Vector Analysis and Quaternions

Project Gutenberg 's Vector Analysis and Quaternions . by Alexander Macf arlane

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever. You may copy it . give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www . eutenber^. net

Title: Vector Analysis and Quaternions

Author: Alexander Macf arlane

Release Date: October 5, 2004 [EBook #13609]

Language : English

Character set encoding ; TeX

*** START OF THIS PROJECT GUTEHBERG EBOOK VECTOR ANALYSIS AND QUATERNIONS "*"

Produced by David Starner, Joshua Hutchinson, John Hagerson, and the
Project Gutenberg Dn-line Distributed Proofreaders.

MATHEMATICAL MONOGRAPHS.

EDITED BY

MANSFIELD MERRIMAN and ROBERT S. WOODWARD.

No. 8.

VECTOR ANALYSIS

QUATERNIONS.

ALEXANDER MACFARLANE,

SECR.ETAR.Y OF InTERXATIOWAL ASSOCIATION FOR. PrOALOTJWG THE StUDY OF Qu ATERWIONS.

NEW YORK:

JOHN WILEY ^ SONS.

London: CHAPMAN Sz HALL, Limited.

1906.

THANSCRlHEJi's XOTES This material uias originally published iii a book by Klerriman and Wood-
ward titled Higher Mathematio. J believe that some of the page number cross-references have
been retained from that presentation of this material.

! did my best to recreate the index.

MATHEMATICAL MONOGRAPHS.

EDITED BY

Mansfield Merriman and Robert S. Woodv^ard.
Octavo. Cloth. Si, 00 each.

No. 1. History of Modern Mathematics,

By David Eugexe S^[ith.

No, 3, Synthetic Projective Geometry.

By George Cruce Halsted.

No, 3, Determinants.

By Laenas Gifford Weld.

No. 4. Hyperbolic Functions,

By .James McMahox

No, 5, Harmonic Functions.

By WiLLEAU E. BVERLY.

No, 6, Grassmann^s Space Analysis.

By Edward W. Hyde.

No, 7. Probability and Theory of Errors,

By Robert S. Woodward

No, 8, Vector Analysis and Quaternions.

By Alexander Macfarlaxe.

No, 0. DifFerential Equations,

By William Woolsey Johnsow.

No. 10, The Solution of Equations,

By Maxsfield Merrimaw.

No. 11, Functions of a Complex Variable-

Bv Thom \s S. Fiske.

PUBLISHED BY

JOHN WILEY & SONS, Inc., NEW YORK.
CHAPMAN & HALL, Limited, LONDON.

Editors' Preface

The volume called Higher Mathematics, the first edition of which was pub-
lished in 1896, contained eleven chapters by eleven authors, each chapter being
independent of the others, but all supposing the reader to have at least a math-
ematical training equivalent to that given in classical and engineering colleges.
The publication of that volume is now discontinued and the chapters are issued
in separate form. In these reissues it will generally be found that the mono-
graphs are enlarged by additional articles or appendices which either amplify
the former presentation or record recent ad^'ances. This plan of publication has
been arranged in order to meet the demand of teachers and the convenience
of classes, but it is also thought that it may prove advantageous to readers in
special lines of mathematical literature.

It is the intention of the publishers and editors to add other monographs to
the series from time to time, if the call for the same seems to warrant it. Among
the topics which are under consideration are those of elliptic functions, the the-
ory of numbers, the group theory, the calculus of variations, and non-Euclidean
geometry; possibly also monographs on branches of astronomy, mechanics, and
mathematical physics may be included. It is the hope of the editors that this
form of publication may tend to promote mathematical study and research over
a wider field than that which the former volume has occupied.

December, 1905.

Author's Preface

Since this Introduction to Vector Analysis and Qnateinions was first published
in 1S9G. the study of the subject has become much more general: and whereas
some reviewers then regarded the analysis as a luxury, it is now recognized as a
necessity for the exact student of physics or engineering. In America, Professor
Hathaw^ay has pubhshed a Primer of Quaternions (New York, 1890), and Dr.
Wilson has amplified and extended Professor Gibbs' lectures on vector analysis
into a text-book for the use of students of mathematics and physics (New^ York,
1901). In Great Britain, Professor Henrici and Mr. Turner haA^e published a
manual for students entitled Vectors and Rotors (London. 1903): Dr. Knott
has prepared a new edition of Kelland and Tait's Introduction to Quaternions
(London, 1904): and Professor Joly has realized Hamilton's idea of a Manual of
Quaternions (London, 1905). In Germany Dr. Bucherer has published Elemente
der Vektoranalysis (Leipzig. 1903) which has now reached a second edition.

Also the writings of the great masters have been rendered more accessible.
A new edition of Hamilton's classic, the Elements of Quaternions, has been pre-
pared by Professor Joly (London, 1899. 1901) ; Tait's Scientific Papers have been
reprinted in collected form (Cambridge, 1898, 1900): and a complete edition of
Grassmann's mathematical and physical works has been edited by Friedrich En-
gel with the assistance of several of the eminent mathematicians of Germany
(Leipzig, 1S94-). In the same interval many papers, pamphlets, and discussions
have appeared. For those who desire information on the literature of the subject
a Bibliography has been published by the Association for the promotion of the
study of Quaternions and Allied Mathematics (Dublin, 1904).

There is still much variety in the matter of notation, and the relation of
Vector Analysis to Quaternions is still the subject of discussion (see Journal of
the Deutsche Mathematiker-Vereinigung for 1904 and 1905).

ChATHA^[. OxTARIO. CaXAD.A, December, 1905.

Contents

Editors' Preface iii

Author's Preface iv

1 Introduction.. 1

2 Addition of Coplanar Vectors. 3

3 Products of Coplanar Vectors. 9

4 Coaxial Quaternions. 16

5 Addition of Vectors in Space. 21

6 Product of Two Vectors. 23

7 Product of Three Vectors. 28

8 Composition of Quantities. 32

9 Spherical Trigonometry. 37

10 Composition of Rotations. 44
Index 47

11 PROJECT GUTENBERG "SMALL PRINT"

Article 1

Introduction,

By "Vector Analysis'^ is meant a space analysis in which the vector is the funda-
mental idea; by "Quaternions" is meant a space-analysis in which the quaternion
is the fundamental idea. They are in truth complementary parts of one whole:
and in this chapter they will be treated as such, and developed so as to har-
monize with one another and with the Cartesian Analysis . The subject to be
treated is the analysis of quantities in space, whether they are vector in nature,
or quaternion in nature, or of a still different nature, or are of such a kind that
they can be adequately represented by space quantities.

Every proposition about quantities in space ought to remain true when re-
stricted to a plane; just as propositions about quantities in a plane remain true
when restricted to a straight line. Hence in the following articles the ascent
to the algebra of space is made through the intermediate algebra of the plane.
Arts. 2-4 treat of the more restricted analysis, while Arts. 5-10 treat of the
general analysis.

This space analysis is a universal Cartesian analysis, in the same manner as
algebra is a universal arithmetic. By providing o^ explicit notation for directed
quantities, it enables their general properties to be investigated independently
of any particular system of coordinates, whether rectangular, cylindrical, or
polar. It also has this advantage that it can express the directed quantity by a
linear function of the coordinates, instead of in a roundabout way by means of
a quadratic function.

The different views of this extension of analysis which have been held by
independent writers are briefly indicated by the titles of their works:

• Argand, Essai sur une maniere de representer les quantites imaginaires dans les
constructions geometriques, 1B06.

• Warren. Treatise on the geometrical representation of the square roots of nega-
tive quantities. 1828.

• Moebius. Der barycentrische Calcul. 1827.

• BellavitiS; Calcolo delle Equip ollenze. 1835.

For a discussion of the (elation of Vector Analysis to Quoteznions, see Nature, 1391— 13S3.

ARTICLE 1. INTRODUCTION. 2

• Grassmann. Die lineale Ausdehnungslehre. 1844.

• De Morgan. Trigonometry and Double Algebra. 1849.

• O'Brien, Symbolic Form.s derived from, the conception of the translation of a
directed magnitude. Philosophical Transactions, 1851.

• Hamilton. Lectures on Quaternions. 1853. and Elements of Quaternions. 18GG.

• Tait. Elementary Treatise on Quaternions. 1867.

• Hankel. Vorlesungen liber die complexen Zahlen und ihre Functionen. 18G7.

• Schlegel, System der Raumlehre. 1872.

• Hoiiel. Theorie des quantites complexes. 1874.

• Gibbs. Elements of Vector Analysis. 1881—4.

• Peano, Calcolo geometrico. 1888.

• Hyde, The Directional Calculus. 1890.

• Heaviside. Vector Analysis, in "Reprint of Electrical Papers,'' 1885—92.

• Macfarlane, Principles of the Algebra of Physics. 1891. Papers on Space Analy-
sis. 1891-3.

An excellent synopsis is given by Hagen in the second volume of his "Synopsis der
hbheren Mathematik."

Article 2

Addition of Coplanar
Vectors.

By a "vector" is meant a quantity which has magnitude and direction. It is
graphically represented by a line whose length represents the magnitude on
some convenient scale, and whose direction coincides with or represents the
direction of the vector. Though a vector is represented by a line, its physical
dimensions may be different from that of a line. Examples are a linear velocity
which is of one dimension in length, a directed area which is of two dimensions
in length, an axis which is of no dimensions in length.

A vector will be denoted by a capital italic letter, as B} its magnitude
by a small italic letter, as b, and its direction by a small Greek letter, as P.
For example. B = 6,^, R = rp. Sometimes it is necessary to introduce a dot
or a mark / to separate the specification of the direction from the expression
for the magnitude: but in such simple expressions as the above, the difference
is sufficiently indicated by the difference of type. A system of three mutually
rectangular axes will be indicated, as usual^ by the letters i, j. k.

