 
which is the primary object of exposition here. In this 
respect these questions are similar to questions of rigor. 



THE INTEGRAL CALCULUS OF VECTORS 205 

The Integrating Operators. The Potential 

86.] Hitherto there have been considered line, surface, 
and volume integrals of functions both scalar and vector. 
There exist, however, certain special volume integrals which, 
owing to their intimate connection with the differentiating 
operators V, V, Vx, and owing to their especially frequent 
occurrence and great importance in physics, merit especial 
consideration. Suppose that 

^0** Vv *a) 
is a scalar function of the position in space of the point 



For the sake of definiteness V may be regarded as the 
density of matter at the point (# 2 , y v 2 2 ). In a homogeneous 
body V is constant. In those portions of space in which no 
matter exists V is identically zero. In non-homogeneous dis 
tributions of matter V varies from point to point; but at 
each point it has a definite value. 

The vector 

r 2 = z 2 i + y 2 j + * 2 k, 

drawn from any assumed origin, may be used to designate 
the point (# 2 , y 2 , z 2 ). Let 

On yi. *i) 

be any other fixed point of space, represented by the vector 



drawn from the same origin. Then 

r 2 - r x = O 2 - !>! + (y 2 - yi ) j + (z 2 - *j) k 

is the vector drawn from the point (x v y v Zj) to the point 
(#2> IJy 2 2)- A S ^ s vec ^or occurs a large number of times 
in the sections immediately following, it will be denoted by 

r i2 = r 2 ~~ r i- 



206 VECTOR ANALYSIS 

The length of r 12 is then r 12 and will be assumed to be 
positive. 



-i2 = V r 12 r 12 = V (* 2 - x^ + (y 2 - ^) 2 + 2 - ^) 2 . 
Consider the triple integral 



The integration is performed with respect to the variables 
^2> ^2> ^ 2 that is, with respect to the body of which V 
represents the density (Fig. 35). During 
the integration the point (x v y v z^ re 
mains fixed. The integral / has a definite 
value at each definite point (x v y v zj. 

It is a function of that point. The in- 
FIG. 3o. . 

terpretation of this integral / is easy, if 

the function V be regarded as the density of matter in space. 
The element of mass dm at (# 2 , y 2 , z 2 ) is 

dm V (# 2 , y 2 , 2! 2 ) dx^ dy z dz% = Vdv. 

The integral / is therefore the sum of the elements of mass 
in a body, each divided by its distance from a fixed point 



r 
J 



dm 



This is what is termed the potential at the point (x v y v 
due to the body whose density is 



The limits of integration in the integral / may be looked at 
in either of two ways. In the first place they may be 
regarded as coincident with the limits of the body of which 
V is the density. This indeed might seem the most natural 
set of limits. On the other hand the integral / may be 



THE INTEGRAL CALCULUS OF VECTORS 207 

regarded as taken over all space. The value of the integral 
is the same in both cases. For when the limits are infinite 
the function V vanishes identically at every point (# 2 , y 2 , 2 2 ) 
situated outside of the body and hence does not augment 
the value of the integral at all. It is found most convenient 
to consider the limits as infinite and the integral as extended 
over all space. This saves the trouble of writing in special 
limits for each particular case. The function Vot itself then 
practically determines the limits owing to its vanishing iden 
tically at all points unoccupied by matter. 

87.] The operation of finding the potential is of such 
frequent occurrence that a special symbol, Pot, is used for it. 

Pot r=fff V ^ y " * 2> rf* 2 dy^ dz y (22) 

The symbol is read "the potential of V." The potential, 
Pot V, is a function not of the variables #. 2 , y v z 2 with 
regard to which the integration is performed but of the point 
(x v y^ Zj) which is fixed during the integration. These 
variables enter in the expression for r 12 . The function V 
and Pot V therefore have different sets of variables. 

It may be necessary to note that although V has hitherto 
been regarded as the density of matter in space, such an 
interpretation for V is entirely too restricted for convenience. 
Whenever it becomes necessary to form the integral 



i " < 22 > 



of any scalar function V, no matter what V represents, that 
integral is called the potential of V. The reason for calling 
such an integral the potential even in cases in which it has 
np connection with physical potential is that it is formed 
according to the same formal law as the true potential and 



208 VECTOR ANALYSIS 

by virtue of that formation has certain simple rules of opera 
tion which other types of integrals do not possess. 

Pursuant to this idea the potential of a vector function 

W O 2 , y 2 , z 2 ) 
may be written down. 

Pot W = W ( * 2 y * * 2) dx, rfy. rf, r (23) 






In this case the integral is the sum of vector quantities 
and is consequently itself a vector. Thus the potential of a 
vector function W is a vector function, just as the potential 
of a scalar function V was seen to be a scalar function of posi 
tion in space. If W be resolved into its three components 

W O 2 , 2/ 2 , z 2 ) = i X O 2 , y v z 2 ) + j T <> 2 , y v z 2 ) 

+ kZ <> 2 , y v z 2 ) 

Pot W = i Pot X + j Pot Y+ k Pot Z. (24) 

The potential of a vector function W is equal to the vector 
sum of the potentials of its three components X, Y, Z. 

The potential of a scalar function V exists at a point 
(x v y v z p ) when and only when the integral 



taken over all space converges to a definite value. If, 
for instance, V were everywhere constant in space the in 
tegral would become greater and greater without limit as 
the limits of integration were extended farther and farther 
out into space. Evidently therefore if Jhe potential is to exist 
F must approach zero as its limit as the point (# 2 , y v 3 2 ) 
recedes indefinitely. A few important sufficient conditions 
for the convergence of the potential may be obtained by 
transforming to polar coordinates. Let 



THE INTEGRAL CALCULUS OF VECTORS 209 

x = r sin 6 cos fa 
yr sin 6 sin fa 

z = r cos 0, 
dv = r 2 sm0 dr dO d<f>. 

Let the point (x v y v ^) which is fixed for the integration 
be chosen at the origin. Then 

r i2 = r 
and the integral becomes 



or simply PotF= CCCVrsmff dr d0 dfa 

If the function V decrease so rapidly that the product 

Vr* 

remains finite as r increases indefinitely, then the integral con 
verges as far as the distant regions of space are concerned. 
For let 



r = 00 



dr d0d<f> 



r = 00 



dr d0 d<f> 



= QO 



Hence the triple integral taken over all space outside of a 
sphere of radius R (where R is supposed to be a large quan 
tity) is less than %TT* K jR, and consequently converges as far 
as regions distant from the origin are concerned. 

14 



210 VECTOR ANALYSIS 

If the function V remain finite or if it become infinite so 

weakly that the product 

Vr 

remains finite when r approaches zero, then the integral converges 
as far as regions near to the origin are concerned. For let 

Vr<K 
f CCrrsmddr d0 d<f> < C C fadr d0 d<f>. 



r = 

C C C 



dO d<t> = 



Hence the triple integral taken over all space inside a sphere 
of radius R (where R is now supposed to be a small quantity) 
is less than 2 Tr 2 K R and consequently converges as far as 
regions near to the origin which is the point (x v y v Zj) are 
concerned. 

If at any point (x 2 , y 2 , z 2 ) not coincident with the origin, 
i. e. the point (x x , y v z x ), the function V becomes infinite so 
weakly that the product of the value 0/V at a point near to 
( X 2> J2> Z 2) ty the square of the distance of that point from 
(x 2 , y 2 , z 2 ) remains finite as that distance approaches zero, then 
the integral converges as far as regions near to the point (x 2 , y 2 , z 2 ) 
are concerned. The proof of this statement is like those given 
before. These three conditions for the convergence of the 
integral Pot V are sufficient. They are by no means neces 
sary. The integral may converge when they do not hold. 
It is however indispensable to know whether or not an integral 
under discussion converges. Unless the tests given above 
show the convergence, more stringent ones must be resorted 
to. Such, however, will not be discussed here. They belong 
to the theory of integration in general rather than to the 



THE INTEGRAL CALCULUS OF VECTORS 



211 



theory of the integrating operator Pot. The discussion of 
the convergence of the potential of a vector function W re 
duces at once to that of its three components which are scalar 
functions and may be treated as above. 

88.] The potential is a function of the variables x v y v z l 
which are constant with respect to the integration. Let the 
value of the potential at the point (x v y v z^ be denoted by 



The first partial derivative of the potential with respect to x l 
is therefore 

LIM ^[ 



The value of this limit may be determined by a simple 
device (Fig. 36). Consider 
the potential at the point 



due to a certain body T. This 
is the same as the potential at 
the point 



FlG - 36 - 



due to the same body T displaced in the negative direction by 
the amount A x r For in finding the potential at a point P 
due to a body T the absolute positions in space of the body 
T and the point P are immaterial. It is only their positions 
relative to each other which determines the value of the poten 
tial. If both body and point be translated by the same 
amount in the same direction the value of the potential is un 
changed. But now if T be displaced in the negative direction 
by the amount A#, the value of Fat each point of space is 
changed from 

v C*2> y* **) to v 0*2 

where A# 2 = A x r 



212 VECTOR ANALYSIS 

Hence 

[Pot V(x v y v z^ + AX,, yt , *, = [Pot F<> 2 + A a; 2 ,y 2 , 



Hence LlM j [ Pot HX. + A ..,,, . t - [Pot 

A #! = / 



It will be found convenient to introduce the limits of 
integration. Let the portion of space originally filled by the 
body T be denoted by M ; and let the portion filled by the 
body after its translation in the negative direction through 
the distance A x l be denoted by M . The regions M and M 1 
overlap. Let the region common to both be M ; and let the 
remainder of M be m; the remainder of M 1 , m 1 . Then 



Pot V (a, + A * y r * 2 ) f ^ 






d rrr 

J J J m 



^ 



Pot 



/// "\F ( W (II * *\ /*/*/* 1^ f V tl 9 ^ 

I I I r I O/ft t/n ^O/, ill *V* / O1 V91 " <) J 1 

= / / / I * y rft>,+ / / / ^_Mrf r2 . 

J J J M ^j 2 J J Jm T YL 

Hence (25) becomes, when A ^j is replaced *by its equal Aic 2 , 

t As all the following potentials are for the point ar lf yi, i the bracket and 
indices have been dropped. 



THE INTEGRAL CALCULUS OF VECTORS 213 



+ 

Or, 



my(* 






C f r 
J J J 







A 2 



, 

" 



A a; 2 ==0 



LIM 



^, 



r 12 



v yy g 2 



= rrr 
jJJ j 



LIM ( 



r 12 



^ 



9 X 






when A ^! approaches zero as its limit the regions mand m ; , 
which are at no point thicker than A #, approach zero ; M 
and Jf both approach -Jf as a limit. 



t There are cases in which this reversal of the order in which the two limits 
are taken gives incorrect results. This is a question of double limits and leads to 
the mazes of modern mathematical rigor. 

J If the derivative of Fis to exist at the surface bounding T the values of the 
function V must diminish continuously to zero upon the surface. If Fchanged 
suddenly from a finite value within the surface to a zero value outside the de 
rivative QVlS^i would not exist and the triple integral would be meaningless. 
For the same reason V is supposed to be finite and continuous at every point 
within the region T. 



214 VECTOR ANALYSIS 

Then if it be assumed that the region T is finite and that V 
vanishes upon the surface bounding T 



T/nvr rrr V(Y <>/ z \ 

i <t\ jvi I I . I V \ ^o* fo ^9y 7 /\ 

A^ojjj OT riaAa;2 Ji = 

Consequently the expression for the derivative of the poten 
tial reduces to merely 

3 Pot F r r r i 3 F 3 F 

= 1/1 dv* = Pot 

d x l J J J M r 12 3# 2 3^2 

^%^ partial derivative of the potential of a scalar function V 
is equal to the potential of the partial derivative of V. 

The derivative V of the potential ofVis equal to the potential 
of the derivative V V. 

VPotF=PotVF (27) 

This statement follows immediately from the former. As 
the V upon the left-hand side applies to the set of vari 
ables x v y^ Zj, it may be written V r In like manner the 
V upon the right-hand side may be written V 2 to call atten 
tion to the fact that it applies to the variables # 2 , y 2 , z 2 of F. 

Then V a Pot F= Pot V 2 F (27) 

To demonstrate this identity V may be expanded in terms of 

.3 PotF .3 PotF . 3 PotF 

I i I lr 

* ^ J * T- * ^ 



SV 
+jPot -l- 

3 



THE INTEGRAL CALCULUS OF VECTORS 215 

As i, j, k are constant vectors they may be placed under 
the sign of integration and the terms may be collected. Then 
by means of (26) 



The curl V X and divergence V of the potential of a vector 
function W are equal respectively to the potential of the curl and 
divergence of that function. 

V, x Pot W = Pot V 2 x W, 

(28) 
or curl Pot W = Pot curl W 

and Vj Pot W = Pot V 2 W, 

or div Pot W = Pot div W. 

These relations may be proved in a manner analogous to the 
above. It is even possible to go further and form the dels 
of higher order 

v v Pot r= Pot v VF; (30) 

Uf>iac<3* 

V- V Pot W = Pot V V W, (31) 

V V Pot W = Pot VV W, (32) 

V x V x Pot W= Pot V x V x W. (33) 

The dels upon the left might have a subscript 1 attached to 
show that the differentiations are performed with respect to 
the variables x v y v z v and for a similar reason the dels upon 
the right might have been written with a subscript 2. The 
results of this article may be summed up as follows: 

Theorem: The differentiating operator V and the integrating 
operator Pot are commutative. 

*89.] In the foregoing work it has been assumed that the 
region T was finite and that the function Fwas everywhere 
finite and continuous inside of the region T and moreover 
decreased so as to approach zero continuously at the surface 
bounding that region. These restrictions are inconvenient 



216 VECTOR ANALYSIS 

and may be removed by making use of a surface integral. 
The derivative of the potential was obtained (page 213) in 
essentially the form 



otr_ r r r I SV 
x l J J J ,f r 12 2x 2 



-LJ-LJjl J. * \^o i * V 2 ^2 ** 2^ 7 

a V 2 

12 

LIM 1 rrr T r (g a> y ff g a ) fgr 

r i2 

Let d a be a directed element of the surface $ bounding the 
region J!f. The element of volume dv z in the region m r is 
therefore equal to 

dt? 2 =A# 2 i da. 

Hence - I I I ,^2^-^2^2112) 



L f f f 

2 J J Jm- 

= T f V 

J J 



r !2 

The element of volume d v 2 ^ n *^ e re gi n m ^ s equal to 
di, 2 = -Aa; 2 i.da. 



Hence * /Tf 

J\X^J J Jm 



> 



Consequently 



i-da. (34) 

^ r !2 



THE INTEGRAL CALCULUS OF VECTORS 217 

The volume integral is taken throughout the region M with 
the understanding that the value of the derivative of V at 
the surface S shall be equal to the limit of the value of that 
derivative when the surface is approached from the interior 
of M. This convention avoids the difficulty that arises in 
connection with the existence of the derivative at the surface 
S where V becomes discontinuous. The surface integral is 
taken over the surface S which bounds the region. 

Suppose that the region M becomes infinite. By virtue of 
the conditions imposed upon V to insure the convergence of 

the potential 

Vr* < K. 

Let the bounding surface S be a sphere of radius , a quan 
tity which is large. 

i d a < R 2 d 6 d<f>. 

<//*-. 

s 

The surface integral becomes smaller and smaller and ap 
proaches zero as its limit when the region M becomes infinite. 
Moreover the volume integral 

JLJT^ 

remains finite as M becomes infinite. Consequently provided 
V is such a function that Pot V exists as far as the infinite 
regions of space are concerned, then the equation 



= 



holds as far as those regions of space are concerned. 

Suppose that V ceases to be continuous or becomes infinite 
at a single point (x^ y v z^) within the region T. Surround 



218 VECTOR ANALYSIS 

this point with a small sphere of radius R. Let S denote the 
surface of this sphere and M all the region T not included 
within the sphere. Then 

r r r i 9V r r v 

=JJj^^ dv * + JJ*-^ 1 

By the conditions imposed upon V 

Vr<K 
V . 



//>" <//.* 



d6 d^ 



Consequently when the sphere of radius R becomes smaller 
and smaller the surface integral may or may not become zero. 
Moreover the volume integral 

1 3V . 



may or may not approach a limit when E becomes smaller 
and smaller. Hence the equation 

SPotF SV 



has not always a definite meaning at a point of the region 
T at which V becomes infinite in such a manner that the 
product Vr remains finite. 

If, however, V remains finite at the point in question so 
that the product Vr approaches zero, the constant K is zero 
and the surface integral becomes smaller and smaller as R 
approaches zero. Moreover the volume integral 



THE INTEGRAL CALCULUS OF VECTORS 219 

approaches a definite limit as R becomes infinitesimal. Con 
sequently the equation 

5 Pot V _ p dV 

7\ A Ob 



holds in the neighborhood of all isolated pointe at which V 
remains finite even though it be discontinuous. 

Suppose that V becomes infinite at some single point 
(iC 2 , y 2 , 2 2 ) not coincident with (x^ y v z^). According to the 
conditions laid upon V 

VI* < K, 

where I is the distance of the point (z 2 , y 2 , z 2 ) from a point 
near to it. Then the surface integral 

V . 

r !2 

need not become zero and consequently the equation 
5PotF SV 

= Pot TT 

need not hold for any point (a? r y v z^) of the region. But 
if V becomes infinite at # 2 , y 2 , z 2 in such a manner that 

VI <K, 

then the surface integral will approach zero as its limit and 
the equation will hold. 

Finally suppose the function V remains finite upon the 
surface S bounding the region jT, but does not vanish there. 
In this case there exists a surface of discontinuities of V. 
Within this surface V is finite ; without, it is zero. The 
surface integral 

F. 



220 VECTOR ANALYSTS 

does not vanish in general. Hence the equation 

SPotF 9V 

= --- = Pot -^r 

dX 1 v%i 

cannot hold. 

Similar reasoning may be applied to each of the three 
partial derivatives with respect to x v y v z r By combining 
the results it is seen that in general 

Vj PotF= Pot V 2 F+ f f Z da. (35) 






Let F be any function in space, and let it be granted that 
Pot F exists. Surround each point of space at which V 
ceases to be finite by a small sphere. Let the surface of the 
sphere be denoted by S. Draw in space all those surfaces 
which are surfaces of discontinuity of V. Let these sur 
faces also be denoted by S. Then the formula (35) holds 
where the surface integral is taken over all the surfaces 
which have been designated by S. If the integral taken 
over all these surfaces vanishes when the radii of the spheres 
above mentioned become infinitesimal, then 



^ (27) 

This formula 

V 1 PotF=PotV 2 F. 

will surely hold at a point (x x , y v Zj) if V remains always 
finite or becomes infinite at a point (x 2 , y 2 , z 2 ) so that the 
product V 1 remains finite, and if V possesses no surfaces of 
discontinuity, and if furthermore the product V r 3 remains finite 
as r becomes infinite. 1 In other cases special tests must be 
applied to ascertain whether the formula (27) can be used 
or the more complicated one (35) must be resorted to. 

1 For extensions and modifications of this theorem, see exercises. 



THE INTEGRAL CALCULUS OF VECTORS 221 

The relation (27) is so simple and so amenable to trans 
formation that V will in general be assumed to be such a 
function that (27) holds. In cases in which V possesses a 
surface S of discontinuity it is frequently found convenient 
to consider V as replaced by another function V which has 
in general the same values as Fbut which instead of possess 
ing a discontinuity at S merely changes very rapidly from 
one value to another as the point (# 2 , y 2 , 2 2 ) passes from one 
side of S to the other. Such a device renders the potential 
of V simpler to treat analytically and probably conforms to 
actual physical states more closely than the more exact 
conception of a surface of discontinuity. This device prac 
tically amounts to including the surface integral in the 
symbol Pot VF: 

In fact from the standpoint of pure mathematics it is 
better to state that where there exist surfaces at which the 
function V becomes discontinuous, the full value of Pot V V 
should always be understood as including the surface integral 



//. 



in addition to the volume integral 

>VF 



SSSr- 

U *J *J 10 



2 

12 

In like manner Pot V W, Pot V X W, New V W and other 
similar expressions to be met in the future must be regarded 
as consisting not only of a volume integral but of a surface 
integral in addition, whenever the vector function W possesses 
a surface of discontinuities. 

It is precisely this convention in the interpretation of 
formulae which permits such simple formulae as (27) to hold 
in general, and which gives to the treatment of the integrat 
ing operators an elegance of treatment otherwise unobtainable. 



222 VECTOR ANALYSIS 

The irregularities which may arise are thrown into the inter 
pretation, not into the analytic appearance of the formulse. 
This is the essence of Professor Gibbs s method of treatment. 
90.] The first partial derivatives of the potential may also 
be obtained by differentiating under the sign of integration. 1 

Q 2 > 3/21*2) _ , , , 



CCC 

= Jj J *>- 



^ rrr (*,-*!> r^y,,*,) 
i "^^^ V[(* a -* 1 )H(y 2 ^ 1 ) 2 +(v-^)T 8 ( 3 7) 



In like manner for a vector function W 
S Pot W /* /* /* *, ~ .. ~ ~ 

p ^ ""I I / . /r/- \9 i /.. .. \2 i /^. ~ \2^ia 2 y ! 

Or 



and ^!W= / / / *"-,- " " d, r (38) 

12 



But 2 -! 2 - x 2 - ! = 12 . 

1 If an attempt were made to obtain the second partial derivatives in the same 
manner, it would be seen that the volume integrals no longer converged. 



THE INTEGRAL CALCULUS OF VECTORS 223 

Hence V Pot F = /// ^f- d v r (39) 
In like manner 

^,, (40) 



and V Pot W = ^ * 



= fff 






These three integrals obtained from the potential by the 
differentiating operators are of great importance in mathe 
matical physics. Each has its own interpretation. Conse 
quently although obtained so simply from the potential each 
is given a separate name. Moreover inasmuch as these 
integrals may exist even when the potential is divergent, 
they must be considered independent of it. They are to 
be looked upon as three new integrating operators defined 
each upon its own merits as the potential was defined. 

Let, therefore, 



(42) 



12 



12 



.3 

r 12 



= Max W. (44) 



If the potential exists, then 

V Pot F= New F 

VxPotW = LapW (45) 

V-PotW = MaxW. 
The first is written New V and read The Newtonian of V! 9 



224 VECTOR ANALYSIS 

The reason for calling this integral the Newtonian is that if 
V represent the density of a body the integral gives the force, 
of attraction at the point (x^ y v Zj) due to the body. This 
will be proved later. The second is written Lap W and 
read "the Laplacian of W." This integral was used to a 
considerable extent by Laplace. It is of frequent occurrence 
in electricity and magnetism. If W represent the current 
C in space the Laplacian of C gives the magnetic force at the 
point (x v y v zj due to the current. The third is written 
Max W and read " the Maxwellian of W." This integral was 
used by Maxwell. It, too, occurs frequently in electricity 
and magnetism. For instance if W represent the intensity 
of magnetization I, the Maxwellian of I gives the magnetic 
potential at the point (x^ y v z^) due to the magnetization. 

To show that the Newtonian gives the force of attraction 
according to the law of the inverse square of the distance. 
Let dm<i be any element of mass situated at the point 
f rce at ( x v Vv z i) due to dm is equal to 



in magnitude and has the direction of the vector r 12 from the 
point (x v y v zj to the point (# 2 , y 2 , z 2 ). Hence the force is 



Integrating over the entire body, or over all space according 
to the convention here adopted, the total force is 



where V denotes the density of matter. 



THE INTEGRAL CALCULUS OF VECTORS 225 
The integral may be expanded in terms of i, j, k, 



12 



The three components may be expressed in terms of the po 
tential (if it exists) as 



12 
(42) 



It is in this form that the Newtonian is generally found in 
books. 

To show that the Laplacian gives the magnetic force per 
unit positive pole at the point (x v y v z^) due to a distribution 
W (# 2 , y<p z 2 ) f electric flux. The magnetic force at (x v y v x ) 
due to an element of current d C 2 is equal in magnitude to 
the magnitude d C% of that element of current divided by the 
square of the distance r 12 ; that is 

dC* 

*2 

T 12 

The direction of the force is perpendicular both to the vector 
element of current dC 2 and to the line r 12 joining the points. 
The direction of the force is therefore the direction of the 
vector product of r 12 and dC 2 . The force is therefore 



3 
12 



r 
T 

15 



226 VECTOR ANALYSIS 

Integrating over all space, the total magnetic force acting at 
the point (x^ y v z^) upon a unit positive pole is 

c r r r i2 x d C 2 r r r* x w 7 

/// J V J -J J J ^ ".- 

This integral may be expanded in terms of i, j, k. Let 



W (x v y v * 2 ) = i X(x v y v z^ + j Y (x^ y v z^) 

4- k^O 2 , y v z%). 

r i2=(^ 2 -^i) i+(y-yi)J+ (a-*i)k- 
The i, j, k components of Lap W are respectively 

C.-^^ 

(43) 






In terms of the potential (if one exists) this may be written 

3 Pot Z S Pot Y 
i Lap W = g g 

r= lP^_a|2t^ (43) ,, 



To show that if I be the intensity of magnetization at the 
point (x v %>*2)> that is, if I be a vector whose magnitude is 
equal to the magnetic moment per unit volume and whose 



THE INTEGRAL CALCULUS OF VECTORS 227 

direction is the direction of magnetization of the element d v% 
from south pole to north pole, then the Maxwellian of I is the 
magnetic potential due to the distribution of magnetization. 
The magnetic moment of the element of volume d t> 2 is I d v%. 
The potential at (x v y v 24) due to this element is equal to its 
magnetic moment divided by the square of the distance r 12 
and multiplied by the cosine of the angle between the direc 
tion of magnetization I and the vector r 12 . The potential is 

therefore 

r 12 I dv% 



Integrating, the total magnetic potential is seen to be 



12 

This integral may also be written out in terms of x, y, z. 
Let- 

*ia I = O a - x i) A + (y a ~ Vi) B + (*2 - *i) & 

If instead of x v y v z l the variables x^ y, z; and instead of 
x v y& z z ^ e variables %, ?;, f be used 1 the expression takes 
oq the form given by Maxwell. 



According to the notation employed for the Laplacian 

Max w -fff (*.-i 



(44) 



1 Maxwell : Electricity and Magnetism, Vol. II. p. 9. 



228 VECTOR ANALYSIS 

The Maxwellian of a vector function is a scalar quantity. 
It may be written in terms of the potential (if it exists) as 



SPotF 

Max W = -= -- + = - + = -- (44)" 
dx l 3y l 9z l 

This form of expression is much used in ordinary treatises 
upon mathematical physics. 

The Newtonian, Laplacian, and Maxwellian, however, should 
not be associated indissolubly with the particular physical 
interpretations given to them above. They should be looked 
upon as integrating operators which may be applied, as the 
potential is, to any functions of position in space. The New 
tonian is applied to a scalar function and yields a vector 
function. The Laplacian is applied to a vector function 
and yields a function of the same sort. The Maxwellian 
is applied to a vector function and yields a scalar function. 
Moreover, these integrals should not be looked upon as the 
derivatives of the potential. If the potential exists they 
are its derivatives. But they frequently exist when the 
potential fails to converge. 

91.] Let V and W be such functions that their potentials 
exist and have in general definite values. Then by (27) and 
(29) 

V V PotF= V Pot VF = Pot V VF. 

But by (45) V Pot V = New F, 

and V.Pot VF=Max VF. 

Hence V. V PotF= V. NewF= Max VF 

= PotV.VF (46) 

By (27) and (29) V V Pot W = V Pot V. W= Pot V V W. 
But by (45) V Pot W = Max W, 

and by (45) V Pot V W = New V. W. 



THE INTEGRAL CALCULUS OF VECTORS 229 

Hence V V Pot W = V Max W = New V W 

= Pot VV.W (47) 

By (28) V x V x Pot W = V x Pot V x W 

= Pot V x V x W. 

But by (45) V x Pot W = Lap W, 

and V x Pot V x W = Lap V x W. 

Hence V x V x Pot W = V x Lap W = Lap V x W 

= Pot V x V x W. (48) 

By (56), Chap. III. V - V x Pot W = 0, 
or V Pot V x W = 0. 

Hence V Lap W = Max V x W = 0. (49) 

And by (52), Chap. III. V x V PotF= 0, 
or VxPotVF=0. 

Hence V x New V = Lap V V = 0. (50) 

And by (58), Chap. III. V x V x W = VV W - V V W, 
V.VW = VV-W VxVxW. 

Hence V V Pot W New V W Lap V X W, (51) 
or V V Pot W = V Max W V X Lap W. 

These formulae may be written out in terms of curl and 
div if desired. Thus 

div New V = Max V F, (46) 

V Max W = New div W (47) 

curl Lap W = Lap curl W (48) 

div Lap W = Max curl W = (49) 

curl New V = Lap V F = (50) 

V V Pot W = New div W Lap curl W. (51) 



230 VECTOR ANALYSIS 

Poisson s Equation 
92.] Let V "be any function in space such that the potential 

PotF 
has in general a definite value. Then 

V V PotF= - 4 TrF, (52) 



c> 2 PotF 3 2 PotF 3 2 PotF 



This equation is known as Poisson s Equation. 

The integral which has been defined as the potential is a 
solution of Poisson s Equation. The proof is as follows. 






V x . V x Pot r= Vj . New F= Max V 2 r= T f C * ^* V dv v 

The subscripts 1 and ^ have been attached to designate 
clearly what are variables with respect to which the differen 
tiations are performed. 

V 1 .V 1 PotF=V 1 .NewF=ff TVJ--. V 2 Fdv a . 

But Vj = - V 2 

r vt r u 

and V 2 (v V 2 ^ = V 2 V 2 F+ V V a . V a 



THE INTEGRAL CALCULUS OF VECTORS 231 
Hence - V 2 - V 2 V = V V 2 V 2 - V 2 . ( V V 2 \ 

r !2 r !2 \ r !2/ 

m v, .v.r=rv. . v 2 + v..(V v V 

y 13 r !2 \ W 

Integrate : 



But V 2 V 2 = 0. 



That is to say satisfies Laplace s Equation. And by (8) 



Hence Vj V x Pot V = f f f V x - - V 2 Vd v 2 (53) 

=rr ^ v t .rfa. 

J J s 7*12 

The surface integral is taken over the surface which bounds 
the region of integration of the volume integral. This is 
taken " over all space." Hence the surface integral must be 
taken over a sphere of radius R, a large quantity, and R must 
be allowed to increase without limit. At the point (x r y^z^)^ 
however, the integrand of the surface integral becomes in 
finite owing to the presence of the term 



232 VECTOR ANALYSIS 

Hence the surface S must include not only the surface of the 
sphere of radius J2, but also the surface of a sphere of radius 
R , a small quantity, surrounding the point (x^y^z^) and B f 
must be allowed to approach zero as its limit. 

As it has been assumed that the potential of V exists, it is 
assumed that the conditions given (Art. 87) for the existence 
of the potential hold. That is 



< -fiT, when r is large 
Vr < K, when r is small. 

Introduce polar coordinates with the origin at the point 
(#i> #i i) Then r 12 becomes simply r 

and V x = - V a = -^ * 

l ii 12 ** 

Then for the large sphere of radius R 

1 r 

V, . da = r r 2 sm0 d0 dd>. 

*3 



!2 



4*3 



Hence the surface integral over that sphere approaches zero 
as its limit. For 



Hence when -R becomes infinite the surface integral over the 
large sphere approaches zero as its limit. 
For the small sphere 

1 r 

V t d a = -- 5 r 2 sin d d 6. 

<r 7*v 

r !2 

Hence the integral over that sphere becomes 



THE INTEGRAL CALCULUS OF VECTORS 233 

Let V be supposed to be finite and continuous at the point 
( x vl/v z i) which has been selected as origin. Then for the 
surface integral V is practically constant and equal to its 
value 

V (*ii Vv *i) 

at the point in question. 

sintf d 



f fs 



- f /Vsi 



Hence - sintf d8 d< = - 



when the radius R f of the sphere of integration approaches 
zero as its limit. Hence 



v > v 



- ff, rv - < =- 4 * F < 68)1 



and V- VPotF=-47rF. (52) 

In like manner if W is a vector function which has in 
general a definite potential, then that potential satisfies Pois- 
son s Equation. 

V V Pot W = - 4 TT W. (52) 

The proof of this consists in resolving W into its three com 
ponents. For each component the equation holds. Let 



v- 

V. VPotF=-47r F, 

V V Pot Z = 4 TT Z. 
Consequently 

V V Pot (JTi + Fj + Zk) = - 4 TT (JTi + Fj + 



234 VECTOR ANALYSIS 

Theorem : If V and W are such functions of position in space 
that their potentials exist in general, then for all points at which 
V and W are finite and continuous those potentials satisfy 
Poisson s Equation, 

V- VPot r=-4irF; (52) 

V V Pot W = - 4 TT W. (52) 

The modifications in this theorem which are to be made at 
points at which V and W become discontinuous will not be 
taken up here. 

93.] It was seen (46) Art. 91 that 



V VPotF = V- NewT=Max VF1 
Hence V New V = - 4 TT V (53) 

or Max VF=-47rF. 

In a similar manner it was seen (51) Art. 91 that 

V V Pot W = V Max W V x Lap W 

= New V W Lap V x W. 

Hence V Max W - V x Lap W = - 4 TT W, (54) 
or New V . W - Lap V x W = - 4-rr W. (54) 

By virtue of this equality W is divided into two parts. 

W = -7 Lap V x W 7 New V- W. (55) 

4-7T 4?T 

Let W = W! + W 2 , 

where W t = -r Lap V x W = - Lap curl W (56) 
4-rr 4?r 



-: NewV- W = 7 

4-7T 4-7T 



and W = -: NewV- W = 7 Newdiv W. (57) 

- - 



THE INTEGRAL CALCULUS OF VECTORS 235 

Equation (55) states that any vector function W multiplied 
by 4 TT is equal to the difference of the Laplacian of its curl 
and the Newtonian of its divergence. Furthermore 

V W, = V Lap V x W = -7 V- V x Lap W r 
4?r 4-7T 

But the divergence of the curl of a vector function is zero. 
Hence V.W 1 = divW 1 = (58) 

V x W 2 = - j Vx New V W 2 = - VxV Max W 2 . 

But the curl of the derivative of a scalar function is zero. 
Hence V x W 2 = curl W 2 = 0. (59) 

Consequently any vector function W which has a potential 
may be divided into two parts of which one has no divergence 
and of which the other has no curl. This division of W into 
two such parts is unique. 

In case a vector function has no potential but both its curl 
and divergence possess potentials, the vector function may be 
divided into three parts of which the first has no divergence ; 
the second, no curl; the third, neither divergence nor curl. 

Let W = Lap V x W - New V W + W. (55) 

4 7T 4 7T 

As before 

V Lap V x W = T V V x Pot V x W = 
4?r 4-7T 

1 1 

and - V x New V W - VxV Pot V W = 0. 

4-7T 4 7T 

The divergence of the first part and the curl of the second 
part of W are therefore zero. 



236 VECTOR ANALYSIS 

V x Lap VxW = VxVxPotVxW 
4?r 4-7T 

= VV Pot V x W - ~ V V Pot V X W. 

4 7T 4?T 



-- VV P ot V x W = -- V P o t V V x W = , 
4?r 



for V V x W = 0. 

Hence ^ V VPot V x W = V x W. 

4-7T 

Hence V x Lap VxW = VxW = VxW l . 
4?r 

The curl of W is equal to the curl of the first part 
-r Lap V x W 

47T 

into which W is divided. Hence as the second part has no 
curl, the third part can have none. Moreover 

- V New V W = V W V W 2 . 

T: 7T 

Thus the divergence of W is equal to the divergence of 
the second part 

- New V W. 
4?r 

into which W is divided. Hence as the first part has no 
divergence the third can have none. Consequently the third 
part W 3 has neither curl nor divergence. This proves the 
statement. 

By means of Art. 96 it may be seen that any function W 3 
which possesses neither curl nor divergence, must either 



THE INTEGRAL CALCULUS OF VECTORS 237 

vanish throughout all space or must not become zero at 
infinity. In physics functions generally vanish at infinity. 
Hence functions which represent actual phenomena may be 
divided into two parts, of which one has no divergence and 
the other no curl. 

94.] Definition : A vector function the divergence of which 
vanishes at every point of space is said to be solenoidal. A 
vector function the curl of which vanishes at every point of 
space is said to be irrotational. 

In general a vector function is neither solenoidal nor irrota 
tional. But it has been shown that any vector function which 
possesses a potential may be divided in one and only one 
way into two parts W v W 2 of which one is solenoidal and 
the other irrotational. The following theorems may be stated. 
They have all been proved in the foregoing sections. 