The analysis of a vector here supposed is that into magnitude and direction.
According to Hamilton and Tait and other writers on Quaternions, the vector
is analyzed into tensor and unit-vector, which means that the tensor is a m.ere
ratio destitute of dimensions, w^hile the unit-vector is the physical magnitude.
But it will be found that the analysis into magnitude and direction is much
more in accord with physical ideas, and explains readily many things which are
difficult to explain by the other analysis.

A vector quantity may be such that its components have a common point
of application and are applied simultaneously; or it may be such that its com-
ponents are applied in succession, each component starting from the end of its

This notation is found convenient by electrical writers in order to harmonize u^'ith the
Hospitalier system of symbols and obbieviotions.

"The dot ■vv'QS used for this purpose in the outhor's Note on Plane Algebra, 1833; Kennelly
has since used fL for the same purpose in his electrical papers.

ARTICLE 2. ADDITION OF COPLANAR VECTORS. 4

predecessor. An example of the former is found in two forces applied simul-
taneously at the same point, and an example of the latter in two rectilinear
displacements made in succession to one another.

Composition of Components havin°; a common Point of Application. — Let
OA and OB represent two vectors of the same kind simultaneously applied at
the point O. Draw BC parallel to OA, and AC parallel to OB. and join OC.
The diagonal OC represents in magnitude and direction and point of application
the resultant of OA and OB. This principle was discovered with reference to
force, but it applies to any vector quantity coming under the above conditions.

Take the direction of OA for the initial direction: the direction of any other
vector will be sufficiently denoted by the angle round which the initial direction
has to be turned in order to coincide with it. Thus O A may be denoted by /i/O,
OB by /2/^2 1 OC by f/6. From the geometry of the figure it follows that

and

/j sin O2

tanS =

/i + /2 00562 '
hence

OC = V f{ + fi + 2/1 /a cose^ / tan-

/o sin 02

2 ■

Exajnple. — Let the forces applied at a point be 2/0° and 3/G0°. Then the
resultant is J4 + 9 + 12 x ^ /tan"^ ^^ = 4,30/36^ 30'.

If the first component is given as f\/Oi , then we have the more symmetrical
formula

^ /_ /iCOsgi +/2COsfc^2

When the components are equal, the direction of the resultant bisects the
angle formed by the vectors; and the magnitude of the resultant is twice the
projection of either component on the bisecting line. The aboA'e formula reduces
to

O2 /O2
OC = 2/xcos^/^.

ARTICLE 2. ADDITION OF COPLANAR VECTORS. 5

Example. — The resultant of two equal alternating electromotive forces which
differ 120° in pha.se is equal in magnitude to either and has a phase of 00"^.

Given a vector and one component, to find the other component. — Let DC
represent the resultant, and OA the component. Join AC and draw OB equal
and parallel to AC. The line OB represents the component required, for it is
the only line which combined with OA gives OC as resultant. The line OB is
identical with the diagonal of the parallelogram formed by OC and OA reversed:
hence the rule is, "Reverse the direction of the component, then compound it
w^itb. the given resultant to find the required component.'' Let f /O be the vector
and /i/O one component: then the other component is

/2/^= yjr-+fi -2//icosy tan-^

/sinS

-/l+/cos(

Given the resultant and the directions of the two components, to find the
magnitude of the components, — The resultant is represented by OC. and the
directions by OX and OY . From C draw CA parallel to OY . and CB parallel
to OX\ the lines OA and OB cut off represent the required components. It is
evident that OA aiid OB when compounded produce the given resultant OC^
and there is only one set of two components which produces a given resultant:
hence they are the only pair of components having the given directions.

Let f /O be the vector and /Oj and /S2 the given directions. Then

h cos(e2 - e^) + h = fcos{e2-e).,

from which it follows that

^{cos{0 -Oi) -cos(^2 -^)cos(^2 -^1}}

/i = /-

1 -C0S^{^2 -^1)

ARTICLE 2. ADDITION OF COPLANAR VECTORS.

For example, let 100/60*', /30°, and /QO"* be given; then

cos 30"

/i = 100,

Composition of any Number of Vectors applied at a common Point. — The
resultant may be found by the following graphic construction: Take the vectors
in any order, as A, B^ G. From the end of A draw B equal and parallel
to B. and from the end of B draw C equal and parallel to G\ the vector
from the beginning of A to the end of C" is the resultant of the given vectors.
This follows by continued application of the parallelogram construction. The
resultant obtained is the same, whatever the order; and as the order is arbitrary,
the area enclosed has no physical meaning.

The result may be obtained analytically as follows:

Given

Now

Similarly

and

Hence

/i/Si+^/Sa + Za/^s + ■■■ + /„/&„.

/iM = h cos ei/0 + h sin ^i/ J ■

/2/^ = /2 cos ej/O + /2 sin 92/ J .

fr^jOn = in COsg^/O +/„ Smg„/| .

i:{/M = {i:/-^^}/^+{E/E

'TV

-ve/-»)'+(E/.-)"--'|^-

ARTICLE 2. ADDITION OF COPLANAR VECTORS. 7

In the case of a sum of simultaneous vectors applied at a common point,
the ordinary rule about the transposition of a term in an equation holds good.
For example, if A -\- B -\- C = 0, then A -\- B = -C , and A -\- C = -B , and
5 -|-C = —A. etc. This is permissible because there is no real order of succession
among the given components.

A

I

t

B.f

f
I

Composition of Successive Vectors, — The composition of successive vectors
partakes more of the nature of multiplication than of addition. Let A be a
vector starting from the point O, and B a vector starting from the end of A.
Draw the third side OP ^ and from O draw a vector equal to B. and from its
extremity a vector equal to A. The line OP is not the complete equivalent
of A + B ; if it were so, it would also be the complete equivalent oi B -\- A.
But A -\- B and B -\- A determine different paths; and as they go oppositely
around, the areas they determine with OP have different signs. The diagonal
OP represents A -\- B only so far as it is considered independent of path. For
any number of successive vectors, the sum so far as it is independent of path is
the vector from the initial point of the first to the final point of the last. This is
also true when the successive vectors become so small as to form a continuous
curve. The area between the curve OPQ and the vector OQ depends on the
path, and has a physical meaning.

Prob. 1. The resultant vector is 123/45 . and one component is 100/0 : find the other
component.

Prob. 2. The velocity of a body in a given plane is 200/75 . and one component is

100/25 ; find the other component.

Prob. 3. Three alternating magnetomotive forces are of equal virtual "value, but each pair
differs in phase by 120°; find the resultant. [Ans. Zero.)

Prob. A. Find the components of the vector 100/70^ in the directions 20° and 100^.
Prob. 5. Calculate the resultant vector of 1/10^, 2/20°. 3/30°, 4/40°.

Prob. 6. Compound the following magnetic fluxes: hsinnt -\- hsinijit — 120 )/120 +
/isin(nt- 240°)/240°. (Ans. '^h/nt.)

Tills does not hold true of □. sum of vectors having a real order of succession. It is a
mistake to attempt to found space-analysis upon arbitrary formal laivs: the fundamental rules
must be made to express universal properties of the thing denoted In this chapter no attempt
is made to apply formal laws to directed quantities. What is attempted is an analysis of these
quantities.

ARTICLE 2. ADDITION OF COPLANAR VECTORS. 8

Prob. 7. Compound two alternating magnetic fluxes at a point acosnt/Q and asinnt/^.
{Ans. a/nt.)

Prob. 8. Find the resultant of two simple alternating electromotive forces 100/20° and
50/75'*.

Prob. 9. Prove that a uniform circular motion is obtained by compounding two equal
simple harmonic motions which have the space-phase of their angular positions
equal to the supplement of the time-phase of their motions.

Article 3

Products of Coplanar
Vectors.

When all the vectors considered are confined to a common plane, each may
be expressed as the sum of two rectangular components. Let i and j denote
two directions in the plane at right angles to one another; then A = aii + flojj
B = hii + 62 J 1 R = xi -\- yj- Here i and j are not unit-vectors, but rather signs
of direction.

Product of two Vectors. — Let A = a^i + a2J and B = h^i -\- h2J be any

two vectors, not necessarily of the same kind physically. We assume that their
product is obtained by applying the distributive law. but we do not assume that
the order of the factors is indifferent. Hence

AB = (ui? + a2J){bii + 62 J } = <^i^i n + 112^2 J J + i^i^2^J + <J2^2i'-

If we assume, as suggested by ordinary algebra, that the square of a sign of
direction is +, and further that the product of two directions at right angles to

one another is the direction normal to both, then the above reduces to

AB = aibi + a^t^ + (<^i^2 ~ a2bi)k.

Thus the complete product breaks up into two partial products, namely,
ai^i + a^^E which is independent of direction, and (^162 — a^^i )fc which has the

axis of the plane for direction.

A common explanation which is given of ij ^ k \s that i is an operator; j an operand,
and k the result. The kind of operator which i is supposed to denote is a quadrant of turning
round the axis i; it is supposed not to be an axis, but a quadrant of rotation round an axis.
This explains the result ij = k. but unfortunately it does not explain ii = +; for it Un'ould give
ii = i.

ARTICLE 3. PRODUCTS OF COPLANAR VECTORS.

10

Scalar Product of two Vectors. — By a scalar quantity is meant a quantity
which has magnitude and may be positive or negative but is destitute of direc-
tion. The former partial product is so called because it is of such a nature. It
is denoted by SAB where the symbol S, being in Roman type, denotes, not a
vector, but a function of the vectors A and B . The geometrical meaning of SAB
is the product of A and the orthogonal projection of B upon A. Let OP and
OQ represent the vectors A and B: draw QM and NL perpendicular to OP.
Then

(OP)iOM) = {OP){OL) + {OP){LM'),

?1— + ^2 —

a a

Corollary 1. — SB A = SAB. For instance, let A denote a force and B the
velocity of its point of application; then SAB denotes the rate of working of the
force. The result is the same whether the force is projected on the velocity or
the velocity on the force.

Example 1. — A force of 2 pounds East -I- 3 pounds North is moved with a
velocity of 4 feet East per second + 5 feet North per second; find the rate at
which work is done.

2x4 + 3x5 = 23 foot-pounds per second.

Corollary 2. — A = a^ -h aj = a . The square of any vector is independent

of direction; it is an essentially positive or signless quantity: for whatever the
direction of A, the direction of the other A must be the same; hence the scalar
product cannot be negative.

Example 2. — A stone of 10 pounds mass is moving with a velocity 64 feet
down per second -h 100 feet horizontal per second. Its kinetic energy then is

— (64^ -h 100") foot-poundals,

a quantity which has no direction. The kinetic energy due to the downward

velocity is 10 x

64

10

and that due to the horizontal velocity is — x 100 ; the

2 ^ 2

ARTICLE 3. PRODUCTS OF COPLANAR VECTORS.

11

whole kinetic energy is obtained, not by vector, but by simple addition, when
the components are rectangular.

?' +

Vector Product of two Vectors. — The other partial product from its nature
is called the vector product, and is denoted by V AB . Its geometrical meaning is
the product of A and the projection of B which is perpendicular to A, that is, the
area of the parallelogram formed upon A and B. Let OP and OQ represent the
vectors A aiid B. and draw the lines indicated by the figure. It is then evident
that the area of the triangle OPQ = aib2 — 2^2*^2 ~ 2^1 ^2 ~ 2^^'^ ~^'i)(^2~*^2) =

Thus {^162 — a-2bi)k denotes the magnitude of the parallelogram formed by
A and B and also the axis of the plane in which it lies.

It follows that YBA = —YAB. It is to be observed that the coordinates of
A and B are mere component vectors^ whereas A and B themselves are taken
in a real order.

Example. — Let A = [lOi + llj) inches and B = (5i -I- 12j) inches, then
V AB = (120 — 55)A; square inches; that is, 65 square inches in the plane which
has the direction k for axis.

If A is expressed as act and B as 6/3, then SAB = afccosa.J, where a3
denotes the angle between the directions g and ,J.

Example. — The effective electromotive force of 100 volts per inch /90* along
a conductor 8 inch /45'' is SAB = 8x100 cos/45°/90° volts, that is, 800 cos 45°
volts. Here /45° indicates the direction a and /QO*^ the direction /3, and /45°/90^
means the angle between the direction of 45° and the direction of 90" .

Also y AB = ab sin q3 ■ aP, where a,5 denotes the direction which is normal
to both a and j3, that is. their pole.

Example. — At a distance of 10 feet /30° there is a force of 100 pounds /SO^
The moment is Y AB

= 10 X 100sin/30''/G0'' pound-feet 90°//90°

= 1000 sin 30^ pound feet 90<>//90°

Here 90°/ specifies the plane of the angle and /90° the angle. The two
together written as above specify the normal k.

ARTICLE 3. PRODUCTS OF COPLANAR VECTORS. 12

Reciprocal of a Vector. — By the reciprocal of a A^ector is meant tlie vector
which, combined with the original vector produces the product +1. The recip-
rocal of A is denoted by A~ . Since AB = ai(coso:/3 + sin a/? ■ a/3) , b must
equal a~ and P must be identical with a in order that the product may be 1.

It follows that

I 1 aa aii + ixoj

< —s — '■! X

The reciprocal and opposite vector is —A~. In the figure let OP = 23

be the given vector; then OQ = ^13 is its reciprocal, and OR = ^{—P) is its
reciprocal and opposite."

10

Example.— If ^ = 10 feet East + 5 feet North. A~^ = feet East +

^ ■ 125

5 , 10 5

feet North and -A~^ = feet East feet North.

125 125 125

Product of the reciprocal of a vector and oiiother vector. —

A-'^B = \aB,
a-

b

= — (cosct/^-h sin Q!3 ■ a3).
a

Hence SA~^ B = -cosct/? and V A~^ B = - sin a,d ■ q/3.

"Writeis who identify a vector with a quadrantal versor ore logically led to define the
reciprocal of a vector as being opposite in direction as well as reciprocal in magnitude.

ARTICLE 3. PRODUCTS OF COPLANAR VECTORS. 13

Product of three Coplanar Vectors. — Let A = a^i -\- ^2 J i ^ = ^i* + ^2j)
C = c\i + coj denote any three vectors in a common plane. Then

{AB)C = {{aibi +^2^2) + (^1^2 -a2bi)k}{cii-\-C2J)

= {aibi + a-2b2){cii + cej) + (0162 - a26i)(-C2i + cij).

The former partial product means the vector C multiplied by the scalar
product of A and B ; while the latter partial product means the complementary
vector of C multiplied by the magnitude of the vector product of A and B.
If these partial products (represented by OP and OQ) unite to form a total
product, the total product will be represented by Oi?, the resultant of OP and
OQ.

The former product is also expressed by SAB ■ C, where the point separates
the vectors to which the S refers; ajid more analytically by abc cos q3 ■ 7.

The latter product is also expressed by {V AB)C , which is equivalent to
V(VA5)C, because Y AB is at right angles to C. It is also expressed by

abc sin ay3 - 0,57. where q3~( denotes the direction which is perpendicular to
the perpendicular to a and 3 and 7.

If the product is formed after the other mode of association we have

A{BC) = {aii + a2J)(5iCi +63^2) + {a^i -^ a2J){hiC2 -62^1)^^
= {bici + b2C2){aii + a2J) + (61 C2 -&2^i)(a2i - ^ij)
= SBC ■A-\-YA(yBC).

The vector 031 — aij is the opposite of the complementary vector of aii-\-a2J.
Hence the latter partial product differs with the mode of association.

Example.— Let A = 1/0^ + 2/90°, B = 3/0° + 4/90°, C = 5/0° + 6/90°.
The fourth proportional to A.B.C is

1x3+2x4

-G/0° -\-rj/9oA

1x4-2x3
"^ 1^+2^
= 13.4/0° + 11.2/90°.

Square of a Binomial of Vectors. — If A -\- B denotes a sum of non-successive
vectors, it is entirely equivalent to the resultant vector C. But the square of

ARTICLE 3. PRODUCTS OF COPLANAR VECTORS. 14

any vector is a positive scalar, hence the square oi A -\- B must be a positive
scalar. Since A and B are in reality components of one vector, the square must
be formed after the rules for the products of rectangular components (p. 432).
Hence

{A-\-B)^ = {A + B)(A + B),

= A^ -\-AB -\-BA-\-B-,

= A"^ -\-B^ -\-SAB -^SBA -\-VAB -^VBA,

= A^- -\-B^ -\-2SAB.

This may also be written in the form

a^ -\-b^ -\- 2abcosa3.

But when A-\- B denotes a sum of successive vectors, there is no third vector
C which is the complete equiA^alent: and consequently we need not expect the
square to be a scalar quantity. We observe that there is a real order, not of the
factors, but of the terms in the binomial: this causes both product terms to be
AB , givino;

{A-\-B)- = A^ -^2AB + B-

= A^ -\-B- -^2SAB -\-2\AB.

The scalar pai't ^ives the square of the length of the third side, while the
vector part gives four times the area included between the path and the third
side.

Square of a Trinomial of Coplanar Vectors. — Let A -h B -\- C denote a sum
of successive vectors. The product terms must be formed so as to preserve the
order of the vectors in the trinomial; that is, A is prior to B and C, and B is
prior to C. Hence

(A -\- B -\- C)^ = A^- -\- B^ -\- C^ -\- 2AB + 2AC + 2BC
,^ + B^ + C^ + 2(SAB + SAC + SB
+ 2(yAB + VAC + VBC) . (2)

= A^ + B^ + C^ + 2{SAB + SAC + SBC), (1)

Hence

and

S(^ + 5 + C)^ = (1)
= a~ -\- b -\- c + 2ab cos oP + 2ac cos a'/ + 2bc cos ,^7

V{A + 5 + C)^ = (2)
= \2ab sin a3 + 2ac sin 07 + 2bc sin ,^7} ■ aj3

ARTICLE 3. PRODUCTS OF COPLANAR VECTORS. 15

The scalar part ^jves the square of the vector from the beginning of A to
the end of C and is all that exists when the vectors are non-successive. The
vector part is four times the area included between the successive sides and the
resultant side of the polygon.

Note that it is here assumed that V(^ -\- B)C = \' AC -\-YBC. which is the
theorem of moments. Also that the product terms are not formed in cyclical
order, but in accordance with the order of the vectors in the trinomial.

Example.— Let A = 3/0°. B = 5/30^, C = Ij^^"; find the area of the
polygon.

-V(AB -V AC -VBC) = -{15sin/0730° + 21 sin /0745° + 35/30745°},
= 3.75 + 7.42 + 4.53 = 15.7.

Prob. 10. At a distance of 25 centimeters /20° there is a force of 1000 dynes /80^; find
the moment.

Prob. 11. A conductor in an armature has a velocity of 240 inches per second /300 and
the magnetic flux is 50.000 lines per square inch /O ; find the vector product.
(Ans. 1.04 X 10 lines per inch per second.)

Prob. 12. Find the sine and cosine of the angle between the directions 0.8141 E. + 0.580TN.J
and 0.5060 E. + 0.8625 N.

Prob. 13. When a force of 200 pounds /2rO^ is displaced by 10 feet /30", what is the work
done (scalar product)? What is the meaning of the negative sign in the scalar

product?

Prob. 14. A mass of 100 pounds is moving with a velocity of 30 feet E. per second + 50
feet SE. per second; find its kinetic energy.

Prob. 15. A force of 10 pounds /45" is acting at the end of 8 feet /200°; find the torque,
or vector product.

Prob. IG. The radius of curvature of a curve is 2/0 + 5/90 ; find the curvature.
(Ans. .03/0° + .17/90°)

Prob. 17. Find the fourth proportional to 10/0° + 2/90^. 8/0° -3/90°, and G/0° + 5/90°.