With respect to a solenoidal function W v the operators 

Lap and V X or curl 
4?r 

are inverse operators. That is 

Lap V x Wi = V x -j Lap Wi = W r (60) 

4?r 4-rr 

Applied to an irrotational function W 2 either of these opera 
tors gives zero. That is 

Lap W 2 = , V x W 2 = 0. (61) 

With respect to an irrotational function W 2 , the operators 

- New and v or div 
4?r 

are inverse operators. That is 

_ _L New V W 2 = - V -i- New W 2 = W 2 . (62) 

4 7T 4 7T 



238 VECTOR ANALYSIS 

With respect to a scalar function V the operators 

V or div and - New, 

4-7T 

and also -= Max and V 

4?r 

are inverse operators. That is 

-V.-i NewF= V (63) 

4 7T 



and ~--Max VF= V. 

4?r 

TFttA respect to a solenoidal function W x the operators 
- Pot and V x V x or curl curl 

47T 

are inverse operators. That is 

Pot V x V x W x = V x V x Pot Wi = W r (64) 
4?r 4?r 

With respect to an irrotational function W 2 the operators 

Pot and VV 
4?r 

are inverse operators. That is 

_ _L Pot VV . W 2 = - VV . -L Pot W 2 = W 2 . (65) 

With respect to any scalar or vector function V, W the 
operators 

Pot and V V 

4-7T 

are inverse operator*. That is 



THE INTEGRAL CALCULUS OF VECTORS 239 

_ JL Pot v v v= - v v -i- Pot F= v 

4?r 4?r 

and - , Pot V V W = - V V -^- Pot W = W. (66) 
4?r 4?r 

With respect to a solenoidal function W x the differentiating 
operators of the second order 

V V and V X V x 
are equivalent 

- V V W x = V x V x W r (67) 

With respect to an irrotational function W 2 the differentiat 
ing operators of the second order 

V V and V V 
are equivalent That is 

V- VW 2 = V V-W 2 . (68) 

By integrating the equations 

4^^=- V.NewF 
and 4 TT W = V x Lap W - V Max W 

by means of the potential integral Pot 

4<7rPotF=:-Pot V New F= - Max New F (69) 
4 TT Pot W = Pot V x Lap W - Pot V Max W 

4 TT Pot W = Lap Lap W - New Max W. (70) 
Hence for scalar functions and irrotational vector functions 

- New Max 

47T 

is an operator which is equivalent to Pot. For solenoidal vector 
functions the operator ^ 

- Lap Lap 



240 VECTOR ANALYSIS 

gives the potential. For any vector function the first operator 
gives the potential of the irrotational part; the second^ the 
potential of the solenoidal part. 

*95.] There are a number of double volume integrals which 
are of such frequent occurrence in mathematical physics as 
to merit a passing mention, although the theory of them will 
not be developed to any considerable extent. These double 
integrals are all scalar quantities. They are not scalar func 
tions of position in space. They have but a single value. 
The integrations in the expressions may be considered for 
convenience as extended over all space. The functions by 
vanishing identically outside of certain finite limits deter 
mine for all practical purposes the limits of integration in 
case they are finite. 

Given two scalar functions Z7, V of position in space. 
The mutual potential or potential product, as it may be called, 
of the two functions is the sextuple integral 



Pot 

(71) 

One of the integrations may be performed 

, yi ,^) PotVdv, 



( * 2 y* * 2> Pot Ud ** (T2) 

In a similar manner the mutual potential or potential product 
of two vector functions W, W" is 



(71) 

This is also a scalar quantity. One integration may be car 
ried out 



THE INTEGRAL CALCULUS OF VECTORS 241 
Pot (W, W") =w (x v y v ,) . Pot W" dv t 



The mutual Laplacian or Laplacian product of two 
vector functions W , W" of position in space is the sextuple 

integral 

Lap(W ,W") 

=ffffff w (*! yi *i) ;nr x w " (** y* *) <*i <*" 2 - 

(73) 

One integration may be performed. 

Lap (W, W") = f ( f W" (^ 2 , ya , * 2 ) Lap W rf v a 

(T4) 

v y i *i) La P w " d r 

The Newtonian product of a scalar function F, and a vector 
function W of position in space is the sextuple integral 

rf* 2 . 

(75) 
By performing one integration 

New ( F, W) =///W (* 2 , y 2 , * 2 ) New Frf t, a . (76) 

In like manner the Maxwellian product of a vector function 
W and a scalar function F of position in space is the 
integral 

Max (W,F) =/////JV(*i^*i) J- W0r 2 ,2/ 2 ,* 2 )rf W 

(77) 

16 



242 VECTOR ANALYSIS 

One integration yields 

Max (W, F) =fff V(x v y v zj Max W d v 1 = - New ( F, W). 

(78) 
By (53) Art. 93. 

4?r UPotr = - (V New CO PotF. 

V [New U Pot F] = (V New V) Pot F + (New IT) V Pot F. 
-(V.NewOPotF=-V.[NewPTotF]+NewtT.NewF 
Integrate : 

47r f f |VpotFdi>=- f f fv. [Ne 

+ C f CtfewU- NewFdv. 
4-Tr Pot IT, F)= f NewT. NewFdv 



, F)= f f f 
- T f 



Pot F New Z7 rf a. (79) 

The surface integral is to be taken over the entire surface S 
bounding the region of integration of the volume integral. 
As this region of integration is " all space," the surface S may 
be looked upon as the surface of a large sphere of radius R. 
If the functions U and F vanish identically for all points out 
side of certain finite limits, the surface integral must vanish. 
Hence 

4 TT Pot ( U, F) = f f fNew U New Vd v. (79) 

By (54) Art. 93, 

47rW". PotW = V x Lap W" Pot W 
- V Max W" Pot W . 



THE INTEGRAL CALCULUS OF VECTORS 243 

But V . [Lap W" x Pot W ] = Pot W V x Lap W" 

- Lap W" V x Pot W , 

and V [Max W" Pot W] = Pot W V Max W" 

+ Max W" V Pot W. 

Hence V x Lap W" Pot W = V [Lap W" x Pot W] 

+ Lap W" Lap W , 

and V Max W" Pot W = V - [Max W" Pot W] 

- Max W" Max W . 

Hence substituting: 

4 ?r W" Pot W = Lap W Lap W + Max W Max W" 
+ V [Lap W" X Pot W ] 

-V [Max W" Pot W ]. 
Integrating .- 

4 TT Pot (W, W") = ff f Lap W Lap W" dv 

r c r 

+ / / I Max W Max W d v (80) 
J J J 

I I PotW x Lap W" da / / Max W"PotWWa. 

If now W and W" exist only in finite space these surface 
integrals taken over a large sphere of radius B must vanish 
and then 

4 TT Pot (W, W") = f f fLap W Lap W" d v 

+ 11 fMax W Max W" d v. (80) 
J J J 

* 96.] There are a number of useful theorems of a function- 
theoretic nature which may perhaps be mentioned here owing 



244 VECTOR ANALYSIS 

to their intimate connection with the integral calculus of 
vectors. The proofs of them will in some instances be given 
and in some not. The theorems are often useful in practical 
applications of vector analysis to physics as well as in purely 
mathematical work. 

Theorem : If V (#, y, z) be a scalar function of position 
in space which possesses in general a definite derivative V V 
and if in any portion of space, finite or infinite but necessarily 
continuous, that derivative vanishes, then the function V is 
constant throughout that portion of space. 

Given VF=0. 

To show F= const. 

Choose a fixed point (# 15 y v zj in the region. By (2) page 
180 

y> * V F. d r = V(x, y,z)-V (x v y v zj. 

u ft* *i 

But fvr.dr =f<) . dr = 0. 

Hence F(#, y, z) = V (x v y v zj = const. 

Theorem : If F" (#, y, 2;) be a scalar function of position 
in space which possesses in general a definite derivative V V ; 
if the divergence of that derivative exists and is zero through 
out any region of space, 1 finite or infinite but necessarily 
continuous ; and if furthermore the derivative V V vanishes 
at every point of any finite volume or of any finite portion of 
surface in that region or bounding it, then the derivative 
vanishes throughout all that region and the function V re 
duces to a constant by the preceding theorem. 

1 The term throughout any region of space must be regarded as including the 
boundaries of the region as well as the region itself. 



THE INTEGRAL CALCULUS OF VECTORS 245 

Given V V V= for a region T, 

and V F= for a finite portion of surface S. 

To show J^= const. 

Since V Evanishes for the portion of surface S, Vis certainly 
constant in S. Suppose that, upon one side of S and in the 
region T, V were not constant. The derivative V V upon 
this side of S has in the main the direction of the normal to 
the surface S. Consider a sphere which lies for the most 
part upon the outer side of S but which projects a little 
through the surface S. The surface integral of VF over 
the small portion of the sphere which projects through the 
surface S cannot be zero. For, as V V is in the main normal 
to S 9 it must be nearly parallel to the normal to the portion 
of spherical surface under consideration. Hence the terms 

VT- da, 

in the surface integral all have the same sign and cannot 
cancel each other out. The surface integral of V V over 
that portion of S which is intercepted by the spherical sur 
face vanishes because V V is zero. Consequently the surface 
integral of V V taken over the entire surface of the spherical 
segment which projects through S is not zero. 

But f r vr-da= f r fv. vrd*=o. 

Hence f /Vr da = 0. 

It therefore appears that the supposition that V is not 
constant upon one side of S leads to results which contradict 
the given relation V V V 0. The supposition must there 
fore have been incorrect and V must be constant not only in 
S but in all portions of space near to $ in the region T. By 



246 VECTOR ANALYSIS 

an extension of the reasoning V is seen to be constant 
throughout the entire region T. 

Theorem : If V (x, y, z) be a scalar function of position in 
space possessing in general a derivative V V and if through 
out a certain region 1 T of space, finite or infinite, continuous 
or discontinuous, the divergence V V V of that derivative 
exists and is zero, and if furthermore the function V possesses 
a constant value c in all the surfaces bounding the region 
and V (x, y, z) approaches c as a limit when the point (x, y, z) 
recedes to infinity, then throughout the entire region T the 
function V has the same constant value c and the derivative 
W vanishes. 

The proof does not differ essentially from the one given 
in the case of the last theorem. The theorem may be gen 
eralized as follows : 

Theorem: If V(x,y, z) be any scalar function of position 
in space possessing in general a derivative W; if U (x, y, z) 
be any other scalar function of position which is either posi 
tive or negative throughout and upon the boundaries of a 
region T, finite or infinite, continuous or discontinuous; if 
the divergence V [ U V V~\ of the product of U and V V 
exists and is zero throughout and upon the boundaries of T 
and at infinity ; and if furthermore V be constant and equal 
to c upon all the boundaries of T and at infinity ; then the 
function V is constant throughout the entire region T and 
is equal to c. 

Theorem : If V (#, y, z) be any scalar function of position 
in space possessing in general a derivative V V ; if through 
out any region T of space, finite or infinite, continuous or 
discontinuous, the divergence V V V of this derivative exists 
and is zero ; and if in all the bounding surfaces of the region 
T the normal component of the derivative VF" vanishes and 
at infinite distances in T (if such there be) the product 

1 The region includes its boundaries. 



THE INTEGRAL CALCULUS OF VECTORS 247 

r 2 9 Vj 3 r vanishes, where r denotes the distance measured 
from any fixed origin ; then throughout the entire region T 
the derivative V Evanishes and in each continuous portion 
of T V is constant, although for different continuous portions 
this constant may not be the same. 

This theorem may be generalized as the preceding one 
was by the substitution of the relation V ( U V F) = for 
V-VF=Oand Ur*3V/3r = for r^SV/Sr = 0. 

As corollaries of the foregoing theorems the following 
statements may be made. The language is not so precise 
as in the theorems themselves, but will perhaps be under 
stood when they are borne in mind. 

If V U = V V, then U and V differ at most by a 
constant. 

If V-V7=V.VF and if VZ7 = VF in any finite 
portion of surface S, then V U = V V at all points and V 
differs from V only by a constant at most. 

If V.VJ7= V- VF and if V= V in all the bounding 
surfaces of the region and at infinity (if the region extend 
thereto), then at all points 7 and Fare equal. 

If V V 7 = V V F and if in all the bounding surfaces 
of the region the normal components of VZ7 and VFare 
equal and if at infinite distances r 2 (3 U/Sr 9 F/5r) is 
zero, then V ?7and V Fare equal at all points of the region 
and U differs from F only by a constant. 

Theorem : If W and W" are two vector functions of position 
in space which in general possess curls and divergences ; if 
for any region I 7 , finite or infinite but necessarily continuous, 
the curl of W is equal to the curl of W" and the divergence 
of W is equal to the divergence of W"; and if moreover 
the two functions W and W" are equal to each other at 
every point of any finite volume in T or of any finite surface 
in Tor bounding it; then W is equal to W" at every point 
of the region T. 



248 VECTOR ANALYSIS 

Since V x W = V x W", V x (W - W") = 0. A vec 
tor function whose curl vanishes is equal to the derivative * 
of a scalar function V (page 197). Let VF=W W". 
Then V V V= owing to the equality of the divergences. 
The theorem therefore becomes a corollary of a preceding one. 

Theorem : If W and W" are two vector functions of posi 
tion which in general possess definite curls and divergences ; 
if throughout any aperiphractic* region T, finite but not 
necessarily continuous, the curl of W is equal to the curl of 
W" and the divergence of W is equal to the divergence of 
W"; and if furthermore in all the bounding surfaces of the 
region T the tangential components W 7 and W" are equal; 
then W ; is equal to W" throughout the aperiphractic region T. 

Theorem: If W and W" are two vector functions of posi 
tion in space which in general possess definite curls and 
divergences ; if throughout any acyclic region T, finite but not 
necessarily continuous, the curl of W is equal to the curl 
W" and the divergence of W is equal to the divergence of 
W"; and if in all the bounding surfaces of the region T the 
normal components of W and W" are equal ; then the func 
tions W and W" are equal throughout the region acyclic T. 

The proofs of these two theorems are carried out by means 
of the device suggested before. 

Theorem: If W and W" are two vector functions such 
that V V W and V V W" have in general definite values 
in a certain region T, finite or infinite, continuous or discon 
tinuous ; and if in all the bounding surfaces of the region 
and at infinity the functions W and W" are equal ; then W 
is equal to W" throughout the entire region T. 

The proof is given by treating separately the three com 
ponents of W and W". 

1 The region T may have to be made acyclic by the insertion of diaphragms. 

2 A region which encloses within itself another region is said to be periphrac- 
tic. If it encloses no region it is aperiphractic. 



THE INTEGRAL CALCULUS OF VECTORS 249 



SUMMARY OF CHAPTER IV 

The line integral of a vector function W along a curve C is 
defined as 

f Wdr=f [Widx + W^dy + W.dz]. (1) 
J c J c 



The line integral of the derivative V V of a scalar function 
V along a curve C from r to r is equal to the difference 
between the values of V at the points r and r and hence the 
line integral taken around a closed curve is zero ; and con 
versely if the line integral of a vector function W taken 
around any closed curve vanishes, then W is the derivative 
V V of some scalar function V. 



f ri 

/ TO 



f 

J 



(2) 
(3) 



and if C W dr = 0, then W = VF. 

Jo 



Illustration of the theorem by application to mechanics. 

The surface integral of a vector function W over a surface 
S is defined as 



= ff 



Gauss s Theorem| : The surface integral of a vector func- 
tiorTtaken over a closed surface is equal to the volume 
integral of the divergence of that function taken throughout 
the volume enclosed by that surface 



250 VECTOR ANALYSIS 

= f f {Xdydz+ Ydz dx + Zdxdy], (8) 

if X, I 7 , Z be the three components of the vector function W. 
Stokes s Theorem: The surface integral of the curl of a 
vector function taken over any surface is equal to the line 
integral of the function taken around the line bounding the 
surface. And conversely if the surface integral of a vector 
function TJ taken over any surface is equal to the line integral 
of a function W taken around the boundary, then U is the 
curl of W. 

//,VxW.*.=/ o W.*r, (11) 

and if ffjj da =f W rfr, then TI = V x W. (12) 

Application of the theorem of Stokes to deducing the 
equations of the electro-magnetic field from two experimental 
facts due to Faraday. Application of the theorems of Stokes 
and Gauss to the proof that the divergence of the curl of 
a vector function is zero and the curl of the derivative of 
a scalar function is zero. 

Formulae analogous to integration by parts 

I w V v di = \u v~] T / v V u rf r, (14) 

f f r r r 

J J 8 t/O J J S 

cc ^ c r 

I t vwXv / yaa=i o i6v yar = I v V u a r, (16) 

*/ */ 8 */ t/O 



THE INTEGRAL CALCULUS OF VECTORS 251 
I C CuV *vdv = I I uv d&- ii f V u*vdv, (17) 

/Y vu x v . da=- r r A/!* . v x * dv. (i8> 

Green s Theorerii: 
/ / I V u V v dv = / f ttV-y -da f T TwV -V v dv 



= if v V ^ d a I I I vV *V udv, (19) 
.Vi;-i;V.Vw)rfi;= f C (uVv t?Vw).rfa. (20) 



__ - 
Kelvin s generalization: 

i I Tw^7u^vdv= I I w^Vv-rfa // / ^ 

= / / i? w V i^ da T T TV V [w V w] rf v. (21) 

The integrating operator known as the potential is defined 
by the equation 

Pot r= V(xy v Z ^ dxt dy 2 dz y (22) 






Pot w =*? yy * ^^2 ^y 2 ^^- (23) 

VPot T=PotVF; (27) 

V x Pot W = Pot V x W, (28) 

V Pot W = Pot V W, (29) 

V V Pot F= Pot V VF, (30) 



252 VECTOR ANALYSIS 

V V Pot W = Pot V V W, (31) 

VV Pot W = Pot VV W, (32) 

V x V x Pot W = Pot V x V x W. (33) 

The integrating operator Pot and the differentiating operator 
V are commutative. 

The three additional integrating operators known as the 
Newtonian, the Laplacian, and the Maxwellian. 



__ __ f f *19 \ Ay * fO? ^9 / ^ / i ***. 

New T= / / / -^ ^- ^-^- rf^ 2 dy 2 dz 2 . (42) 

f* f* f* Y ^ \XT ^7* ?y 2 ^ 

Lap W = I I / o 2 2 2 ^^ 2 ^2/2 dz v ( 43 ) 

j J J r 12 

Max W= I I I q 2> 2 * 2 rf^ 2 rfy d^ 2 . 






If the potential exists these integrals are related to it as fol 

lows: 

V Pot F= New V, 

V x Pot W = Lap W, (45) 



The interpretation of the physical meaning of the Newtonian 
on the assumption that V is the density of an attracting 
body, of the Laplacian on the assumption that W is electric 
flux, of the Maxwellian on the assumption that W is the 
intensity of magnetization. The expression of these integrals 
or their components in terms of a?, y, % ; formulae (42) , (43) , 
(44) and (42)", (43)", (44)". 

V New F= Max V F, (46) 

V Max W = New V W, (47) 

V x Lap W = Lap V x W, (48) 



THE INTEGRAL CALCULUS OF VECTORS 253 

V Lap W = Max V x W = 0, (49) 

V x New V = Lap V V= 0, (50) 
V V Pot W = New V W Lap V X W 

= V Max W - V x Lap W. (51) 

The potential is a solution of Poisson s Equation. That is, 

V. VPotF = -47rF; (52) 

and V. VPotF=-47rW. (52) 

F= V.NewF, (53) 



W = - A Lap V x W - New V W. (55) 

4-7T 4 7T 

Hence W is divided into two parts of which one is 
solenoidal and the other irrotational, provided the potential 
exists. In case the potential does not exist a third term W 3 
must be added of which both the divergence and the curl 
vanish. A list of theorems which follow immediately from 
equations (52), (52) , (53), (55) and which state that certain 
integrating operators are inverse to certain differentiating 
operators. Let V be a scalar function, W x a solenoidal vector 
function, and W 2 an irrotational vector function. Then 

Lap V x Wj = V x Lap W l = W r (60) 

47T 4 7T 

Lap W 2 = 0, V x W 2 = (61) 

4 7T 

-- New V. W, = - V New W 2 = W 2 . (62) 

47T 4?T 



254 VECTOR ANALYSIS 

f_V- - A -NewF = V 

l (63) 

- Max VF= V. 

4 7T 

- Pot V x V x W t = V x V x Pot W x = W, (64) 

47T 

-- 1 - Pot VV W 2 = - V V - - Pot W 2 = W 2 . (65) 

4 7T 4 7T 



(66) 



[--- Pot v. vr=- v-v - 

4?r 4?r 

1 ! 

L- -r - Pot V . V W = - V V Pot W = W. 
4?r 4?r 



-V-VWj^VxVxWj (67) 

V V W 2 = VV . W 2 (68) 

4 TT Pot V = - Max New V (69) 

4 TT Pot W = Lap Lap W - New Max W. (70) 

Mutual potentials Newtonians, Laplacians, and Maxwellians 
may be formed. They are sextuple integrals. The integra 
tions cannot all be performed immediately ; but the first three 
may be. Formulae (71) to (80) inclusive deal with these inte 
grals. The chapter closes with the enunciation of a number 
of theorems of a function-theoretic nature. By means of 
these theorems certain facts concerning functions may be 
inferred from the conditions that they satisfy Laplace s equa 
tion and have certain boundary conditions. 

Among the exercises number 6 is worthy of especial atten 
tion. The work done in the text has for the most part assumed 
that the potential exists. But many of the formulce connecting 
Newtonians, Laplacians, and Maxwellians hold when the poten 
tial does not exist. These are taken up in Exercise 6 referred to. 



THE INTEGRAL CALCULUS OF VECTORS 255 



EXERCISES ON CHAPTER IV 

I. 1 If V is a scalar function of position in space the line 
integral 



is a vector quantity. Show that 



That is ; the line integral of a scalar function around a 
closed curve is equal to the skew surface integral of the deriv 
ative of the function taken over any surface spanned into 
the contour of the curve. Show further that if V is constant 
the integral around any closed curve is zero and conversely 
if the integral around any closed curve is zero the function V 
is constant. 

Hint : Instead of treating the integral as it stands multiply 
it (with a dot) by an arbitrary constant unit vector and thus 
reduce it to the line integral of a vector function. 

2. If W is a vector function the line integral 



=/w 

J c 



x dr 



is a vector quantity. It may be called the skew line integral 
of the function W. If c is any constant vector, show that if 
the integral be taken around a closed curve 

H c = / / (cVW cVW) da = c/ Wxdr, 

1 The first four exercises are taken from Foppl s Einfiihrung in die Max- 
well sche Theorie der Electricitat where they are worked out. 



256 VECTOR ANALYSIS 

and H-c = c. ]JJ 8 V Wda- J J s V (W d a) j 



In case the integral is taken over a plane curve and the 
surface S is the portion of plane included by the curve 



Show that the integral taken over a plane curve vanishes 
when W is constant and conversely if the integral over any 
plane curve vanishes W must be constant. 

3. The surface integral of a scalar function V is 



This is a vector quantity. Show that the surface integral 
of V taken over any closed surface is equal to the volume 
integral of W taken throughout the volume bounded by 
that surface. That is 



Hence conclude that the surface integral over a closed sur 
face vanishes if V be constant and conversely if the surface 
integral over any closed surface vanishes the function V must 
be constant. 

4. If W be a vector function, the surface integral 

T= f C d&x W 



may be called the skew surface integral. It is a vector 
quantity. Show that the skew surface integral of a vector 



THE INTEGRAL CALCULUS OF VECTORS 257 

function taken over a closed surface is equal to the volume 
integral of the vector function taken throughout the volume 
bounded by the surface. That is 



Hence conclude that the skew surface integral taken over 
any surface in space vanishes when and only when W is an 
irrotational function. That is, when and only when the line 
integral of W for every closed circuit vanishes. 

5. Obtain some formulae for these integrals which are 
analogous to integrating by parts. 

6. The work in the text assumes for the most part that the 
potentials of Fand W exist. Many of the relations, however, 
may be demonstrated without that assumption. Assume that 
the Newtonian, the Laplacian, the Maxwellian exist. For 
simplicity in writing let 



Then New V = V t Pn V(x v y y * 2 ) d t> 2 , (81) 



Lap W =i^ 12 X W (x v y v 2 ) dv v (82) 

Max W =fffv i p u W (z 2 , y v z^dv v (83) 

(84) 



c c r 

-JJJr^rdvr 



17 



258 VECTOR ANALYSIS 

By exercise ( 3 )/// V 2 (Pit v ) d v* 

It can be shown that if V is such a function that New V 
exists, then this surface integral taken over a large sphere of 
radius R and a small sphere of radius R* approaches zero 
when R becomes indefinitely great; and R f , indefinitely 
small. Hence 



or NewF=PotVF. (85) 

Prove in a similar manner that 

Lap W = Pot V x W, (86) 

Max W = Pot V W. (87) 

By means of (85), (86), (87) it is possible to prove that 
V x Lap W = Lap V x W, 
V-New F=Max VF, 

V Max W = New V W. 
Then prove 

/*/*/* f* f* f* 

VxLapW=i / I^ 12 VV-W di? 2 I l f^ 12 V-VWdi 
%} <J *J *J J *J 

and V Max W = f/JJPii V V W d v v 

Hence V x Lap W - V Max W = -ffffv V V W d v 
Hence V x Lap W - V Max W = 4 TT W. (88) 

7. An integral used by Helmholtz is 



THE INTEGRAL CALCULUS OF VECTORS 259 
or if W be a vector function 

H (W) =/// W d "2" < 9 ) 

Show that the integral converges if V diminishes so rapidly 

that 

K 



when r becomes indefinitely great. 

Vtf(F) = #(VF) = New(r 2 F), (91) 

V # (W) = # (V W) = Max (r 2 W), (92) 

V x H (W) = 5" (V x W) = Lap (r 2 W), (93) 

=J ff (V. VP) = Max(r 2 VF) = 2 Pot F (94) 

. (95) 



H ( F) = - -L Pot Pot PI (96) 

J 7T 

^ (W) = - -?- Pot Pot W. (97) 

2 7T 

~2W = VxVx^T(W) + VV.^r (W). (98) 

8. Give a proof of Gauss s Theorem which does not depend 
upon the physical interpretation of a function as the flux of a 
fluid. The reasoning is similar to that employed in Art. 51 
and in the first proof of Stokes s Theorem. 

9. Show that the division of W into two parts, page 235, 
is unique. 

10. Treat, in a manner analogous to that upon page 220, 
the case in which V has curves of discontinuities. 



CHAPTER V 

LINEAR VECTOR FUNCTIONS 

97.] AFTER the definitions of products had been laid down 
and applied, two paths of advance were open. One was 
differential and integral calculus ; the other, higher algebra 
in the sense of the theory of linear homogeneous substitutions. 
The treatment of the first of these topics led to new ideas 
and new symbols to the derivative, divergence, curl, scalar 
and vector potential, that is, to V, V, Vx, and Pot with the 
auxiliaries, the Newtonian, the Laplacian, and the Maxwellian. 
The treatment of the second topic will likewise introduce 
novelty both in concept and in notation the linear vector 
function, the dyad, and the dyadic with their appropriate 
symbolization. 

The simplest example of a linear vector function is the 
product of a scalar constant and a vector. The vector r 

T = CT (1) 

is a linear function of r. A more general linear function 
may be obtained by considering the components of r individ 
ually. Let i, j, k be a system of axes. The components of 

r are 

i r, j r, k r. 

Let each of these be multiplied by a scalar constant which 
may be different for the different components. 

c l i r, c 2 j r, c 3 k r. 



LINEAR VECTOR FUNCTIONS 261 

Take these as the components of a new vector r 

r = i (Cji-^ + j (c a j-r) + k (c 8 k-r). (2) 

The vector r is then a linear function of r. Its components 
are always equal to the corresponding components of r each 
multiplied by a definite scalar constant. 

Such a linear function has numerous applications in geom 
etry and physics. If, for instance, i, j, k be the axes of a 
homogeneous strain and c v c 2 , c 3 , the elongations along these 

axes, a point 

r = ix + j y + bz 

becomes r = i c l x -f j c 2 y + k c 3 z, 

or r = i c l i r + j <? 2 j r + k c 3 k r. 

This sort of linear function occurs in the theory of elasticity 
and in hydrodynamics. In the theory of electricity and 
magnetism, the electric force E is a linear function of the 
electric displacement D in a dielectric. For isotropic bodies 
the function becomes merely a constant 



But in case the body be non-isotropic, the components of the 
force along the different axes will be multiplied by different 
constants k v & 2 , & 3 . Thus 

E = i% 1 i*D + j 2 j .D + k&gk-D. 

The linear vector function is indispensable in dealing with 
the phenomena of electricity, magnetism, and optics in non- 
isotropic bodies. 

98.] It is possible to define a linear vector function, as has 
been done above, by means of the components of a vector. 
The most general definition would be 



262 VECTOR ANALYSIS 

Definition : A vector r is said to be a linear vector func 
tion of another vector r when the components of r along 
three non-coplanar vectors are expressible linearly with scalar 
coefficients in terms of the components of r along those same 
vectors. 

If r = XB, + yb + zc, where [abc] ^ 0, 

and r = # a + y b + z c, 

and if x f = a l x + b l y + c l z f 

y r = a^x + 6 2 y + c 2 z, (3) 

z f = a z x + l z y + c 3 z, 

then r is a linear function of r. (The constants a^ l v c v 
etc., have no connection with the components of a, b, c par 
allel to i, j, k.) Another definition however is found to be 
more convenient and from it the foregoing may be deduced. 
Definition : A continuous vector function of a vector is 
said to be a linear vector function when the function of the 
sum of any two vectors is the sum of the functions of those 
vectors. That is, the function /is linear if 

/(r 1 + r 2 )=/(r 1 )+/(r a ). (4) 

Theorem : If a be any positive or negative scalar and if / 
be a linear function, then the function of a times r is a times 
the function of r. 

/0r) = a/(r), (5) 

And hence 

/(a 1 r 1 + a 2 r 2 + a 3 r 3 + .-) 

= i f<Ji) + <**f (r a )+ 8 /(*8) + (5) 

The proof of this theorem which appears more or less 
obvious is a trifle long. It depends upon making repeated 
use of relation (4). 



LINEAR VECTOR FUNCTIONS 263 

Hence /(2r) = 2/(r). 

In like manner / (n r) = nf (r) 

where n is any positive integer. 

Let m be any other positive integer. Then by the relation 
just obtained 



Hence / (.i) =/( i r )=-?./ (,). 

\ w / \ m / m 



That is, equation (5) has been proved in case the constant a 
is a rational positive number. 

To show the relation for negative numbers note that 

/(0)=/(0 + 0) = 2/(0). 
Hence /(0) = 0. 

But /(O) =/(r-r) =/( r +(-r)) =/(r) 
Hence r= 



To prove (5) for incommensurable values of the constant 
a, it becomes necessary to make use of the continuity of the 
function /. That is 



Let x approach the incommensurable number a by passing 
through a suite of commensurable values. Then 



Hence *****. + ( xi} = a 

x = a J v ~ 



264 VECTOR ANALYSIS 

LlM (ar)=ar. 

# = a v 

Hence /(") = / 00 

which proves the theorem. 

Theorem: A linear vector f unction /(r) is entirely deter 
mined when its values for three non-coplanar vectors a, b, c are 
known. 

Let l=/(a), 

m=/(b), 
n=/(c). 

Since r is any vector whatsoever, it may be expressed as 

r = #a + yb + 3C. 
Hence / (r) = x 1 + y m + z n. 

99.] In Art. 97 a particular case of a linear function was 
expressed as 

r = i c l i r + j c 2 j r + k c 3 k r. 

For the sake of brevity and to save repeating the vector r 
which occurs in each of these terms in the same way this 
may be written in the symbolic form 



In like manner if a p a 2 , a 8 be any given vectors, and b p b 2 , 
b 3 , another set equal in number, the expression 

r = a! b x r + a 2 b 2 r + a 3 b 3 r + - (6) 

is a linear vector function of r ; for owing to the distributive 
character of the scalar product this function of r satisfies 
relation (4). For the sake of brevity r may be written sym 
bolically in the form 

r = ( ai b x + a a b 2 + a 3 b 3 + .) r. (6) 



LINEAR VECTOR FUNCTIONS 265 

No particular physical or geometrical significance is to be 
attributed at present to the expression 

(a^ + a^ + agbg + .) (7) 

It should be regarded as an operator or symbol which -con 
verts the vector r into the vector r and which merely 
affords a convenient and quick way of writing the relation 
(6). 

Definition : An expression a b formed by the juxtaposition 
of two vectors without the intervention of a dot or a cross is 
called a dyad. The symbolic sum of two dyads is called a 
dyadic binomial ; of three, a dyadic trinomial ; of any num 
ber, a dyadic polynomial. For the sake of brevity dyadic 
binomials, trinomials, and polynomials will be called simply 
dyadics. The first vector in a dyad is called the antecedent ; 
and the second vector, the consequent. The antecedents of a 
dyadic are the vectors which are the antecedents of the 
individual dyads of which the dyadic is composed. In like 
manner the consequents of a dyadic are the consequents of 
the individual dyads. Thus in the dyadic (7) a p a 2 , a 3 are 
the antecedents and b r b 2 , b 3 - the consequents. 

Dyadics will be represented symbolically by the capital 
Greek letters. When only one dyadic is present the letter 
will generally be used. In case several are under consid 
eration other Greek capitals will be employed also. With 
this notation (7) becomes 



and (6) may now be written briefly in the form 

r = d> r. (8) 

By definition r = aj b x r + a 2 b 2 r + a 3 b 3 r + 

The symbol <P-r is read dot r. It is called the direct 
product of into r because the consequents bj, b 2 , b 3 - are 



266 VECTOR ANALYSIS 

multiplied into r by direct or scalar multiplication. The 
order of the factors and r is important. The direct 
product of r into is 

r <P = r . (a a ^ + a 2 b 2 + a 3 b 3 + . ) 

= r . a x bj + r a 2 b 2 + r - a 3 b 3 + . . . (9) 

Evidently the vectors r and r are in general different. 

Definition : When the dyadic is multiplied into r as r, 
is said to be a pref actor to r. When r is multiplied in as 
r <#, is said to be a post/actor to r. 

A dyadic used either as a pref actor or as a postf actor to a 
vector r determines a linear vector function of r. The two linear 
vector functions thus obtained are in general different from 
one another. They are called conjugate linear vector func 
tions. The two dyadics 

^ajbj + ajbg + agbg + ... 

and = b x a x + b 2 a 2 + b 3 a 3 + , 

each of which may be obtained from the other by inter 
changing the antecedents and consequents, are called conjitr 
gate dyadics. The fact that one dyadic is the conjugate of 
another is denoted by affixing a subscript C to either. 

Thus = C = c . 

Theorem: A dyadic used as a postf actor gives the same 
result as its conjugate used as a prefactor. That is 

r = C r. (9) 

100.] Definition : Any two dyadics and W are said to 
be equal 

when r = W r for all values of r, 

or when r = r W for all values of r, (10) 

or when B r = B W r for all values of s and r. 



LINEAR VECTOR FUNCTIONS 267 

The third relation is equivalent to the first. For, if the 
vectors r and W r are equal, the scalar products of any 
vector s into them must be equal. And conversely if the 
scalar product of any and every vector s into the vectors r 
and *T are equal, then those vectors must be equal. In 
like manner it may be shown that the third relation is equiva 
lent to the second. Hence all three are equivalent. 

Theorem : A dyadic is completely determined when the 

values 0.a, 0.b, 0.c, 

where a, b, c are any three non-coplanar vectors, are known. 
This follows immediately from the fact that a dyadic defines 
a linear vector function. If 



. r = 0.(#a + 2/b + zc)==# a + ?/*b-Mc, 

Consequently two dyadics and W are equal provided equa 
tions (10) hold for three non-coplanar vectors r and three 
non-coplanar vectors s. 

Theorem : Any linear vector function / may be represented 
by a dyadic to be used as a prefactor and by a dyadic , 
which is the conjugate of 0, to be used as a postfactor. 

The linear vector function is completely determined when 
its values for three non-coplanar vectors (say i, j, k) are 
known (page 264). Let 



/ 



Then the linear function / is equivalent to the dyadic 



to be used as a postfactor; and to the dyadic 

= <P (7 = ia + jb + kc, 
to be used as a prefactor. 



268 VECTOR ANALYSIS 

The study of linear vector functions therefore is identical 
with the study of dyadics. 

Definition : A dyad a b is said to be multiplied by a scalar 
a when the antecedent or the consequent is multiplied by 
that scalar, or when a is distributed in any manner between 
the antecedent and the consequent. If a = a a 11 

a (ab) = (a a) b = a (a b) = (a a) (a" b). 

A dyadic is said to be multiplied by the scalar a when 
each of its dyads is multiplied by that scalar. The product 

is written 

a or <Pa. 

The dyadic a $ applied to a vector r either as a prefactor or 
as a postfactor yields a vector equal to a times the vector 
obtained by applying to r that is 

(a 0) r = a (0 r). 

Theorem : The combination of vectors in a dyad is distrib 
utive. That is 

(a + b) c = a c + b c ... 

and a (b + c) = ab + ac. 

This follows immediately from the definition of equality of 
dyadics (10). For 

[(a + b) c] r = (a + b) c r = a c r + b c r = (a c + b c) r 

and 

[a(b + c)] r = a (b + c) r = ab -r + ac- r = (ab + ac) r. 