Prob. 18. Find the area of the polygon whose successive sides are 10/30°. 9/100^, 8/180°.
7/225°.

Article 4

Coaxial Quaternions,

By a "quateinion"' is meant the operator whicli changes one vector into another.
It is composed of a magnitude and a turning factor. The magnitude may or may
not be a mere ratio, that is. a quantity destitute of physical dimensions: for the
two vectors may or may not be of the same physical kind. The turning is in a
plane, that is to say. it is not conical. For the present all the vectors considered
lie in a common plane; hence all the quaternions considered have a common

Let A and R be two coinitial vectors: the direction normal to the plane may
be denoted by 3. The operator which changes A into R consists of a scalar
multiplier and a turning round the axis 3. Let the former be denoted by r and

the latter by 3 , where denotes the angle in radians. Thus R = r!3 A and

1 d 1 d 1 1 fl

reciprocally A = -d~^R. Also —R = r 3^ and —A = -/3"^.
T A R r

The turning factor 3 may be expressed as the sum of two component op-
erators, one of which has a zero angle and the other an angle of a quadrant.
Thus

3^ = cosO -.J^ + sin^ -3^.

When the angle is naught, the turning- factor may be omitted; but the above
form shows that the equation is homogeneous, and expresses nothing but the

The ideo of the "quaternion" is due to Hamilton Its importance moy be judged from
the fact that it has made solid trigonometrical analysis possible. It is the most important
key to the extension of analysis to space. Ety mo logically '"quaternion" means defined by four
elements; which is true in space; in plane analysis it is defined by t'vs'o.

16

ARTICLE 4. COAXIAL QUATERNIONS. 17

equivalence of a gh^en quaternion to two component quaternions.

Hence

r,3^ = rcos^ + rsin^ ■ ,3^
= p + g . jf

and

rp^A = pA-\-q3^A

= pa ■ a -\- qa • 3^ a..

The relations between r and S. and p and g. are ^jven by

5

Example. — Let E denote a sine alternating electromotive force in magnitude
and phase, and / the alternating current in magnitude and phase, then

E = [t + 27ni^ ■3'^) I,

where r is the resistance. I the self-induction, n the alternations per unit of
time, and 3 denotes the axis of the plane of representation. It follows that
E = rl -\- 2-Knl ■ 3^1; also that

I~^E = T + 27i-nZ-,J^,

that is, the operator which changes the current into the electromotive force
is a quaternion. The resistance is the scalar part of the quaternion, and the

inductance is the vector part.

Components of the Reciprocal of a Quaternion. — Given

R= [p + q-Q^) A,

then

p + g ■ , J ^

p-q- 3\

[p-\-q-3^) [p-q-p'-

-R

9

p2 _^ g2 p2 + g

3^>R

"En the method of complex nmnbers j3 T is expressed by i, which stands for y— 1. The
advantages of using the above notation are tbat it is capable of being apphed to space, and

that it also serves to specify the general turning factor as well as the quadrantal turning
factor fB'^.

ARTICLE 4. COAXIAL QUATERNIONS. 18

Example. — Take the same application as above. It is important to obtain /

in terms of E. By the above we deduce that from E = (j -\- 2nit! ■ Q^)I

Addition of Coaxial Quaternions. — If the ratio of each of several vectors to
a constant vector A is ^ven, the ratio of their resultant to the same constant
vector is obtained by taking the sum of the ratios. Thus^ if

Ri = (pi +91 ■,^^)^^

Rn = {Pn+qn -0')^..

then

and reciprocally

i:^= E^+eo-^^M-

Example. — In the case of a compound circuit composed of a number of simple
circuits in parallel

ri - 27rn/i ■ /3^ 7-2 - 27rn/2 ■ /5t

therefore,

r

- 27rn/

■,5M

r2

+ (27r7i)2i2j
r

= ^^[^^TjLw) ~''''"^r^- + (Lnr-p-^"\^'

and reciprocally

r^+(2iT^)^^V '^^'"■' V^" r' + (27r7i)^?'

This theorem Uh'qs discovered by Lord Rayleigh; PhiloBophLca.1 IXIogazine, May, 1830. See
also Bedell & Crehore s Alternating Currents, p. 233.

ARTICLE 4. COAXIAL QUATERNIONS. 19

Product of Coaxial Quaternions. — If the quaternions whicli change A to i?,
and R to R\ are °;iven, the quaternion which chan°;es A to R' is obtained by
taking the product of the given quaternions.

Given

E = r,3^A = (p-\-q -3^) A
and

R' = r'3^'A= (p +q' -3^) R,
then

R' = rr',3^+^'A = {{pp' - qq') + (pq + p'q) ■ B'^} A,

Note that the product is formed by taking the product of the magnitudes,
and likewise the product of the turning factors. The angles are summed because
they are indices of the common base P.

Quotient of two Coaxial Quaternions. — If the given quaternions are those
which change A to /?. and A to R' , then that which changes R to R' is obtained
by taking the quotient of the latter by the former.

Given

R = r3^A = (p + q ■ 6'^)A

and

then

R' = t',3'^'a= {p +q ■,3^)A,

R'=-f-'R,

T

p- + r

p^ -\-q^

Prob. 19. The impressed alternating electromotive force is 200 volts, the resistance of the
circuit is 10 ohms, the self-induction is j^ henry, and there are 60 alternations
per second; required the current. (Ans. 18.7 amperes / — 20° 42'.)

Pvlany writers, such as Hayward in "Vector Algebra and Trigonometry," and Stringham
in "Uniplanar Algebra," treat this product of coaxial quaternions as if it inuere the product of
vectors This is the fundamental error in the Argand method.

ARTICLE 4. COAXIAL QUATERNIONS. 20

Prob. 20. If in the above circuit the current is 10 amperes, find the impressed voltage.

Prob. 21. If the electromotive force is 110 volts /9 and the current is 10 amperes /d — ■rJ''.
find the resistance and the self-induction, there bein^ 120 alternations per sec-
ond.

Prob. 22. A number of coils having resistances rj . r2- etc.. and self-inductions li, I2- etc..
are placed in series; find the impressed electromotive force in terms of the cur-
rent, and reciprocally.

Article 5

Addition of Vectors in
Space.

A vector in space can be expressed in terms of three independent components,
and when these form a rectangular set the directions of resolution are expressed
by i, J, k. Any variable vector R may be expressed 0.5 E = rp = xi -\- yj + zk^
and any constant vector B may be expressed as

B = b^3 = bji + b2J + b^k.

In space the symbol p for the direction involves two elements. It may be
specified as

xi -\- yj -\- zk

P =

1^ + y^ -V z^

where the three squares are subject to the condition that their sum is unity. Or
it may be specified by this notation, 4>II0^ a generalization of the notation for a

plane. The additional angle 4>/ is introduced to specify the plane in which the
angle from the initial line lies.

If we are given R in the form Tif>j/0 , then we deduce the other form thus:

B = r cosO ■ i -\- T sin cos(p ■ j -\- r sin sin <j> • k.

If R is given in the form z? + yj + zk. we deduce

R= v/:c^ + y2

z- tan — // tan

For example,

5 = 10 30''// 45''

= 10 cos 45'' ■» + 10sin45°cos30'' ■ j + 10 sin 45"^ sin 30° - k.

21

ARTICLE 5. ADDITION OF VECTORS IN SPACE. 22

Ao;ain, from C = 3i + 4j + bk we deduce

C = \/9 + IG + 25 tan"^ - // tan

■^ // _i ^41

4// 3

= 7.07 bl°A//GA°.9.

To find the resultant of any number of component vectors applied at a com-
mon point, let i?i. R2, . ■ ■ Rn represent the n vectors or,

J?l = :rii + Vii + El A:,

R^ = Xj,i + y„j + ^nfc;

then

and

E^={j:^h^[EyUHE^-^

z ")+(!:») +£

tan <b = == — and tan =

V^(E?^)' + (E^)'

Ey E^

Successive Addition. — When the successive vectors do not lie in one plane,
the several elements of the area enclosed will lie in different planes, but these
add by vector addition into a resultant directed area.

Prob. 23. Express A = Ai - rjj -\- Gk and S = Si + Gj - Tk in the form r^y/fG

(Ans. 8.8 130° //G3° and lO.S 311°//Gl" .S.)

Prob. 24. Express C = 123 5TV/142° and D = 456 65V/200^ in the form xi + yj -\- zk.
Prob. 25. Express E = 100 — /'— and F = 1000 - /7 — in the form xi -h !/:J + zk.

Prob. 26. Find the resultant of 10 20V/30^, 20 30^>//40°. and 30 A^" j/bO" .

Prob. 27. Express in the form r <p//8 the resultant vector of li + '2j — 3k. Ai — 5?" + ^k and

-7z-h8? + 9jt.

Article 6

Product of Two Vectors,

Rules of Signs for Vectors in Space. — By the rules ?" = +, j = -\-^ ij = fc,
and ji = —k we obtained (p. 432) a product of two vectors containing two
partial products, each of which has the highest importance in mathematical
and physical analysis. Accordingly, from the symmetry of space we assume that
the following rules are true for the product of two vectors in space:

r = +,

f = +,

fc^ = +,

ij = k,

jk = i,

ki = j ,

ji = —k,

kj = -i,

ik = -j

The square combinations give results which are independent of direction,
and consequently are summed by simple addition. The area vector determined
by * and j can be represented in direction by A:, because k is in tri-dimensional
space the axis which is complementary to i and j. We also observe that the
three rules ij = k, jk = i. ki = j are derived from one another by cyclical
permutation; likewise the three rules ji = —k. kj = — i, ik = —j. The figure
shows that these rules are made to represent the relation of the advance to the
rotation in the right-handed screw. The physical meaning of these rules is made
clearer by an application to the dynamo and the electric motor. In the dynamo
three principal vectors have to be considered: the velocity of the conductor at
any instant, the intensity of magnetic fiux, and the vector of electromotive force.
Frequently all that is demanded is. given two of these directions to determine

23

ARTICLE 6. PRODUCT OF TWO VECTORS. 24

the third. Suppose that the direction of the velocity is i, and that of the flux j,
then the direction of the electromotive force is k. The formula ij = k becomes

velocity flux = electromotive-force^

from which we deduce

flux electromotive- force = velocity,

and

electromotive-force velocity = flux.