Hence it follows that a dyad which consists of two factors, 
each of which is the sum of a number of vectors, may be 
multiplied out according to the law of ordinary algebra 
except that the order of the factors in the dyads must be 
maintained. 



LINEAR VECTOR FUNCTIONS 269 



bn+ ... (11) 
+ cl-f cm-f cn+ 



The dyad therefore appears as a product of the two vectors of 
which it is composed, inasmuch as it obeys the characteris 
tic law of products the distributive law. This is a justifi 
cation for writing a dyad with the antecedent and conse 
quent in juxtaposition as is customary in the case of products 
in ordinary algebra. 

The N onion Form of a Dyadic 

10L] From the three unit vectors i, j, k nine dyads may 
be obtained by combining two at a time. These are 

ii, ij, ik, 

ji, jj, jk, (12) 

ki, kj, kk. 

If all the antecedents and consequents in a dyadic be ex 
pressed in terms of i, j, k, and if the resulting expression be 
simplified by performing the multiplications according to the 
distributive law (11) and if the terms be collected, the dyadic 
may be reduced to the sum of nine dyads each of which is 
a scalar multiple of one of the nine fundamental dyads given 
above. 

= a n ii + a 12 ij + a 13 ik 

+ 2 iJi +a 22 jj + a23 jk (13) 

+ a 31 ki + a 32 kj + a 33 kk. 

This is called the nonion form of 0. 

Theorem : The necessary and sufficient condition that two 
dyadics 4> and W be equal is that, when expressed in nonion 



270 VECTOR ANALYSIS 

form, the scalar coefficients of the corresponding dyads be 
equal. 

If the coefficients be equal, then obviously 

<P. r= W . r 

for any value of r and the dyadics by (10) must be equal. 
Conversely, if the dyadics and W are equal, then by (10) 

s r = s W r 

for all values of s and r. Let s and r each take on the values 

i,j,k. Then (14) 

i . d> . i = i - i, i . . j = i . W j, i k = i iT k 

j . 0.i=j. W.i, j. </.j =j. . j, j. <P.k = j. ?T.k 

k. <P-i = k. ?F.i, k- 0- j = k- ?F.j, k 0- k = k r.k. 

But these quantities are precisely the nine coefficients in the 
expansion of the dyadics and W. Hence the corresponding 
coefficients are equal and the theorem is proved. 1 This 
analytic statement of the equality of two dyadics can some 
times be used to greater advantage than the more fundamental 
definition (10) based upon the conception of the dyadic as 
defining a linear vector function. 

Theorem : A dyadic may be expressed as the sum of nine 
dyads of which the antecedents are any three given non- 
coplanar vectors, a, b, c and the consequents any three given 
non-coplanar vectors 1, m, n. 

Every antecedent may be expressed in terms of a, b, c ; 
and every consequent, in terms of 1, m, n. The dyadic may 
then be reduced to the form 

= a n al + & 12 am + a 13 an 
+ a 21 bl + 22 bin + a 23 bn (15) 

-f- fflai c 1 + a 32 c m + ^33 c n. 

1 As a corollary of the theorem it is evident that the nine dyads (12) are in 
dependent. None of them may be expressed linearly in terms of the others. 



LINEAR VECTOR FUNCTIONS 271 

This expression of <P is more general than that given in 
(13). It reduces to that expression when each set of vectors 
a, b, c and 1, m, n coincides with i, j, k. 

Theorem : Any dyadic <# may be reduced to the sum of 
three dyads of which either the antecedents or the consequents, 
but not both, may be arbitrarily chosen provided they be non- 
coplanar. 

Let it be required to express 4> as the sum of three dyads 
of which a, b, c are the antecedents. Let 1, m, n be any other 
three non-coplanar vectors. may then be expressed as in 
(15). Hence 

= a (a n 1 + 12 m + a 13 n) + b (a 21 1 + 22 m + 23 n) 

+ c Osi 1 + 32 m + a 32 n), 
or <P = aA + bB + cC. (16) 

In like manner if it be required to express $ as the sum of 
three dyads of which the three non-coplanar vectors 1, m, n are 
the consequents 

= Ll + Mm + Nn, (16) 

where L = a n a + a 2l b + a 31 c, 

M = a 12 a + 22 b + a 32 c > 
N = a lB a + a 23 b + a ZB c. 

The expressions (15), (16), (16) for are unique. Two equal 
dyadics which have the same three non-coplanar ante 
cedents, a, b, c, have the same consequents A, B, C - - these 
however need not be non-coplanar. And two equal dyadics 
which have the same three non-coplanar consequents 1, m, n, 
have the same three antecedents. 

102. ] Definition: The symbolic product formed by the juxta 
position of two vectors a, b without the intervention of a dot 
or a cross is called the indeterminate product of the two vectors 
a and b. 



272 VECTOR ANALYSIS 

The reason for the term indeterminate is this. The two 
products a b and a x b have definite meanings. One is a 
certain scalar, the other a certain vector. On the other hand 
the product ab is neither vector nor scalar it is purely 
symbolic and acquires a determinate physical meaning only 
when used as an operator. The product a b does not obey 
the commutative law. It does however obey the distributive 
law (11) and the associative law as far as scalar multiplication 
is concerned (Art 100). 

TJieorem : The indeterminate product a b of two vectors is 
the most general product in which scalar multiplication is 
associative. 

The most general product conceivable ought to have the 
property that when the product is known the two factors are 
also known. Certainly no product could be more general. 
Inasmuch as scalar multiplication is to be associative, that is 

a (ab) = (a a) b = a (a b) = (a* a) (a"b), 

it will be impossible to completely determine the vectors a 
and b when their product a b is given. Any scalar factor 
may be transferred from one vector to the other. Apart from 
this possible transference of a scalar factor, the vectors com 
posing the product are known when the product is known. In 
other words 

Theorem : If the two indeterminate products a b and a b 
are equal, the vectors a and a , b and b must be collinear and 
the product of the lengths of a and b (taking into account the 
positive or negative sign according as a and b have respec 
tively equal or opposite directions to a and b ) is equal to the 
product of the lengths of a and b . 

Let a = a l i + & 2 j + a 3 k, 

b = l l i + & 2 j + 6 3 k, 



LINEAR VECTOR FUNCTIONS 273 

a = a 1 i + a 2 j + a 3 k, 

v = Vi + yj + &, * 

Then &b = a 1 b 1 ii + a^^ ij + a 1 b 3 ik 

a 2 & 3 jj + a 2 6 3 jk 
a,&, kj + a 3 6 3 kk. 
and a V = , &/ ii + a/V ij + aj 6 3 ik 

+ <V ji + , &, jj + a 2 6 3 jk 
+ o 8 6j ki + o, 6 t kj + a 3 & 3 kk. 

Since ab = a b corresponding coefficients are equal. Hence 

a 1 :a 2 :a s = a 1 :a 2 :a 3 , 
which shows that the vectors a and a are collinear. 

And & 1 :,:6 8 = V- / - V. 

which shows that the vectors b and V are collinear. 

But a l b l = a/ &/. 

This shows that the product of the lengths (including sign) 
are equal and the theorem is proved. 

The proof may be carried out geometrically as follows. 
Since ab is equal to a V 

ab r = a b r 

for all values of r. Let r be perpendicular to b. Then b r 
vanishes and consequently Vr also vanishes. This is true 
for any vector r in the plane perpendicular to b. Hence b and 
b are perpendicular to the same plane and are collinear. In 
like manner by using a b as a postf actor a and a are seen 
to be parallel. Also 

ab-b = a b -b, 

which shows that the products of the lengths are the same. 

18 



274 VECTOR ANALYSIS 

The indeterminate product ab imposes Jive conditions upon 
the vectors a and b. The directions of a and b are fixed and 
likewise the product of their lengths. The scalar product 
a b, being a scalar quantity, imposes only one condition upon 
a and b. The vector product a x b, being a vector quantity, 
imposes three conditions. The normal to the plane of a and 
b is fixed and also th e area of the parallelogram of which they 
are the side. The nine indeterminate products (12) of i,j, k 
into themselves are independent. The nine scalar products 
are not independent. Only two of them are different. 



and i.j=j.i=j.k = kj=ki = ik = 0. 

The nine vector products are mot independent either; for 

ixi = jxj = kxk = 0, 

and ixj = jxi, jxk= kxj, kxi ixk. 

The two products a b and a x b obtained respectively from 
the indeterminate product by inserting a dot and a cross be 
tween the factors are functions of the indeterminate product. 
That is to say, when ab is given, a b and a x b are determined. 
For these products depend solely upon the directions of a and b 
and upon the product of the length of a and b, all of which 
are known when ab is known. That is 

if ab = a b , a b = a b and a x b = a x b . (17) 

It does not hold conversely that if a b and a x b are known 
a b is fixed ; for taken together a b and a X b impose upon the 
vectors only four conditions, whereas a b imposes five. Hence 
a b appears not only as the most general product but as the 
most fundamental product. The others are merely functions 
of it. Their functional nature is brought out clearly by the 
notation of the dot and the cross. 



LINEAR VECTOR FUNCTIONS 275 

Definition: A scalar known as the scalar of may be ob 
tained by inserting a dot between the antecedent and conse 
quent of each dyad in a dyadic. This scalar will be denoted 
by a subscript S attached to 0. l 
If <P = a 1 b 1 + a 2 b 2 + a 3 b 3 + ... 

8 = &1 b x + a 2 b 2 + a 3 . b 3 + . . (18) 

In like manner a vector known as the vector of may be 
obtained by inserting a cross between the antecedent and con 
sequent of each dyad in 0. This vector will be denoted by 
attaching a subscript cross to 0. 

X = aj x b x + a 2 x b 2 + a 3 x b 3 + . - (19) 
If be expanded in nonion form in terms of i, j, k, 

s = a n + a^ + a BZ , (20) 

#x = ( 28 - a 3 2 ) * + 0*31 - a ! 3 ) J + (^12 - a 2l) k - ( 21 ) 

Or S = i- 0-i + j- <Pj + k. (?-k, (20) 

<? x =(j . (P-k-k* ^.j) i+ (k- (P-i-i. (P.k) j 

+ (i- 0-j-j.0.i)k. (21) 

In equations (20) and (21) the scalar and vector of are 
expressed in terms of the coefficients of when expanded 
in the nonion form. Hence if and W are two equal 
dyadics, the scalar of is equal to the scalar of and the 
vector of is equal to the vector of . 

If = W, S = s and X = y x . (22) 

From this it appears that S and X are functions of 
uniquely determined when is given. They may sometimes 
be obtained more conveniently from (20) and (21) than from 
(18) and (19), and sometimes not. 

1 A subscript dot might be used for the scalar of * if it were sufficiently distinct 
and free from liability to misinterpretation. 



276 VECTOR ANALYSIS 

Products of Dyddics 

103.] In giving the definitions and proving the theorems 
concerning products of dyadics, the dyad is made the under 
lying principle. What is true for the dyad is true for the 
dyadic in general owing to the fact that dyads and dyadics 
obey the distributive law of multiplication. 

Definition: The direct product of the dyad ab into the 

dyad c d is written , , x , ,. 

(ab) (cd) 

and is by definition equal to the dyad (b c) a d, 

(ab)-(cd) = a(b.c)d = b-c ad. 1 (23) 

That is, the antecedent of the first and the consequent of the 
second dyad are taken for the antecedent and consequent 
respectively of the product and the whole is multiplied by 
the scalar product of the consequent of the first and the 
antecedent of the second. 

Thus the two vectors which stand together in the product 

(ab). (cd) 

are multiplied as they stand. The other two are left to form 
a new dyad. The direct product of two dyadics may be 
defined as the formal expansion (according to the distributive 
law) of the product into a sum of products of dyads. Thus 

*=(a 1 b 1 + a 2 b 2 + a 3 b 3 + ...) 
and r^CCjdj + c 2 d 2 + c 3 d 3 + -..) 

d>. ?T=(a 1 b 1 +a 2 b 2 + a 3 b 3 + ) 

(c^j + c 2 d 2 + C 3 d 3 +) 
= a 1 b 1 c 1 d 1 + a 1 b 1 *e 2 d a + a x b x C 3 d 3 + 
+ a 2 b 2 -c 1 d 1 +a 2 b 2 .c 2 d 2 + a 2 b 2 C 3 d 3 + (23) 

+ agbg-c^ + a 3 b 3 -c 2 d 2 + a 3 b 3 c 3 d 3 H 

+ 

1 The parentheses may be omitted in each of these three expressions. 



LINEAR VECTOR FUNCTIONS 277 

x ajdj + bj c 2 a x d 2 -f b x c 3 a x d 3 + 
l a 2 d x + b 2 -c 2 a 2 d 2 + b 2 c 3 a^j d 3 -f- 



b 3 .c 2 a 3 d 2 + b 3 -c 3 a 3 d 3 



(23)" 



The product of two dyadics and W is a dyadic W. 

Theorem : The product W of two dyadics (P and W when 
regarded as an operator to be used as a prefactor is equiva 
lent to the operator W followed by the operator 0. 

Let =&.. 

To show Q r = d> ( W r), 

or ((? W)*T = 0- (^ 0- (24) 

Let ab be any dyad of <? and c d any dyad of W. 

(ab cd) r = b c (ad r) = (b c) (d r) a, 
ab (c d r) = a b c (d r) = (b c) (d r) a, 
Hence (a b c d) r = a b (c d r). 

The theorem is true for dyads. Consequently by virtue of 
the distributive law it holds true for dyadics in general. 

If r denote the position vector drawn from an assumed origin 
to a point P in space, r = W r will be the position vector of 
another point P , and r" = (^(3 r r) will be the position 
vector of a third point P n . That is to say, W defines a trans 
formation of space such that the points P go over into the 
points P f . defines a transformation of space such that the 
points P f go over into the points P". Hence W followed by 
carries P into P ff . The single operation W also carries 
PintoP". 

Theorem: Direct multiplication of dyadics obeys the dis* 
tributive law. That is 



278 VECTOR ANALYSIS 

( + f ) = . W + . 

and (0 f + 0) W = f . + . W. (25) 

Hence in general the product 

(4>+ 4> + 4>" + ...).( W+ + ?F"+...) 

may be expanded formally according to the distributive law. 

Theorem : The product of three dyadics <P, W, Q is associa- 
tive. Thatis ( t.r). o= t. (ma> (26) 

and consequently either product may be written without 
parentheses, as . V . Q, (26 ) 

The proof consists in the demonstration of the theorem for 
three dyads ab, cd, ef taken respectively from the three 
dyadics 4>, , Q. 

(abcd) ef = (bc) ad ef = (bc) (d- e) af, 
ab (cdef) = (d^e) ab -cf = (d e) (b c) af. 

The proof may also be given by considering 0, W, and Q 
as operators 



Let 
Let 
Again {^.(f. J)} . r = *. [(f. J2).r]. 



Hence {(* F) Q\ -r = {(? (V T)\ r 

for all values of r. Consequently 



LINEAR VECTOR FUNCTIONS 279 

The theorem may be extended by mathematical induction 
to the case of any number of dyadics. The direct product 
of any number of dyadics is associative. Parentheses may 
be inserted or omitted at pleasure without altering the result. 

It was shown above (24) that 

(<P T) r = . ( - r) = <P V r. (24) 

Hence the product of two dyadics and a vector is associative. 
The theorem is true in case the vector precedes the dyadics 
and also when the number of dyadics is greater than two. 
But the theorem is untrue when the vector occurs between 
the dyadics. The product of a dyadic, a vector, and another 
dyadic is not associative. 

(#.r). V 0.(r- ). (27) 

Let ab be a dyad of $, and c d a dyad of . 

(a b r) c d = b r (a c d) = (b r) (a c) d, 
ab (r c d) = ab d (r c) = b d (r c) a 
Hence (ab r) c d ab (r cd). 

The results of this article may be summed up as follows : 

Theorem: The direct product of any number of dyadics 
or of any number of dyadics with a vector factor at either 
end or at both ends obeys the distributive and associative 
laws of multiplication parentheses may be inserted or 
omitted at pleasure. But the direct product of any number 
of dyadics with a vector factor at some other position than at 
either end is not associative parentheses are necessary to 
give the expression a definite meaning. 

Later it will be seen that by making use of the conjugate 
dyadics a vector factor which occurs between other dyadics 
may be placed at the end and hence the product may be 
made to assume a form in which it is associative. 



280 VECTOR ANALYSIS 

104.] Definition: The skew products of a dyad ab into 
a vector r and of a vector r into a dyad ab are defined 
respectively by the equations 

(ab) x r = a(b x r), 
rx(ab) = (r x a)b. 

The skew product of a dyad and a vector at either end is a 
dyad. The obvious extension to dyadics is 



rrrajbj x r + a 2 b 2 xr + a 3 b 3 x r + ... 
r x = r x (a a b a + a 2 b 2 + a 3 b 3 + . . .) (28) 

= r x ajbj + r x a 2 b 2 + r x a 3 b 3 + ... 

Theorem: The direct product of any number of dyadics 
multiplied at either end or at both ends by a vector whether 
the multiplication be performed with a cross or a dot is 
associative. But in case the vector occurs at any other 
position than the end the product is not associative. That is, 

(rx <P) Sr = rx(0.y)=rx <P , 
(<P ?F) xr=(P.(?P xr) = <P.? r xr, 
(r x #) s = r x ( s) = r x <P s, (29) 

r . (0 x s) = (r </>) x s = r <P x s, 
rx($xs) = (rx $)xs = rx $xs, 
but !P (rX^)^(S jr -r) X*. 

Furthermore the expressions 

s r x <P and <P x r s 
can have no other meaning than 

s r x <P = s (r x <P), 



LINEAR VECTOR FUNCTIONS 281 

since the product of a dyadic with a cross into a scalar s r 
is meaningless. Moreover since the dot and the cross may 
be interchanged in the scalar triple product of three vectors 
it appears that 

s r x ^ = (s x r) 0, 

<p x r s = 4> (r x s), (31) 

and 0-(r x 5F) = (</> x r) V. 

The parentheses in the following expressions cannot be 
omitted without incurring ambiguity. 

<p.(r x s) (0-r) x s, 
(sx r). 0*sx(r-0), (31) 

(0-r) x * x(r. >). 

The formal skew product of two dyads a b and c d would be 
(ab) x (cd) = a(b x c)d. 

In this expression three vectors a, b x c, d are placed side 
by side with no sign of multiplication uniting them. Such 

an expression 

rst (32) 

is called a triad ; and a sum of such expressions, a triadic. 
The theory of triadics is intimately connected with the theory 
of linear dyadic functions of a vector, just as the theory of 
dyadics is connected with the theory of linear vector functions 
of a vector. In a similar manner by going a step higher 
tetrads and tetradics may be formed, and finally polyads and 
polyadics. But the theory of these higher combinations of 
vectors will not be taken up in this book. The dyadic 
furnishes about as great a generality as is ever called for in 
practical applications of vector methods. 



282 VECTOR ANALYSIS 

Degrees of Nullity of Dyadics 

105.] It was shown (Art. 101) that a dyadic could always 
be reduced to a sum of three terms at most, and this reduction 
can be accomplished in only one way when the antecedents 
or the consequents are specified. In particular cases it may 
be possible to reduce the dyadic further to a sum of two 
terms or to a single term or to zero. Thus let 

<P = al + bm + cn. 

If 1, m, n are coplanar one of the three may be expressed 
in terms of the other two as 

1 = x m + y n. 

Then $ = a#m + ayn + bm + cn, 

= (a# + b)m + (ay + c)n. 

The dyadic has been reduced to two terms. If 1, m, n were 
all collinear the dyadic would reduce to a single term and if 
they all vanished the dyadic would vanish. 

Theorem : If a dyadic be expressed as the sum of three 
terms 

<p = al + bm + en 

of which the antecedents a, b, c are known to be non-coplanar, 
then the dyadic may be reduced to the sum of two dyads 
when and only when the consequents are coplanar. 

The proof of the first part of the theorem has just been 
given. To prove the second part suppose that the dyadic 
could be reduced to a sum of two terms 

$ = dp + eq 

and that the consequents 1, m, n of were non-coplanar. 
This supposition leads to a contradiction. For let 1 , m , n 
be the system reciprocal to 1, m, n. That is, 

mx n n x 1 1 x m 



_ 
= 



[Tmn] 



LINEAR VECTOR FUNCTIONS 283 

The vectors 1 , m f , n exist and are non-coplanar because 
1, m, n have been assumed to be non-coplanar. Any vector r 
may be expressed in terms of them as 

r = xl f + ym + zn/ 

<p.r = (al + bm + en) (xl 1 + ym + zn ). 
But 1 1 = m m = n n = 1, 

and 1 m = 1 m = m n = m n = n 1 = n 1 = 0. 
Hence $ r = x a + y b + z,e. 

By giving to r a suitable value the vector d> r may be made 
equal to any vector in space. 

But r = (dp + e q) - r = d (p r) + e (q r). 

This shows that r must be coplanar with d and e. Hence 
r can take on only those vector values which lie in the 
plane of d and e. Thus the assumption that 1, m, n are non- 
coplanar leads to a contradiction. Hence 1, m, n must be 
coplanar and the theorem is proved. 

Theorem : If a dyadic be expressed as the sum of three 
terms 



of which the antecedents a, b, c are known to be non-coplanar, 
the dyadic can be reduced to a single dyad when and only 
when the consequents 1, m, n are collinear. 

The proof of the first part was given above. To prove 
the second part suppose <P could be expressed as 



Let 



284 VECTOR ANALYSIS 

From the second equation it is evident that W used as a 
postfactor for any vector 

r = x a + y b + zc , 
where a , V, c is the reciprocal system to a, b, c gives 



From the first expression 

r = 0. 
Hence #lxp+ymxp + znxp 

must be zero for every value of r, that is, for every value of x, 
y, z. Hence 

1 x p = 0, mxp = 0, nxp = 0. 

Hence 1, m, and n are all parallel to p and the theorem has 
been demonstrated. 

If the three consequents 1, m, n had been known to be non- 
coplanar instead of the three antecedents, the statement of 
the theorems would have to be altered by interchanging the 
words antecedent and consequent throughout. There is a fur 
ther theorem dealing with the case in which both antecedents 
and consequents of are coplanar. Then is reducible to 
the sum of two dyads. 

106.] Definition: A dyadic which cannot be reduced to 
the sum of fewer than three dyads is said to be complete. A 
dyadic which may be reduced to the sum of two dyads, but 
cannot be reduced to a single dyad is said to be planar. In 
case the plane of the antecedents and the plane of the con 
sequents coincide when the dyadic is expressed as the sum of 
two dyads, the dyadic is said to be uniplanar. A dyadic 
which may be reduced to a single dyad is said to be linear. 
In case the antecedent and consequent of that dyad are col- 



LINEAR VECTOR FUNCTIONS 285 

linear, the dyadic is said to be unilinear. If a dyadic may be 
so expressed that all of its terms vanish the dyadic is said to 
be zero. In this case the nine coefficients of the dyadic as 
expressed in nonion form must vanish. 

The properties of complete, planar, uniplanar, linear, and 
unilinear dyadics when regarded as operators are as follows. 
Let 

s = r and t = r <P. 

If is complete s and t may be made to take on any desired 
value by giving r a suitable value. 



As is complete 1, m, n are non-coplanar and hence have a 
reciprocal system l f , m , n . 



s = . (xl f + ym f + zn ) =#a + yb + zc. 
In like manner a, b, c possess a system of reciprocals a , V, c . 
yb + zc ) = xl + ym + zn. 



A complete dyadic applied to a vector r cannot give zero 
unless the vector r itself is zero. 

If is planar the vector s may take on any value in the plane 
of the antecedents and t any value in the plane of the consequents 
of ; but no values out of those planes. The dyadic when 
used as a prefactor reduces every vector r in space to a vector 
in the plane of the antecedents. In particular any vector r 
perpendicular to the plane of the consequents of is reduced 
to zero. The dyadic used as a postfactor reduces every 
vector r in space to a vector in the plane of the consequents 
of <P. In particular a vector perpendicular to the plane of 
the antecedents of is reduced to zero. In case the dyadic 
is uniplanar the same statements hold. 

If is linear the vector s may take on any value collinear 
with the antecedent of and t any value collinear with the con- 



286 VECTOR ANALYSIS 

sequent of ; but no other values. The dyadic used as a 
prefactor reduces any vector r to the line of the antecedent 
of 0. In particular any vectors perpendicular to the con 
sequent of are reduced to zero. The dyadic used as a 
postfactor reduces any vector r to the line of the consequent 
of 0. In particular any vectors perpendicular to the ante 
cedent of are thus reduced to zero. 

If is a zero dyadic the vectors s and t are loth zero no 
matter what the value of r may be. 

Definition : A planar dyadic is said to possess one degree of 
nullity. A linear dyadic is said to possess two degrees of 
nullity. A zero dyadic is said to possess three degrees of nul 
lity or complete nullity. 

107.] Theorem : The direct product of two complete dyadics 
is complete; of a complete dyadic and a planar dyadic, 
planar ; of a complete dyadic and a linear dyadic, linear. 

Theorem: The product of two planar dyadics is planar 
except when the plane of the consequent of the first dyadic 
in the product is perpendicular to the plane of the antece 
dent of the second dyadic. In this case the product reduces 
to a linear dyadic and only in this case. 

Let B, I \) I + a 2 b 2 , 

Q = . W. 

The vector s = W r takes on all values in the plane of Cj 
and c 2 



The vector s f = s takes on the values 

g = . s = x (b x c x ) a x + y (b x c 2 ) a x 

+ x ( b 2 C l) a 2 + y 0>2 C ) E 2> 

s = \x (bj c x ) + y (bj c 2 )} a x + {x (b 2 c x ) + y (b 2 c 2 )} a 2 . 



LINEAR VECTOR FUNCTIONS 287 

Let s = x &i + y a 2 , 

where x 1 = x (b x Cj) + y (b l c 2 ), 

and y 1 x (b 2 Cj) 4- y (b 2 c 2 ). 

These equations may always be solved for x and y when 
any desired values x 1 and y are given that is, when s has 
any desired value in the plane of EJ and a 2 unless the 

determinant 

V^ bj-c, 

b 2 c x b 2 c 2 
But by (25), Chap. II., this is merely the product 

0>i x t> 2 ) ( c i x C 2) = - 

The vector \ x b 2 is perpendicular to the plane of the con 
sequents of <P; and c l x c 2 , to the plane of the antecedents of 
. Their scalar product vanishes when and only when the 
vectors are perpendicular that is, when the planes are per 
pendicular. Consequently s may take on any value in the 
plane of a x and a 2 and is therefore a planar dyadic 
unless the planes of b x and b 2 , c x and c 2 are perpendicular. 
If however b x and b 2 , Cj and c 2 are perpendicular s f can take 
on only values in a certain line of the plane of a x and a^ and 
hence <P W is linear. The theorem is therefore proved. 

Theorem : The product of two linear dyadics is linear 
except when the consequent of the first factor is perpen 
dicular to the antecedent of the second. In this case the 
product is zero and only in this case. 

Theorem : The product of a planar dyadic into a linear is 
linear except when the plane of the consequents of the 
planar dyadic is perpendicular to the antecedent of the linear 
dyadic. In this case the product is zero and only in this 
case. 

Theorem: The product of a linear dyadic into a planar 
dyadic is linear except when the consequent of the linear 



288 VECTOR ANALYSIS 

dyadic is perpendicular to the plane of the antecedents of 
the planar dyadic. In this case the product is zero and 
only in this case. 

It is immediately evident that in the cases mentioned the 
products do reduce to zero. It is not quite so apparent that 
they can reduce to zero in only those cases. The proofs are 
similar to the one given above in the case of two planar 
dyadics. They are left to the reader. The proof of the 
first theorem stated, page 286, is also left to the reader. 

The Idemfactor; 1 Reciprocals and Conjugates of Dyadics 

108.] Definition : If a dyadic applied as a pre factor or as 
a postf actor to any vector always yields that vector the 
dyadic is said to be an idemf actor. That is 

if r = r for all values of r, 

or if r = r for all values of r, 

then is an idemfactor. The capital I is used as the sym 
bol for an idemfactor. The idemfactor is a complete dyadic. 
For there can be no direction in which I r vanishes. 
Theorem : When expressed in nonion form the idemfactor is 

I = ii + jj + kk. (33) 

Hence all idemfactors are equal. 

To prove that the idemfactor takes the form (33) it is 
merely necessary to apply the idemfactor I to the vectors 
i, j, k respectively. Let 

1 = a n ii+ 12 ij + a 13 ik 



ki + a 32 kj + a 33 kk. 



1 In the theory of dyadics the idemfactor I plays a role analogous to unity in 
ordinary algebra. The notation is intended to suggest this analogy. 



LINEAR VECTOR FUNCTIONS 289 

I . i = a n i + a 21 j 4- a 31 k. 
If I-i = i, 

a n = 1 and a 21 = a 31 = 0. 

In like manner it may be shown that all the coefficients 
vanish except a n , a 22 , a 33 all of which are unity. Hence 

I = ii + jj + kk. (33) 

Theorem : The direct product of any dyadic and the idem- 
factor is that dyadic. That is, 

I = and 1-0 = 0. 
For (0 I) r = (I r) = r, 

no matter what the value of r may be. Hence, page 266, 



In like manner it may be shown that I = 0. 

Theorem: If a , V, c and a, b, c be two reciprocal systems 
of vectors the expressions 

I = aa + bb + cc , (34) 

I = a a + b b + c c 
are idemfactors. 

For by (30) and (31) Chap. II., 

r = raa + r*bb + r cc , 
and r = ra a + r b b + r.c c. 

Hence the expressions must be idemfactors by definition. 
Theorem : Conversely if the expression 

= al + bm + en 

is an idemfactor 1, m, n must be the reciprocal system of 
a, b, c. 

19 



290 VECTOR ANALYSIS 

In the first place since (P is the idemfactor, it is a complete 
dyadic. Hence the antecedents a, b, c are non-coplanar and 
possess a set of reciprocals a , b , c . Let 

r = #a + y V + 20 . 
By hypothesis r $ = r. 

Then r <P = xl + ym-\-zn = xsi + y b + zc 

for all values of r, that is, for all values of x, y, z. Hence the 
corresponding coefficients must be equal. That is, 



Theorem : If (Pand be any two dyadics, and if the product 
<? W is equal to the idemf actor ; l then the product W 0, 
when the factors are taken in the reversed order, is also 
equal to the idemfactor. 

Let V = L 

To show W = L 



r . (0 . W) - = (r 0) ( W (?) = r 0. 

This relation holds for all values of r. As is complete r 
must take on all desired values. Hence by definition 

W = I. 

If the product of two dyadics is an idemfactor, that product 
may be taken in either order. 

109.] Definition: When two dyadics are so related that 
their product is equal to the idemfactor, they are said to be 

1 This necessitates both the dyadics * and V to be complete. For the product 
of two incomplete dyadics is incomplete and hence could not be equal to the 
idemfactor. 



LINEAR VECTOR FUNCTIONS 291 

reciprocals. 1 The notation used for reciprocals in ordinary 
algebra is employed to denote reciprocal dyadics. That is, 

if 0.y=I, = ?F-i = 1 and 5T= 0-i=L (35) 

W 

Theorem: Reciprocals of the same or equal dyadics are 
equal. 

Let and W be two given equal dyadics, <J>~ 1 and JT" 1 
their reciprocals as defined above. By hypothesis 

0= W, 



and W. ~i = l. 

To show 0- 1 = ~i. 

0. 0-1 = 1= -1. 

As 0=, 0.0~i=0.-\ 



0-1.0 = I, 

I.0-i = 0-i = I. W~i = -1. 
Hence 0-i = ~\ 

The reciprocal of is the dyadic whose antecedents are the 
reciprocal system to the consequents of and whose conse 
quents are the reciprocal system to the antecedents of 0. 

If a complete dyadic be written in the form 

= al + bm + en, 

its reciprocal is 0" 1 = 1 a + m V + n c . (36) 

For (al + bm + cn) (1 a + n V + n c ) =aa + bV + ce . 

Theorem : If the direct products of a complete dyadic 
into two dyadics W and Q are equal as dyadics then W and Q 

1 An incomplete dyadic has no (finite) reciprocal. 



292 VECTOR ANALYSIS 

are equal. If the product of a dyadic into two vectors 
r and s (whether the multiplication be performed with a dot 
or a cross) are equal, then the vectors r and s are equal. 
That is, 

if - = d> J2, then = Q, 

and if r = s, then r = s, (37) 

and if x r = x s, then r = s. 

This may be seen by multiplying each of the equations 
through by the reciprocal of 0, 

0-1.0. W = = 0-i Q = , 
0-i . . r = r = 0~! s = s, 

0-1. 0xr = IXr=0" 1 - 0X8 = 1X8. 

To reduce the last equation proceed as follows. Let t be 
any vector, 

tIxr = tIxs, 

t I = t. 
Hence t x r = t x s. 

As t is any vector, r is equal to s. 

Equations (37) give what is equivalent to the law of can- 
celation for complete dyadics. Complete dyadics may be 
canceled from either end of an expression just as if they 
were scalar quantities. The cancelation of an incomplete 
dyadic is not admissible. It corresponds to the cancelation 
of a zero factor in ordinary algebra. 

110.] Theorem: The reciprocal of the product of any 
number of dyadics is equal to the product of the reciprocals 
taken in the opposite order. 

It will be sufficient to give the proof for the case in which 
the product consists of two dyadics. To show 



LINEAR VECTOR FUNCTIONS 293 



. V 5F- 1 0~ l = ( ?F y- 1 } . 0~ l = 0. 0- l = I. 
Hence (0 ?F) ( JF- 1 0- 1 ) = I. 

Hence ?T and W~ l (P" 1 must be reciprocals. That is, 



The proof for any number of dyadics may be given in the 
same manner or obtained by mathematical induction. 

Definition : The products of a dyadic <P, taken any number 
of times, by itself are called powers of and are denoted in 
the customary manner. 

. = 0*, 

. . = . 02 0^ 

and so forth. 

Theorem : The reciprocal of a power of <P is the power of 
the reciprocal of <P. 

(0)-i = (0- 1 )" = 0- (37) 

The proof follows immediately as a corollary of the preced 
ing theorem. The symbol <P" n may be interpreted as the 
nth power of the reciprocal of or as the reciprocal of 
the nth power of 0. 

If be interpreted as an operator determining a trans 
formation of space, the positive powers of correspond to 
repetitions of the transformation. The negative powers of 
correspond to the inverse transformations. The idemfactor 
corresponds to the identical transformation that is, no trans 
formation at all. The fractional and irrational powers of CP 
will not be defined. They are seldom used and are not 
single-valued. For instance the idemfactor I has the two 
square roots 1. But in addition to these it has a doubly 
infinite system of square roots of the form 

<P = -ii + jj + kk. 



294 VECTOR ANALYSIS 

Geometrically the transformation 



is a reflection of space in the j k-plane. This transformation 
replaces each figure by a symmetrical figure, symmetrically 
situated upon the opposite side of the j k-plane. The trans 
formation is sometimes called perversion. The idemfactor 
has also a doubly infinite system of square roots of the form 



Geometrically the transformation 

r = V.T 

is a reflection in the i-axis. This transformation replaces each 
figure by its equal rotated about the i-axis through an angle 
of 180. The idemfactor thus possesses not only two square 
roots ; but in addition two doubly infinite systems of square 
roots ; and. it will be seen (Art. 129) that these are by no 
means all. 

111.] The conjugate of a dyadic has been defined (Art. 99) 
as the dyadic obtained by interchanging the antecedents and 
consequents of a given dyadic and the notation of a subscript 
C has been employed. The equation 

r . = 4> r (9) 

has been demonstrated. The following theorems concerning 
conjugates are useful. 

Theorem : The conjugate of the sum or difference of two 
dyadics is equal to the sum or difference of the conjugates, 

(d> T) =0 C W c . 

Theorem : The conjugate of a product of dyadics is equal 
to the product of the conjugates taken in the opposite order. 



LINEAR VECTOR FUNCTIONS 295 

It will be sufficient to demonstrate the theorem in case 
the product contains two factors. To show 

(d>.T )c =W c .0 Ct (40) 

(0 . W) c . r = r (0 W) = (r 4>) 5F, 

r . = <P C . r, 

(r . <P) . W = V c . (r <P) = ^ <^. r. 
Hence (4> V) c = c .4> c . 

Theorem : The conjugate of the power of a dyadic is the 
power of the conjugate of the dyadic. 



This is a corollary of the foregoing theorem. The expression 
n c may be interpreted in either of two equal ways. 

Theorem : The conjugate of the reciprocal of a dyadic is 
equal to the reciprocal of the conjugate of the dyadic. 



= ^ (42) 

For (@~ l )c c = (& 0~ 1 )c = Ic = I- 

The idemfactor is its own conjugate as may be seen from 
the nonion form. 