The corresponding formula for the electric motor is

current flux = mechanical-force,

from which we derive by cyclical permutation

flux force = current, and force current = flux.

The formula velocity flux = electromotive-force is much handier than any
thumb-and-finger rule: for it compares the three directions directly with the
right-handed screw.

Example. — Suppose that the conductor is normal to the plane of the paper,
that its velocity is towards the bottom, and that the magnetic flux is towards
the left; corresponding to the rotation from the velocity to the flux in the right-
handed screw we have advance into the paper: that then is the direction of the
electromotive force.

Again, suppose that in a motor the direction of the current along the conduc-
tor is up from the paper, and that the magnetic flux is to the left; corresponding
to current flux we have advance towards the bottom of the page, which therefore
must be the direction of the mechanical force which is applied to the conductor.

Complete Product of two Vectors. — Let A = a^i -h a2J + a^k and B =
bii -h 62J + ^3^^ be any two vectors, not necessarily of the same kind physically.
Their product, according to the rules (p. 444), is

AB = (ui? + aoj + a^k)(bii -h 62 j + ^3^):
= a^biii -h a-2b2Jj + a^^b^kk

+ a2b^jk -h asb2kj -h a^biki -h aib-^ik -h aifc^^J + <^2hj^
= a^bi -h a2b2 + a^b^

+ {a2bz)i + (^361 — aib3)j -h (^1^2 — ^2^1)^

= a^bi -f- (22^2 + <t3^3 +

fll

^2

^5

bi

b2

hz

i

3

k

ARTICLE 6. PRODUCT OF TWO VECTORS.

25

Thus the product breaks up into two partial products, namely, fli^i -\-a-2b2-\-

a^b^^ which is independent of direction, and j^i b-j 65 j , which has the diiec-

\ i j k\
tion normal to the plane of A and B. The former is called the scalar product,
and the latter the vector product.

In a sum of vectors, the vectors are necessarily homogeneous, but in a prod-
uct the vectors may be heterogeneous. By making ag = i:^ = 0. we deduce the
resnlts already obtained for a plane.

Scalar Product of two Vectors. — The scalar product is denoted as before by
SAB . Its geometrical meaning is the product of A and the orthogonal projection
of B upon A. Let OP represent A, and OQ represent B. and let OL, LA/, and
MN be the orthogonal projections upon OP of the coordinates bii. feaj. b^k
respectiA'ely. Then ON is the orthogonal projection of OQ. and

OP X ON = OP X [OL + LM + MN),

( ai ^2 '^5

= a\b-i h 62 h 65 —

•-a a a

= (iih\ + fl2^2 + '^3^3 = SAB .

Example. — Let the intensity of a magnetic flux be 5 = 61? + 62J + h^k.,
and let the area be S = sii + ioj + ^z^\ then the flux through the area is

SSB = bisi-\- b-2S2 -\- bss-j.

Corollary 1.— Hence SBA = SAB. For

61 ai -\-b2a2 -\- b^a^i = aibi +02^2 + ^5^^-

The product of B and the orthogonal projection on it of A is equal to the
product of A and the orthogonal projection on it of B. The product is positive
when the vector and the projection have the same direction, and negative when
they have opposite directions.

Corollary 2. — Hence A^ = ai'

03 = a . The square of A must be

positive: for the two factors have the same direction.

Vector Product of two Vectors. — The vector product as before is denoted by
Y AB . It means the product of A and the component of B which is perpendicular

ARTICLE 6. PRODUCT OF TWO VECTORS.

26

to A, and is represented by the area of the parallelogram formed by A and
B. The orthogonal projections of this area upon the planes of jk^ ki^ and ij
represent the respective components of the product. For, let OP and OQ (see
second figure of Art. 3) be the orthogonal projections of A and B on the plane
of * and j: then the triangle OPQ is the projection of half of the parallelogram
formed by A and B. But it is there shown that the area of the triangle OPQ
is 17(0162 ~ <^2^i)- Thns {aib-2 — a-2bi)k denotes the magnitude and direction of
the parallelogram formed by the projections of A and B on the plane of i and
j. Similarly (^2^:3 ~ 0362)1 denotes in magnitude and direction the projection
on the plane of j and k. and (a^^hi — ajb^jj that on the plane of /: and i.

Corollary 1.— Hence V^.4 = -\' AB .

Example. — Given two lines A = 7i — lOj + 3fc and B = — 9« + 4 j — Gk: to
find the rectangular projections of the parallelogram which they define:

\'AB = (60 - 12)i + (-27 + 42)j + (28 -90)^
= 48i + 15j -G2fc.

Corollary 2. — If A is expressed as aa and B as 5,J, then SAB = abcosaS
and V AB = a6 sin a,J ■ aS. where a3 denotes the direction which is normal to
both a and 3. and drawn in the sense giA'en by the right-handed screw.

Example.— Given A = r'^/0 and B = r'WJlO' . Then

SAB = tt'cos'^/8'^/6'

= rr {cosScos^ + sin sin cos{^ — <^)}.

Product of two Sums of non-successive Vectors. — Let A and B be two com-
ponent vectors, giving the resultant A -\- B ^ and let C denote any other vector
having the same point of application.

A-^U

Let

A = aij -\- a2J -\- a^k.
B = hii-\-b2J-\-b3k.,
C = cii -\- C2J + c^k.

ARTICLE 6. PRODUCT OF TWO VECTORS. 27

Since A and B are independent of order,

A-\-B = {ai +bi}i + (a2 +62)^ + {«3 + ^a)^,
consequently by the principle already established

S{A-\-B)C = (ai +5i}ci + (a^ +62)^2 + {^3 + ^3)^3
= aici + a2C2 + a^cs + 6iCi + fc^co + h^s
= SAC -\- SBC.
Similarly

V{A-\-B)C = {(02 + ^2)^3 - {as -^h)c2}i +etc.

= (a2<:'3 — <13C2)« + (&2<^3 — ^3C2)i + " " "

= VylC -I-V5C.

Hence (A-\-B)C = AC +5C.

In the same way it may be shown that if the second factor consists of two
components, C and D. which are n on- successive in their nature, then

{A -\- B){C -\- D) = AC -\-An -\-BC -\-BD.

When A -\- B is a sum of component vectors

{A-\-Bf = A^ -\-B^ -\-AB -\-BA

= A^ -\-B^ + 2S^B.

Prob. 28. The relative velocity of a conductor is S.W.. and the magnetic flux is N.W.;

what is the direction of the electromotive force in the conductor?
Prob. 29. The direction of the current is vertically downward, that of the magnetic flux is

West; find the direction of the mechanical force on the conductor.
Prob. 30. A body to which a force of 2z + 3j -\- 4fr pounds is applied moves with a velocity

of Si -\- Gj -\- 7k feet per second; flnd the rate at which work is done.
Prob. 31. A conductor 8i H- 9^ -|- 10k inches long is subject to an electromotive force of

1H-|- V2j-\- 13k volts per inch; flnd the difference of potential at the ends. (Ans.

326 volts.)
Prob. 32. Find the rectangular projections of the area of the parallelogram defined by the

vectors A = 12i - 23j - 34fr and B = -45z - bQj -\- Q7k.
Prob. 33. Show that the moment of the velocity of a body with respect to a point is equal

to the sum of the moments of its component velocities with respect to the same

point.
Prob. 34. The arm is 9z + Hj + 13fr feet, and the force applied at either end is 17i-\- 19 j +

23^ pounds weight; find the torque.

Prob. 35. A body of 1000 pounds mass has linear velocities of 50 feet per second 30*'//45
and 60 feet per second 60"//22".5; find its kinetic energy.

Prob. 36. Show that if a system of area-vectors can be represented by the faces of a
polyhedron, their resultant vanishes.

Prob. 37. Show that work done by the resultant velocity is equal to the sum of the works
done by its com.ponent5.

Article 7

Product of Three Vectors,

Complete Product. — Let us take A = a^i -\- a2J -\- (13k. B = bii -\- b^j -\-b^k. and
C = cii H- C2J + c-^k. By the product of A. B, aud C is meaut the product of
the product of A and B with C, accordiu^ to the rules p. 444). Hence

ABC = (aibi + Q262 + a3h3){cii + C2J + c^^k)

+ I (1^2 ^5 — 0362 )i + (<J^&1 — ai6^)j + (aifc^ — a2 6i)A;| (cii + C2J + c^fc)
= (ai^i + a-2b2 + a5i3)(ci? + c^ j + C3A:) (l)

1^2 a^i

1^3 fll|

1^2 b,\

1^3 til

<^i

<^2

i

J

\ai ao

<^3
k

\aj ^2 as
+ 1^1 ^2 ^5 1

|Cl C2 C3I

(2)

(3)

Example.— Let A = li -\- 2j -\- 2k, B = 4i -\- .5j + 6A:, and C = 7i -\- 8j + 9A;.
Then

(1) = {4 + 10 + lS){7i-\-Sj-\-9k) = 32(7i +8j + 9fc).

-3 -3

7 8 9
i j k

(2) =
{3) =

= 78i + 6j - 66A:.

1 2 3

4 5
7 8 9

= 0.

28

ARTICLE 7. PRODUCT OF THREE VECTORS. 29

If we write A = aa ^ B = b3, C = C7, then

ABC = ahccosoL'S-'f (l)

+ abc sin cxQ sin 0,^7 ■ Oi!3^ (2}

+ abc sin (:t/3 cosa/37, (3}

where cos 0,^7 denotes the cosine of the an^le between the directions a,J and 7,
and a/^7 denotes the direction which is normal to both ol3 and 7.
We may also write

ABC = SAB -C -\-Y{yAB)C -\-S{yAB}C
(1) (2) (3)

First Partial Product. — It is merely the third vector multiplied by the scalar
product of the other two, or weighted by that product as an ordinary algebraic
quantity. If the directions are kept constant, each of the three partial products
is proportional to each of the three magnitudes.

Second Partial Product. — The second partial product may be expressed as
the difference of two products similar to the first. For

Y{VAB)C = {-{b2C2 +i^C3)ai + (c^a^ + c^asj^iji

+ {-(^5"^^ + hci)a2 + {c^as + ciai)b2}j

+ { — (^Ki + ''2^2)^5 + (^i<ti + C2a2)b:i]k.

By adding to the first of these components the null term {biciai —ciaibiji we
get —SBC -aii -\- SCA -bii. and by treating the other two components similarly
and adding the results we obtain

Y{yAB)C = -SBC ■A-\-SCA B.

The principle here proved is of great use in solving equations [see p. 455) .

Example. — Take the same three vectors as in the preceding example. Then

y{VAB}C = -{28 + 40 + 54)(li -\-2j-\-3k)
+ (T + IG + 27)(4i + 5j + 0A:)
= 78i + 6j - GGfc.

ARTICLE 7. PRODUCT OF THREE VECTORS. 30

The determmant expression for this partial product may also be written in
the form

It follows that the frequently occurring determinant expression

\ai a-2\\ci C2 I |a2 <t3 | |<^2 ^3 | . |i^3 <^1 1 |'^3 ci I
\bi ^2 k^l f?2 ^^2 ^3 "^2 ^3 \ \^3 ^1 <^3 '^l

means S(yAB){yCD).

Third Partial Product. — From the determinant expression for the third prod-
uct, we know that

S{yAB)C = S(yBC)A = S(yCA)B

= -S{VBA)C = -S(yCB}A = -S{yAC)B.

Hence any of the three former may be expressed by SABC , and any of the three
latter by -SABC.

VAB

The third product S{y AB)C is represented by the vohime of the paral-
lelepiped formed by the vectors A^B^C taken in that order. The line V AB
represents in magnitude and direction the area formed by A and B^ and the
product of V AB with the projection of C upon it is the measure of the volume
in magnitude and sign. Hence the volume formed by the three vectors has no
direction in space, but it is positive or negative according to the cyclical order
of the vectors.

In the expression abc sin q,^ cos 0/^7 it is evident that sin q/^ corresponds
to sin^, and cosa/?-/ to cos i^, in the usual formula for the volume of a paral-
lelepiped.

Example. — Let the velocity of a straight wire parallel to itself he V =

1000/30"^ centimeters per second, let the intensity of the magnetic flux be
B = G000/90° lines per square centimeter, and let the straight wire L = 15

centimeters 60°// 45^. Then YVB = 6000000 sin 60° 90^//90'' lines per centime-
ter per second. Hence S{VVB)L = 15 x 6000000 sin 60° cos <i^ lines per second
where cos (^ = sin 45° sin 60°.

Sum of the Partial Vector Products. — By adding the first and second par-
tial products we obtain the total vector product of ABC, which is denoted by
Y{ABC). By decomposing the second product we obtain

y{ABC) = SAB -C -SBC -A -^SCA B.

ARTICLE 7. PRODUCT OF THREE VECTORS. 31

By removing the common multiplier abc, we get

V(a/?7) = cos a 13 ■ 7 — cos/?-/ ■ a + cos 7a ■ 3.

Simiilaily

ViSfG) = cos ,57 ■ a — cos 70: ■ ,5 + cos q3 ■ 7

and

V(7a:/3) = cos 7a ■ P — cos a3 ■ 7 + cos Pj ■ or.
These three vectors have the same magnitude, for the square of each is
cos" a/5 + cos S', + cos 7a — 2 cos a 3 cos P-^ cos 7a,
that is, 1 -[S{a,3'i)}'^.

They have the directions respectively of o' , P\ 7', which are the corners of
the triangle whose sides are bisected by the corners a. 3. 7 of the given triangle.

Prob. 38. Find the second partial product of 920V/30°' 103D°//dt}'', 11 Ab° //■Hb" . Also
the third partial product.

Prob. 39. Find the cosine of the angle between the plane of lii + mjj -\- mk and i2i +
"^2J + ^2^ and the plane of i^i + sn^j + n^k and [41 -\- jn4J + itik.

Prob. 40. Find the volume of the parallelepiped determined by the vectors lD0i-|-50j + 25t .
50z+ 10j-\- 80fr, and -7r^z -\- 40? - SDk.

Prob. 41. Find the volume of the tetrahedron determined by the extremities of the follow-
ing vectors: 3z - 2j -\- Ik. -Ai -h Sj - 7k. Si — Tj- •2k. Si -\- Aj - 3k.

Prob. 42. Find the voltage at the terminals of a conductor when its velocity is 1500 cen-
timeters per second, the intensity of the magnetic flux is 7000 lines per square
centimeter, and the length of the conductor is 20 centimeters, the angle between
the first and second being 30°. and that between the plane of the first two and
the direction of the third 60°. (Ans. .91 volts.)

Prob. 43. Let a = 20V/10''i 3 = 30° //25° . 7 = 40V/3S°. Find Vq57; and deduce \'i3ja
and W'yaiS.

Article 8

Composition of Quantities,

A number of homogeneous quantities are simultaneously located at different
points; it is required to find how to add or com.pound them.

Addition of a Located Scalar Quantity. — Let m^ deuote a mass m situated
at the extremity of the radius-vector A. A mass jn — jn may be introduced at

the extremity of any radius-vector R, so that

mA = (m - m)R + m^
= mii -\- mA -m^

= TJiR + m(A — R).

Here A — i? is a sim.ultaneous sum, aud denotes the radius-vector from, the
extremity of R to the extremity of A. The product m(A — R) is what Clerk
Maxwell called a mass- vector, and means the directed momeut of ni w^ith respect
to the extremity of R. The equation states that the m.a3s m at the extremity
of the vector A is equivalent to the equal mass at the extremity of i? , together
with the said mass-vector applied at the extremity of R. The equation expresses
a physical of mechanical principle.

Hence for any number of masses, nii at the extremity of Ai . mo ^t the
extremity of A^, etc.,

/ ^ mA = 2_^ ^^ + Z-*i ^^{^ ~ ^

32

ARTICLE 8. COMPOSITION OF QUANTITIES. 33

where the latter term denotes the sum of the mass-vectors treated as simulta-
neous vectors applied at a common point. Since

the resultant moment will vanisli if

J? = =^ . or R} m=} mA

Corollary. — Let

K = x\ -\- yj -\- zh.
and

^ = ^ij + 6ij + cifc;

then the above condition may be written as

X^{77i(ai + fe j + cfc)}
Ti -h yj + eA: = =

^TTJ Yl.'^ Yn\ '

therefore

^ = ^v^ — . y = ^v^ — . 2 =

E?^ ' E?^ ' !]■

Example— Given 5 pounds at 10 feet 45°//30'' and 8 pounds at 7 feet

60''//45°: find the moment when both masses are transferred to 12 feet ZS^Z/GO".

mivli = 50(cos30°*-hsin30°cos45''j-hsin30°3in45''fc),

mi^i = 56(cos45°i-hsin45°cos60''j-|-sin45°sin60''fc).
{mi -hma)/? = 156(cos GO'^i -h sin G0° cos 75"^ -h sin G0° sin 75°A;),
moment = nii^i -h 77i2'42 ~ ("^i + "12)^-

ARTICLE 8. COMPOSITION OF QUANTITIES. 34

Composition of a Located Vector Quantity. — Let F^ denote a force applied
at the extremity of the raxiius-vector A. As a force F — F may introduced at

the extremity of any radius-vector R, we have

Fa = {F-F)+Fa
= Fr +V(A-J?)F.

This equation asserts that a force F apphed at the extremity of A is equiva-
lent to an equal force applied at the extremity of R together with a couple whose
magnitude and direction are given by the vector product of the radius-vector
from the extremity of R to the extremity of A and the force.

Hence for a system of forces applied at different points, such as Fi at Ai,

F2 at Ao . etc., we obtain

H (Fa) = Yl (-Pfi) + H V (^ - i?) F

Since

^\'{A-R)F = J2 V^4i^ - J2 V^^

the condition for no resultant couple is

which requires ^ i^ to be normal to ^VAF.

Example. — Given a force It -\-2j -\-^k pounds weight at 4i -h 5j -\-Gk feet, and
a force of Ti -h 9j -h llA; pounds weight at lOi -h 12 j -h 14fc feet: find the torque
which must be supplied when both are transferred to 2i -I- 5j -I- 3A:, so that the

ARTICLE 8. COMPOSITION OF QUANTITIES.

35

effect may be the same as before.

VAiFi = 3i -Gj + 3A:,
YA2F2 = Gi -12j +6A:,

^ VAF = 9i -ISj -\-9k,

Y^F = Si-\-Uj -\-14k,
Vi?^ F = 3Ti -4j -ISk,

Torque = -2Si - 14j + 27k.

By takino; the vector product of the above equal vectors with the reciprocal
of ^ F we obtain

v{(viJEf)^}=v{(i:v-4f)^}.

By the principle previously established the left member resolves into —R +
Si?= — ' ^ F: and the right member is equivalent to the complete product on

account of the two factors being normal to one another: hence

that is.

-^ = ^E(^'-4n

+ SE

1

Y.f

■E^-

(1)

(2)

The extremity of R lies on a straight line whose perpendicular is the vector
(1) and whose direction is that of the resultant force. The term (2) means the
projection of R upon that line.