I = ii + j j + kk 



Hence (^c)" 1 *c 

Hence C^) 1 = (*"% 

The expression ^ c -1 may therefore be interpreted in either 
of two equivalent ways as the reciprocal of the conjugate 
or as the conjugate of the reciprocal. 

Definition: If a dyadic is equal to its conjugate, it is said 
to be self -conjugate. If it is equal to the negative of its con- 



296 VECTOR ANALYSIS 

jugate, it is said to be anti-self -conjugate. For se//-conjugate 

dyadics. 

r = r, = C . 

For anti-self-conjugate dyadics 

r = r, = 00. 

Theorem : Any dyadic may be divided in one and only one 
way into two parts of which one is self-conjugate and the 
other anti-self-conjugate. 



For 0=5(0+0,) + 2 (0-0c). (43) 

But (& + c ) c = c +<p cc =: 4> c + d>, 

and (0 - 0<,) c = $ - <P CC = 0<, _ 0. 

Hence the part |(0 + &c) is self-conjugate; and the part 
|(^~ 4> c ), anti-self-conjugate. Thus the division has been 
accomplished in one way. Let 



and 



Suppose it were possible to decompose in another way 
into a self-conjugate and an anti-self-conjugate part. Let 
then 

= (0 + J2) + (0"-). 

Where (0 + 0) = (0 + ), = + c = f + ^ 
Hence if (0 ; -f J2) is self-conjugate, fi is self-conjugate. 



Hence if (0" J2) is anti-self-conjugate is anti-self- 
conjugate. 



LINEAR VECTOR FUNCTIONS 297 

Any dyadic which is both self-conjugate and anti-self-conju 
gate is equal to its negative and consequently vanishes. 
Hence Q is zero and the division of into two parts is 
unique. 

Anti-self-conjugate Dyadics. The Vector Product 
112.] In case is any dyadic the expression 



gives the anti-self-conjugate part of 0. If should be en 
tirely antinself-con jugate is equal to 0". Let therefore n 
be any anti-self-conjugate dyadic, 



Suppose <P = al-hbm-fcn, 

$ c = al la + bm mb + cn nc, 
20" r = a 1 r lar-hbmr mbr + cnr n c r. 
But a 1 r 1 a r = (a x 1) x r, 

bm r mb r = (b x m) x r, 
c n r n c r = (cxn)xr. 
Hence 0" r = ~ (a x 1 + b x m 4- c x n) x r. 
But by definition <P x = axl-fbxm4-cxn. 
Hence 0" r = - ~ X x r, 

r 0" = 0" c . r = - 0" r = I0 X x r = - \ r x X . 

The results may be stated in a theorem as follows. 

Theorem : The direct product of any anti-self-conjugate 
dyadic and the vector r is equal to the vector product of 
minus one half the vector of that dyadic and the vector r. 



VECTOR ANALYSIS 



Theorem : Any anti-self-conjugate dyadic <P !f possesses one 
degree of nullity. It is a uniplanar dyadic the plane of 
whose consequents and antecedents is perpendicular to <P X ", 
the vector of <P. 

This theorem follows as a corollary from equations (44). 

Theorem : Any dyadic may be broken up into two parts 
of which one is self-conjugate and the other equivalent to 
minus one half the vector of used in cross multiplication. 

<p . r = <P r r ^ <P X x r, 

or symbolically $ = \ # x X. (45) 

113.] Any vector c used in vector multiplication defines a 
linear vector function. For 

cx(r + s)=cxr + cxs. 

Hence it must be possible to represent the operator c x as a 
dyadic. This dyadic will be uniplanar with plane of its 
antecedents and consequents perpendicular to c, so that it 
will reduce all vectors parallel to c to zero. The dyadic may 
be found as follows 



By (31) I- (c x !) = (! x c) - 1, 

(I x c) r = \ (I x c) I } r = {I - (c x I)} r 

= I (c x I) r = (c x I) r. 
Hence c x r = (I x c) r = (e x I) r, 

and r x c = r (I x c) = r (c x I). (46) 

This may be stated in words. 



LINEAR VECTOR FUNCTIONS 299 

Theorem : The vector c used in vector multiplication with 
a vector r is equal to the dyadic I x c or c X I used in direct 
multiplication with r. If c precedes r the dyadics are to be 
used as prefactors ; if c follows r, as postfactors. The dyadics 
I X c and c X I are anti-self-conjugate. 

In case the vector c is a unit vector the application of the 
operator c X to any vector r in a plane perpendicular to c is 
equivalent to turning r through a positive right angle about 
the axis c. The dyadic c X I or I x c where c is a unit vector 
therefore turns any vector r perpendicular to c through a 
right angle about the line c as an axis. If r were a vector 
lying out of a plane perpendicular to c the effect of the dyadic 
I X c or c x I would be to annihilate that component of r which 
is parallel to c and turn that component of r which is perpen 
dicular to c through a right angle about c as axis. 

If the dyadic be applied twice the vectors perpendicular to 
r are rotated through two right angles. They are reversed in 
direction. If it be applied three times they are turned through 
three right angles. Applying the operator I x c or c X I four 
times brings a vector perpendicular to c back to its original 
position. The powers of the dyadic are therefore 

(I x c) 2 = (c x I) 2 = - (I - cc), 

(I x c) 3 = (c x I) 3 = - I x c = - c x I, 

(47) 
(I x c ) 4 = (c x I) 4 = I - c c, 

(I x c) 5 = (c x I) 6 = I x c = c x I. 

It thus appears that the dyadic I x c or c x I obeys the same 
law as far as its powers are concerned as the scalar imaginary 
V 1 in algebra. 

The dyadic Ixc orcxlisa quadrantal versor only for 
vectors perpendicular to c. For vectors parallel to c it acts 
as an annihilator. To avoid this effect and obtain a true 



300 VECTOR ANALYSIS 

quadrantal versor for all vectors r in space it is merely neces 
sary to add the dyad c c to the dyadic I X c or c X I. 

If X = Ixc + cc = cxI + cc, 

X 2 = I + 2cc, 

X 3 = iXc + cc, (48) 



The dyadic X therefore appears as a fourth root of the 
idemfactor. The quadrantal versor X is analogous to the 
imaginary V 1 of a scalar algebra. The dyadic X is com 
plete and consists of two parts of which I x c is anti-self- 
conjugate ; and c c, self-conjugate. 

114.] If i, j, k are three perpendicular unit vectors 

Ixi = ixl = kj jk, 
I xj-j x I = ik-ki, (49) 

Ixk=k x I=ji ij, 
as may be seen by multiplying the idemfactor 



into i, j, and k successively. These expressions represent 
quadrantal versors about the axis i,j, k respectively combined 
with annihilators along those axes. They are equivalent, 
when used in direct multiplication, to i x, jx, k X respectively, 



jj, 



The expression (I x k) 4 is an idemfactor for the plane of i and 
j, but an annihilator for the direction k. In a similar man 
ner the dyad k k is an idemfactor for the direction k, but an 



LINEAR VECTOR FUNCTION S 301 

annihilate! for the plane perpendicular to k. These partial 
idemfactors are frequently useful. 

If a, b, c are any three vectors and a , V, c the reciprocal 

system, 

aa + bb 

used as a prefactor is an idemfactor for all vectors in the 
plane of a and b, but an annihilator for vectors in the direc 
tion c. Used as a postfactor it is an idemfactor for all vectors, 
in the plane of a and V, but an annihilator for vectors in the 
direction c . In like manner the expression 



cc 



used as a prefactor is an idemfactor for vectors in the direction 
c, but for vectors in the plane of a and b it is an annihilator. 
Used as a postfactor it is an idemfactor for vectors in the 
direction c , but an annihilator for vectors in the plane of a 
and V, that is, for vectors perpendicular of c. 
If a and b are any two vectors 

(a x b) x I = I x (a x b) = ba - ab. (50) 

For 

{(a x b) x I}r = (a x b) x r = bar ab T = (ba ab>r. 

The vector a x b in cross multiplication is therefore equal to 
the dyadic (b a a b) in direct multiplication. If the vector 
is used as a prefactor the dyadic must be so used. 

(a xb) x r = (b a a b) r, 
r x (a x b) = r - (ba - ab). (51) 

This is a symmetrical and easy form in which to remember 
the formula for expanding a triple vector product. 



302 VECTOR ANALYSIS 



Reduction of Dyadics to Normal Form 

115.] Let be any complete dyadic and let r be a unit 
vector. Then the vector r 



is a linear function of r. When r takes on all values consis 
tent with its being a unit vector that is, when the terminus 
of r describes the surface of a unit sphere, the vector r 
varies continuously and its terminus describes a surface. This 
surface is closed. It is in fact an ellipsoid. 1 

Theorem : It is always possible to reduce a complete dyadic 
to a sum of three terms of which the antecedents among 
themselves and the consequents among themselves are mutu 
ally perpendicular. This is called the normal form of 0. 

<P = ai i + bj j + ck k. 
To demonstrate the theorem consider the surface described 

by 

r = 0-r. 

As this is a closed surface there must be some direction of r 
which makes r a maximum or at any rate gives r as great 
a value as it is possible for r to take on. Let this direction 
of r be called i, and let the corresponding direction of r 
the direction in which r takes on a value at least as great as 
any be called a. Consider next all the values of r which 
lie in a plane perpendicular to i. The corresponding values 
of r lie in a plane owing to a fact that (P r is a linear vector 

1 This may be proved as follows : 

r = * r, r^*- 1 - r / = r l .* c - 1 . 
Hence r .r=l=: r . (* e -i.* - 1 )- r = r V r . 

By expressing in nonion form, the equation r r = 1 is seen to be of the second 
degree. Hence r describes a quadric surface. The only closed quadric surface 
is the ellipsoid. 



LINEAR VECTOR FUNCTIONS 303 

function. Of these values of r one must be at least as great 
as any other. Call this b and let the corresponding direction 
of r be called j. Finally choose k perpendicular to i and j 
upon the positive side of plane of i and j. Let c be the 
value of r which corresponds to r = k. Since the dyadic 
changes i, j, k into a, b, c it may be expressed in the form 

<P = ai + bj + ck. 

It remains to show that the vectors a, b, c as determined 
above are mutually perpendicular. 

r = (ai + bj + ck)-r, 

dr f = (ai -f- bj + ck) -dr, 

r dr r = r ai di + r bj di + r - ck- dr. 

When r is parallel to i, r is a maximum and hence must be 
perpendicular to di r . Since r is a unit vector di is always 
perpendicular to r. Hence when r is parallel to i 

r b j-dr + r c kdr = 0. 

If further dr is perpendicular to j, r c vanishes, and if 
dr is perpendicular to k, r b vanishes. Hence when r is 
parallel to i, r is perpendicular to both b and c. But when 
r is parallel to i, r is parallel to a. Hence a is perpendicular 
to b and c. Consider next the plane of j and k and the 
plane of b and c. Let r be any vector in the plane of j and k. 

r = (bj + ck)r, 

dr r = (bj + ck) dr, 

r -dr = r -b j dr -f r c k-dr. 

When r takes the value j, r is a maximum in this plane and 
hence is perpendicular to dr f . Since r is a unit vector it is 



304 VECTOR ANALYSIS 

perpendicular to dr. Hence when r is parallel to j, dr 
is perpendicular to j, and 



Hence r c is zero. But when r is parallel to j, r takes the 
value b. Consequently b is perpendicular to c. 

It has therefore been shown that a is perpendicular to b and 
c, and that b is perpendicular to c. Consequently the three 
antecedents of are mutually perpendicular. They may be 
denoted by i , j , k . Then the dyadic $ takes the form 

4> = ai i +bj j +ck k, (52) 

where a, J, c are scalar constants positive or negative. 

116.] Theorem: The complete dyadic <? may always be 
reduced to a sum of three dyads whose antecedents and 
whose consequents form a right-handed rectangular system 
of unit vectors and whose scalar coefficients are either all 
positive or all negative. 

& = (ai i + fcj j + ck k). (53) 

The proof of the theorem depends upon the statements 
made on page 20 that if one or three vectors of a right-handed 
system be reversed the resulting system is left-handed, but 
if two be reversed the system remains right-handed. If then 
one of the coefficients in (52) is negative, the directions of the 
other two axes may be reversed. Then all the coefficients 
are negative. If two of the coefficients in (52) are negative, 
the directions of the two vectors to which they belong may 
be reversed and then the coefficients in are all positive. 
Hence in any case the reduction to the form in which all 
the coefficients are positive or all are negative has been 
performed. 

As a limiting case between that in which the coefficients 
are all positive and that in which they are all negative comes 



LINEAR VECTOR FUNCTIONS 305 

the case in which one of them is zero. The dyadic then 

takes the form 

<P = ai i +&j j (54) 

and is planar. The coefficients a and b may always be taken 
positive. By a proof similar to the one given above it is 
possible to show that any planar dyadic may be reduced to 
this form. The vectors i andj are perpendicular, and the 
vectors i and j are likewise perpendicular. 

It might be added that in case the three coefficients a, &, c 
in the reduction (53) are all different the reduction can be 
performed in only one way. If two of the coefficients (say 
a and 6) are equal the reduction may be accomplished in an 
infinite number of ways in which the third vector k is always 
the same, but the two vectors i , j to which the equal coeffi 
cients belong may be any two vectors in the plane per 
pendicular to k. In all these reductions the three scalar 
coefficients will have the same values as in any one of them. 
If the three coefficients a, 6, c are all equal when $ is reduced 
to the normal form (53), the reduction may be accomplished 
in a doubly infinite number of ways. The three vectors 
i , j , k may be any right-handed rectangular system in 
space. In all of these reductions the three scalar coefficients 
are the same as in any one of them. These statements will 
not be proved. They correspond to the fact that the ellipsoid 
which is the locus of the terminus of r may have three 
different principal axes or it maybe an ellipsoid of revolution, 
or finally a sphere. 

Theorem : Any self -con jugate dyadic may be expressed in 

the form = aii + &jj + ckk (55) 

where a, &, and c are scalars, positive or negative. 

Let <P = ai i -f Jj j +ck k, (52) 

+ 6jj + ckk , 
20 



306 VECTOR ANALYSIS 

0.0 c =a*i i f + & 2 j j +c a k k 
jj + c 2 kk. 



Since = 0^ 

0* C = C . 0= 0*. 

I * * + j j + k k n + j j 



If i and i were not parallel (& 2 a 2 !) would annihilate 
two vectors i and i and hence every vector in their plane. 
(0* a 2 I) would therefore possess two degrees of nullity 
and be linear. But it is apparent that if a, 6, c are different 
this dyadic is not linear. It is planar. Hence i and i must 
be parallel. In like manner- it may be shown that j and j , 
k and k are parallel. The dyadic therefore takes the form 

0= aii + bjj + ckk 
where a, J, c are positive or negative scalar constants. 

Double Multiplication * 

117.] Definition : The double dot product of two dyads is 
the scalar quantity obtained by multiplying the scalar product 
of the antecedents by the scalar product of the consequents. 
The product is denoted by inserting two dots between the 

ab:cd = a-c bd. (56) 

This product evidently obeys the commutative law 
ab:cd = cd:ab, 

1 The researches of Professor Gibbs upon Double Multiplication are here 
printed for the first time. 



LINEAR VECTOR FUNCTIONS 307 

and the distributive law both with regard to the dyads and 
with regard to the vectors in the dyads. The double dot 
product of two dyadics is obtained by multiplying the prod 
uct out formally according to the distributive law into the 
sum of a number of double dot products of dyads. 

If <p = * l }> l + a 2 b 2 + a 3 b 3 + ... 

and W = G! d x + c 2 d 2 + c 8 d 3 + 



= a 1 b 1 :o 1 d 1 4- a 1 b 1 :c 2 d 2 + a 1 b 1 :c 3 d 3 + 

+ aab^Cjdj + a 2 b 2 :c 2 d 2 + a 2 b 2 :c 3 d 3 + (56) 

+ a 3 b 3 :c 1 d 1 + a 3 b 3 :c 2 d 2 + a 3 b 3 :c 3 d s + . . 



+ a a c 1 bg-d! -f d 2 c 2 b 2 d 2 + a 2 -c 3 b 2 d 3 + 
+ a 3 -c 1 bg.djH-a3.C2 b 3 -d 2 + a 3 .c 3 b 3 .d 3 + --- 
+ ............... (66)" 

Definition: The double cross product of two dyads is the 
dyad of which the antecedent is the vector product of the 
antecedents of the two dyads and of which the consequent is 
the vector product of the consequent of the two dyads. The 
product is denoted by inserting two crosses between the 
dyads 

abcd = axc b x d. (57) 

This product also evidently obeys the commutative law 
ab cd = cd * ab, 



308 VECTOR ANALYSIS 

and the distributive law both with regard to the dyads and 
with regard to the vectors of which the dyads are composed. 
The double cross product of two dyadics is therefore defined 
as the formal expansion of the product according to the 
distributive law into a sum of double cross products of 
dyads. 

If <P = a 1 b 1 + a 2 b 2 + a 3 b 3 + ... 

and *F = c l & 1 + c 2 d 2 + C 3 d 3 + 

* y = (a^ + a 2 b 2 + a 3 b 3 + ) x (c^ + C 2 d 2 

+ c 3 a 3 + ...) 

= a 1 b 1 * Ojdj + a x b x * c 2 d 2 + ajbj * c 3 d 3 + 

+ a 2 b 2 * cjdj + a 2 b 2 c 2 d 2 + a 2 b 2 * c 3 d 3 + ... (57) 

+ a a b 3 x M! + a 3 b 3 * c 2 d 2 + a 3 b 3 ^ c 3 d 3 + 



c! b 2 xd 1 -fa 2 xc 2 b 2 xd 2 + a 2 xc 3 b a 
+ a 3 xc 3 b 3 xd x +a 3 xc 2 bgXdj +a 3 xc 3 b 3 
+ ............ ... (57)" 

Theorem : The double dot and double cross products of 
two dyadics obey the commutative and distributive laws of 
multiplication. But the double products of more than two 
dyadics (whenever they have any meaning) do not obey the 
associative law. 

d> : W :0 

$>*=*$ (58) 

(<P * T) I Q * I (^x)- 

The theorem is sufficiently evident without demonstration. 



LINEAR VECTOR FUNCTIONS 309 

Theorem : The double dot product of two fundamental 
dyads is equal to unity or to zero according as the two 
dyads are equal or different. 



ij:ki = i-k j i = 0. 

Theorem: The double cross product of two fundamental 
dyads (12) is equal to zero if either the antecedents or the 
consequents are equal. But if neither antecedents nor con 
sequents are equal the product is equal to one of the funda 
mental dyads taken with a positive or a negative sign. 

That is 

ij *ik =ix i j x k = 

ij *ki =i x k j x i = +jk. 

There exists a scalar triple product of three dyads in 
which the multiplications are double. Let <P, 5T, Q be any 
three dyadics. The expression 

* WiQ 

is a scalar quantity. The multiplication with the double 
cross must be performed first. This product is entirely in 
dependent of the order in which the factors are arranged or 
the position of the dot and crosses. Let ab, cd, and ef be 

three dyads, 

ab*cd:ef=[ace] [bdf]. (59) 

That is, the product of three dyads united by a double cross 
and a double dot is equal to the product of the scalar triple 
product of the three antecedents by the scalar triple product 
of the three consequents. From this the statement made 
above follows. For if the dots and crosses be interchanged 
or if the order of the factors be permuted cyclicly the two 
scalar triple products are not altered. If the cyclic order of 



310 VECTOR ANALYSIS 

the factors is reversed each scalar triple product changes 
sign. Their product therefore is not altered. 

118.] A dyadic may be multiplied by itself with double 

cross. Let 

<P = al + bm + en 



* = (al + bm + en) * (al-f bm + en) 

ss= a x a 1 x 1 + a x b Ixm + axc Ixn 
i 

+bxa mxl+bxb m x m + b x c mxn 
+ cxa nxl + cxb n x m + c x c n x n. 

The products in the main diagonal vanish. The others are 
equal in pairs. Hence 



0<P = 2(bxc mxn+cxa nxl-faxb Ixm). (60) 
If a, b, c and 1, m, n are non-coplanar this may be written 

+ b/m + c n > (60) 



The product fl> $ is a species of power of 0. It may be re 
garded as a square of The notation $ 2 will be employed 
to represent this product after the scalar factor 2 has been 
stricken out. 

0*0 

2 = ^ = (bxc mxn + cxa nxl + axb Ixm) (61). 
J 

The triple product of a dyadic expressed as the sum of 
three dyads with itself twice repeated is 

</>*$: = 2 $ 2 : <P 

<P 2 :0=(bxc mxn-fexa nxl + axb Ixm) 
: (al + bm + en). 

In expanding this product every term in which a letter is 
repeated vanishes. For a scalar triple product of three vec- 



LINEAR VECTOR FUNCTIONS 311 

tors two of which are equal is zero. Hence the product 
reduces to three terms only 

2 :0=[bca] [mnl] + [cab] [nlm] + [abc] [linn] 
or 2 : = 3 [a b c] [Imn] 

0*0:0 = 6 [abc] pmn]. 

The triple product of a dyadic by itself twice repeated is 
equal to six times the scalar triple product of its antecedents 
multiplied by the scalar triple product of its consequents. 
The product is a species of cube. It will be denoted by 8 
after the scalar factor 6 has been stricken out. 

0*0:0 

(62) 



119.] If 2 be called the second of ; and 8 , the third of 
0, the following theorems may be stated concerning the 
seconds and thirds of conjugates, reciprocals, and products. 

Theorem : The second of the conjugate of a dyadic is equal 
to the conjugate of the second of that dyadic. The third of 
the conjugate is equal to the third of the dyadic. 

<*,).= <.), 



Theorem: The second and third of the reciprocal of a 
dyadic are equal respectively to the reciprocals of the second 
and third. 

<*- ), = (*,)-!=*,* 
(f 1 ). = (*,)->-*.- 

Let = al + bm + cn 

<p- 1 = l a + m b + n c (36) 

a l + b m + c n 



n ] 



812 VECTOR ANALYSTS 

(*) - 1 [a b c ] [! m n ] (1 a + m b + n c) 



[a b c] [1 m n] 
But [a b c ] [a be] = 1 and [1 m n ] [Imn] = 1. 
Hence (0,)-* = (0- 1 )., = 0,-*. 

8 =[abc] [Imn], 



[abc] [Imn] 
C^-Oa = IX W] [1 m n ]. 
Hence (0,)- 1 = (*-), = 0,-*. 

Theorem: The second and third of a product are equal 
respectively to the product of the seconds and the product of 
the thirds. 

(f.f), = *,.*, 

(0. *),= *, ^3- 

Choose any three non-coplanar vectors 1, m, n as consequents 
of and let 1 , m , n be the antecedents of W. 

<P = al + b m + en, 

?T = l d + m e + n f, 

r = ad + be + cf, 

( . W\ =bxc exf + cxa fxd + axb dxe, 

<P 2 = bxc mxn-fcxa nxl + axb Ixm, 
?T 2 = m x n e x f + n x 1 f x d + 1 x m dxe. 
Hence 2 5P* 2 = bxc exf + cxa fxd + axb dxe. 
Hence (# 5T) 2 = <? 2 ?F 2 . 

(^. JT) 8 = [abc] [def] 



LINEAR VECTOR FUNCTIONS 313 

8 = [abc] [Imn], 

r g = P m n ] [defj. 
Hence 8 z = [a be] [def]. 

Hence (0.F),= 8 y,. 

Theorem : The second and third of a power of a dyadic are 
equal respectively to the powers of the second and third of 
the dyadic. 

(*"), = W=0," 

(0 ) 8 = W = 03 B 

Theorem : The second of the idemfactor is the idemfactor. 
The third of the idemfactor is unity. 

I = I 

1=1 (6T) 

lg 1. 

Theorem: The product of the second and conjugate of 
a dyadic is equal to the product of the third and the 
idemfactor. 

a . 0,= 8 I, (68) 

<P 2 = b x c mxn + cxa nxl + axb Ixm, 

C ;= la + mb + nc, 
<P 2 $<, = [1 m n] (b x c a + c x a b + a x b c). 

The antecedents a, b, c of the dyadic may be assumed to 
be non-coplanar. Then 

(b x c a + c x a b + a x b c) = [ab c] (a a + V b + c c) 

= [abc] I. 
Hence 2 <& c = ^> 3 1 . 

120.] Let a dyadic be given. Let it be reduced to the 
sum of three dyads of which the three antecedents are 
non-coplanar. 



314 VECTOR ANALYSIS 

= al + b m + cn, 

2 = b x c mxn + cxa nxl + axb 1 x m, 

[Imn]. 



Theorem: The necessary and sufficient condition that a 
dyadic be complete is that the third of be different from 
zero. 

For it was shown (Art. 106) that both the antecedents and 
the consequents of a complete dyadic are non-coplanar. 
Hence the two scalar triple products which occur in 8 
cannot vanish. 

Theorem: The necessary and sufficient condition that a 
dyadic $ be planar is that the third of shall vanish but the 
second of <P shall not vanish. 

It was shown (Art. 106) that if a dyadic be planar its con 
sequents 1, m, n must be planar and conversely if the conse 
quents be coplanar the dyadic is planar. Hence for a planar 
dyadic <P 8 must vanish. But $ 2 cannot vanish. Since a, 
b, c have been assumed non-coplanar, the vectors b x c, c x a, 
a x b are non-coplanar. Hence if 2 vanishes each of the 
vectors mxn, nxl, Ixm vanishes that is, 1, m, n are col- 
linear. But this is impossible since the dyadic is planar 
and not linear. 

Theorem: The necessary and sufficient condition that a 
non-vanishing dyadic be linear is that the second of 0, and 
consequently the third of 0, vanishes. 

For if be linear the consequents 1, m, n, are collinear. 
Hence their vector products vanish and the consequents of 
<P 2 vanish. If conversely <P 2 vanishes, each of its consequents 
must be zero and hence these consequents of are collinear. 

The vanishing of the third, unaccompanied by the vanish 
ing of the second of a dyadic, implies one degree of nullity. 
The vanishing of the second implies two degrees of nullity. 



LINEAR VECTOR FUNCTIONS 315 

The vanishing of the dyadic itself is complete nullity. The 
results may be put in tabular form. 

8 ^0, is complete. 
<P 3 = 0, # 2 * 0, is planar. (69) 

8 = 0, </> 2 = 0, <P * 0, is linear. 

It follows immediately that the third of any anti-self-conjugate 
dyadic vanishes; but the second does not. For any such 
dyadic is planar but cannot be linear. 

Nonion Form. Determinants. 1 Invariants of a Dyadic 
121.] If be expressed in nonion form 

= a u ii + a 12 i j + a 18 ik (13) 



+ a 81 ki + a 32 kj + a 33 kk. 

The conjugate of <P has the same scalar coefficients as 0, but 
they are arranged symmetrically with respect to the main 
diagonal. Thus 



( 70 ) 



The second of $ may be computed. Take, for instance, one 
term. Let it be required to find the coefficient of ij in C? 2 . 
What terms in can yield a double cross product equal to 
ij? The vector product of the antecedents must be i and 
the vector product of the consequents must be j. Hence the 
antecedents must be j and k ; and the consequents, k and i. 
These terms are 

021J 1 x33 kk = - a 2l a 33 i J 

a 31 k i J a 23 j k = a 31 a 23 i j. 

1 The results hold only for determinants of the third order. The extension to 
determinants of higher orders is through Multiple Algebra. 



316 VECTOR ANALYSIS 

Hence the term in i j in $ 2 is 



This is the first minor of a 19 in the determinant 

a t 



*12 

a i 
a* 



This minor is taken with the negative sign. That is, the 
coefficient of i j in 2 is what is termed the cof actor of the 
coefficient of i j in the determinant. The cofactor is merely 
the first minor taken with the positive or negative sign 
according as the sum of the subscripts of the term whose 
first minor is under consideration is even or odd. The co 
efficient of any dyad in 2 is easily seen to be the cofactor of 
the corresponding term in $. The cofactors are denoted 
generally by large letters. 



is the cofactor of a* 



33 



n 



is the cofactor of a 12 . 



is the cofactor of a 32 . 



With this notation the second of becomes 



ik 
kk 



(71) 



The value of the third of <P may be obtained by writing 
as the sum of three dyads 

= (a n i + a 21 j + a sl k) i + (a 12 i + a 22 j + a 32 k) j 

+ (a 13 i + a 23 j + a 33 k)k 



LINEAR VECTOR FUNCTIONS 317 

^3 = [Oil i + 21 J + "31 k ) (21 * + 22 J + a 33 k ) 



This is easily seen to be equal to the determinant 



a 1 
2 
a Q 



a i 
a z 
a* 



(72) 



For this reason 3 is frequently called the determinant of 
and is written 

<P 3 = I I (72) 

The idea of the determinant is very natural when is 
regarded as expressed in nonion form. On the other hand 
unless be expressed in that form the conception of $ 3 , 
the third of $, is more natural. 

The reciprocal of a dyadic in nonion form may be found 
most easily by making use of the identity 

2 .</> c =0 3 I (68) 



or 



or 



Hence 0" 1 = 



(73) 



318 VECTOR ANALYSIS 

If the determinant be denoted by D 



(73) , 



If is a second dyadic given in nonion form as 



+ 6 31 ki + & 32 kj + & 33 kk, 

the product W of the two dyadics may readily be found 
by actually performing the multiplication 

. = O n 6 U + a 12 6 21 + a 18 6 81 ) ii + (a u 6 12 + a la 6 22 

+ 6 i J + a 6 + a 6 + ik 



6 32) k J + ( a 31 6 12 + a 32 6 23 + a 33 & 33> k k 

: W = a n i n + a 12 6 12 + o 18 J 18 



6 21 



31 



Since the third or determinant of a product is equal to the 
product of the determinants, the law of multiplication of 
determinants follows from (65) and (74). 



LINEAR VECTOR FUNCTIONS 



319 



"11 



"21 



a. 



i 



a 22 a 23 

Ojrtn dinn 



a n 6 19 + io &, 



"12 



22 



a 32 6 22 



"11 



"31 



23 



&12 13 
&22 6 23 
& 32 6 33 



!3 



"11 "13 
*21 & 13 
K 31 6 13 



"21 
^31^ 

a 



*32 ( 



4- a 



33 



12 



22 



32 



!3 



23 



(76) 



The rule may be stated in words. To multiply two deter 
minants form the determinant of which the element in the 
mth row and nth column is the sum of the products of the 
elements in the rath row of the first determinant and nth 
column of the second. 

If = al 



<? 2 = bxc mxn + cxa nxl + axb Ixm. 

Then 

I 2 I =(^ 2 ) 3 = [bxc cxa axb] [mxn nxl Ixm] 
Hence I <P 2 I = (<P 2 ) 3 = [a b c] 2 [1 m n] 2 = <P 3 2 . 



Hence 



n 



22 



33 



*ia 



"22 



a 2 
a 



(77) 



The determinant of the cofactors of a given determinant of 
the third order is equal to the square of the given determinant. 
122.] A dyadic has three scalar invariants that is 
three scalar quantities which are independent of the form in 
which ^ is expressed. These are 



the scalar of <P, the scalar of the second of <P, and the third 
or determinant of 0. If be expressed in nonion form these 
quantities are 



320 



VECTOR ANALYSIS 



(78) 



*11 

hi 



32 



*18 



33 



No matter in terms of what right-handed rectangular system 
of these unit vectors may be expressed these quantities are 
the same. The scalar of is the sum of the three coefficients 
in the main diagonal. The scalar of the second of is the 
sum of the first minors or cofactors of the terms in the 
main diagonal. The third of is the determinant of the 
coefficients. These three invariants are by far the most 
important that a dyadic possesses. 

Theorem : Any dyadic satisfies a cubic equation of which 
the three invariants S , 0%& (P 3 are the coefficients. 

By (68) (0-xI\*(0-xY) c = (0-xl\ 

#n x a 12 a 13 

21 #22 X #23 

#31 #32 #33 X 

Hence (# x I) 3 = Z x 2S + x 2 S x* 
as may be seen by actually performing the expansion. 

(<p __ x i) 2 . f<p _ x !)<; = Z x 0^ + x 2 S x*. 

This equation is an identity holding for all values of the 
scalar x. It therefore holds, if in place of the scalar x, the 
dyadic which depends upon nine scalars be substituted. 
That is 



But the terms upon the left are identically zero. Hence 



LINEAR VECTOR FUNCTIONS 321 

This equation may be called the Hamilton-Cayley equation. 
Hamilton showed that a quaternion satisfied an equation 
analogous to this one and Cayley gave the generalization to 
matrices. A matrix of the Tith order satisfies an algebraic 
equation of the nth degree. The analogy between the theory 
of dyadics and the theory of matrices is very close. In fact, 
a dyadic may be regarded as a matrix of the third order and 
conversely a matrix of the third order may be looked upon as 
a dyadic. The addition and multiplication of matrices and 
dyadics are then performed according to the same laws. A 
generalization of the idea of a dyadic to spaces of higher 
dimensions than the third leads to Multiple Algebra and the 
theory of matrices of orders higher than the third. 

SUMMARY OF CHAPTER V 

A vector r is said to be a linear function of a vector r 
when the components of r are linear homogeneous functions 
of the components of r. Or a function of r is said to be a 
linear vector function of r when the function of the sum of 
two vectors is the sum of the functions of those vectors. 

(ri + r a )=f(r 1 ) + f(r a ). (4) 

These two ideas of a linear vector function are equivalent. 
A sum of a number of symbolic products of two vectors, 
which are obtained by placing the vectors in juxtaposition 
without intervention of a dot or cross and which are called 
dyads, is called a dyadic and is represented by a Greek 
capital. A dyadic determines a linear vector function of 
a vector by direct multiplication with that vector 

= &1 b x + a 2 b 2 + a 3 b 3 + - (7) 

- r = a x bj r + a 2 b 2 r + a 3 b 3 r H (8) 

21 



322 VECTOR ANALYSIS 

Two dyadics are equal when they are equal as operators 
upon all vectors or upon three non-coplanar vectors. That 
is, when 

<P i = W r for all values or for three non- 

coplanar values of r, (10) 

or r = r for all values or for three non- 

coplanar values of r, 

or s r = s W r for all values or for three non- 
coplanar values of r and s. 

Any linear vector function may be represented by a dyadic. 

Dyads obey the distributive law of multiplication with 
regard to the two vectors composing the dyad 



(a + b + c+ ) (1 + m + n + ...) = al + am + an+ 

+ bl + bm + bn + 
+ cl + cm + en + 



(11) 

Multiplication by a scalar is associative. In virtue of these 
two laws a dyadic may be expanded into a sum of nine terms 
by means of the fundamental dyads, 

ii, ij, ik, 

ji, Jj, Jk, (12) 

ki, kj, kk, 
as = a n ii + a 12 i j + a 18 ik, 

= ai J * + <*22 J J + <*23 J k ( 13 ) 

= a 31 k i + a 82 k j + a 33 k k. 

If two dyadics are equal the corresponding coefficients in 
their expansions into nonion form are equal and conversely 



LINEAR VECTOR FUNCTIONS 323 

Any dyadic may be expressed as the sura of three dyads of 
which the antecedents or the consequents are any three 
given non-coplanar vectors. This expression of the dyadic is 
unique. 

The symbolic product ab known as a dyad is the most 
general product of two vectors in which multiplication by a 
scalar is associative. It is called the indeterminate product. 
The product imposes five conditions upon the vectors a and 
b. Their directions and the product of their lengths are 
determined by the product. The scalar and vector products 
are functions of the indeterminate product. A scalar and 
a vector may be obtained from any dyadic by inserting a dot 
and a cross between the vectors in each dyad. This scalar 
and vector are functions of the dyadic. 

0* = i *! + a a b a + a 8 b 8 + (18) 

X = &1 x bj + a 2 x b 2 + a 3 x b 3 + (19) 

0, = i-0.i + j*0-j + k 0*k (20) 

= a n -f a 22 + #339 

X = (j . . k - k j) i + (k i - i k) j 
+ (i- 0-j -j 0i) k (21) 



The direct product of two dyads is the dyad whose ante 
cedent and consequent are respectively the antecedent of the 
first dyad and the consequent of the second multiplied by 
the scalar product of the consequent of the first dyad and 
the antecedent of the second. 

JL 

(ab) (c d) = (b . c) a/ *T (23) 



The direct product of two dyadics is the formal expansion, 
according to the distributive law, of the product into the 



324 VECTOR ANALYSIS 

sum of products of dyads. Direct multiplication of dyadics 
or of dyadics and a vector at either end or at both ends obeys 
the distributive and associative laws of multiplication. Con 
sequently such expressions as 

Q.W.T, s.0-? 7 *, s.^.^.r, $>.W.Q (24)-(26) 

may be written without parentheses; for parentheses may 
be inserted at pleasure without altering the value of the 
product. In case the vector occurs at other positions than 
at the end the product is no longer associative. 