ARTICLE 8. COMPOSITION OF QUANTITIES. 36

The condition for the central axis is that the resultant force and the resultant
couple should have the same direction: hence it is given by

that is

V (Vfl ^ f) ^ F = V {J2 AF) J2 F.

By expanding the left member according to the same principle as above, we
obtain

- (H ^) ' ^ + s^ I] ^ ■ E ^ = ^'" (Z -4 J^) I] ^ ;

therefore

This is the same straight line as before, only no relation is now imposed on
the directions of ^ F and J^ YAF; hence there always is a central axis.

Example. — Find the central axis for the system of forces in the previous
example. Since X^ F = 8i + llj + 14fc, the direction of the line is

Si + lli + 14A;
V64 + 121 + 19G'

1 8i + 11J + 14A; ^

Since == — = and > \ AF = 9i — lS }-\-9k, the perpendicular

Y.F 381 ^ J ^ y y

to the line is

Si + Uj -\-lAk 1

V 9i -ISj -\-9k= {351i + 54j - 243A:}.

3S1 381

Prob. 44. Find the moment at 90"//2r0^ of 10 pounds at 4 feet 10''//20° and 20 pounds

at 5 feet 30V/1^0''.

Prob. 45. Find the torque for 4z + 3:j"-h2fr pounds weight at ■2i-3j-\-lk feet, and2z-i;:-U
pounds weight at —3i -\- 4?" + 5k feet when transferred to — 3z — 2j — Ak feet.

Prob. 46. Find the central axis in the above case.

Prob. 47. Prove that the mass- vector drawn from any origin to a mass equal to that of the
whole system placed at the center of mass of the system is equal to the sum of
the mass-vectors drawn from the same origin to all the particles of the system.

Article 9

Spherical Trigonometry.

Let i. J, k denote three mutually perpendicular axes. In order to distinguish
clearly between an axis and a quadrantal version round it, let *^ , J^ , k'^ denote
quadrantal versions in the positive sense about the axes i, j, k respectively. The
directions of positive version are indicated by the arrows.

By i'^i'^ is meant the product of two quadrantal versions round i; it is
equiA'alent to a semicircular version round i; hence *TiT = j" = — . Similarly
j'^JT means the product of two quadrantal versions round j. and j'^j'^ = j'^ =
— . Similarly k'^ k'^ = k^ = —.

By i'^jT is meant a quadrant round i followed by a quadrant round j; it is
equiA'alent to the quadrant from j to i, that is, to —k'^. But j^ i^ is equivalent
to the quadrant from —i to — j, that is^ to k^ . Similarly for the other two pairs
of products. Hence we obtain the following

Rules for Versors.

i^j^ = -k^ , j^i^ = k^ ,

j^k^ = — ?^, k^ j^ = i^
k'^i'^ = —j ^ , i T fc T = J T ,

37

ARTICLE 9. SPHERICAL TRIGONOMETRY.

38

The meaning of these rules will be seen from the following application. Let
li + tnj + nk denote any axis, then (li + nij + nk)'^ denotes a quadrant of
an^le round that axis. This quadrantal version can be decomposed into the
three rectangular components li^ . mj'^ .. nk'^: and these components are not
successive versions, but the parts of one version. Similarly any other quadrantal
version {I'i + ni'j -h n'j) ^ can be resolved into I i^ y m j'^ ^ n'k'^ . By applying
the above rules, we obtain

{li -\-mj + nk)^{l'i + m'j -\-n'k)^

= {li^ + mj^ + nk^){l'i^ + m'j^ + n'k^)

= —(II + mnt + nn ) — {■mn — tn n)i^ — {nl — n l)j'^ — [lin — I m.)k^

= —{II' + mm + nn') — | [mn — nf.'n)i + {nV — n'l)j + {hn — l'm)k^ ^ .

Product of Two Spherical Versors. — Let Q denote the axis and h the ratio
of the spherical versor PA^ then the versor itself is expressed by ,J , Similarly
let 7 denote the aa:is and c the ratio of the spherical versor AQ , then the versor
itself is expressed by 7*^.

Now

3^ = cos6 + sin6 ■ p^ ,
and

7 = cos c + sin c ■ 7^ ;

therefore

Q 7^^= (cosfeH- sin i ■ /3 ?")(cos c + sin c ■ 7^)

= cos6cosc + cos6 sin c-73" + cosc sin h -3^ + sin h sin c ■ ,J'^7'^ .

But from the preceding paragraph

^T-^Y = — cos ,^7 — sin ,^7 ■ ,i37 ^ :

ARTICLE 9. SPHERICAL TRIGONOMETRY. 39

therefore

/^ 7^ = cos 6 cos c — sin b sin c cos/?-/ (l)

+ {cos 6 sin c ■ 7 + cose sin b ■ 13 — sin b sin c sin Pf ■ ,^7}^ . (2)

The first term gives the cosine of the product versor; it is equivalent to the

fundamental theorem of spherical trigonometry, namely.

cosa= cos6cosc + sin b sin c cos A^

where A denotes the external angle instead of the angle included by the sides.
The second term is the directed sine of the angle; for the square of (2) is
equal to 1 minus the square of (1). and its direction is normal to the plane of
the product angle.

Example.— Let 3 = 30^//45° and 7 = OO^Z/SO". Then
cos i3', = cos 45° cos 30^ + sin 45^ sin 30"^ cos 30°,
and

sin,/37 ■'3j = ¥,^7:
but

f3 = cos45°i + sin45°cos30''j + sin 45° sin 30°A:,
and

', = cos30°i + sin30°cos00^j + sin 30° sin 60°A::
therefore

V/37 = {sin 45° cos 30° sin 30° sin OO"" - sin 45° sin 30° sin 30° cosGO""}!

+ {sin 45'' sin 30^ cos 30° - cos 45'' sin 30° sin 60°}j
+ {cos45°sin30°cos60° - sin 45'' cos 30° cos 30° }jt-

Quotient of Two Spherical Versors. — The reciprocal of a given versor is
derived by changing the sign of the index: 7"*^ is the reciprocal of 7*^. As
3 = cos 6 + sin b ■ /5^, and y~*^ = cose — sin c ■ 7"^ ,

/? 7 = cos b cos c + sin b sin c cos ,^7

+ {cos c sin h • 3 — cos b sin c ■ -/ + sin h sin c sin Q-^ ■ ,^7}^ .

Principles of Elliptic and Hyperbolic AnolysiSj p. 2.

ARTICLE 9. SPHERICAL TRIGONOMETRY. 40

Product of Three Spherical Versors. — Let a** denote the versor PQ, 3 the

versor QR, and 7^ the versor RS\ then a°,jS^ denotes PS. Now a^/^S^

= (cos a + sin a ■ ct ?")(cos5 + sin h ■ ,J'^)(cosc + sin c ■ 7"^)

= cos a cos 6 cose (l)

+ cos a cos 5 sin c - 7"^ + cos a cose sin b • i^^ +cos6cosc sin a • ol^ (2)

+ cos a sin h sin c ■ ,^"^7"^ + cos6 sin a sin c ■ a "^7"^

+ cose sin a sin b ■ a'^ 3'^ (3}

+ sin a sin 6 sin c ■ ct "^ ,J "^ 7 "^ (4}

The versors in (3) are expanded by the rule already obtained, namely,

P'2-fY = — cos ,57 — sin ,i?7 ■ ,^7 ^ .
The versor of the fourth term is

a^3^f^ = —{cos 0.3 + sinet/3 ■ a,'? ' )7^

= — cos a,J ■ "/ ' + sin a:/3 cos 0,^7 + sin a 3 sin a,J7 ■ 0,^7 .

Now sin a3 sin a:/37 ■ a,57 = cos 07 -5 — cos 3 J ■ o: (p. 451) , hence the last
term of the product, when expanded, is

sin a sin b sin c | — cosa,5 ■ 7"^ + cos 07 ■3'^ — cos ,^7 ■ q^ + cos Ci3'-i\ -

ARTICLE 9. SPHERICAL TRIGONOMETRY. 41

Hence

cos g'^3 "f' = cos a cos 5 cose — cos a sin h sin c cos /^7

— cos 5 sin a sin c cos 0:7 — cose sin a sin h cosa/3
+ sin a sin h sin c sin olQ cos gPj.

and, letting Sin denote the directed sine,

Sin g'^3 -f^ = cos a cos 5 sin c ■ 7 + cosacosc sin b ■ 3

+ cos h cos c sin a • c^ — cos a sin 6 sin c sin ,^7 ■ 3'f

— cos 5 sin (X sin c sin a 7 ■ 07

— cos c sin a sin & sin a/3 - a,J

— sin (X sin h sin c {cos a/? ■ 7 — coso:7 ■/? + cos 3-^ ■ a} .

Extension of the Exponential Theorem to Spherical Trigonometry. — It has
been shown (p. 458) that

cos ,J 7 = cos 5 cose — sin h sin c cos/37
and

(sin /3 7^j ^ = cose sin b • 3^ -\- cos b sin c ■ 7"^ — sin b sin e sin ,^7 ■ 3") ^ .

Now

6^ b^ b'

cos5 = l -- + ---+ etc.

and

Sin b = b 1 etc .

3! 5!

Substitute these series for cos fe, sin fc, cos c, and sin c in the above equations,
multiply out, and group the homogeneous terms together. It will be found that

cos,/3^7'^ = 1 -{b-^ +26ccos/?7 + c^}

+ —{5'* +46^ccos/57 + G6^e^ -\- 4bc^ cos i3j -\- c^}

^{5^ +06^ccos/37 + 156^e^ -\-20b^c^cosi3',

+ l^b'^c^ -\- GbrJ cos ,3j -\- rj \ -\- ...,

In the above coae the three Qxes of the successive angles are not perfectly independent, for
the third ongle must begin where the second leaves off. But the theorem remains true when
the axes aie independent; the factors are then quaternions in the most general sense

ARTICLE 9. SPHERICAL TRIGONOMETRY. 42

where the coefficients are those of the binomial theorem, the only difference
being that cos/?"/ occurs in all the odd terms as a factor. Similarly, by expanding
the terms of the sine, we obtain

{Sin i3^Y)^ = 6-/5^ + c-7^ -6c sin,/?-/ -3^/'

{b~^ ■ ,3^ + U^c ■ 7^ + 35c^ ■ ,J^ + c^ ■ 7^}

•J .