The skew product of a dyad and a vector may be defined 

by the equation 

(ab) x r = a b x r, 

r x (ab) = r x a b. (28) 

The skew product of a dyadic and a vector is equal to the 
formal expansion of that product into a sum of products of 
dyads and that vector. The statement made concerning the 
associative law for direct products holds when the vector is 
connected with the dyadics in skew multiplication. The 
expressions 

r x ?F, ^ x r, r x $ s, r $ x s, r x <P x s (29) 

may be written without parentheses and parentheses may be 
inserted at pleasure without altering the value of the product. 
Moreover 

s (r x <P) = (s x r) - <P, (<P x r) s = (r x s), 

<p.(rx ?P) = (0 x r) W. (31) 

But the parentheses cannot be omitted. 

The necessary and sufficient condition that a dyadic may 
be reduced to the sum of two dyads or to a single dyad or 
to zero is that, when expressed as the sum of three 
dyads of which the antecedents (or consequents) are known 



LINEAR VECTOR FUNCTIONS 325 

to be non-coplanar, the consequents (or antecedents) shall 
be respectively coplanar or collinear or zero. A complete 
dyadic is one which cannot be reduced to a sum of fewer 
than three dyads. A planar dyadic is one which can be 
reduced to a sum of just two dyads. A linear dyadic is one 
which can be reduced to a single dyad. 

A complete dyadic possesses no degree of nullity. There 
is no direction in space - for which it is an annihilator. A 
planar dyadic possesses one degree of nullity. There is one 
direction in space for which it is an annihilator when used as 
a prefactor and one when used as a postfactor. A linear 
dyadic possesses two degrees of nuljity. There are two 
independent directions in space for which it is an annihilator 
when used as a prefactor and two directions when used as a 
postfactor. A zero dyadic possesses three degrees of nullity 
or complete nullity. It annihilates every vector in space. 

The products of a complete dyadic and a complete, planar, 
or linear dyadic are respectively complete, planar, or linear. 
The products of a planar dyadic with a planar or linear dyadic 
are respectively planar or linear, except in certain cases where 
relations of perpendicularity between the consequents of the 
first dyadic and the antecedents of the second introduce one 
more degree of nullity into the product. The product of a 
linear dyadic by a linear dyadic is in general linear ; but in 
case the consequent of the first is perpendicular to the ante 
cedent of the second the product vanishes. The product of 
any dyadic by a zero dyadic is zero. 

A dyadic which when applied to any vector in space re 
produces that vector is called an idemfactor. All idemfactors 
are equal and reducible to the form 

I = ii + jj + kk. (33) 

Or I = aa + bb + cc . (34) 

The product of any dyadic and an idemfactor is that dyadic. 



326 VECTOR ANALYSIS 

If the product of two complete dyadics is equal to the idem- 
factor the dyadics are commutative and either is called 
the reciprocal of the other. A complete dyadic may be 
canceled from either end of a product of dyadics and vectors 
as in ordinary algebra ; for the cancelation is equivalent to 
multiplication by the reciprocal of that dyadic. Incomplete 
dyadics possess no reciprocals. They correspond to zero in 
ordinary algebra. The reciprocal of a product is equal to the 
product of the reciprocals taken in inverse order. 

(0. 5F)- 1 = 5F- 1 0-i. (38) 

The conjugate of a dyadic is the dyadic obtained by inter 
changing the order of the antecedents and consequents. The 
conjugate of a product is equal to the product of the con 
jugates taken in the opposite order. 

(0. 9%= W c . C . (40) 

The conjugate of the reciprocal is equal to the reciprocal of 
the conjugate. A dyadic may be divided in one and only 
one way into the sum of two parts of which one is self- 
conjugate and the other anti-self-conjugate. 



Any anti-self-conjugate dyadic or the anti-self-conjugate 
part of any dyadic, used in direct multiplication, is equivalent 
to minus one-half the vector of that dyadic used in skew 
multiplication. 



T=-j0 x xr, 



(44) 



A dyadic of the form c X I or I x c is anti-self-conjugate and 
used in direct multiplication is equivalent to the vector o 
used in skew multiplication. 



LINEAR VECTOR FUNCTIONS 327 

Also c x r = (I x c) r = (c x I) r, (46) 

c x <P = (I x c) = (c x I) 0. 

The dyadic c X I or I x c, where c is a unit vector is a quad- 
ran tal versor for vectors perpendicular to c and an annihilator 
for vectors parallel to c. The dyadic Ixc + ccisa true 
quadrantal versor for all vectors. The powers of these dyadics 
behave like the powers of the imaginary unit V^l, as ma y 
be seen from the geometric interpretation. Applied to the 
unit vectors i, j, k 

I x i = i x I = kj - j k, etc. (49) 

The vector a x b in skew multiplication is equivalent to 
(a x b) X I in direct multiplication. 

(ax b) x 1 = 1 x (ax b)=ba-ab (50) 
(a x b) x r = (b a a b) r 
r x (a x b) = r (b a - ab). (51) 

A complete dyadic may be reduced to a sum of three 
dyads of which the antecedents among themselves and the 
consequents among themselves each form a right-handed 
rectangular system of three unit vectors and of which the 
scalar coefficients are all positive or all negative. 

0= (ai i + ftj j + ck k). (53) 

This is called the normal form of the dyadic. An incom 
plete dyadic may be reduced to this form but one or more of 
the coefficients are zero. The reduction is unique in case 
the constants a, 6, c are different. In case they are not 
different the reduction may be accomplished in more than 
one way. Any self-conjugate dyadic may be reduced to 

the normal form 

4> = aii + 6jj + ckk, (55) 

in which the constants a, S, c are not necessarily positive. 



328 VECTOR ANALYSIS 

The double dot and double cross multiplication of dyads 
is defined by the equations 

ab:cd = ac b.d, (56) 

abcd = axc bxd. (57) 

The double dot and double cross multiplication of dyadics 
is obtained by expanding the product formally, according to 
the distributive law, into a sum of products of dyads. The 
double dot and double cross multiplication of dyadics is com 
mutative but not associative. 

One-half the double cross product of a dyadic by itself 
is called the second of 0. If 



<P 2 =i <Px <P = b xc mxn + cxa nxl+axb Ixm. (61) 

One-third of the double dot product of the second of and 
is called the third of and is equal to the product of the 
scalar triple product of the antecedents of and the scalar 
triple product of the consequent of 0. 

0a = \0$ 0: <P=[abc] [Imn]. (62) 

The second of the conjugate is the conjugate of the second. 
The third of the conjugate is equal to the third of the 
original dyadic. The second and third of the reciprocal are 
the reciprocals of the second and third of the second and 
third of a dyadic. The second and third of a product are the 
products of the seconds and thirds. 

(*c\ = (*.)* 



(65) 



LINEAR VECTOR FUNCTIONS 329 

The product of the second and conjugate of a dyadic is equal 
to the product of the third and the idemfactor. 

^^c=^ 1 (68) 

The conditions for the various degrees of nullity may be 
expressed in terms of the second and third of 0. 

4> 3 * 0, is complete 
8 = 0, <P 2 * 0, is planar (69) 

<P 3 = 0, $ 2 = 0, * 0, is linear. 

The closing sections of the chapter contain the expressions 
(70)-(78) of a number of the results in nonion form and the 
deduction therefrom of a number of theorems concerning 
determinants. They also contain the cubic equation which is 
satisfied by a dyadic 4>. 

03 _ Q a 02 + 0^ 03 + ^ [ _ (79) 

This is called the Hamilton-Cayley equation. The coeffi 
cients S , <P<i S , and 3 are the three fundamental scalar in 
variants of <P. 

EXERCISES ON CHAPTER V 

1. Show that the two definitions given in Art. 98 for 
a linear vector function are equivalent 

2. Show that the reduction of a dyadic as in (15) can be 
accomplished in only one way if a, b, c, 1, m, n, are given. 

3. Show (<P x a) c = - a x (1> C . 

4. Show that if <Pxr= XT for any value of r different 
from zero, then must equal ?P unless both and are 
linear and the line of their consequents is parallel to r. 

5. Show that if r = for any three non-coplanar values 
of r, then = 0. 



330 VECTOR ANALYSIS 

6. Prove the statements made in Art. 106 and the con 
verse of the statements. 

7. Show that if Q is complete and if Q = W Q , then 
<P and W are equal. Give the proof by means of theory 
developed prior to Art. 109. 

8. Definition : Two dyadics such that ? r = that 
is to say, two dyadics that are commutative are said to be 
homologous. Show that if any number of dyadics are homo 
logous to one another, any other dyadics which may be obtained 
from them by addition, subtraction, and direct multiplication 
are homologous to each other and to the given dyadics. Show 
also that the reciprocals of homologous dyadics are homolo 
gous. Justify the statement that if ~ l or ~ l (P, 
which are equal, be called the quotient of by ?F, then the 
rules governing addition, subtraction, multiplication and 
division of homologous dyadics are identical with the rules 
governing these operations in ordinary algebra it being 
understood that incomplete dyadics are analogous to zero, 
and the idemfactor, to unity. Hence the algebra and higher 
analysis of homologous dyadics is practically identical with 
that of scalar quantities. 

9. Show that (I X c) c X $ and (c X I) & = c X #. 

10. Show that whether or not a, b, c be coplanar 

abxc+bcxa+caxb = [abc]I 
and bxca+cxab+axbc=[abc]L 

11. If a, b, c are coplanar use the above relation to prove 
the law of sines for the triangle and to obtain the relation 
with scalar coefficients which exists between three coplanar 
vectors. This may be done by multiplying the equation by a 
unit normal to the plane of a, b, and c. 

12. What is the-condition which must subsist between the 
coefficients in the expansion of a dyadic into nonion form if 



LINEAR VECTOR FUNCTIONS 331 

the dyadic be self -con jugate ? What, if the dyadic be anti- 
self-conjugate ? 

13. Prove the statements made in Art. 116 concerning the 
number of ways in which a dyadic may be reduced to its 
normal form. 

14. The necessary and sufficient condition that an anti- 
self-conjugate dyadic be zero is that the vector of the 
dyadic shall be zero. 

15. Show that if be any dyadic the product <P C is 
self-conjugate. 

16. Show how to make use of the relation $ x = to 
demonstrate that the antecedents and consequents of a self 
conjugate dyadic are the same (Art. 116). 

17. Show that 2 <P 2 = 2 & 3 

and (0 + W\ = </> 2 + 4>*V + ^ 

18. Show that if the double dot product : of a dyadic 
by itself vanishes, the dyadic vanishes. Hence obtain the 
condition for a linear dyadic in the f orin <P 2 : 2 = 0. 

19. Show that (<P + ef) 3 = <P 3 + e- 2 f. 

20. Show that (0 + ?T) 3 = 8 + <P 2 : V + d> : ?F 2 + V* 

21. Show that the scalar of a product of dyadics is un 
changed by cyclic permutation of the dyadics. That is 



CHAPTER VI 

ROTATIONS AND STRAINS 

123.] IN the foregoing chapter the analytical theory of 
dyadics has been dealt with and brought to a state of 
completeness which is nearly final for practical purposes. 
There are, however, a number of new questions which present 
themselves and some old questions which present themselves 
under a new form when the dyadic is applied to physics 
or geometry. Moreover it was for the sake of the applica 
tions of dyadics that the theory of them was developed. It is 
then the object of the present chapter to supply an extended 
application of dyadics to the theory of rotations and strains 
and to develop, as far as may appear necessary, the further 
analytical theory of dyadics. 

That the dyadic $ may be used to deuote a transformation 
of space has already been mentioned. A knowledge of the 
precise nature of this transformation, however, was not needed 
at the time. Consider r as drawn from a fixed origin, and r 
as drawn from the same origin. Let now 

r = 0-r. 

This equation therefore may be regarded as defining a trans 
formation of the points P of space situated at the terminus of 
r into the point P , situated at the terminus of r . The origin 
remains fixed. Points in the finite regions of space remain in 
the finite regions of space. Any point upon a line 

r = b + x a 
becomes a point r f = $ b + # $ * 



ROTATIONS AND STRAINS 333 

Hence straight lines go over into straight lines and lines 
parallel to the same line a go over by the transformation into 
lines parallel to the same line a. In like manner planes 
go over into planes and the quality of parallelism is invariant. 
Such a transformation is known as a homogeneous strain. 
Homogeneous strain is of frequent occurrence in physics. For 
instance, the deformation of the infinitesimal sphere in a fluid 
(Art. 76) is a homogeneous strain. In geometry the homo 
geneous strain is generally known by different names. It is 
called an affine collineation with the origin fixed. Or it is 
known as a linear homogeneous transformation. The equa 
tions of such a transformation are 

x 1 = a x + 



n l2 13 



y< = 



124.] Theorem : If the dyadic gives the transformation 
of the points of space which is due to a homogeneous strain, 
2 , the second of 0, gives the transformation of plane areas 
which is due to that strain and all volumes are magnified by 
that strain in the ratio of 3 , the third or determinant of 
to unity. 

Let <P = al + bm + cn 

r = <P.r = al-r-f bm r -f cnr. 

The vectors 1 , m , n are changed by into a, b, c. Hence 
the planes determined by m and n , n and 1 , 1 and m are 
transformed into the planes determined by b and c, c and a, 
a and b. The dyadic which accomplishes this result is 

$ 2 =r b x c mxn + cxa nxl + axb Ixm. 

Hence if s denote any plane area in space, the transformation 
due to replaces s by the area s such that 



334 VECTOR ANALYSIS 

It is important to notice that the vector s denoting a plane 
area is not transformed into the same vector s as it would 
be if it denoted a line. This is evident from the fact that in 
the latter case acts on s whereas in the former case <P 2 acts 
upon s. 

To show that volumes are magnified in the ratio of <P Z to 
unity choose any three vectors d, e, f which determine the 
volume of a parallelepiped [d e f]. Express with the vec 
tors which form the reciprocal system to d, e, f as consequents. 



The dyadic <P changes d, e, f into a, b, c (which are different 
from the a, b, c above unless d, e, f are equal to 1 , m , n ). 
Hence the volume [d e f ] is changed into the volume [a b c]. 

8 = [abc][dVf] 

[d e fr^Cdef]. 

Hence [a b c] = [d e f] $ 3 . 

The ratio of the volume [a b cj to [d e f] is as <P 3 is to unity. 
But the vectors d, e, f were any three vectors which deter 
mine a parallelepiped. Hence all volumes are changed by 
the action of in the same ratio and this ratio is as 3 is to 1. 

Eotations about a Fixed Point. Versors 

125.] Theorem : The necessary and sufficient condition that 
a dyadic represent a rotation about some axis is that it be 
reducible to the form 

= i i+j j + k k (1) 

where i , j , k and i, j, k are two right-handed rectangular 
systems of unit vectors. 

Let r = #i-f-f-3k 



ROTATIONS AND STRAINS 335 

Hence if C? is reducible to the given form the vectors i, j, k 
are changed into the vectors i , j , k and any vector r is 
changed from its position relative to i, j, k into the same posi 
tion relative to i ,j ,k . Hence by the transformation no 
change of shape is effected. The strain reduces to a rotation 
which carries i, j, k into i , j , k . Conversely suppose the 
body suffers no change of shape that is, suppose it subjected 
to a rotation. The vectors i, j, k must be carried into another 
right-handed rectangular system of unit vectors. Let these 
be i , j , k . The dyadic <P may therefore be reduced to the 

form 

= i i + j j+k k. 

Definition : A dyadic which is reducible to the form 
i i + j j + k k 

and which consequently represents a rotation is called a 
versor. 

Theorem: The conjugate and reciprocal of a versor are 
equal, and conversely if the conjugate and reciprocal of a 
dyadic are equal the dyadic reduces to a versor or a versor 
multiplied by the negative sign. 

Let = i i+j j + k k, 



Hence the first part of the theorem is proved. To prove the 

second part let 

= ai + b j + ck, 

<p c = i*+j b + kc, 



If 4>-i =<P C , 

Hence aa4-bb 



336 VECTOR ANALYSIS 

Hence (Art. 108) the antecedents a, b, c and the consequents 
a, b, c must be reciprocal systems. Hence (page 87) they 
must be either a right-handed or a left-handed rectangular 
system of unit vectors. The left-handed system may be 
changed to a right-handed one by prefixing the negative 
sign to each vector. Then 

#.*rff,tnt). (iy 

The third or determinant of a versor is evidently equal to 
unity ; that of the versor with a negative sign, to minus one. 
Hence the criterion for a versor may be stated in the form 

$ = I. 3> n = I I = 1 (%\ 

{/ 3 \ / 

Or inasmuch as the determinant of is plus or minus one 
if (P* (P C =I, it is only necessary to state that if 

C 3 \. / 

$ is a versor. 

There are two geometric interpretations of the transforma 
tion due to a dyadic such that 

9 @ __. j = | 1 = 1 (3) 

(J/ 1 _j_ j j _j_ k k) . 

The transformation due to is one of rotation combined with 
reflection in the origin. The dyadic i i+j j + k k causes a 
rotation about a definite axis it is a versor. The negative 
sign then reverses the direction of every vector in space and 
replaces each figure by a figure symmetrical to it with respect 
to the origin. By reversing the directions of i and j the 
system i , j , k still remains right-handed and rectangular, 
but the dyadic takes the form 

= i i+j j-k k, 

or <P = (i i +j j -k k ) .(i i + j j + k k). 



ROTATIONS AND STRAINS 337 

Hence the transformation due to is a rotation due to 
i i+j j + k k followed by a reflection in the plane of i and 
j . For the dyadic i i + j j k k causes such a transfor 
mation of space that each point goes over into a point sym 
metrically situated to it with respect to the plane of i and j . 
Each figure is therefore replaced by a symmetrical figure. 

Definition : A transformation that replaces each figure by 
a symmetrical figure is called a perversion and the dyadic 
which gives the transformation is called a perversor. 

The criterion for a perversor is that the conjugate of a 
dyadic shall be equal to its reciprocal and that the determi 
nant of the dyadic shall be equal to minus one. 

4>.<P C = I, I0I=-1. (3) 

Or inasmuch as if C? c = I, the determinant must be plus 
or minus one the criterion may take the form 

- C = I, I I < 0, (3) 

is a perversor. 

It is evident from geometrical considerations that the prod 
uct of two versors is a versor ; of two perversors, a versor ; 
but of a versor and a perversor taken in either order, a 
perversor. 

. 126.] If the axis of rotation be the i-axis and if the angle 
of rotation be the angle q measured positive in the positive 
trigonometric direction, then by the rotation the vectors 
i, j, k are changed into the vectors i ,j ,k such that 

i = i 

j = j cos q + k sin y, 
k = j sin q + k cos q. 

The dyadic $ = i i + j j + k k which accomplishes this rota 
tion is 



338 VECTOR ANALYSIS 

= ii + cos q (jj + kk) + sin q (k j - jk). (4) 
jj +kk = I-ii, 
kj-jk = I x i. 
Hence = i i + cos q (I i i) + sin q I x i. ,( 5 ) 

If more generally in place of the i-axis any axis denoted 
by the unit vector a be taken as the axis of rotation and if as 
before the angle of rotation about that axis be denoted by q, 
the dyadic which accomplishes the rotation is 

= a a + cos q (I a a) + sin q I x a. (6) 

To show that this dyadic actually does accomplish the 
rotation apply it to a vector r. The dyad a a is an idemfactor 
for all vectors parallel to a; but an annihilator for vectors 
perpendicular to a. The dyadic I a a is an idemfactor 
for all vectors in the plane perpendicular to a; but an 
annihilator for all vectors parallel to a. The dyadic I x a 
is a quadrantal versor (Art. 113) for vectors perpendicular 
to a; but an annihilator for vectors parallel to a. If then 

r be parallel to a 

0.r = aar = r. 

Hence leaves unchanged all vectors (or components of 
vectors) which are parallel to a. If r is perpendicular to a 

. r = cos q r + sin q a x r. 

Hence the vector r has been rotated in its plane through the 
angle q. If r were any vector in space its component parallel 
to a suffers no change ; but its component perpendicular to a 
is rotated about a through an angle of q degrees. The whole 
vector is therefore rotated about a through that angle. 
Let a be given in terms of i, j, k as 



a l a z ik 



ROTATIONS AND STRAINS 339 

-r a 2 a x ji + a 2 2 j j + a 2 a 3 jk 
+ a z a l ki + 8 2 kj + a 3 2 kk, 

I = ii + jj + kk, 
I X a = 0ii-a 3 ij + 2 ik, 



~a 2 ki + a x kj + Okk. 
Hence 

$ = {&J 2 (1 cos #) + cos #} i i 

+ S a i a 2 (1 cos 2) ~~ a 3 S i n 2} lj 

+ { a i a s (1 c s ?) + a a sin ^^ ik 
+ { 2 a 1 (1 cos^) + a 3 sin q} ji 

+ { 2 2 C 1 - cos 2) + cos q} j j 

+ ( a 2 a 3 (1 "" COS 2) a l S l n 2l J * 

+ { 3 ! (1 cos ^) a 2 sin q} ki 

+ {^3^2 (1 cos q) 4- ajsin^} kj 

+ {3 2 (1 cosg) + cosg} kk. (7) 

127.] If be written as in equation (4) the vector of <P 
and the scalar of may be found. 

X = i x i + cos q (j x j + k x k) + sin q (k x j - j x k) 

<P X = 2 sin q i 

<2> s = i - i + cosg (j j+k -k) +sing (k j j -k), 
a = 1 + k cos q. 

The axis of rotation i is seen to have the direction of <P X , 
the negative of the vector of 0. This is true in general. 
The direction of the axis of rotation of any versor is the 
negative of the vector of (P. The proof of this statement 
depends on the invariant property of $ x . Any versor 
may be reduced to the form (4) by taking the direction of i 



340 VECTOR ANALYSIS 

coincident with the direction of the axis of rotation. After 
this reduction has been made the direction of the axis is seen 
to be the negative of <P X . But <P X is not altered by the 
reduction of <P to any particular form nor is the axis of 
rotation altered by such a reduction. Hence the direction of 
the axis of rotation is always coincident with $ x , the direc 
tion of the negative of the vector of <?. 

The tangent of one-half the angle of version q is 



sin q * x , ON 

(8) 




1 + cos q 1 + 4> 



s 



The tangent of one-half the angle of version is therefore 
determined when the values of <# x and <P S are known. The 
vector $ x and the scalar (P s , which are invariants of <P, deter 
mine completely the versor <?. Let ft be a vector drawn 
in the direction of the axis of rotation. Let the magnitude 
of ft be equal to the tangent of one-half the angle q of 
version. 



The vector ft determines the versor <P completely, ft will be 
called the vector semi-tangent of version. 

By (6) a versor $ was expressed in terms of a unit vector 
parallel to the axis of rotation. 

<p = a a + cos q (I a a) + sin q I x a. 
Hence if ft be the vector semi-tangent of version 



There is a more compact expression for a versor in terms 
of the vector semi-tangent of version. Let c be any vector in 
space. The version represented by ft carries 

c ft x c into c + ft X c. 



ROTATIONS AND STRAINS 341 

It will be sufficient to show this in case c is perpendicular to 
ft. For if c (or any component of it) were parallel to ft the 
result of multiplying by ft x would be zero and the statement 
would be that c is carried into c. In the first place the mag 
nitudes of the two vectors are equal. For 

(c ft x c) (c ft x c) = c c +ftxc-ftxc 2c-ftxc 

(c + ft x c) (c + ft x c)-= cc-hftxc ftxc + 2cftxc 

cc + ftxcftxc = cc + ftft c c ft c ft.c. 

Since ft and c are by hypothesis perpendicular 

c-c + ftxc.ftxc=:c 2 (l + tan 2 \ q). 

The term c ft X c vanishes. Hence the equality. In the 
second place the angle between the two vectors is equal to q. 

(c ftxc)(c + ftxc)_cc ft x c ft x c 

c 2 (1 + tan 2 - q) c* (1 + tan 2 i j) 

2 2 



= cos q 



c 2 (1 + tan 2 i q) 

(c ft x c) x (c + ft x c) _ 2 c x (ft x c) 
c 2 (1 + tan 2 1 2 ) c 2 (1 + tan 2 I j) 

2 * 



2 c 2 tan i 2 

= sin j. 



Hence the cosine and sine of the angle between c ft X c 
and c + ft x c are equal respectively to the cosine and sine of 
the angle q : and consequently the angle between the vectors 
must equal the angle q. Now 



342 VECTOR ANALYSIS 

C ftXC=(I Ixft)-C 
and (c + a x c) = (I + I x ft) c 

(I + I x Q) (I - I x tt)- 1 -(I-Ixft) = I + Ixft. 
Multiply by c 

(I + I x a) (I - I x Q)- 1 (c - Q x c) = c + a x c. 
Hence the dyadic 

= (I + I x tt) (I - I X Q)- 1 (10) 

carries the vector c ft x c into the vector c + ft X c no matter 
what the value of c. Hence the dyadic determines the 
version due to the vector semi-tangent of version ft. 

The dyadic I + 1 x ft carries the vector c ft x c into 
(I + ft.ft)c. 

(I + I x ft) (c ftxc) = c + ftxc ftxc ftx(ftxc) 

(I + I X Q) (c Q X c) = c + Q Q c = (1 + Q Q) C . 
Hence the dyadic 



1 + ftft 

carries the vector c ft x c into the vector c, if c be perpen 
dicular to ft as has been supposed. Consequently the dyadic 

(I + Ixft) 2 
1 + ft-ft 

produces a rotation of all vectors in the plane perpendicular 
to ft. If, however, it be applied to a vector x ft parallel to ft 
the result is not equal to x ft. 

+ IXQ)-(I + IXQ) (I + IXQ) . Q *Q 

i + O-O *V-* I + Q.Q v "l + Q- 



ROTATIONS AND STRAINS 343 

To obviate this difficulty the dyad Q, ft, which is an annihilator 
for all vectors perpendicular to ft, may be added to the nu 
merator. The versor (P may then be written 

ftft+CI + IXft)* 
1 + ft-ft 

(i + 1 x ft) (i + 1 x ft) = i + 2 1 x ft + (i x ft) . (i x ft) 

(Ixft)-(I xft) = (I xft) x ft = l.ftft-ft.ftl. 
Hence substituting : 

^(l-ft.ft)I + 2ftft + 2Ixft 

1 + ft ft 

This may be expanded in nonion form. Let 



(11) 



128. ] If a is a unit vector a dyadic of the form 

= 2aa-I (12) 

is a liquadrantal versor. That is, the dyadic turns the 
points of space about the axis a through two right angles. 
This may be seen by setting q equal to TT in the general 
expression for a versor 

= a a 4- cos q (I a a) + sin q I x a, 

or it may be seen directly from geometrical considerations. 
The dyadic <P leaves a vector parallel to a unchanged but re 
verses every vector perpendicular to a in direction. 

Theorem: The product of two biquadrantal versors is a 
versor the axis of which is perpendicular to the axes of the 



344 VECTOR ANALYSIS 

biquadrantal versors and the angle of which is twice the 
angle from the axis of the second to the axis of the first 
Let a and b be the axes of two biquadrantal versors. The 

product 

=(2bb-I).(2aa-I) 

is certainly a versor; for the product of any two versors 
is a versor. Consider the common perpendicular to a and b. 
The biquadrantal versor 2 a a I reverses this perpendicular 
in direction. (2bb I) again reverses it in direction and con 
sequently brings it back to its original position. Hence the 
product Q leaves the common perpendicular to a and b un 
changed. Q is therefore a rotation about this line as axis. 



The cosine of the angle from a to Q a is 
a Q a = 2 b - a b a - a . a = 2 (b a) 2 - 1 = cos 2 (b, a). 

Hence the angle of the versor Q is equal to twice the angle 
from a to b. 

Theorem : Conversely any given versor may be expressed 
as the product of two biquadrantal versors, of which the axes 
lie in the plane perpendicular to the axis of the given versor 
and include between them an angle equal to one half the 
angle of the given versor. 

For let Q be the given versor. Let a and b be unit vectors 
perpendicular to the axis J? x of this versor. Furthermore 
let the angle from a to b be equal to one half the angle of 
this versor. Then by the foregoing theorem 

J2=(2bb-I).(2aa-I). (14) 

The resolution of versors into the product of two biquad 
rantal versors affords an immediate and simple method for 
compounding two finite rotations about a fixed point. Let 
d> and be two given versors. Let b be a unit vector per- 



ROTATIONS AND STRAINS 345 

pendicular to the axes of and W. Let a be a unit vector 
perpendicular to the axis of <P and such that the angle from 
a to b is equal to one half the angle of 0. Let c be a unit 
vector perpendicular to the axis of W and such that the angle 
from b to c is equal to one half the angle of . Then 

</> = (2bb-I).(2aa-I) 

$T=(2ec-I).(2bb-I) 

V. = (2 cc - I) (2 bb - 1)2. (2 aa - I). 

But (2 bb I) 2 is equal to the idemfactor, as may be seen from 
the fact that it represents a rotation through four right angles 
or from the expansion 

(2bb-I).(2bb-I) = 4b.b bb-4bb + I = I. 
Hence W <P = (2 c c - I) (2 a a - I). 

The product of W into is a versor the axis of which is 
perpendicular to a and c and the angle of which is equal to 
one half the angle from a to c. 

If and W are two versors of which the vector semi- 
tangents of version are respectively QJ and ft^ the vector 
semi-tangent of version Q 3 of the product <P is 

q 1 + a 2 +a 2 xa 1 
a ~ i-a.-a, 

Let 0=(2bb-I) (2aa-I) 

and = (2 c c - I) . (2 bb - I). 

. <P = (2cc-I) (2aa-I). 

iff (V <?) 

* ~ y x 



ba -2aa -2b b 
x = 4a b b X a, 



346 VECTOR ANALYSIS 

5 = 4(a.b) 2 -l, 
?T = 4 c b cb 2 b b - 2 c c + 1, 
r x = 4 c b c x b, 
?r 5 = 4(c.b) 2 -l 
JF <p = 4 c a ca 2 c c 2 a a + I 

(?F. <p) x = 4 ca c x a, 
(ST. 0)^ = 4 (c-a) 2 -l. 
axb bx c axe 



Hence 



t = - -, ^ = T , 3 = - 
a b b c a c 

(bxc) x (axb) [abc] b 

J 



x Q = 



a b b c a b b c 

But [abc] r = bxc a r + c x a b r + a x b c r, 

bxc axb axe 

b c a b a b b c 

Hence Q 2 x Qj = C^ ft 2 + 8 






Q = 



a b b - c 
(a x b) (b x c) _ a b b c a c b b 



2 " a-bb-c abbc a*bb*c 



Hence r-^r = 1 ft Q r 

a* b b c 



Q . = . . . 



ROTATIONS AND STRAINS 847 

This formula gives the composition of two finite rotations. 
If the rotations be infinitesimal ftj and Q^ are both infinitesi 
mal. Neglecting infinitesimals of the second order the for 
mula reduces to 



The infinitesimal rotations combine according to the law of 
vector addition. This demonstrates the parallelogram law for 
angular velocities. The subject was treated from different 
standpoints in Arts. 51 and 60. 



icSy Right Tensors, Tonics, and Cyclotonics 

129.] If the dyadic <P be a versor it may be written in the 
form (4) 

= ii + cos q (jj + kk) + sin q (kj - jk). 

The axis of rotation is i and the angle of rotation about that 
axis is q. Let be another versor with the same axis and 
an angle of rotation equal to q . 

= ii + cos q f (j j + kk) + sin q r (kj jk). 
Multiplying : 

. y = = i i + cos (g + ? ) (j j + k k) 

+ Bin(j+ 9 )(kj-jk). (16) 

This is the result which was to be expected the product of 
two versors of which the axes are coincident is a versor with 
the same axis and with an angle equal to the sum of the 
angles of the two given versors. 

If a versor be multiplied by itself, geometric and analytic 
considerations alike make it evident that 

2 = i i + cos 2q (j j + kk) + sin 2 q (k j - j k), 
and 4> = ii + cos nq (j j + kk) + sin nq (kj j k). 



348 VECTOR ANALYSIS 

On the other hand let 4> l equal jj + kk; and <P 2 equal 
kj-jk. Then 

<p = (i i + cos q l + sin q $ 2 ) n . 

The product of ii into either l or <P 2 is zero and into itself is 

ii. Hence 

4> n = ii + (cos q d> l + sin q 2 ) n 

n = ii + cos n q (PS + n cos 11 ^ 1 q sin # fl^"" 1 <P 2 + . 

The dyadic ^ raised to any power reproduces itself. (Pf = <P r 
The dyadic <P 2 raised to the second power gives the negative 
of <#! ; raised to the third power, the negative of <P 2 ; raised 
to the fourth power, l ; raised to the fifth power, <P Z and so 
on (Art. 114). The dyadic l multiplied by 2 is equal to 
<P 2 . Hence 

<p n = i i + cos n q l + n cos n ~ l q sin q <P% 

nfnl) 
V ; - 2 



But & n = i i + cos n q l + sin n q <P y 

Equating coefficients of <P l and $ 2 in these two expressions 

for n 

n (n 1) 
cos n q = cos n q ~~^TI - COS>1 " ? sin 2 q + 



71 (71-1) (71-2) 



n " 3 



sm 7i q = TI cos "^ j sin q -- : - cos n "# sm^ + 

o ! 

Thus the ordinary expansions for cos nq and sin 715 are 
obtained in a manner very similar to the manner in which 
they are generally obtained. 

The expression for a versor may be generalized as follows. 
Let a,b, c be any three non-coplanar vectors ; and a , V, c , the 
reciprocal system. Consider the dyadic 

<p = aa 4- cos q (bb + cc ) + sin q (cb be ). (17) 



ROTATIONS AND STRAINS 349 

This dyadic leaves vectors parallel to a unchanged. Vectors 
in the plane of b and c suffer a change similar to rotation. 

Let 

r = cos p b + sin p c, 

r = <P r = cos (p + q) b + sin (p + q) c. 

This transformation may be given a definite geometrical 
interpretation as follows. The vector r, when p is regarded 
as a variable scalar parameter, describes an ellipse of which 
b and c are two conjugate semi-diameters (page 117). Let 
this ellipse be regarded as the parallel projection of the 

unit circle 

r = cos p i + sin q j. 

That is, the ellipse and the circle are cut from the same 
cylinder. The two semi-diameters i and j of the circle pro 
ject into the conjugate semi-diameters a and b of the ellipse. 
The radius vector r in the ellipse projects into the radius vector 
f in the unit circle. The radius vector r in the ellipse which 
is equal to r, projects into a radius vector r in the circle 

such that 

f = cos (p + q) i + sin (jp + q) j. 

Thus the vector r in the ellipse is so changed by the applica 
tion of as a prefactor that its projection f in the unit circle 
is rotated through an angle q. 

This statement may be given a neater form by making use 
of the fact that in parallel projection areas are changed in a 
definite constant ratio. The vector r in the unit circle may 
be regarded as describing a sector of which the area is to the 
area of the whole circle as q is to 2 TT. The radius vector f 
then describes a sector of the ellipse. The area of this sector 
is to the area of the whole ellipse as q is to 2 TT. Hence the 
dyadic $ applied as a prefactor to a radius vector r in an ellipse 
of which b and c are two conjugate semi-diameters advances 
that vector through a sector the area of which is to the area of 



350 VECTOR ANALYSIS 

the whole ellipse as q is to 2-Tr. 1 Such a displacement of the 
radius vector r may be called an elliptic rotation through a 
sector q from its similarity to an ordinary rotation of which 
it is the projection. 

Definition : A dyadic of the form 

= aa + cos q (bb + cc ) + sin q (c V - be ) (17) 

is called a cyclic dyadic. The versor is a special case of a 
cyclic dyadic. 

It is evident from geometric or analytic considerations that 
the powers of a cyclic dyadic are formed, as the powers of a 
versor were formed, by multiplying the scalar q by the power 
to which the dyadic is to be raised. 

n = a a + cos nq (b b + c c ) + sin nq (c V b c ). 
If the scalar q is an integral sub-multiple of 2 TT, that is, if 

27T 

= m, 
1 

it is possible to raise the dyadic to such an integral power, 
namely, the power w, that it becomes the idemfactor 



may then be regarded as the mth root of the idemfactor. 
In like manner if q and 2 TT are commensurable it is possible 
to raise to such a power that it becomes equal to the idem- 
factor and even if q and 2 TT are incommensurable a power of 
d> may be found which differs by as little as one pleases from 
the idemfactor. Hence any cyclic dyadic may be regarded as 
a root of the idemfactor. 

1 It is evident that fixing the result of the application of < to all radii vectors 
in an ellipse practically fixes it for all vectors in the plane of b and c. For any 
vector in that plane may be regarded as a scalar multiple of a radius vector of 
the ellipse. 



ROTATIONS AND STRAINS 351 

130.] Definition: The transformation represented by the 

<Z> = ii + &jJ+ckk (18) 

where a, 6, c are positive scalars is called a ^rare strain. The 



dyadic itself is called a rt^Atf tensor. 

A right tensor may be factored into three factors 



The order in which these factors occur is immaterial. The 
transformation 



is such that the i and j components of a vector remain un 
altered but the k-component is altered in the ratio of c to 1. 
The transformation may therefore be described as a stretch or 
elongation along the direction k. If the constant c is greater 
than unity the elongation is a true elongation : but if c is less 
than unity the elongation is really a compression, for the ratio 
of elongation is less than unity. Between these two cases 
comes the case in which the constant is unity. The lengths 
of the k-components are then not altered. 

The transformation due to the dyadic may be regarded 
as the successive or simultaneous elongation of the com 
ponents of r parallel to i, j, and k respectively in the ratios 
a to 1, b to 1, c to 1. If one or more of the constants a, 6, c 
is less than unity the elongation in that or those directions 
becomes a compression. If one or more of the constants is 
unity, components parallel to that direction are not altered. 
The directions i, j, k are called the principal axes of the strain. 
Their directions are not altered by the strain whereas, if the 
constants #, &, c be different, every other direction is altered. 
The scalars a, 6, c are known as the principal ratios of 
elongation. 

In Art. 115 it was seen that any complete dyadic was 
reducible to the normal form 



352 VECTOR ANALYSIS 

where a, J, c are positive constants. This expression may be 
factored into the product of two dyadics. 

0= (ai i + ftj j + ck k ) (i i + j j + k k), (19) 
or 0= (i i+j j + k k) (aii + 6jj + ckk). 
The factor i i + j j + k k 

which is the same in either method of factoring is a versor. 
It turns the vectors i, j, k into the vectors i , j , k . The vector 
semi-tangent of the versor 

ixi +j xj + k xk 

i>i + ^ + k kls i + i.i- +j .j + k.k" 

The other factor 

ai i + l j j + ck k , 

or aii 

is a right tensor and represents a pure strain. In the first 
case the strain has the lines i , j , k for principal axes: in 
the second, i, j, k. In both cases the ratios of elongation are 
the same, a to 1, b to 1, c to 1. If the negative sign occurs 
before the product the version and pure strain must have 
associated with them a reversal of directions of all vectors in 
space that is, a perversion. Hence 

Theorem: Any dyadic is reducible to the product of a 
versor and a right tensor taken in either order and a positive 
or negative sign. Hence the most general transformation 
representable by a dyadic consists of the product of a rota 
tion or version about a definite axis through a definite angle 
accompanied by a pure strain either with or without perver 
sion. The rotation and strain may be performed in either 
order. In the two cases the rotation and the ratios of elonga 
tion of the strain are the same ; but the principal axes of the 
strain differ according as it is performed before or after the 



ROTATIONS AND STRAINS 353 

rotation, either system of axes being derivable from the other 
by the application of the versor as a prefactor or postfactor 
respectively. 

If a dyadic be given the product of and its conjugate 
is a right tensor the ratios of elongation of which are the 
squares of the ratios of elongation of (P and the axes of which 
are respectively the antecedents or consequents of accord 
ing as C follows or precedes in the product. 

4> (ai i + 6 j j + ck k), 
C = (aii + 6 jj + ckk ), 
. C = a i i + 6 2 j j + c 2 k k , (20) 

c 2 kk. 



The general problem of finding the principal ratios of elonga 
tion, the antecedents, and consequents of a dyadic in its 
normal form, therefore reduces to the simpler problem of find 
ing the principal ratios of elongation and the principal axes 
of a pure strain. 

131.] The natural and immediate generalization of the 
right tensor 



is the dyadic <P = aaa + &bb + ccc (21) 

where a, 6, c are positive or negative scalars and where a, b, c 
and a , b , c are two reciprocal systems of vectors. Neces 
sarily a, b, c and a , b , c are each three non-coplanar. 
Definition : A dyadic that may be reduced to the form 



(21) 
is called a tonic. 

The effect of a tonic is to leave unchanged three non- 
coplanar directions a, b, c in space. If a vector be resolved 
into its components parallel to a, b, c respectively these 

23 



354 VECTOR ANALYSIS 

components are stretched in the ratios a to 1, & to 1, c to 1. 
If one or more of the constants a, &, c are negative the com 
ponents parallel to the corresponding vector a, b, c are re 
versed in direction as well as changed in magnitude. The 
tonic may be factored into three factors of which each 
stretches the components parallel to one of the vectors a, b, c 
but leaves unchanged the components parallel to the other 
two. 

cc ) (aa + &bb + ccXa 



The value of a tonic is not altered if in place of a, b, c 
any three vectors respectively collinear with them be sub 
stituted, provided of course that the corresponding changes 
which are necessary be made in the reciprocal system a , b , c . 
But with the exception of this change, a dyadic which is 
expressible in the form of a tonic is so expressible in only 
one way if the constants a, 6, c are different. If two of the 
constants say J and c are equal, any two vectors coplanar 
with the corresponding vectors b and c may be substituted 
in place of b and c. If all the constants are equal the tonic 
reduces to a constant multiple of the idemfactor. Any three 
non-coplanar vectors may be taken for a, b, c. 

The product of two tonics of which the axes a, b, c are the 
same is commutative and is a tonic with these axes and 
with scalar coefficients equal respectively to the products of 
the corresponding coefficients of the two dyadics. 

= a x a a + \ b V + ^ c c 



c 2 cc 



0. y = <? = a 1 a 2 aa + ^^bV-f c^cc . (22) 

The generalization of the cyclic dyadic 

a a + cos q (b V + c c ) + sin q (c b b c ) 
is = a aa -1- 1 (b V + cc ) + c (c V - be ), (23) 



ROTATIONS AND STRAINS 355 

where a, b, c are three non-coplanar vectors of which a r , V, c 
is the reciprocal system and where the quantities a, 6, c, are 
positive or negative scalars. This dyadic may be changed 
into a more convenient form by determining the positive 
scalar p and the positive or negative scalar q (which may 
always be chosen between the limits TT) so that 



and c=psinq. (24) 

That is, 



and tan 2 =. (24 y 

Then 

+ cc ) + p sin q (cV be ). (25) 



This may be factored into the product of three dyadics 
0= (aaa + bV + cc ) (a a + p bV + jpcc ) 
{aa + cos q (b b 4- o c ) + sin q (cV - be )}. 

The order of these factors is immaterial. The first is a tonic 
which leaves unchanged vectors parallel to b and c but 
stretches those parallel to a in the ratio of a to 1. If a is 
negative the stretching must be accompanied by reversal 
in direction. The second factor is also a tonic. It leaves 
unchanged vectors parallel to a but stretches all vectors in 
the plane of b and c in the ratio p to 1. The third is a 
cyclic factor. Vectors parallel to a remain unchanged ; but 
radii vectors in the ellipse of which b and c are conjugate 
semi-diameters are rotated through a sector such that the 
area of the sector is to the area of the whole ellipse as q to 
2 TT. Other vectors in the plane of b and c may be regarded 
as scalar multiples of the radii vectors of the ellipse. 



356 VECTOR ANALYSIS 

Definition : A dyadic which is reducible to the form 
<P = a aa + p cos q (bb + cc ) + p sin q (c V be ), (25) 

owing to the fact that it combines the properties of the 
cyclic dyadic and the tonic is called a cyclolonic. 

The product of two cyclotonics which have the same three 
vectors, a, b, c as antecedents and the reciprocal system 
a , b , c for consequents is a third cyclotonic and is com 
mutative. 



cc ) + p l sinq l (cb f be ) 
5F = a 2 aa +jp 2 cos j 2 (bb f + cc ) + jp a sin q 2 (cb be ) 
0. 5P*= W* <? = a 1 a 2 aa + p l p 2 cos (q l + j a ) (bb + cc ) 
+ Pi P* sin ( 2l + & ) (c b - b c ). (26) 

Reduction of Dyadics to Canonical Forms 

132.] Theorem : In general any dyadic may be reduced 
either to a tonic or to a cyclotonic. The dyadics for which 
the reduction is impossible may be regarded as limiting cases 
which may be represented to any desired degree of approxi 
mation by tonics or cyclotonics. 

From this theorem the importance of the tonic and cyclo 
tonic which have been treated as natural generalizations of 
the right tensor and the cyclic dyadic may be seen. The 
proof of the theorem, including a discussion of all the 
special cases that may arise, is long and somewhat tedious. 
The method of proving the theorem in general however is 
patent. If three directions a, b, c may be found which are 
left unchanged by the application of $ then <P must be a 
tonic. If only one such direction can be found, there exists 
a plane in which the vectors suffer a change such as that due 
to the cyclotonic and the dyadic indeed proves to be such. 



ROTATIONS AND STRAINS 357 

The question is to find the directions which are unchanged 
by the application of the dyadic 0. 
If the direction a is unchanged, then 

a = a a (27) 

or (0 al).a = 0. 

The dyadic a I is therefore planar since it reduces vectors 
in the direction a to zero. In special cases, which are set 
aside for the present, the dyadic may be linear or zero. In 
any case if the dyadic 

<P-aI 

reduces vectors collinear with a to zero it possesses at least 
one degree of nullity and the third or determinant of <P 
vanishes. 

(0-aI) 8 = 0. (28) 

Now (page 331) (0 + W) z = <P B + <P 2 : W + : W^ + z . 
Hence (4> - a I) 8 = <P Z - a <Z> 2 : 1 + a 2 : ^ - a 3 1 8 

I 2 = I and I 3 = 1. 
But : 1 = 



Hence the equation becomes 

a 3 - a 2 a + a 0^ -0 3 = 0. (29) 

The value of a which satisfies the condition that 



is a solution of a cubic equation. Let x replace a. The 
cubic equation becomes 

x* - x* d> 3 + x 2S - 8 = 0. (29) 



\ 

358 VECTOR ANALYSIS 

Any value of x which satisfies this equation will be such 

that 

(*-aI), = 0. (28) 

That is to say, the dyadic x I is planar. A vector per 
pendicular to its consequents is reduced to zero. Hence 
leaves such a direction unchanged. The further discussion 
of the reduction of a dyadic to the form of a tonic or a cyclo- 
tonic depends merely upon whether the cubic equation in x 
has one or three real roots. 

133.] Theorem : If the cubic equation 

x* - x* 4> s + x 2 * - 8 = (29) 

has three real roots the dyadic <P may in general be reduced 
to a tonic. 

For let x = a, x = &, x = c 

be the three roots of the equation. The dyadics 
<P a I, 61, <P cl 

are in general planar. Let a, b, c be respectively three 
vectors drawn perpendicular to the planes of the consequents 
of these dyadics. 



b = 0, (30) 

(0-cI).c = 0. 
Then <P a = a a, 

</>-b = &b, (30) 

<p . c = cc. 

If the roots a, &, c are distinct the vectors a, b, c are non- 
coplanar. For suppose 

c = ma + ?ib 



ROTATIONS AND STRAINS 359 

m $ a raca-ffl>0b n c b = 0. 
But a = a a, b = 6 b. 

Hence m (a c) a + n (b c) b = 0, 

and m(a c) = 0, n(b c) = 0. 

Hence m = or a = c, TI = or b = c. 

Consequently if the vectors a, b, c are coplanar, the roots are 
not distinct; and therefore if the roots are distinct, the 
vectors a, b, c are necessarily non-coplanar. In case the roots 
are not distinct it is still always possible to choose three 
non-coplanar vectors a, b, c in such a manner that the equa 
tions (30) hold. This being so, there exists a system a , b , c 
reciprocal to a, b, c and the dyadic which carries a, b, c into 
a a, b b, c c is the tonic 



Theorem : If the cubic equation 

x* - x* 4> a + x d> 2S - 3 = (29) 

has one real root the dyadic may in general be reduced to 
a cyclotonic. 

The cubic equation has one real root. This must be posi 
tive or negative according as <P B is positive or negative. Let 
the root be a. Determine a perpendicular to the plane of 
the consequents of 4> a I. 

(<P-aI) .a = 0. 
Determine a also so that 

a . (0- a I) = 

and let the lengths of a and a be so adjusted that a a = l. 
This cannot be accomplished in the special case in which a 



360 VECTOR ANALYSIS 

and a ; are mutually perpendicular. Let b be any vector in 
the plane perpendicular to a . 

a (0 - a I) - b = 0. 

Hence (<P al)b is perpendicular to a . Hence <Pb is 
perpendicular to a . In a similar manner <P 2 b, $ 3 b, and 
<P~ l b, 0~ 2 b, etc., will all be perpendicular to a and lie in 
one plane. The vectors <P b and b cannot be parallel or 
would have the direction b as well as a unchanged and 
thus the cubic would have more than one real root. 

The dyadic changes a, b, b into a, </> 2 b, <P b re 
spectively. The volume of the parallelepiped 

[<p.a </> 2 b </>.b] = </> 3 [a <P-b b]. (31) 
But $a = aa. 

Hence a a (<P 2 b) x (0 b) = <P 3 a (0 b) x b. (31) 

The vectors <0 2 b, $ b, b all lie in the same plane. Their 
vector products are parallel to a and to each other. Hence 

a (0 2 - b) x (</>.b) = 3 ><Pb xb. (31)" 

Inasmuch as a and <P 3 have the same sign, let 

^ = a-i* s . (32) 

Let also b 1 =;r 1 #-b b 2 = /r 2 # 2 b ? etc. (33) 

and b_! p (&- 1 b b_ 2 = p 2 #~ 2 b, etc. 

b 2 X b x b x X b, 
or (b 2 + b) x b x = 0. 

The vectors b 2 + b and b x are parallel. Let 

b 2 + b = 27ib r (34) 

Then b 3 + b 1 = 27ib 2 b 1 + b 2 = 2nb 3 etc., 

b x -f b_! = 2 n b b_! + b_ 2 = 2 n b_ x etc. 



ROTATIONS AND STRAINS 861 

Lay off from a common origin the vectors 

b, bj, b 2 , etc., b_ j, b_2, etc. 

Since is not a tonic, that is, since there is no direction in 
the plane perpendicular to a which is left unchanged by 
these vectors b OT pass round and round the origin as m takes 
on all positive and negative values. The value of n must 
therefore lie between plus one and minus one. Let 

n = cos q. (36) 

Then b-j + bj = 2 cos q b. 

Determine c from the equation 

b x = cos q b + sin q c. 
Then b_j = cos q b sin q c. 

Let a , b f , c be the reciprocal system of a, b, c. This is pos 
sible since a was so determined that a a = 1 and since 
a, b, c are non-coplanar. Let 

= cos q (bV + ccO + sin q (c V - be ). 
Then ra = 0, ?F.b = b 1 , .})_ l = b. 
Hence (a *& + p ) & = a a = $ a, 

(a aa + p W) b = p b x = - b, 
(a aa + p ) b_ a =p b = d> . b_ r 

The dyadic a a a + p W changes the vectors a, b and b^ into 
the vectors - a, b, and b_ x respectively. Hence 

= (a aa + p W) = a aa + ^ cos j (bb + cc 7 ) 

4- ^? sin q (c V b o ). 

The dyadic in case the cubic equation has only one real 
root is reducible except in special cases to a cyclotonic. 
The theorem that a dyadic in general is reducible to a tonic 
or cyclotonic has therefore been demonstrated. 



362 VECTOR ANALYSIS 

134.] There remain two cases 1 in which the reduction 
is impossible, as can be seen by looking over the proof. In 
the first place if the constant n used in the reduction to cyclo- 
tonic form be 1 the reduction falls through. In the second 
place if the plane of the antecedents of 



and the plane of the consequents are perpendicular the 
vectors a and a used in the reduction to cyclotonic form are 
perpendicular and it is impossible to determine a such that 
a a shall be unity. The reduction falls through. 

If n=l, b_ 1 + b 1 = 2b. 

Let b_ 1 + b 1 = 2b. 

Choose c = b 1 b = b b_ r 

Consider the dyadic W = a aa + p (bV + co ) 4- p o V 
y.a = aa=<P.a, 

*P b pb + pc pbi = <P b, 
?p*.o=jt)c= - pb 1 JP b = c. 
Hence <P = a aa + p (b V + cc ) + p cb r (37) 

The transformation due to this dyadic may be seen best by 
factoring it into three factors which are independent of the 
order or arrangement 



.(aa + bb + cc + cb 7 ). 

1 In these cases it will be seen that the cubic equation has three real roots. 
In one case two of them are equal and in the other case three of them. Thus 
these dyadics may be regarded as limiting cases lying between the cyclotonic in 
which two of the roots are imaginary and the tonic in which all the roots are real 
and distinct. The limit may be regarded as taking place either by the pure 
imaginary part of the two imaginary roots of the cyclotonic becoming zero or by 
two of the roots of the tonic approaching each other. 



ROTATIONS AND STRAINS 863 

The first factor represents an elongation in the direction a in a 
ratio a to 1. The plane of b and c is undisturbed. The 
second factor represents a stretching of the plane of b and c in 
the ratio t p to 1. The last factor takes the form 

I + cb . 
(I + oV) a = #a, 

(I + c V) x b = x b + x o, 
(I + c V) x c = x c. 

A dyadic of the form I + cb leaves vectors parallel to a and c 
unaltered. A vector #b parallel to b is increased by the vec 
tor c multiplied by the ratio of the vector # b to b. In other 
words the transformation of points in space is such that the 
plane of a and c remains fixed point for point but the points 
in planes parallel to that plane are shifted in the direction c 
by an amount proportional to the distance of the plane in 
which they lie from the plane of a and c. 
Definition : A dyadic reducible to the form 

I + cb 

is called a shearing dyadic or shearer and the geometrical 
transformation which it causes is called a shear. The more 
general dyadic 

<P = a aa + p (b V + c c ; ) + o V (37) 

will also be called a shearing dyadic or shearer. The trans 
formation to which it gives rise is a shear combined with 
elongations in the direction of a and is in the plane of b and c. 
If n = 1 instead of n = +1, the result is much the same. 
The dyadic then becomes 

$ = a aa -,p (bV + c<0 - c V (37) 

$ = (a aa + bb r + cc ) {aa f -p (b D + cc )> (I + cV). 



364 VECTOR ANALYSIS 

The factors are the same except the second which now repre 
sents a stretching of the plane of b and c combined with a 
reversal of all the vectors in that plane. The shearing dyadic 
then represents an elongation in the direction a, an elonga 
tion combined with a reversal of direction in the plane of 
b and c, and a shear. 

Suppose that the plane of the antecedents and the plane of 
the consequents of the dyadic 0al are perpendicular. Let 
these planes be taken respectively as the plane of j and k and 
the plane of i and j. The dyadic then takes the form 

<p a I A j i + B j j + C k i + D k j. 
The coefficient B must vanish. For otherwise the dyadic 

j Bk) 



is planar and the scalar a + B is a root of the cubic equation. 
With this root the reduction to the form of a tonic may be 
carried on as before. Nothing new arises. But if B vanishes 
a new case occurs. Let 



This may be reduced as follows to the form 

ab + bc 

where a V = a c = b c = and b V = 1. 
Square W W 2 = A D ki = ac . 

Hence a must be chosen parallel to k ; and c , parallel to i. 
The dyadic W may then be transformed into 



Then =AD*, V= Ci + Di 

A D 

b = A j c = i. 



ROTATIONS AND STRAINS 365 

With this choice of a, b, V, c the dyadic reduces to the 
desired form ab + be and hence the dyadic <P is reduced to 

= al + ab + bc (38) 

or = aaa + abb + ace + aV + be . 

This may be factored into the product of two dyadics the 
order of which is immaterial. 



The first factor al represents a stretching of space in all 
directions in the ratio a to 1. The second factor 



represents what may be called a complex shear. For 
r = IT + ab r+ bc -r= r-t-aV-r + bc -r. 

If r is parallel to a it is left unaltered by the dyadic Q. If 
r is parallel to b it is changed by the addition of a term 
which is in direction equal to a and in magnitude propor 
tional to the magnitude of the vector r. In like manner 
if r is parallel to c it is changed by the addition of a term 
which in direction is equal to b and which in magnitude is 
proportional to the magnitude of the vector r. 

-zb = (I + ab -f bc ).zb= zb + a a 
Q *xc = (I + ab + be ) xc = xc 4- #b. 

Definition : A dyadic which may be reduced to the form 
<P = aI + ab + bc (38) 

is called a complex shearer. 

The complex shearer as well as the simple shearer men 
tioned before are limiting cases of the cyclotonic and tonic 
dyadics. 



366 VECTOR ANALYSIS 

135.] A more systematic treatment of the various kinds 
of dyadics which may arise may be given by means of the 
Hamilton-Cayley equation 



03 _ a 02 + 0^ _ 3 i = 
and the cubic equation in x 

x* - S x* + <P 25 x - 8 = 0. (29) 

If a, &, c are the roots of this cubic the Hamilton-Cayley 
equation may be written as 

(0 - al) (<P - JI) (0 - <?I) = 0. (40) 

If, however, the cubic has only one root the Hamilton-Cayley 
equation takes the form 

(0_al).(0 2 - 2^0082 4> + p*I) = 0. (41) 

In general the Hamilton-Cayley equation which is an equa 
tion of the third degree in is the equation of lowest degree 
which is satisfied by 0. In general therefore one of the above 
equations and the corresponding reductions to the tonic or 
cyclotonic form hold. In special cases, however, the dyadic 
may satisfy an equation of lower degree. That equation 
of lowest degree which may be satisfied by a dyadic is called 
its characteristic equation. The following possibilities occur. 

I. (<P - a I) - (0 - b I) ( - c I) = 0. 

II. (0-aI) 

III. ( 

IV. (<P - a !).(</>- 61) = 0. 
V. (0-aI) 3 = 0. 

VI. (<P al) 2 = 0. 

VII. (<P-aI) = 0. 



ROTATIONS AND STRAINS 367 

In the first case the dyadic is a tonic and may be reduced 

to the form 

6bb + ccc . 



In the second case the dyadic is a cyclotonic and may be 
reduced to the form 

d> = a a a + p cos q (bb + cc ) + p sin q (eb be ). 

In the third case the dyadic is a simple shearer and may be 
reduced to the form 

</> = aaa + 6 (bb + cc ) + cb . 

In the fourth case the dyadic is again a tonic. Two of the 
ratios of elongation are the same. The following reduction 
may be accomplished in an infinite number of ways. 

= aaa + b (bb + cc ). 

In the fifth case the dyadic is a complex shearer and may be 

so expressed that 

0= al + ab -f be . 

In the sixth case the dyadic is again a simple shearer which 
may be reduced to the form 

4> = al + cb =a (aa 4- bb + cc ) + cb . 

In the seventh case the dyadic is again a tonic which may be 
reduced in a doubly infinite number of ways to the form 

= al = a(aa / + bb + cc ). 

These seven are the only essentially different forms which a 
dyadic may take. There are then only seven really different 
kinds of dyadics three tonics in which the ratios of elonga 
tion are all different, two alike, or all equal, and the cyclo 
tonic together with three limiting cases, the two simple and 
the one complex shearer. 



368 VECTOR ANALYSIS 

Summary of Chapter VI 

The transformation due to a dyadic is a linear homogeneous 
strain. The dyadic itself gives the transformation of the 
points in space. The second of the dyadic gives the trans 
formation of plane areas. The third of the dyadic gives the 
ratio in which volumes are changed. 



The necessary and sufficient condition that a dyadic repre 
sent a rotation about a definite axis is that it be reducible to 

the form 

= i i + j j + k k (1) 

or that 4> c = I <P 3 = + 1 (2) 

or that c = I 8 > 

The necessary and sufficient condition that a dyadic repre 
sent a rotation combined with a transformation of reflection 
by which each figure is replaced by one symmetrical to it is 

that 

= -(i i + j j + k k) (iy 

or that $ <P C = I* ^3 = 1 

or that 0.00 = 1, 3 <0. (3) 

A dyadic of the form (1) is called a versor ; one of the form 
(1) , a perversor. 

If the axis of rotation of a versor be chosen as the i-axis 
the versor reduces to 

= ii + cos q (j j + kk) + sin q (kj - j k) (4) 
or = ii + cos q (I ii) + sin q I x i. (5) 

If any unit vector a is directed along the axis of rotation 

<p = a a + cos q (I a a) + sin q 1 x a (6) 
The axis of the versor coincides in direction with X . 



ROTATIONS AND STRAINS 369 

If a vector be drawn along the axis and if the magnitude of 
the vector be taken equal to the tangent of one-half the angle 
of rotation, the vector determines the rotation completely. 
This vector is called the vector semi-tangent of version. 



2 (9) 

In terms of Q the versor <P may be expressed in a number of 



was. 

a 



dft / a a \ 

<P = + cos q (I - ) + sin q I x 

a-ft \ a-ay 



(10) 



or <D = (I + I x ft) (I - I x Q)- 1 (10) 

J^ + axQ) (loy , 



tf = 

1 + Q-ft 

If a is a unit vector a dyadic of the form 

<P = 2aa-I (11) 

is a biquadrantal versor. Any versor may be resolved into 
the product of two biquadrantal versors and by means of 
such resolutions any two versors may be combined into 
another. The law of composition for the vector semi-tangents 
of version is 



A dyadic reducible to the form 
<P = aa + cos q (bb + cc ) + sin q (cb -W) (17) 

is called a cyclic dyadic. It produces a generalization of 
simple rotation an elliptic rotation, so to speak. The pro- 

24 



370 VECTOR ANALYSIS 

duct of two cyclic dyadics which have the same antecedents 
a, b, c and consequents a b c is obtained by adding their 
angles q. A cyclic dyadic may be regarded as a root of the 
idemfactor. A dyadic reducible to the form 

= aii +bjj + ckk (18) 

where #, &, c are positive scalars is called a right tensor. It 
represents a stretching along the principal axis i, j, k in the 
ratio a to 1, b to 1, c to 1 which are called the principal ratios 
of elongation. This transformation is a pure strain. 

Any dyadic may be expressed as the product of a versor, 
a right tensor, and a positive or negative sign. 

= (a i i + & j j + c k k ) (i i + j j + k k) 
or <P= (i i + j j + k k).(aii + Jjj + ckk). (19) 



Consequently any linear homogeneous strain may be regarded 
as a combination of a rotation and a pure strain accompanied 
or unaccompanied by a perversion. 

The immediate generalizations of the right tensor and the 
cyclic dyadic is to the tonic 

= aaa + &bb + ccc (21) 

and cyclotonic 

cc ) + c(cV-bc) (23) 



or <P = aaa + p cos q (bb + cc )+^sing (cV be ) (25) 

where p = + V 6 2 + c 2 and tan I q = - -. (24) 

* 2? + 

Any dyadic in general may be reduced either to the form 
(21), and is therefore a tonic, or to the form (25), and is 
therefore a cyclotonic. The condition that a dyadic be a 
tonic is that the cubic equation 

+ 0^ x - <J> 3 = (29) 



ROTATIONS AND STRAINS 371 

shall have three real roots. Special cases in which the 
reduction may be accomplished in more ways than one arise 
when the equation has equal roots. The condition that a 
dyadic be a cyclotonic is that this cubic equation shall have 
only one real root. There occur two limiting cases in which 
the dyadic cannot be reduced to cyclotonic form. In these 
cases it may be written as 

4> =aaa +jp (bb + cc ) + cb (37) 

and is a simple shearer, or it takes the form 

= al + ab + bc (38) 

and is a complex shearer. Dyadics may be classified accord 
ing to their characteristic equations 

(<P-aI).(0-&I).(<P-cI) =0 tonic 

(# a I) (<P 2 2 p cos q + jp 2 1) = cyclotonic 

(0 a I) (# & I) 2 = simple shearer 

(0_ <*!)($ &I) = special tonic 

(0 a I) 8 = complex shearer 

(0 a I) 2 = special simple shearer 

(0 a I) = special tonic. 



CHAPTER VII 

MISCELLANEOUS APPLICATIONS 

Quadric Surfaces 
136.] If be any constant dyadic the equation 

r . . r = const. (1) 

is quadratic in r. The constant, in case it be not zero, may 
be divided into the dyadic and hence the equation takes 
the form 

r r = 1, 



or r r = 0. (2) 

The dyadic may be assumed to be self -conjugate. For if 
W is an anti-self-conjugate dyadic, the product r W r is 
identically zero for all values of r. The proof of this state 
ment is left as an exercise. By Art. 116 any self-conjugate 
dyadic is reducible to the form 



- t L l JJ 

- 



If 



Hence the equation r r = 1 

represents a quadric surface real or imaginary. 

The different cases which arise are four in number. If the 
signs are all positive, the quadric is a real ellipsoid. If one 
sign is negative it is an hyperboloid of one sheet; if two are 



QUADRIC SURFACES 373 

negative, a hyperboloid of two sheets. If the three signs are 
all negative the quadric is imaginary. In like manner the 

equation 

r r = 

is seen to represent a cone which may be either real or 
imaginary according as the signs are different or all alike. 

Thus the equation 

r 0- r = const. 

represents a central quadric surface. The surface reduces to 
a cone in case the constant is zero. Conversely any central 
quadric surface may be represented by a suitably chosen self- 
conjugate dyadic in the form 

r d> r = const. 

This is evident from the equations of the central quadric 
surfaces when reduced to the normal form. They are 

# 2 7/ 2 z 2 
- = const. 
a 2 6 J c 2 

The corresponding dyadic <Pis $ = . 

a* o* c* 

The most general scalar expression which is quadratic in 
the vector r and which consequently when set equal to a con 
stant represents a quadric surface, contains terms like 

r r, (r a) (b r) , r c, d e, 

where a, b, c, d, e are constant vectors. The first two terms 
are of the second order in r ; the third, of the first order ; and 
the last, independent of r. Moreover, it is evident that these 
four sorts of terms are the only ones which can occur in a 
scalar expression which is quadratic in r. 

But r r = r I r, 

and (r a) (b r) = r a b r. 



374 VECTOR ANALYSIS 

Hence the most general quadratic expression may be reduced 
to 



where is a constant dyadic, A a constant vector, and 
a constant scalar. The dyadic may be regarded as self- 
conjugate if desired. 

To be rid of the linear term r A, make a change of origin 
by replacing r by r t. 

(r -t). 0- (r -t) + (r -t) A+ C=0 

r . <P * r t $ r r <P t + t <P t 

+ r -A-t- A + (7=0. 

Since is self-conjugate the second and third terms are 
equal. Hence 

r r + 2 r (J A - t) + C f = 0. 

If now is complete the vector t may be chosen so that 

IA = 0-t or t = 5 0- 1 - A. 

L L 

Hence the quadric is reducible to the central form 
r r = const. 

In case is incomplete it is unt planar or unilinear because 
is self -con jugate. If A lies in the plane of or in the line 
of as the case may be the equation 



is soluble for t and the reduction to central form is still pos 
sible. But unless A is so situated the reduction is impossible. 
The quadric surface is not a central surface. 

The discussion and classification of the various non-central 
quadrics is an interesting exercise. It will not be taken up 
here. The present object is to develop so much of the theory 



QUADRIC SURFACES 375 

of quadric surfaces as will be useful in applications to mathe 
matical physics with especial reference to non-isotropic 
media. Hereafter therefore the central quadrics and in par 
ticular the ellipsoid will be discussed. 

137.] The tangent plane may be found by differentiation. 

r <P r = 1. 

di r + r <t> di = 0. 
Since <P is self-conjugate these two terms are equal and 

dr.0-r = 0. (5) 

The increment d r is perpendicular to <P r. Hence r is 
normal to the surface at the extremity of the vector r. Let 
this normal be denoted by K and let the unit normal be n. 

BT = <P r (6) 

r r 



n = 



r) (0 r) Vr # 2 r* 



Let p be the vector drawn from the origin perpendicular to 
the tangent plane, p is parallel to n. The perpendicular 
distance from the origin to the tangent plane is the square 
root of p p. It is also equal to the square root of r p. 

r p = r cos (r, p) p = p 2 . 
Hence r p = p p. 

Or Ll! = , . JL = L 

p.p p.p 

But r0r = rH = l. 

Hence inasmuch as p and IT are parallel, they are equal. 

0.r = !T=-^-. (T) 

p.p 



376 VECTOR ANALYSIS 

On page 108 it was seen that the vector which has the direc 
tion of the normal to a plane and which is in magnitude equal 
to the reciprocal of the distance from the origin to the plane 
may be taken as the vector coordinate of that plane. Hence 
the above equation shows that <P r is not merely normal to 
the tangent plane, but is also the coordinate of the plane. 
That is, the length of <P r is the reciprocal of the distance 
from the origin to the plane tangent to the ellipsoid at 
the extremity of the vector r. 

The equation of the ellipsoid in plane coordinates may be 
found by eliminating r from the two equations. 

( r r = 1, 



Hence r r = H 0- 1 0- 1 If = JT 0" 1 H. 
Hence the desired equation is 

H-0-i-H = l. (8) 

*4 +y+" 

c 2 kk. 



Let r = #i-t-yj+3k, 

and N = ui + v j + wk, 

where u, v, w are the reciprocals of the intercepts of the 

plane N upon the axes i, j, k. Then the ellipsoid may be 

written in either of the two forms familiar in Cartesian 
geometry. 



or K 0- 1 .N = a 2 w a + Z> 2 v 2 + c 2 w 2 = 1. (10) 



QUADRIC SURFACES 377 

138.] The locus of the middle points of a system of 
parallel chords in an ellipsoid is a plane. This plane is 
called the diametral plane conjugate with the system of 
chords. It is parallel to the plane drawn tangent to the 
ellipsoid at the extremity of that one of the chords which 
passes through the center. 

Let r be any radius vector in the ellipsoid. Let n be the 
vector drawn to the middle point of a chord parallel to a. 

Let r = s + x a. 

If r is a radius vector of the ellipsoid 

r r = (B + x a) <D (s + x a) = 1. 
Hence s $ s + 2 # s . a + 2? a # a = 1. 

Inasmuch as the vector s bisects the chord parallel to a the 
two solutions of x given by this equation are equal in mag 
nitude and opposite in sign. Hence the coefficient of the 

linear term x vanishes. ,. 

s . . a = 0. 

Consequently the vector s is perpendicular to a. The 
locus of the terminus of s is therefore a plane passed through 
the center of the ellipsoid, perpendicular to a, and parallel 
to the tangent plane at the extremity of a. 

If b is any radius vector in the diametral plane conjugate 

with a, _ A 

b a = 0. 

The symmetry of this equation shows that a is a radius 
vector in the plane conjugate with b. Let c be a third radius 
vector in the ellipsoid and let it be chosen as the line of 
intersection of the diametral planes conjugate respectively 

with a and b. Then 

a . d> . b = 0, 

b . c = 0, (11) 

e a = 0. 



378 VECTOR ANALYSIS 

The vectors a, b, c are changed into <P a, <D b, <P c by 
the dyadic 0. Let 

a = a, V = b, c - b c. 
The vectors a , b , c form the system reciprocal to a, b, c. 
For a a = a a = 1, b V = b b = 1, 

c o = c o = 1, 
and a V = a b = 0, b c = b c = 0, 

c a = c a = 0. 

The dyadic may be therefore expressed in the forms 

= a a + b b + cV, (12) 

and 0" 1 = aa + bb + cc. 

If for convenience the three directions a, b, c, be called a 
system of three conjugate radii vectors, and if in a similar 
manner the three tangent planes at their extremities be called 
a system of three conjugate tangent planes, a number of 
geometric theorems may be obtained from interpreting the 
invariants of 0. A system of three conjugate radii vectors 
may be obtained in a doubly infinite number of ways. 

The volume of a parallelepiped of which three concurrent 
edges constitute a system of three conjugate radii vectors is 
constant and equal in magnitude to the rectangular parallele 
piped constructed upon the three semi-axes of the ellipsoid. 

For let a, b, c be any system of three conjugate axes. 

0- 1 = aa + bb + cc. 

The determinant or third of 0" 1 is an invariant and inde 
pendent of the form in which is expressed. 

3 -i=[abc] 2 . 



QUADRIC SURFACES 379 

But if 0-!:=a 2 ii + & 2 jj + c 2 kk, 



Hence [a b c] = a 6 c. 

This demonstrates the theorem. In like manner by inter 
preting <P 3 , <Ps~~\ and S it is possible to show that: 

The sum of the squares of the radii vectors drawn to an 
ellipsoid in a system of three conjugate directions is constant 
and equal to the sum of the squares of the semi-axes. 

The volume of the parallelepiped, whose three concurrent 
edges are in the directions of the perpendiculars upon a system 
of three conjugate tangent planes and in magnitude equal to 
the reciprocals of the distances of those planes from the 
center of the ellipsoid, is constant and equal to the reciprocal 
of the parallelepiped constructed upon the semi-axes of the 
ellipsoid. 

The sum of the squares of the reciprocals of the three per 
pendiculars dropped from the origin upon a system of three 
conjugate tangent planes is constant and equal to the sum of 
the squares of the reciprocals^ the semi-axes. 

If i, j, k be three mutually perpendicular unit vectors 

4> s = i* </> i + j <P j + k # k, 
tf^-i = i . 0-i . i + j . 0-i . j + k 0" 1 k. 

Let a, b, c be three radii vectors in the ellipsoid drawn 
respectively parallel to i, j, k. 

a . . a = b 

i i j 

Hence <P a = - - - + * 
a a 

But the three terms in this expression are the squares of the 
reciprocals of the radii vectors drawn respectively in the i, j, 
k directions. Hence : 



380 VECTOR ANALYSIS 

The sum of the squares of the reciprocals of three mutually 
perpendicular radii vectors in an ellipsoid is constant. And 
in a similar manner: the sum of the squares of the perpen 
diculars dropped from the origin upon three mutually perpen 
dicular tangent planes is constant. 

139.] The equation of the polar plane of the point deter 
mined by the vector a is l 

s a = 1. (13) 

For let s be the vector of a point in the polar plane. The 
vector of any point upon the line which joins the terminus of 

s and the terminus of a is 

y s + #a 

x + y 
If this point lies upon the surface 



8*0*8+ 



x + y x + y 

2 x y 



If the terminus of s lies in the polar plane of a the two values 
of the ratio x:y determined by this equation must be equal 
in magnitude and opposite in sign. Hence the term in x y 
vanishes. 

Hence s a = 1 

is the desired equation of the polar plane of the terminus 
of a. 

Let a be replaced by z a. The polar plane becomes 

s . (p . z a = 1, 

1 
or s (P a = - 

z 

1 It is evidently immaterial whether the central quadric determined by * be 
real or imaginary, ellipsoid or hyperboloid. 



QUADRIC SURFACES 381 

When z increases the polar plane of the terminus of z a 
approaches the origin. In the limit when z becomes infinite 
the polar plane becomes 

s a = 0. 

Hence the polar plane of the point at infinity in the direction 
a is the same as the diametral plane conjugate with a. This 
statement is frequently taken as the definition of the diame 
tral plane conjugate with a. In case the vector a is a radius 
vector of the surface the polar plane becomes identical with 
the tangent plane at the terminus of a. The equation 

s <P a = 1 or s IT = 1 

therefore represents the tangent plane. 

The polar plane may be obtained from another standpoint 
which is important. If a quadric Q and a plane P are given, 



and P = r c C = 0, 

the equation (r r - 1) + k (r c - C) 2 = 

represents a quadric surface which passes through the curve 
of intersection of Q and P and is tangent to Q along that 
curve. In like manner if two quadrics Q and Q f are given, 

Q = T r 1 = 
Q = T* -r-l = 0, 
the equation (r r 1) + k (r # r - 1) = 

represents a quadric surface which passes through the curves 
of intersection of Q and Q and which cuts Q and Q f at no 
other points. In case this equation is factorable into two 
equations which are linear in r, and which consequently rep 
resent two planes, the curves of intersection of Q and Q r 
become plane and lie in those two planes. 



382 VECTOR ANALYSIS 

If A is any point outside of the quadric and if all the tangent 
planes which pass through A are drawn, these planes envelop 
a cone. This cone touches the quadric along a plane curve 
the plane of the curve being the polar plane of the point A. 
For let a be the vector drawn to the point A. The equation 
of any tangent plane to the quadric is 

s . . r = 1. 

If this plane contains A, its equation is satisfied by a. Hence 
the conditions which must be satisfied by r if its tangent 
plane passes through A are 

a . r = 1, 
r <P r = 1. 

The points r therefore lie in a plane r (<P a) = 1 which 
on comparison with (13) is seen to be the polar plane of A. 
The quadric which passes through the curve of intersection 
of this polar plane with the given quadric and which touches 
the quadric along that curve is 

(r r - 1) + k (a r - I) 2 = 0. 
If this passes through the point -4, 

(a . . a - 1) + k (a a - I) 2 = 0. 
Hence (r r - 1) (a a - 1) - (a r - I) 2 = 0. 

By transforming the origin to the point A this is easily seen 
to be a cone whose vertex is at that point 

140.] Let be any self-conjugate dyadic. It is expres 
sible in the form 



where A, 5, C are positive or negative scalars. Further- 
more let A<B<G 

- Bl = (<7- B) kk - (5 - A) ii. 



QUADRIC SURFACES 383 

Let V C B k = c and V B A i = a. 

Then 0- Bl = cc-aa = \ j(c + a)(c-a)+(c-a)(c+a) J. 
Let c + a = p and c a = q. 

Then <P = 51 + -(pq + qp). (14) 



The dyadic $ has been expressed as the sum of a constant 
multiple of the idemfactor and one half the sum 

pq + qp. 

The reduction has assumed tacitly that the constants -4, B, 
are different from each other and from zero. 

This expression for <P is closely related to the circular 
sections of the quadric surface 

r r = 1. 

Substituting the value of $, r r = 1 becomes 

5 r r + r p q r = 1. 
Let r p = n 

be any plane perpendicular to p. By substitution 
B r.r-ftt q r 1 = 0. 

This is a sphere because the terms of the second order all 
have the same coefficient B. If the equation of this sphere 
be subtracted from that of the given quadric, the resulting 
equation is that of a quadric which passes through the inter 
section of the sphere and the given quadric. The difference 

q r (r p n) = 0. 

Hence the sphere and the quadric intersect in two plane 
curves lying in the planes 

q . r = and r p = n. 



384 VECTOR ANALYSIS 

Inasmuch as these curves lie upon a sphere they are circles. 
Hence planes perpendicular to p cut the quadric in circles. 
In like manner it may be shown that planes perpendicular to 
q cut the quadric in circles. The proof may be conducted as 

follows : 

.5 r r + r p q r = 1. 

If r is a radius vector in the plane passed through the center 
of the quadric perpendicular to p or q, the term r p q r van 
ishes. Hence the vector r in this plane satisfies the equation 

B r-r = l 

and is of constant length. The section is therefore a circular 
section. The radius of the section is equal in length to the 
mean semi-axis of the quadric. 

For convenience let the quadric be an ellipsoid. The con 
stants A, B, C are then positive. The reciprocal dyadic (P" 1 
may be reduced in a similar manner. 



B 



B \B C \A B 



Let 1 = 1-* and d = -i. 

Then 1 - -^ I = f f - dd = \ j (f + d) (f - d) 

+ (f-d)(f + d)j 
Let + d = u and d = v. 

Then 0-i = 4 1 + I O v + vu). (15) 

> * 



QUADRIC SURFACES 385 

The vectors u and v are connected intimately with the cir 
cular cylinders which envelop the ellipsoid 

r r = 1 or N (t>~ 1 N = 1. 

For - N N + N u v N = 1. 

z> 

If now N be perpendicular to u or v the second term, namely, 
N u v N, vanishes and hence the equation becomes 

N - N = B. 
That is, the vector N is of constant length. But the equation 



is the equation of a cylinder of which the elements and tan 
gent planes are parallel to u. If then N N is constant the 
cylinder is a circular cylinder enveloping the ellipsoid. The 
radius of the cylinder is equal in length to the mean semi-axis 
of the ellipsoid. 

There are consequently two planes passing through the 
origin and cutting out circles from the ellipsoid. The normals 
to these planes are p and q. The circles pass through the 
extremities of the mean axis of the ellipsoid. There are also 
two circular cylinders enveloping the ellipsoid. The direction 
of the axes of these cylinders are n and v. Two elements of 
these cylinders pass through the extremities of the mean axis 
of the ellipsoid. 

These results can be seen geometrically as follows. Pass 
a plane through the mean axis and rotate it about that 
axis from the major to the minor axis. The section is an 
ellipse. One axis of this ellipse is the mean axis of the 
ellipsoid. This remains constant during the rotation. The 
other axis of the ellipse varies in length from the major to the 
minor axis of the ellipsoid and hence at some stage must pass 
through a length equal to the mean axis. At this stage of 

25 



386 VECTOR ANALYSIS 

the rotation the section is a circle. In like manner consider 
the projection or shadow of the ellipsoid cast upon a plane 
parallel to the mean axis by a point at an infinite distance 
from that plane and in a direction perpendicular to it. As the 
ellipsoid is rotated about its mean axis, from the position in 
which the major axis is perpendicular to the plane of projec 
tion to the position in which the minor axis is perpendicular 
to that plane, the shadow and the projecting cylinder have the 
mean axis of the ellipsoid as one axis. The other axis changes 
from the minor axis of the ellipsoid to the major and hence at 
some stage of the rotation it passes through a value equal to 
the mean axis. At this stage the shadow and projecting 
cylinder are circular. 

The necessary and sufficient condition that r be the major 
or minor semi-axis of the section of the ellipsoid r $ r = 1 
by a plane passing through the center and perpendicular to a 
is that a, r, and r be coplanar. 

Let r <P r = 1 

and r a = 0. 

Differentiate : d r $ r = 0, 

d r a = 0. 
Furthermore d r r = 0, 

if r is to be a major or minor axis of the section; for r is a 
maximum or a mininum and hence is perpendicular to dr. 
These three equations show that a, r, and r are all ortho 
gonal to the same vector dr. Hence they are coplanar. 

[a r 4> r] = 0. (16) 

Conversely if [a r <P r] = 0, 

dr may be chosen perpendicular to their common plane. 
" 



QUADRIC SURFACES 387 

Hence r is a maximum or a minimim and is one of the prin 
cipal semi-axes of the section perpendicular to a. 

141.] It is frequently an advantage to write the equation 
of an ellipsoid in the form 

r ?T 2 r = 1, (17) 

instead of r <P r = 1. 

This may be done ; because if 

ii jj kk 

* = - 2 + ^ + ir> 



is a dyadic such that W* is equal to <P. may be regarded as 
a square root of <P and written as $*. But it must be re 
membered that there are other square roots of <P for 
example, 



and 



For this reason it is necessary to bear in mind that the square 
root which is meant by <P* is that particular one which has 
been denoted by . 

The equation of the ellipsoid may be written in the form 



or .r. .r = . 

Let r be the radius vector of a unit sphere. The equation of 

the sphere is 

r r = 1. 



388 VECTOR ANALYSIS 

If r = ?Trit becomes evident that an ellipsoid may be 
transformed into a unit sphere by applying the operator 
to each radius vector r, and vice versa, the unit sphere may 
be transformed into an ellipsoid by applying the inverse oper 
ator ~ l to each radius vector r . Furthermore if a, b, c are 
a system of three conjugate radii vectors in an ellipsoid 

a- ?F 2 a = b F 2 b = c ^.0 = 1, 
a 2 b = b * c = c 2 a = 0. 

If for the moment a , b , c denote respectively W a, W b, 

W c, 

a a = V V = c c = 1, 

a . V = V c = c a = 0. 

Hence the three radii vectors a , b , c of the unit sphere into 
which three conjugate radii vectors in the ellipsoid are trans 
formed by the operator W ~ 1 are mutually orthogonal. They 
form a right-handed or left-handed system of three mutually 
perpendicular unit vectors. 

Theorem : Any ellipsoid may be transformed into any other 
ellipsoid by means of a homogeneous strain. 

Let the equations of the ellipsoids be 

r <P r = 1, 
and r r = 1. 

By means of the strain 0* the radii vectors r of the first 
ellipsoid are changed into the radii vectors r of a unit sphere 

r = 01. r , r .r = l. 

By means of the strain ~l the radii vectors r of this unit 
sphere are transformed in like manner into the radii vectors f 
of the second ellipsoid. Hence by the product r is changed 

into f . 

f = r- . . r. (19) 



QUADRIC SURFACES 389 

The transformation may be accomplished in more ways 
than one. The radii vectors r of the unit sphere may be 
transformed among themselves by means of a rotation with or 
without a perversion. Any three mutually orthogonal unit 
vectors in the sphere may be changed into any three others. 
Hence the semi-axes of the first ellipsoid may be carried over 
by a suitable strain into the semi-axes of the second. The 
strain is then completely determined and the transformation 
can be performed in only one way. 

142.] The equation of a family of confocal quadric sur 
faces is 



--ir- 
a* n o* n c* n 

If r r = 1 and r W r = 1 are two surfaces of the 
family, 



2 n l 6 2 TI I c 2 7& 

kk 



- 



.-.. ^ Tin C 7l* 2 

0-1 = (a 2 - 71^11+ (& 2 -tti)jj + ((^-w^kk, 
y-i = (a 2 - 7i 2 ) i i + (& 2 - n 2 ) j j + (c 2 - n 2 ) k k. 
Hence 0- 1 - r- 1 ^ (7i 2 - 74) (ii + j j + kk) 

The necessary and sufficient condition that the two quadrics 

r r = 1 
and r r = 1 

be confocal, is that the reciprocals of <P and differ by a 
multiple of the idemfactor 



390 VECTOR ANALYSIS 

If two confocal quadrics intersect, they do so at right angles. 
Let the quadrics be r <P r = 1, 
and r r = 1. 

Let s = <P r and s = r, 

r = 0- 1 . s and r = ~ l s . 
Then the quadrics may be written in terms of s and s as 

s 0- 1 .s = l, 
and s W~ l . s = 1, 

where by the confocal property, 

0-i_ W~* = xl. 

If the quadrics intersect at r the condition for perpendicularity 
is that the normals d> r and r be perpendicular. That is, 

s s = 0. 
But r = W~ l s = 4> rl s = ( r- 1 + x I) s 

= ?r-i . s + # s, 

x s s = s W~ l s - s r- 1 * s = 1 - s 5T- 1 s . 
In like manner 
r = 0- 1 = F- 1 s = (0- 1 - I) s = 0- 1 s - x s . 

X 8 S f = S (P"" 1 S S <P~ l S = S <P~ l S 1. 

Add: 2 a s s = s (0~ l - P" 1 ) s = x s s . 
Hence s s = 0, 

and the theorem is proved. 

If the parameter n be allowed to vary from oo to + oo the 
resulting confocal quadrics will consist of three families of 
which one is ellipsoids ; another, hyperboloids of one sheet ; 
and the third, hyperboloids of two sheets. By the foregoing 



QUADRIC SURFACES 391 

theorem each surface of any one family cuts every surface 
of the other two orthogonally. The surfaces form a triply 
orthogonal system. The lines of intersection of two families 
(say the family of one-sheeted and the family of two-sheeted 
hyperboloids) cut orthogonally the other family the family 
of ellipsoids. The points in which two ellipsoids are cut by 
these lines are called corresponding points upon the two ellip 
soids. It may be shown that the ratios of the components of 
the radius vector of a point to the axes of the ellipsoid 
through that point are the same for any two corresponding 
points. 

For let any ellipsoid be given by the dyadic 



The neighboring ellipsoid in the family is represented by the 

dyadic 

11 

" JJ kk 



= 

a 2 d n b 2 dn c 2 dn 

y-\ = $-i ldn. 

Inasmuch as and are homologous (see Ex. 8, p. 330) 
dyadics they may be treated as ordinary scalars in algebra. 
Therefore if terms of order higher than the first in dn be 
omitted, 0+&dn. 

The two neighboring ellipsoids are then 

r r = 1, 

and r (# + # 2 d n) f 1, 

By (19) f (0 + <Z> 2 d n)-i # r, 

r= I + 



f (I I<Pdn) T r ~ 



392 VECTOR ANALYSIS 

The vectors r and r differ by a multiple of r which is 
perpendicular to the ellipsoid 0. Hence the termini of r and 
r are corresponding points, for they lie upon one of the lines 
which cut the family of ellipsoids orthogonally. The com 
ponents of r and f in the direction i are r i = x and 

dn . dn x 

f i = x = r i i . *r = x -. 
2 2 a 2 

/> ft fj\ 

The ratio of these components is - = 1 



- - 

X A a 



The axes of the ellipsoids in the direction i are Va 2 d n and 

a. Their ratio is 

i dn 

A/a 2 dn a - i .. dn x 

= " l ~* 



T ,., V& 2 dn y j V^ 2 dn z 

In like manner = - and = -. 

by c z 

Hence the ratios of the components of the vectors r and r 
drawn to corresponding points upon two neighboring ellip 
soids only differ at most by terms of the second order in d n 
from the ratios of the axes of those ellipsoids. It follows 
immediately that the ratios of the components of the vectors 
drawn to corresponding points upon any two ellipsoids, sepa 
rated by a finite variation in the parameter n, only differ at 
most by terms of the first order in dn from the ratios of the 
axes of the ellipsoids and hence must be identical with them. 
This completes the demonstration. 

The Propagation of Light in Crystals 1 

143.] The electromagnetic equations of the ether or of any 
infinite isotropic medium which is transparent to electromag 
netic waves may be written in the form 

1 The following discussion must be regarded as mathematical not physical. 
To treat the subject from the standpoint of physics would be out of place here. 



THE PROPAGATION OF LIGHT IN CRYSTALS 393 
d 2 V 

Pot + .FD + VF=O, V.D = O (i) 

where D is the electric displacement satisfying the hydrody- 
namic equation V D = 0, E a constant of the dielectric meas 
ured in electromagnetic units, and V F the electrostatic force 
due to the function F. In case the medium is not isotropic the 
constant E becomes a linear vector function 0. This function 
is self-conjugate as is evident from physical considerations. 
For convenience it will be taken as 4 TT <D. The equations 
then become 



-4-7T0.D + VF=0, V-D=0. (2) 

U/ (/ 

Operate by V x V x. 

V x V x Pot + 47rVxVx0.D = 0. (3) 

CL t 

The last term disappears owing to the fact that the curl of 
the derivative VF vanishes (page 167). The equation may 
also be written as 

Pot V x V x -r-y + 47rVxVx<P.D = 0. (3) 

But VxVx=VV.-V.V. 

Remembering that V D and consequently V and 

-n _ (t t 

V -V-TT vanish and that Pot V V is equal to 4 TT the 
a t 2 

equation reduces at once to 

,72 Tk 

0.D V V $ D, VD = 0. (4) 



dt* 

Suppose that the vibration D is harmonic. Let r be the 
vector drawn from a fixed origin to any point of space. 



394 VECTOR ANALYSIS 

Then D = A cos (m r n f) 

where A and m are constant vectors and n a constant scalar 
represents a train of waves. The vibrations take place in 
the direction A. That is, the wave is plane polarized. The 
wave advances in the direction m. The velocity v of that ad 
vance is the quotient of n by m, the magnitude of the vector 
m. If this wave is an electromagnetic wave in the medium 
considered it must satisfy the two equations of that medium. 
Substitute the value of D in those equations. 

The value of V D, V V $ D, and VV D may be 
obtained most easily by assuming the direction i to be coinci 
dent with m. m r then reduces to m i r which is equal to 
m x. The variables y and z no longer occur in D. Hence 

D = A cos (m x n f) 

3D 

V D = i -z = i A m sin (m x n f) 

d X 

V V d> D = m 2 A cos (m x n f) 

V V # D = m 2 i i- 4> A cos (mx nf). 

Hence V D = m A sin (m r nf) 

V V d> D = m m D 

VV* 0.D = -mm. </>.D. 



Moreover -j- -^ = 7i a D. 

Hence if the harmonic vibration D is to satisfy the equa 
tions (4) of the medium 

n 2 D = m-m <P D m m <P D (5) 

and m A = 0. (6) 



THE PROPAGATION OF LIGHT IN CRYSTALS 395 

The latter equation states at once that the vibrations must 
be transverse to the direction m of propagation of the waves. 
The former equation may be put in the form 



D = <P D - 4> D. (5) 

n 2 7i 2 

Introduce s = - 

n 

The vector s is in the direction of advance m. The magnitude 
of s is the quotient of m by n. This is the reciprocal of the 
velocity of the wave. The vector s may therefore be called 
the wave-slowness. 

D s - s D s s D. 
This may also be written as 

D = (s x s x D) = s x (0 D) x s. 
Dividing by the scalar factor cos (m x n t\ 

A = sx(0A)xs = ss A S A. (7) 

It is evident that the wave slowness s depends not at all 
upon the phase of the vibration but only upon its direction. 
The motion of a wave not plane polarized may be discussed by 
decomposing the wave into waves which are plane polarized. 

144.] Let a be a vector drawn in the direction A of the 
displacement and let the magnitude of a be so determined 

that a d> a = 1. (8) 

The equation (7) then becomes reduced to the form 

a = sx (<2>*a) Xs = s-s #-a ss-#-a (9) 

a a = 1. (8) 

These are the equations by which the discussion of the velocity 
or rather the slowness of propagation of a wave in different 
directions in a non-isotropic medium may be carried on. 

a a = s s a a = s s. (10) 



396 VECTOR ANALYSIS 

Hence the wave slowness s due to a displacement in the 
direction a is equal in magnitude (but not in direction) to the 
radius vector drawn in the ellipsoid a a = 1 in that 
direction. 

axa = = ss a x <P a a x s s <P . a 
= s s(aX # a)- # a aXs # a s # a. 
But the first term contains <P a twice and vanishes. Hence 
a x s a = [a s a] = 0. (11) 

The wave-slowness s therefore lies in a plane with the 
direction a of displacement and the normal a drawn to the 
ellipsoid a <P a = 1 at the terminus of a. Since s is perpen 
dicular to a and equal in magnitude to a it is evidently com 
pletely determined except as regards sign when the direction 
a is known. Given the direction of displacement the line of 
advance of the wave compatible with the displacement is com 
pletely determined, the velocity of the advance is likewise 
known. The wave however may advance in either direction 
along that line. By reference to page 386, equation (11) is seen 
to be the condition that a shall be one of the principal axes of 
the ellipsoid formed by passing a plane through the ellipsoid 
perpendicular to s. Hence for any given direction of advance 
there are two possible lines of displacement. These are the 
principal axes of the ellipse cut from the ellipsoid a a = 1 
by a plane passed through the center perpendicular to the 
line of advance. To these statements concerning the deter- 
minateness of s when a is given and of a when s is given just 
such exceptions occur as are obvious geometrically. If a and 
a are parallel s may have any direction perpendicular to a. 
This happens when a is directed along one of the principal 
axes of the ellipsoid. If s is perpendicular to one of the 
circular sections of the ellipsoid a may have any direction in the 
plane of the section. 



THE PROPAGATION OF LIGHT IN CRYSTALS 397 

When the direction of displacement is allowed to vary the 
slowness s varies. To obtain the locus of the terminus of s, a 
must be eliminated from the equation 

a = s s <P a SB a 
or (I - s s + s s (P) a = 0. (12) 

The dyadic in the parenthesis is planar because it annihilates 
vectors parallel to a. The third or determinant is zero. This 
gives immediately 

(I + #) 8 = 0, 

or (0- 1 - s s 1 + ss) 3 = 0. (13) 

This is a scalar equation in the vector s. It is the locus of 
the extremity of s when a is given all possible directions. A 
number of transformations may be made. By Ex. 19, p. 331, 

(<P + ef) 8 = <P S + e <P a f = 8 + e 00- 1 f </> 3 . 
Hence 



Dividing out the common factor and remembering that $ is 
self-conjugate. 

1 + s- (CM-s- si)- 1 -8 = 0. 



1 + 

8-1.8 



+ s - -- -r -8 = 



S S I 8 8 

8 S 



Hence s - - -= s = 0. (14) 

1 8*8 (P 



Let 



398 VECTOR ANALYSIS 

1 /^_W/_J_\ jj + / i u k 

i-i.i*-^_-jj [i_g JJ+ [ir^J 

Let s = xi + yj + zk and s 2 = # 2 + y* + z 2 . 
Then the equation of the surface in Cartesian coordinates is 

20 o 

^ j?/ 2 z* 

-72 = 0. (14) 



l-fl i-_ 



a 2 



The equation in Cartesian coordinates may be obtained 

rp n f 1 v f rr\ m 



directly from 

The determinant of this dyadic is 

a 2 s 2 + x 2 x y x z 

x y 6 2 s 2 + y^ y z 

x z y z c 2 s 2 



= 0. (13) 



By means of the relation s 2 = x 2 + y 2 + z 2 this assumes the 
forms 

n n I o i o ~T~ n n ~~" -^ 



+ 



"2 2 

s 2 c 2 



or 



This equation appears to be of the sixth degree. It is how 
ever of only the fourth. The terms of the sixth order cancel 
out. 

The vector s represents the wave-slowness. Suppose that a 
plane wave polarized in the direction a passes the origin at a 



THE PROPAGATION OF LIGHT IN CRYSTALS 399 

certain instant of time with this slowness. At the end of a 
unit of time it will have travelled in the direction s, a distance 
equal to the reciprocal of the magnitude of s. The plane will 
be in this position represented by the vector s (page 108). 

If s = ui + vj + wit 

the plane at the expiration of the unit time cuts off intercepts 
upon the axes equal to the reciprocals of u, v, w. These 
quantities are therefore the plane coordinates of the plane. 
They are connected with the coordinates of the points in the 
plane by the relation 

ux + vy + wz = \. 

If different plane waves polarized in all possible different 
directions a be supposed to pass through the origin at the 
same instant they will envelop a surface at the end of a unit 
of time. This surface is known as the wave-surface. The 
perpendicular upon a tangent plane of the wave-surface is the 
reciprocal of the slowness and gives the velocity with which 
the wave travels in that direction. The equation of the wave- 
surface in plane coordinates u, v, w is identical with the equa 
tion for the locus of the terminus of the slowness vector s. 
The equation is 



= 



(15) 



where s 2 = u 2 -f v 2 + w 2 . This may be written in any of the 
forms given previously. The surface is known as FresneVs 
Wave-Surface. The equations in vector form are given on 
page 397 if the variable vector s be regarded as determining a 
plane instead of a point. 

145.] In an isotropic medium the direction of a ray of 
light is perpendicular to the wave-front. It is the same as 
the direction of the wave s advance. The velocity of the ray 



400 VECTOR ANALYSIS 

is equal to the velocity of the wave. In a non-isotropic 
medium this is no longer true. The ray does not travel per 
pendicular to the wave-front that is, in the direction of the 
wave s advance. And the velocity with which the ray travels 
is greater than the velocity of the wave. In fact, whereas the 
wave-front travels off always tangent to the wave-surface, the 
ray travels along the radius vector drawn to the point of tan- 
gency of the wave-plane. The wave-pknes envelop the 
wave-surface; the termini of the rays are situated upon it. 
Thus in the wave-surface the radius vector represents in mag 
nitude and direction the velocity of a ray and the perpen 
dicular upon the tangent plane represents in magnitude and 
direction the velocity of the wave. If instead of the wave- 
surface the surface which is the locus of the extremity of the 
wave slowness be considered it is seen that the radius vector 
represents the slowness of the wave; and the perpendicular 
upon the tangent plane, the slowness of the ray. 

Let v be the velocity of the ray. Then s v = 1 because 
the extremity of v lies in the plane denoted by s. Moreover 
the condition that v be the point of tangency gives d v per 
pendicular to s. In like manner if a r be the slowness of the 
ray and v the velocity of the wave, s v = 1 and the condition 
of tangency gives d s perpendicular to v. Hence 

s v = 1 and s - v = 1, (16) 

and s d v = 0, v d s = 0, v - d s = 0, s d v = 0, 
v may be expressed in terms of a, s, and as follows. 

a = s s <P a s s <P a, 
da = 2s.tfs<P.a s- ^arfs + ss<?-rfa 
sds- & - a. s s # d a. 

Multiply by a and take account of the relations a s = and 
a 4> . d a and a a = s s. Then 



THE PROPAGATION OF LIGHT IN CRYSTALS 401 

s d s a d s B a = 0, 
or d s (s a s <P a) = 0. 

But since v d s = 0, v and s a s <P a have the same 

direction. 

v = x (s a s a), 

s v = # (s s s a s a) = x s s. 

, s a s . . a 
Hence v = , (17) 

8*8 

s . op . a a $ a s $ a 

v <P a = = 0. 

s s 

Hence the ray velocity v is perpendicular to a, that is, the 
ray velocity lies in the tangent plane to the ellipsoid at the 
extremity of the radius vector a drawn in the direction of the 
displacement. Equation (17) shows that v is coplanar with 
a and s. The vectors a, s, a, and v therefore lie in one 
plane. In that plane s is perpendicular to a ; and v ; , to a. 
The angle from s to v is equal to the angle from a to $ a. 

Making use of the relations already found (8) (9) (11) 
(16) (17), it is easy to show that the two systems of vectors 

a, v , a x v and a, s, (<P a) x s 

are reciprocal systems. If a be replaced by a the equa 
tions take on the symmetrical form 

s . a = B s = a a a a = 1, 

v f -a f = v .v =a -a s . v = 1, 

a = s x a x s a = v x a x v (18) 

s = a x v x a v = a x s x a 

a a = 1 a 0- 1 a = 1. 

Thus a dual relation exists between the direction of displace 
ment, the ray-velocity, and the ellipsoid on the one hand ; 

26 



402 VECTOR ANALYSIS 

and the normal to the ellipsoid, the wave-slowness, and the 
ellipsoid ~ l on the other. 

146.] It was seen that if s was normal to one of the cir 
cular sections of the displacement a could take place in any 
direction in the plane of that section. For all directions in 
this plane the wave-slowness had the same direction and the 
same magnitude. Hence the wave-surface has a singular 
plane perpendicular to s. This plane is tangent to the surface 
along a curve instead of at a single point. Hence if a wave 
travels in the direction s the ray travels along the elements of 
the cone drawn from the center of the wave-surface to this 
curve in which the singular plane touches the surface. The 
two directions s which are normal to the circular sections of 
are called the primary optic axes. These are the axes of equal 
wave velocities but unequal ray velocities. 

In like manner v being coplanar with a and a 

[4> a v a] = [a v <P~ l a ] = 0. 

The last equation states that if a plane be passed through 
the center of the ellipsoid <P~ l perpendicular to V, then a 
which is equal to a will be directed along one of the prin 
cipal axes of the section. Hence if a ray is to take a definite 
direction a may have one of two directions. It is more con 
venient however to regard v as a vector determining a plane. 
The first equation 

[0 . a v a] = 

states that a is the radius vector drawn in the ellipsoid to 
the point of tangency of one of the principal elements of the 
cylinder circumscribed about parallel to v : if by a principal 
element is meant an element passing through the extremities 
of the major or minor axes of orthogonal plane sections 
of that cylinder. Hence given the direction v of the ray, the 
two possible directions of displacement are those radii vectors 



VARIABLE DYADICS 403 

of the ellipsoid which lie in the principal planes of the cylin 
der circumscribed about the ellipsoid parallel to v . 

If the cylinder is one of the two circular cylinders which 
may be circumscribed about the direction of displacement 
may be any direction in the plane passed through the center 
of the ellipsoid and containing the common curve of tangency 
of the cylinder with the ellipsoid. The ray-velocity for all 
these directions of displacement has the same direction and 
the same magnitude. It is therefore a line drawn to one 
of the singular points of the wave-surface. At this singular 
point there are an infinite number of tangent planes envelop 
ing a cone. The wave-velocity may be equal in magnitude 
and direction to the perpendicular drawn from the origin to 
any of these planes. The directions of the axes of the two 
circular cylinders circumscriptible about the ellipsoid are 
the directions of equal ray-velocity but unequal wave-velocity. 
They are the radii drawn to the singular points of the wave- 
surface and are called the secondary optic axes. If a ray 
travels along one of the secondary optic axes the wave planes 
travel along the elements of a cone. 



Variable Dyadics. The Differential and Integral Calculus 

147.] Hitherto the dyadics considered have been constant. 
The vectors which entered into their make up and the scalar 
coefficients which occurred in the expansion in nonion form 
have been constants. For the elements of the theory and for 
elementary applications these constant dyadics suffice. The 
introduction of variable dyadics, however, leads to a simplifica 
tion and unification of the differential and integral calculus of 
vectors, and furthermore variable dyadics become a necessity 
in the more advanced applications for instance, in the theory 
of the curvature of surfaces and in the dynamics of a rigid 
body one point of which is fixed. 



404 VECTOR ANALYSIS 

Let W be a vector function of position in space. Let r be 
the vector drawn from a fixed origin to any point in space. 



d r = dx i + dy j + dz k, 

3W , 5W 5W 

^dx-^ + dy + dz-Ti . 

$# c?y c/ z 

( SW 3W 5W) 

Hence d W = d r H - -- h j -= + k - > . 
( dx dy dz ) 

The expression enclosed in the braces is a dyadic. It thus 
appears that the differential of W is a linear function of c?r, 
the differential change of position. The antecedents are i, j, k, 
and the consequents the first partial derivatives of W with re 
spect of x, y, z. The expression is found in a manner precisely 
analogous to del and will in fact be denoted by V W. 



=i- + j + k-. (1) 



Then dW = dr.VW. (2) 

This equation is like the one for the differential of a scalar 

function F. 

dV=dr VF. 

It may be regarded as defining VW. If expanded into 
nonion form VW becomes 



. 
VW = 11 

3x 



.5X .9Y 3Z 

+ ki ls -+kj T - + kk-^--, 
dz 3 z <y z 

if W 



VARIABLE DYADICS 405 

The operators V and V x which were applied to a vector 
function now become superfluous from a purely analytic 
standpoint. For they are nothing more nor less than the 
scalar and the vector of the dyadic V W. 

div W = V W = (V W)* (4) 

curl W = V x W = (V W) x . (5) 

The analytic advantages of the introduction of the variable 
dyadic VW are therefore these. In the first place the oper 
ator V may be applied to a vector function just as to a scalar 
function. In the second place the two operators V and V x 
are reduced to positions as functions of the dyadic V W. On 
the other hand from the standpoint of physics nothing is to 
be gained and indeed much may be lost if the important in 
terpretations of V W and V x W as the divergence and curl 
of W be forgotten and their places taken by the analytic idea 
of the scalar and vector of VW. 

If the vector function W be the derivative of a scalar 
function V^ 

dW = dVF=e?r VVF", 

where V V F= i i 75- + i j = =- + i k ^ , 

<y x <y x c/ y <y x c/ z 

Qty 32 y 3 2 F" 



**- TT o T 

dy dx &y z dy d z 



+ kj - + k j -g- + k k 

dzdx 9zSy 

The result of applying V twice to a scalar function is seen to 
be a dyadic. This dyadic is self -conjugate. Its vector V x V V 
is zero ; its scalar V V V is evidently 

3 2 F 3 2 V 9 2 V 

V-VF= (VVF)*= 0-2 + T-2 + TT 

2 * * 



406 VECTOR ANALYSIS 

If an attempt were made to apply the operator V symboli 
cally to a scalar function V three times, the result would be a 
sum of twenty-seven terms like 

* * 

,etc. 



r r, ^r = ;r- 

c/ x 6 v x d y & z 

This is a triadic. Three vectors are placed in juxtaposition 
without any sign of multiplication. Such expressions will 
not be discussed here. In a similar manner if the operator V 
be applied twice to a vector function, or once to a dyadic func 
tion of position in space, the result will be a triadic and hence 
outside the limits set to the discussion here. The operators 
V x and V may however be applied to a dyadic to yield 
respectively a dyadic and a vector. 

S 50 30 

V x = i x ^- + j x ^- + k x ^-, (7) 

dx Sy 9z 

30 30 30 

V- = i._ + j . + k- T -. (8) 

3x dy 3z 

If = u i + v j + w k, 

where u, v, w are vector functions of position in space, 

Vx $ = V x u i + Vxvj + Vx w k, (7) f 
and V 0= V u i + V v j + V w k. (8) 

Or if = i u + j v -f k w, 



X)v 

T r x* / w W ^ V \ I* ** ^ W 

Vx 0= i{- ^-) + j^ =- 



* -++ < 8 >" 

In a similar manner the scalar operators (a V) and (V V) 
may be applied to 0. The result is in each case a dyadic, 



VARIABLE DYADICS 407 

30 30 5d> 

(a.V)<P = a . 1 ^ + a 2 ^- + a 8 ^, (9) 

32 32 $ 32 

(V . V) <P = ^ f + + - (10) 

o/ z 2 c? y* d z 2 

The operators a V and V V as applied to vector func 
tions are no longer necessarily to be regarded as single oper 
ators. The individual steps may be carried out by means of 
the dyadic VW. 

(a - V) W = a (V W) = a V W, 
(V V) W = V (V W) = V V W. 

But when applied to a dyadic the operators cannot be inter 
preted as made up of two successive steps without making use 
of the triadic V 0. The parentheses however may be removed 
without danger of confusion just as they were removed in 
case of a vector function before the introduction of the dyadic. 
Formulae similar to those upon page 176 may be given for 
differentiating products in the case that the differentiation 
lead to dyadics. 

V (u v) = >V u v + u V v, 

V(vxw)=Vvxw Vwxv, 

Vx (v x w) = w V v V v w v V w + V w v, 

V (v w) = V v w + V w v, 

V (v w) = V v w + v V w. 

Vx (v w) = V x v w v x V w, 

V . (u #) = V u <P + u V 0, 

VxVx <P = VV. <P V V <P, etc. 



The principle in these and all similar cases is that enun 
ciated before, namely : The operator V may be treated sym- 



408 VECTOR ANALYSIS 

bolically as a vector. The differentiations which it implies 
must be carried out in turn upon each factor of a product 
to which it is applied. Thus 

V x (vw) = [V x (v w)] v + [V x (vw)]^ 

[Vx (vw)] w = V xvw, 
[V x (v w)] v = - [v x V w] v = - v x V w. 
Hence Vx (v w) = V x v w v x V w. 

Again V (v x w) = [V (v x w)] T + [V (v x w)]^ 

[V (v x w)] w = V v x w, 
[V (v x w)] v = [ V (w x v)] v = V w x v. 
Hence V(vxw) = Vvxw v w x v. 

148.] It was seen (Art. 79) that if C denote an arc of a 
curve of which the initial point is r and the final point is r 
the line integral of the derivative of a scalar function taken 
along the curve is equal to the difference between the values 
of that function at r and r . 



r* VF= F(r)- F(r ). 

In like manner Cd r . V W = W (r) - W (r ), 
J c 

and Cd r V W = 0. 

Jo 

It may be well to note that the integrals 

fdr.VW and fvw dr 

are by no means the same thing. VW is a dyadic. The 
vector dx cannot be placed arbitrarily upon either side of it. 



VARIABLE DYADICS 409 

Owing to the fundamental equation (2) the differential di 
necessarily precedes V W. The differentials must be written 
before the integrands in most cases. For the sake of uni 
formity they always will be so placed. 

Passing to surface integrals, the following formulae, some 
of which have been given before and some of which are new, 
may be mentioned. 



ff 



ax VW= fdr W 
ff da. Vx W= fdr* 

r/daVx0= I dr < 



The line integrals are taken over the complete bounding curve 
of the surface over which the surface integrals are taken. In 
like manner the following relations exist between volume and 
surface integrals. 



fff 



dv VW=rfa W 



/// 



<* V x *- 



410 VECTOR ANALYSIS 

The surface integrals are taken over the complete bounding 
surface of the region throughout which the volume integrals 
are taken. 

Numerous formulae of integration by parts like those upon 
page 250 might be added. The reader will rind no difficulty in 
obtaining them for himself. The integrating operators may 
also be extended to other cases. To the potentials of scalar 
and vector functions the potential, Pot </>, of a dyadic may be 
added. The Newtonian of a vector function and the Lapla- 
cian and Maxwellian of dyadics may be defined. 



Pot <? = 

New W = // r ^^I^> dV 

d , 



Max * = 






The analytic theory of these integrals may be developed as 
before. The most natural way in which the demonstrations 
may be given is by considering the vector function W as the 
sum of its components, 

W = Xi+ Fj + ^k 

and the dyadic as expressed with the constant consequents 
i, j, k and variable antecedents u, v, w, or vice versa, 



These matters will be left at this point. The object of en 
tering upon them at all was to indicate the natural extensions 
which occur when variable dyadics are considered. These ex 
tensions differ so slightly from the simple cases which have 



THE CURVATURE OF SURFACES 411 

gone before that it is far better to leave the details to be worked 
out or assumed from analogy whenever they may be needed 
rather than to attempt to develop them in advance. It is suffi 
cient merely to mention what the extensions are and how they 
maybe treated. 

The Curvature of Surfaces 1 

149. ] There are two different methods of treating the cur 
vature of surfaces. In one the surface is expressed in para- 
metic form by three equations 

x =/i <X v ) y =/a O> *0 * =/8 <X ")> 
or r = f (u, v). 

This is analogous to the method followed (Art. 57) in dealing 
with curvature and torsion of curves and it is the method 
employed by Fehr in the book to which reference was made. 
In the second method the surface is expressed by a single 
equation connecting the variables x,y,z thus 



, z) = 0. 

The latter method of treatments affords a simple application of 
the differential calculus of variable dyadics. Moreover, the 
dyadics lead naturally to the most important results connected 
with the elementary theory of surfaces. 

Let r be a radius vector drawn from an arbitrary fixed 
origin to a variable point of the surface. The increment d r 
lies in the surface or in the tangent plane drawn to the surface 
at the terminus of r. 



Hence the derivative V^is collinear with the normal to the 
surface. Moreover, inasmuch as F and the negative of F when 

1 Much of what follows is practically free from the use of dyadics. This is 
especially true of the treatment of geodetics, Arts. 155-157. 



412 VECTOR ANALYSIS 

equated to zero give the same geometric surface, V F may be 
considered as the normal upon either side of the surface. In 
case the surface belongs to the family defined by 

F (#, y, z) = const. 

the normal V F lies upon that side upon which the constant 
increases. Let V F be represented by N the magnitude of 
which may be denoted by N, and let n be a unit normal drawn 
in the direction of IT. Then 



(1) 






If s is the vector drawn to any point in the tangent plane at 
the terminus of r, s r and n are perpendicular. Consequently 
the equation of the tangent plane is 



(s-r) 
and in like manner the equation of the normal line is 

(s-r)x VjF=0, 
or s = r + & V JP 

where k is a variable parameter. These equations may be 
translated into Cartesian form and give the familiar results. 

150.] The variation dn of the unit normal to a surface 
plays an important part in the theory of curvature, dn is 
perpendicular to n because n is a unit vector. 



THE CURVATURE OF SURFACES 413 



-* 

N iv 2 

The dyadic I nn is an idemf actor for all vectors perpen 
dicular to n and an annihilator for vectors parallel to n. 
Hence 

dn (I n n) = d n, 

and V^.(I-nn)=0, 



N J N N 

Hence rf n = d r VV .F (I nn). 

But d r = d r (I n n). 

Hanco .-*,. <* ~"> V ^ P-"). (2) 
Let > = (I-.. 



Then dn = dr <P. (4) 

In the vicinity of any point upon a surface the variation d n of 
the unit normal is a linear function of the variation of the 
radius vector r. 

The dyadic is self -con jugate. For 

N4> c = (I - nn), (VV F) c (I - nn)^ 

Evidently (I - n ri) c = (I - n n) and by (6) Art. 147 VV F 
is self-conjugate. Hence <P C is equal to 0. When applied to 
a vector parallel to n, the dyadic produces zero. It is there 
fore planar and in fact uniplanar because self-conjugate. The 
antecedents and the consequents lie in the tangent plane to 



414 VECTOR ANALYSIS 

the surface. It is possible (Art. 116) to reduce to the 
form 

4> = a i i + b j j (5) 

where i and j are two perpendicular unit vectors lying in the 
tangent plane and a and b are positive or negative scalars. 

dn = dr (a i i + b j j ). 

The vectors i , j and the scalars a, 6 vary from point to point 
of the surface. The dyadic C? is variable. 

151.] The conic r r = 1 is called the indicatrix of the 
surface at the point in question. If this conic is an ellipse, 
that is, if a and b have the same sign, the surface is convex at 
the point ; but if the conic is an hyperbola, that is, if a and b 
have opposite signs the surface is concavo-convex. The curve 
r . r = 1 may be regarded as approximately equal to the 
intersection of the surface with a plane drawn parallel to the 
tangent plane and near to it. If r r be set equal to zero 
the result is a pair of straight lines. These are the asymp 
totes of the conic. If they are real the conic is an hyperbola ; 
if imaginary, an ellipse. Two directions on the surface which 
are parallel to conjugate diameters of the conic are called con 
jugate directions. The directions on the surface which coin 
cide with the directions of the principal axes i , j of the 
indicatrix are known as the principal directions. They are a 
special case of conjugate directions. The directions upon the 
surface which coincide with the directions of the asymptotes 
of the indicatrix are known as asymptotic directions. In case 
the surface is convex, the indicatrix is an ellipse and the 
asymptotic directions are imaginary. 

In special cases the dyadic may be such that the coeffi 
cients a and b are equal. may then be reduced to the 

form 

= a(i i + j j ) (5) 



THE CURVATURE OF SURFACES 415 

in an infinite number of ways. The directions i and j may be 
any two perpendicular directions. The indicatrix becomes a 
circle. Any pair of perpendicular diameters of this circle 
give principal directions upon the surface. Such a point is 
called an umbilic. The surface in the neighborhood of an 
umbilic is convex. The asymptotic directions are imaginary. 
In another special case the dyadic $ becomes linear and redu 
cible to the form <p = a i i . (5)" 

The indicatrix consists of a pair of parallel lines perpendicular 
to i . Such a point is called a parabolic point of the surface. 
The further discussion of these and other special cases will be 
omitted. 

The quadric surfaces afford examples of the various kinds 
of points. The ellipsoid and the hyperboloid of two sheets 
are convex. The indicatrix of points upon them is an ellipse. 
The hyperboloid of one sheet is concavo-convex. The in 
dicatrix of points upon it is an hyperbola. The indicatrix 
of any point upon a sphere is a circle. The points are all 
umbilies. The indicatrix of any point upon a cone or cylinder 
is a pair of parallel lines. The points are parabolic. A sur 
face in general may have upon it points of all types elliptic, 
hyperbolic, parabolic, and umbilical. 

152.] A line of principal curvature upon a surface is a 
curve which has at each point the direction of one of the prin 
cipal axes of the indicatrix. The direction of the curve at a 
point is always one of the principal directions on the surface at 
that point. Through any given point upon a surface two per 
pendicular lines of principal curvature pass. Thus the lines 
of curvature divide the surface into a system of infinitesi 
mal rectangles. An asymptotic line upon a surface is a curve 
which has at each point the direction of the asymptotes of the 
indicatrix. The direction of the curve at a point is always 
one of the asymptotic directions upon the surface. Through 



416 VECTOR ANALYSIS 

any given point of a surface two asymptotic lines pass. These 
lines are imaginary if the surface is convex. Even when real 
they do not in general intersect at right angles. The angle 
between the two asymptotic lines at any point is bisected by 
the lines of curvature which pass through that point. 

The necessary and sufficient condition that a curve upon a 
surface be a line of principal curvature is that as one advances 
along that curve, the increment of d n, the unit normal to the 
surface is parallel to the line of advance. For 

rfn= 0. dr = (a i i + b j j ) dr 
dr x i + yj . 

Then evidently d n and d r are parallel when and only when 
dr is parallel to i or j . The statement is therefore proved. 
It is frequently taken as the definition of lines of curvature. 
The differential equation of a line of curvature is 

dnxdr = 0. (6) 

Another method of statement is that the normal to the surface, 
the increment d n of the normal, and the element d r of the 
surface lie in one plane when and only when the element d r 
is an element of a line of principal curvature. The differential 
equation then becomes 

[n dn rfr] = 0. (7) 



The necessary and sufficient condition that a curve upon a 
surface be an asymptotic line, is that as one advances along 
that curve the increment of the unit normal to the surface is 
perpendicular to the line of advance. For 

dn = dr <P 
dn dr = dr dr. 

If then d n d r is zero d r tf> d r is zero. Hence d r is an 
asymptotic direction. The statement is therefore proved. It 



THE CURVATURE OF SURFACES 417 

is frequently taken as the definition of asymptotic lines. The 
differential equation of an asymptotic line is 

d n d r = 0. (8) 

153.] Let P be a given point upon a surface and n the 
normal to the surface at P. Pass a plane p through n. This 
plane p is normal to the surface and cuts out a plane section. 
Consider the curvature of this plane section at the point P. 
Let n be normal to the plane section in the plane of the 
section, n coincides with n at the point P. But unless the 
plane p cuts the surface everywhere orthogonally, the normal 
n to the plane section and the normal n to the surface will not 
coincide, d n and d n will also be different. The curvature 
of the plane section lying in p is (Art. 57). 



____ 
ds d s 2 

As far as numerical value is concerned the increment of the 
unit tangent t and the increment of the unit normal n are 
equal. Moreover, the quotient of d r by d s is a unit vector 
in the direction of d n . Consequently the scalar value of C is 

d n 1 dr dn f dr 
ds ds ds 2 

By hypothesis n = n at P and ndr = n -dr = 0, 



d (V d r) = d n d r + n - d 2 r = 0. 
Hence d n d r + n d 2 r = d n d r + n d 2 r. 
Since n and n are equal at P, 



dn*dr dr <P dr dr <P dr 
Hence C = ^ = - j-^ - = 3 - 3 - ( 9 ) 
ds 2 ds 2 dr dr 

27 



418 VECTOR ANALYSIS 



. 
C7 = a - -- + b 



dr ar ar ar 
Hence tf= a cos 2 (i , dr) + & cos 2 Q , dr), 
or (7= a cos 2 (i , rfr) + b sin 2 (i , dr). (10) 

The interpretation of this formula for the curvature of a 
normal section is as follows : When the plane p turns about 
the normal to the surface from i to j , the curvature C of the 
plane section varies from the value a when the plane passes 
through the principal direction i , to the value b when it 
passes through the other principal direction j . The values 
of the curvature have algebraically a maximum and minimum 
in the directions of the principal lines of curvature. If a and 
b have unlike signs, that is, if the surface is concavo-convex 
at Pj there exist two directions for which the curvature of a 
normal section vanishes. These are the asymptotic directions. 

154.] The sum of the curvatures in two normal sections 
at right angles to one another is constant and independent of 
the actual position of those sections. For the curvature in 
one section is 



C l = a cos 2 (i , dr) + b sin 2 (i , dr), 
and in the section at right angles to this 

(7 2 = a sin 2 (i , di) + b cos 2 (i , dr). 
Hence O l + C 2 = a + b = 4> a (11) 

which proves the statement. 

It is easy to show that the invariant $% s is equal to the pro 
duct of the curvatures a and b of the lines of principal curv 

ature. 

4>t S = ab 

Hence the equation x* <P a x + 0^ 3 = (12) 



THE CURVATURE OF SURFACES 419 

is the quadratic equation which determines the principal curv 
atures a and I at any point of the surface. By means of this 
equation the scalar quantities a and b may be found in terms 
of F(x, y, z). 

. (I-nn) 



N 



N 
(nn VV.F- nn)^ = (nn nn VV^ 7 )^ = (nn 

(VV^) (nn^ 
Hence 9. = - -^ 



(nn. VV^)^ = nn: VVJ^=n. VV F n. 

V.V^ VFVF:WF 
Hence <^^ = -- (13) 



. 

** T --- ^i -- CIS) 

These expressions may be written out in Cartesian coordinates, 
but they are extremely long. The Cartesian expressions for 
2/5 are even longer. The vector expression may be obtained 
as follows: 



(I nn) 2 = nn. 
Hence 



S-IA\ 



155.] Given any curve upon a surface. Let t be a unit 
tangent to the curve, n a unit normal to the surface and m a 



420 VECTOR ANALYSIS 

vector defined as n x t. The three vectors n, t, m constitute 
an i, j, k system. The vector t is parallel to the element d r. 
Hence the condition for a line of curvature becomes 

t x d n = 0. (15) 

Hence m d n = 

d (m n) = = m dn + n d m. 
Hence n d m = 0. 

Moreover m d m = 0. 

Hence t x d m = 0, (16) 

or dmxdn = Q. (16) 

The increments of m and of n and of r are all parallel in case of 
a line of principal curvature. 

A geodetic line upon a surface is a curve whose osculating 
plane at each point is perpendicular to the surface. That the 
geodetic line is the shortest line which can be drawn between 
two points upon a surface may be seen from the following 
considerations of mechanics. Let the surface be smooth and 
let a smooth elastic string which is constrained to lie in the 
surface be stretched between any two points of it. The string 
acting under its own tensions will take a position of equili 
brium along the shortest curve which can be drawn upon the 
surface between the two given points. Inasmuch as the 
string is at rest upon the surface the normal reactions of the 
surface must lie in the osculating plane of the curve. Hence 
that plane is normal to the surface at every point of the curve 
and the curve itself is a geodetic line. 

The vectors t and d t lie in the osculating plane and deter 
mine that plane. In case the curve is a geodetic, the normal 
to the osculating plane lies in the surface and consequently is 
perpendicular to the normal n. Hence 



THE CURVATURE OF SURFACES 421 

n*tx<2t = 0, 

n x t d i = (17) 

or m d t = 0. 

The differential equation of a geodetic line is therefore 

[n dr d 2 r] =0. (18) 

Unlike the differential equations of the lines of curvature 
and the asymptotic line, this equation is of the second order. 
The surface is therefore covered over with a doubly infinite 
system of geodetics. Through any two points of the surface 
one geodetic may be drawn. 

As one advances along any curve upon a surface there is 
necessarily some turning up and down, that is, around the 
axis m, due to the fact that the surface is curved. There may 
or may not be any turning to the right or left. If one advances 
along a curve such that there is no turning to the right or 
left, but only the unavoidable turning up and down, it is to be 
expected that the advance is along the shortest possible route 
that is, along a geodetic. Such is in fact the case. The 
total amount of deviation from a straight line is d t. Since n, 
t, m form an i, j, k system 

I = tt + nn + mm. 
Hence dt = tt*dt + nndt + mm*dt. 

Since t is a unit vector the first term vanishes. The second 
term represents the amount of turning up and down; the 
third term, the amount to the right or left. Hence m d t is 
the proper measure of this part of the deviation from a 
straightest line. In case the curve is a geodetic this term 
vanishes as was expected. 

156.] A curve or surface may be mapped upon a unit 
sphere by the method of parallel normals. A fixed origin is 
assumed, from which the unit normal n at the point P of a 



422 VECTOR ANALYSIS 

given surface is laid off. The terminus P r of this normal lies 
upon the surface of a sphere. If the normals to a surface at all 
points P of a curve are thus constructed from the same origin, 
the points P r will trace a curve upon the surface of a unit 
sphere. This curve is called the spherical image of the given 
curve. In like manner a whole region T of the surface may 
be mapped upon a region T 1 the sphere. The region T 1 upon 
the sphere has been called the hodogram of the region T upon 
the surface. If d r be an element of arc upon the surface the 
corresponding element upon the unit sphere is 

dn= dr. 

If da be an element of area upon the surface, the corre 
sponding element upon the sphere is d*! where (Art. 124). 

d a = <P 2 d a. 
= a i i + & j j 
<P 2 = a6 i x j i f xj f = ab nn. 
Hence dd = ab nn d a. (19) 

The ratio of an element of surface at a point P to the area of 
its hodogram is equal to the product of the principal radii of 
curvature at P or to the reciprocal of the product of the prin 
cipal curvatures at P. 

It was seen that the measure of turning to the right or left 
is m d t. If then is any curve drawn upon a surface the 
total amount of turning in advancing along the curve is the 
integral. 

r m dt. (20) 



c 



For any closed curve this integral may be evaluated in a 
manner analogous to that employed (page 190) hi the proof 
of Stokes s theorem. Consider two curves C and near 



THE CURVATURE OF SURFACES 423 

together. The variation which the integral undergoes when 
the curve of integration is changed from C to C is 

S f m- dt. 

S fm.dt= Cs (m - di)= fSm - dt+ Cm*Sdt 

d(m-8t) = dmSt-hmd 8 1 
S Cm* dt= C Sm* dt- C dm* St + C d (m Si). 

The integral of the perfect differential d (m S t) vanishes 
when taken around a closed curve. Hence 

S I mdt= / Sm dt I dm Sk 

The idemfactor is I = tt + nn + mm, 

8m (2t = m I e2t = m nn dt, 

f or t c t and S m m vanish. A similar transformation may 
be effected upon the term dm S t. Then 



S / mdt= /(Smn n^t rfmn n-St). 

By differentiating the relations m n = and n t = it is 
seen that 



c?mn=:--m^n n . dt = dn t. 
Hence 8 /m^t=/ (m Sn t*dn m dn t Sn) 

Sim rft=/(mxt-8nx^n)= I n Sn x dn. 



424 VECTOR ANALYSIS 

The differential Sn x dn represents the element of area in 
the hodogram upon the unit sphere. The integral 



/ n 8 n x <J n = / n d a 



represents the total area of the hodogram of the strip of 
surface which lies between the curves C and C f . Let the 
curve C start at a point upon the surface and spread out to 
any desired size. The total amount of turning which is re 
quired in making an infinitesimal circuit about the point is 
2 TT. The total variation in the integral is 



f 8 fin. rft=f 



m.dt-27T. 



But if H denote the total area of the hodogram. 
Hence / m d t = 2 TT -JET, 

or iT=27r fm rft, (22) 

or H+ Cm* dt = 27r. 

The area of the hodogram of the region enclosed by any 
closed curve plus the total amount of turning along that curve 
is equal to 2 TT. If the surface in question is convex the area 
upon the sphere will appear positive when the curve upon the 
surface is so described that the enclosed area appears positive. 
If, however, the surface is concavo-convex the area upon the 
sphere will appear negative. This matter of the sign of the 
hodogram must be taken into account in the statement made 
above. 



THE CURVATURE OF SURFACES 425 

157.] If the closed curve is a polygon whose sides are 
geodetic lines the amount of turning along each side is zero. 
The total turning is therefore equal to the sum of the exterior 
angles of the polygon. The statement becomes : the sum of 
the exterior angles of a geodetic polygon and of the area of 
the hodogram of that polygon (taking account of sign) is 
equal to 2 TT. Suppose that the polygon reduces to a triangle. 
If the surface is convex the area of the hodogram is positive 
and the sum of the exterior angles of the triangle is less than 
2 TT. The sum of the interior angles is therefore greater than 
TT. The sphere or ellipsoid is an example of such a surface. 
If the surface is concavo-convex the area of the hodogram is 
negative. The sum of the interior angles of a triangle is in 
this case less than TT. Such a surface is the hyperboloid of one 
sheet or the pseudosphere. There is an intermediate case in 
which the hodogram of any geodetic triangle is traced twice in 
opposite directions and hence the total area is zero. The sum 
of the interior angles of a triangle upon such a surface is equal 
to TT. Examples of this surface are afforded by the cylinder, 
cone, and plane. 

A surface is said to be developed when it is so deformed that 
lines upon the surface retain their length. Geodetics remain 
geodetics. One surface is said to be developable or applicable 
upon another when it can be so deformed as to coincide with 
the other without altering the lengths of lines. Geodetics 
upon one surface are changed into geodetics upon the other. 
The sum of the angles of any geodetic triangle remain un 
changed by the process of developing. From this it follows 
that the total amount of turning along any curve or the area 
of the hodogram of any portion of a surface are also invariant 
of the process of developing. 



426 VECTOR ANALYSIS 



Harmonic Vibrations and Bivectors 

158.] The differential equation of rectilinear harmonic 
motion is 



The integral of this equation may be reduced by a suitable 
choice of the constants to the form 

x = A sin n t. 

This represents a vibration back and forth along the X-axis 
about the point x = 0. Let the displacement be denoted by 
D in place of x. The equation may be written 

D = i A sin n t. 

Consider D = i A sin n t cos m x. 

This is a displacement not merely near the point x = 

Or, 

but along the entire axis of x. At points x = - , where 

in 

k is a positive or negative integer, the displacement is at all 
times equal to zero. The equation represents a stationary 
wave with nodes at these points. At points midway between 
these the wave has points of maximum vibration. If the 
equation be regarded as in three variables x, y, z it repre 
sents a plane wave the plane of which is perpendicular to 
the axis of the variable x. 

The displacement given by the equation 

D x = i A l cos (m x n f) (1) 

is likewise a plane wave perpendicular to the axis of x but 
not stationary. The vibration is harmonic and advances 
along the direction i with a velocity equal to the quotient of 



HARMONIC VIBRATIONS AND BIVECTORS 427 

n by m. If v be the velocity; p the period; and / the wave 

length, 

n 2?r 2 TT / 

v = -, ^p = , I = , v = -. (2) 

m n m p 

The displacement 

D 2 = j A 2 cos (m x nt) 

differs from Dj in the particular that the displacement takes 
place in the direction j, not in the direction i. The wave as 
before proceeds in the direction of x with the same velocity. 
This vibration is transverse instead of longitudinal. By a 
simple extension it is seen that 

D = A cos (m x n t) 

is a displacement in the direction A. The wave advances 
along the direction of x. Hence the vibration is oblique to 
the wave-front. A still more general form may be obtained 
by substituting m r for m x. Then 

D = A cos (m r n t). (3) 

This is a displacement in the direction A. The maximum 
amount of that displacement is the magnitude of A. The 
wave advances in the direction m oblique to the displace 
ment; the velocity, period, and wave-length are as before. 
So much for rectilinear harmonic motion. Elliptic har 
monic motion may be defined by the equation (p. 117). 



The general integral is obtained as 

r = A cos n t + B sin n t. 

The discussion of waves may be carried through as pre 
viously. The general wave of elliptic harmonic motion 
advancing in the direction m is seen to be 



428 VECTOR ANALYSIS 

D = A cos (m r n t) B sin (m r n t). (4) 

dV ( } 

= n | A sin (m r n t) + B cos (m r n t) j (5) 

is the velocity of the displaced point at any moment in the 
ellipse in which it vibrates. This is of course entirely differ 
ent from the velocity of the wave. 

An interesting result is obtained by adding up the dis 
placement and the velocity multiplied by the imaginary 
unit V 1 and divided by n. 



D H -- = A cos (m r n t) B sin (m r n f) 



+ V 1 \ A sin (m r n t) + B cos (m r n t) }. 



The expression here obtained, as far as its form is concerned, 
is an imaginary vector. It is the sum of two real vectors of 
which one has been multiplied by the imaginary scalar V 1. 
Such a vector is called a bivector or imaginary vector. The 
ordinary imaginary scalars may be called biscalars. The use 
of bivectors is found very convenient in the discussion of 
elliptic harmonic motion. Indeed any undamped elliptic har 
monic plane wave may be represented as above by the pro 
duct of a bivector and an exponential factor. The real part 
of the product gives the displacement of any point and the 
pure imaginary part gives the velocity of displacement 
divided by n. 

159.] The analytic theory of bivectors differs from that of 
real vectors very much as the analytic theory of biscalars 
differs from that of real scalars. It is unnecessary to have 
any distinguishing character for bivectors just as it is need- 



HARMONIC VIBRATIONS AND BIVECTORS 429 

less to have a distinguishing notation for biscalars. The bi- 
vector may be regarded as a natural and inevitable extension 
of the real vector. It is the formal sum of two real vectors 
of which one has been multiplied by the imaginary unit V 1- 
The usual symbol i will be maintained for V 1. There is 
not much likelihood of confusion with the vector i for the 
reason that the two could hardly be used in the same place 
and for the further reason that the Italic i and the Clarendon 
i differ considerably in appearance. Whenever it becomes 
especially convenient to have a separate alphabet for bivec- 
tors the small Greek or German letters may be called upon. 
A bi vector may be expressed in terms of i, j, k with com 
plex coefficients. 

If r = TJ + i r 2 

and r i = x i * 

r = # i 



or r = #i + yj + z. 

Two bivectors are equal when their real and their imaginary 
parts are equal. Two bivectors are parallel when one is the 
product of the other by a scalar (real or imaginary). If 
a bivector is parallel to a real vector it is said to have a real 
direction. In other cases it has a complex or imaginary 
direction. The value of the sum, difference, direct, skew, 
and indeterminate products of two bivectors is obvious with 
out special definition. These statements may be put into 
analytic form as follows. 

Let r = TJ + i r 2 and s = s 2 + i s 2 . 

Then if r = s, r l = B I and r 2 = s 2 

if r || s r = x s = (x l + i # 3 ) s, 



480 VECTOR ANALYSIS 

r + s = (r 1 + s 1 ) + i(r 2 + s 2 ), 
r . s = <>! B! - r 2 s 2 ) + i (r l s 2 + r 2 Sl ), 
r x s = (r l x B I r 2 x s 2 ) + i (r x x s a + r a x s x ) 
rs = (r l s l + r 2 s 2 ) + i (T I s 2 + r 2 Sj). 

Two bivectors or biscalars are said to be conjugate when 
their real parts are equal and their pure imaginary parts 
differ only in sign. The conjugate of a real scalar or vector 
is equal to the scalar or vector itself. The conjugate of any 
sort of product of bivectors and biscalars is equal to the pro 
duct of the conjugates taken in the same order. A similar 
statement may be made concerning sums and differences. 


Oi + i r 2 ) ( r i - * r 2 ) = r x TJ + r t r 2 , 

Oi + * * 2 ) X (r x - i r 2 ) = 2 i r 2 x T V 

Ol + * r 2> ( r l ~ * F 2> = ( r l r l + F 2 r 2> + * ( r 2 F l - r l r 2>- 

If the bivector r = TJ 4- i r 2 be* multiplied by a root of unity 
or cyclic factor as it is frequently called, that is, by an imagi 
nary scalar of the form 

cos q + i sin q = a + ib, (7) 

where a 2 + & 2 = 1, 

the conjugate is multiplied by a i 6, and hence the four 
products 



are unaltered by multiplying the bivector r by such a factor. 
Thus if 

r = r / + i r 2 = (a + iV) (r x + i r 2 ), 

TI TI + *z r 2 = r x T! + r 2 r 2 , etc. 



HARMONIC VIBRATIONS AND BIVECTORS 431 

160.] A closer examination of the effect of multiplying a 
bivector by a cyclic factor yields interesting and important 
geometric results. Let 

r i + * r a = ( cos ? + * sin 2) ( r i + i r a)- ( 8 ) 
Then r x = i l cos ^ r 2 sin , 

r a = r 2 cos q + T I sin j. 

By reference to Art. 129 it will be seen that the change pro 
duced in the real and imaginary vector parts of a bivector by 
multiplication with a cyclic factor, is precisely the same as 
would be produced upon those vectors by a cyclic dyadic 

d> = a a + cos q (bb + c c ) - sin q (c b - be ) 

used as a prefactor. b and c are supposed to be two vectors 
collinear respectively with r x and r 2 . a is any vector not in 
their plane. Consider the ellipse of which TJ and r 2 are a 
pair of conjugate semi-diameters. It then appears that r^ 
and r 2 are also a pair of conjugate semi-diameters of that 
ellipse. They are rotated in the ellipse from r 2 toward r 1$ by 
a sector of which the area is to the area of the whole ellipse 
as q is to 2 ?r. Such a change of position has been called an 
elliptic rotation through the sector q. 

The ellipse of which T I and r 2 are a pair of conjugate semi- 
diameters is called the directional ellipse of the bivector r. 
When the bivector has a real direction the directional ellipse 
reduces to a right line in that direction. When the bivector 
has a complex direction the ellipse is a true ellipse. The 
angular direction from the real part T I to the complex part r 2 
is considered as the positive direction in the directional 
ellipse, and must always be known. If the real and imagi 
nary parts of a bivector turn in the positive direction in the 
ellipse they are said to be advanced ; if in the negative direc 
tion they are said to be retarded. Hence multiplication of a 



432 VECTOR ANALYSIS 

bivector by a cyclic factor retards it in its directional ellipse by 
a sector equal to the angle of the cyclic factor. 

It is always possible to multiply a bivector by such a cyclic 
factor that the real and imaginary parts become coincident 
with the axes of the ellipse and are perpendicular. 

r = (cos q + i sin q) (a + i b) where a b = 0. 
To accomplish the reduction proceed as follows : Form 

r r = (cos 2 q + i sin 2 q) (a + i b) (a + i b). 
If a b = 0, 

r r = (cos 2 q + i sin 2 q) (a a b b). 

Let r r = a + i 6, 

and tan 2 q = -. 

a 

With this value of q the axes of the directional ellipse are 
given by the equation 

a -f i b = (cos q i sin q) r. 

In case the real and imaginary parts a and b of a bivector 
are equal in magnitude and perpendicular in direction both a 
and b in the expression for r r vanish. Hence the angle 
q is indeterminate. The directional ellipse is a circle. A 
bivector whose directional ellipse is a circle is called a circu 
lar bivector. The necessary and sufficient condition that a 
non-vanishing bivector r be circular is 

r r = 0, r circular. 
If r = zi + 2/j + *k, 

r . r = x* + y* + z 2 = 0. 

The condition r r = 0, which for real vectors implies r = 0, 
is not sufficient to ensure the vanishing of a bivector. The 



HARMONIC VIBRATIONS AND BIVECTORS 433 

bivector is circular, not necessarily zero. The condition that 
a bivector vanish is that the direct product of it by its con 
jugate vanishes. 

Oi + i r 2 ) (r x - t r 2 ) = r x r x + r 2 r a = 0, 
then F! = r 2 = and r = 0. 

In case the bivector has a real direction it becomes equal to 
its conjugate and their product becomes equal to r r. 

161.] The condition that two bivectors be parallel is that 
one is the product of the other by a scalar factor. Any bi- 
scalar factor may be expressed as the product of a cyclic 
factor and a positive scalar, the modulus of the biscalar. If 
two bivectors differ by only a cyclic factor their directional 
ellipses are the same. Hence two parallel vectors have their 
directional ellipse similar and similarly placed the ratio of 
similitude being the modulus of the biscalar. It is evident 
that any two circular bivectors whose planes coincide are 
parallel. A circular vector and a non-circular vector cannot 
be parallel. 

The condition that two bivectors be perpendicular 

is r s = 0, 

or r t ! r 2 83 = r x s 2 + r 2 B I = 0. 

Consider first the case in which the planes of the bivectors 
coincide. Let 

r = a (TJ + i r 2 ), s = I (s 1 + i g 2 ). 

The scalars a and b are biscalars. r x may be chosen perpen 
dicular to r 2 , and s l may be taken in the direction of r a . The 
condition r s = then gives 

r a 82 = and r x s 2 + r a n l = 0. 

28 



434 VECTOR ANALYSIS 

The first equation shows that r 2 and s 2 are perpendicular and 
hence s l and s 2 are perpendicular. Moreover, the second 
shows that the angular directions from r x to r 2 and from s 1 to 
s 2 are the same, and that the axes of the directional ellipses 
of r and s are proportional. 

Hence the conditions for perpendicularity of two bivectors 
whose planes coincide are that their directional ellipses are 
similar, the angular direction in both is the same, and the 
major axes of the ellipses are perpendicular. 1 If both vectors 
have real directions the conditions degenerate into the per 
pendicularity of those directions. The conditions therefore 
hold for real as well as for imaginary vectors. 

Let r and s be two perpendicular bivectors the planes of 
which do not coincide. Resolve T I and r 2 each into two com 
ponents respectively parallel and perpendicular to the plane 
of s. The components perpendicular to that plane contribute 
nothing to the value of r s. Hence the components of r x 
and r 2 parallel to the plane of s form a bivector r which is 
perpendiqular to s. To this bivector and s the conditions 
stated above apply. The directional ellipse of the bivector r 
is evidently the projection of the directional ellipse of r upon 
the plane of s. 

Hence, if two bivectors are perpendicular the directional 
ellipse of either bivector and the directional ellipse of the 
other projected upon the plane of that one are similar, have 
the same angular direction, and have their major axes per 
pendicular. 

162.] Consider a bivector of the type 



where A and m are bivectors and TI is a biscalar. r is the 
position vector of a point in space. It is therefore to be con- 

1 It should be noted that the condition of perpendicularity of major axes is not 
the same as the condition of perpendicularity of real parts and imaginary parts 



HARMONIC VIBRATIONS AND BIVECTORS 435 

sidered as real, t is the scalar variable time and is also to 
be considered as real. Let 

A = Aj + i A*p 
m = ni + i mj 



D = 

As has been seen before, the factor (A x + i Aj) e <(mt * r ~ nif) 
represents a train of plane waves of elliptic harmonic vibra 
tions. The vibrations take place in the plane of Aj and A 2 , 
in an ellipse of which A x and A% are conjugate semi-diam 
eters. The displacement of the vibrating point from the 
center of the ellipse is given by the real part of the factor. 
The velocity of the point after it has been divided by nj 
is given by the pure imaginary part. The wave advances 
in the direction m r The other factors in the expres 
sion are dampers. The factor ""* is a damper in the 
direction m 2 . As the wave proceeds in the direction m^ it 
dies away. The factor e*** is a damper in time. If n a is 
negative the wave dies away as time goes on. If n 2 is posi 
tive the wave increases in energy as time increases. The 
presence (for unlimited time) of any such factor in an ex 
pression which represents an actual vibration is clearly inad 
missible. It contradicts the law of conservation of energy. 
In any physical vibration of a conservative system n a is ne 
cessarily negative or zero. 

The general expression (9) therefore represents a train of 
plane waves of elliptic harmonic vibrations damped in a 
definite direction and in time. Two such waves may be com 
pounded by adding the bivectors which represent them. If 
the exponent m r n t is the same for both the resulting 
train of waves advances in the same direction and has the 



436 VECTOR ANALYSIS 

same period and wave-length as the individual waves. The 
vibrations, however, take place in a different ellipse. If the 
waves are 



the resultant is (A + B) **. 

By combining two trains of waves which advance in opposite 
directions but which are in other respects equal a system of 
stationary waves is obtained. 



A e - m *- 1 V (mi - lr - n0 + A e~ m * r 
Ae-^- e-"" (e <mi * r + $-"i") = 2Acos (n^ r) e- m * p e^ int 

The theory of bivectors and their applications will not be 
carried further. The object in entering at all upon this very 
short and condensed discussion of bivectors was first to show 
the reader how the simple idea of a direction has to give way 
to the more complicated but no less useful idea of a directional 
ellipse when the generalization from real to imaginary vectors 
is made, and second to set forth the manner in which a single 
bivector D may be employed to represent a train of plane 
waves of elliptic harmonic vibrations. This application of bi 
vectors may be used to give the Theory of Light a wonderfully 
simple and elegant treatment. 1 

1 Such use of bivectors is made by Professor Gibbs in his course of lectures on 
" The Electromagnetic Theory of Light" delivered biannually at Yale University. 
Bivectors were not used in the second part of this chapter, because in the opinion 
of the present author they possess no essential advantage over real vectors until 
the more advanced parts of the theory, rotation of the plane of polarization by 
magnets and crystals, total and metallic reflection, etc., are reached. 



249005