+ —{bc^ -\-b^c}sin3'f -Pj^
•J .

-\ {b'^ ■ i3^ + rjb'^c ■ 7^ + lOb^c'^ ■ ,3^

5!

+ 105^c^ -'i^ + 56c^ -3^ +c^ -7^

- — \b^c-\- '-^b^c^ +6c^l sin57 -^^ - ...
5! L 2-3 J ■ ' '

By adding these two expansions together we get the expansion for ,5 7*^,
namely,

- —{6^ + 26c(cos/57 + sin,/37 -P^') + c^}

_ —[b^ . /?T + U^c ■ 7^ + 3ic- ■ ,J^ + c^ ■ 7^}
•J .

+ —{6^ + 46^c(cos,/37 + sin,/37 ■'3j^) + Gfe^c^
+ 45c^(cos,57 + sin,57 -^^j +c^} + .. .

By restoring the minus, we find that the terms on the second line can be
thrown into the form

— {b- ■ ,3'' -\-2bc -3^-/^ +c- -7^} ,
and this is equal to

where we have the square of a sum of successive terms. In a similar manner the
terms on the third line can be restored to

b^ -3'-^ +36^c-,5'^7^ + 36c^ -,5^7^ + c^ "7^'^',
that is^

ARTICLE 9. SPHERICAL TRIGONOMETRY. 43

Hence

3^^^' = I -\- b ■ ,3^ -\- c ■ j^ -\- — {b ■ 3^ -\- c -7^}^

■^ —{b ■ 3^ -\-c--f^} -^ —{b ■ 3^ -\-c-j^} +...
•J . ~t .

Extension of the Binomial Theorem. — We have proved above that e ^ e^'' ^ =
^bfS 2 +C-1 2 pI■Qvi^j^^j that the powers of the binomial are expanded as due to a
successive sum , that is. the order of the terms in the binomial must be preserved.
Hence the expansion for a power of a successive binomial is given by

{t-^3^ -\-c- 7^}" = 6" ■ /?"^ + nb^-'c ■ ,5("-^)(T)-^f

1-2 ■ '

Example.— Let b = ^ and c = \, i3 = 30^/45", 7 = eO^//30''.

(t .^T + c ■ 7^)^ = -{b^ +c^ -\- 2bc cos 3j -\- 2bc(sin3';)^}

Substitute the calculated values of cos ,^7 and sin ,^7 (page 459) .

\/3 1 1
Prob. 48. Find the equivalent of a quadrantal version round i-\- 7=i -\- ^t followed

2 2V2 2V2

1 ^3 3

by a quadrantal version round —i -\- j -\- —k.

Prob. 49. In the example on p. 459 let b = 25 and c = 50 : calculate out the cosine and
the directed sine of the product angle.

Prob. 50. In the above example calculate the cosine and the directed sine up to and in-
clusive of the fourth power of the binomiaL (Ans. cos = .9735.)

Prob. 51. Calculate the first four terms of the series when b = -^, c = TTfni /^ = 0°//0 .

Prob. 52. From the fundamental theorem of spherical trigonometry deduce the polar the-
orem with respect to both the cosine and the directed sine.
Prob. 53. Prove that if a ,,3 ,7 denote the three versors of a spherical triangle, then

sin ,57 sin 70 sin a3

sin a sin 6 sine

diplnnar quaternions we cannot expect to find that the sum of the logarithms of any two
proposed factors shall be generally equal to the logarithm of the product; but for the simpler
and earlier cose of coplanar quaternionSj thot algebraic property may be considered to exist,
with due modification for multiplicity of value." He was led to this vieiv by not distinguishing
bet'vv'een vectors and quadrnntQl quaternions and between simultaneous and successive addi-
tion. The above demonstration was first given in my paper on "The Fundamental Theorems of
Analysis generalized for Space." It forms the key to the higher development of space analysis.

Article 10

Composition of Rotations,

A version refers to the change of direction of a. hne, but a. rotation refers
to a rigid body. The composition of rotations is a different matter from the
composition of Aversions.

Effect of a Finite Rotation on a Line. — Suppose that a rigid body rotates
radians round the axis 3 passing through the point O. and that R is the radius
vector from O to som.e particle. In the diagram OB represents the axis P, and
OP the vector R. Draw OK and OL, the rectangular components of R.

i3^R = (cos^ + sin£^ • '?^)rp

= r(cos^sin^ ■ /5^)(cos,J/j - t3 + sin,/?/? ■ 3p'3)

= r{cos Qp ■ Q + cos^ sin 3p ■ 3p!3 + sin sin 3p ■ 3p}.

When cos 3p = 0, this reduces to

P^R = cosOR-\- sin eVipR).

The general result may be written

,3^R = S3R ■ 3-\-cose{V3R),3-\-sinOV3R.

44

ARTICLE 10. COMPOSITION OF ROTATIONS.

45

Note that {V3R),3 is equal to Y(VpR)3 because S3R,3 is 0, for it involves
two coincident directions.

Example. — Let 3 = li-\-trtj -\-nk^ where l' -\-tn- -\-n = 1 and R = xi-^yj-\-zk]
then S3R = Ix -\- fny + nz

Y(J3R).3 =

mz—iiy nx—lz ly — mx

I 771 n

i j k

and

V,Ji? =\x y z\ .

\i j k\

Hence

/3 = {It + tny + nz){li + mj + nk)

\7jiz — ny nx —Iz ly — mx

+ cos t

I

\l tn n
-\- sin \z y z

\i j k

m
j

n

k

To prove that /?/^ coincides with the axis of /? ? p^ 3-^ . Take the more general
versoi p . Let OP represent the axis /3, AB the versor 3^>~ ^ BC the versor p .
Then {AB)(BC) = AC = DA. therefore (AB){BC)(AE) = (D A){AE) = DE.
Now DE has the same angle as BC, but its axis has been rotated round P by
the angle b. Hence if ^ = y, the axis of Q^~ p^ Q2 will coincide with 3 p-

The exponential expression for P^~ p^ Q^ is e"?''^ ""'■?"'' ^ ~'~t''p ^ which may
be expanded according to the exponential theorem, the successive powers of the

This theorem inuaB discovered by Cayley. It indicates that quaternion multiplication in the
most general sense has its physical meaning in the composition of rotations.

ARTICLE 10. COMPOSITION OF ROTATIONS. 46

trinomial being formed according to the multinomial theorem, the order of the
factors being preserved.

Composition of Finite Rotations round Axes which Intersect. — Let /3 and 7
denote the two axes in space round which the successive rotations take place,
and let ,J denote the first and 7'^ the second. Let /? x 7^ denote the single
rotation which is equivalent to the two given rotations applied in succession:
the sign X is introduced to distinguish from, the product of versors. It has been
shown in the preceding paragraph that

and as the result is a line, the same principle applies to the subsequent rotation.
Hence

because the factors in a product of versors can be associated in any manner.
Hence, reasoning backwards,

b c

Let fu denote the cosine of ,^^7^. namely.

be be

cos — cos sin — sin — ,

2 2 2 2^

and n ■ zy their directed sine, namely,

then

be c b be — —

cos — sin — ■ 7 + cos — sin — ■ J — sin — sin — sin P7 ■ J7;

2 2^ 22' 22 ' ' ''

r,h € 2 2 , ^

p X7 =771 — n + luin ■ I/.

ARTICLE 10. COMPOSITION OF ROTATIONS. 47

Observation. — The expression (,j3^7^) is not, as mi^ht be supposed, identi-
cal with j3 7^. The former reduces to the latter only when 3 and 7 are the same
or opposite. In the figure ,5 is represented by PQ. j'^ by QR^ (3 7^ by PR,
■5? 7^ by ST . and [3'^^'^) by SU . which is twice ST. The cosine of SU differs
from the cosine of PR by the term —(sin ^ sin ^ sin P'f) It is evident from the
figure that their axes are also different.

Corollary. — When b and c are infinitesimals, cos 3 xy^ = 1 , and Sin 3 xj'^ =
6 ■ ,J + c ■ 7, which is the parallelogram rule for the composition of infinitesimal

rotations.

Prob. 54. Let 3 = 30^^// 45°, ^ = f . and R = 2i - 3j -h 4fc; calculate 3^ R .

Prob. 55. Let i3 = 90° //90° ._ G = ^. R = -i -\- 2j - 3k- calculate i3^R.

Prob. 56. Prove by multiplying out that 3 ^ p-^ f3^ = {jS p} ^ .

Prob. 57. Prove by means of the exponential theorem that ■^~'^,3 "f'^ has an angle h. and

that its axis is 7"'^ ,5.

Prob. 58. Prove that the cosine of (,5?" ■7?' )" differs from the cosine of /3 7 by
— (sin ij-sin ^ sin|37)".

Prob. 59. Compare the axes of (,5^7^)^ and /?^7^.

Prob. 60. Find the value oi ^^ X Y~ when = OV/^O" and 7 = 9OV/90".

Prob. 61. Find the single rotation equivalent toz'^" X j'^ X k'^ .

Prob. 62. Prove that successive rotations about radii to two corners of a spherical triangle
and through angles double of those of the triangle are equivalent to a single
rotation about the radius to the third corner, and through an angle double of
the external angle of the triangle.

Index

Algebra

of space, 1

of the plane, 1
Algebraic imaginary, 17
Argand method, 19
Association of three vectors, 13

Bibliography, iv, 1

Binomial theorem in spherical analy-
sis, 41

Cartesian analysis, 1

Cayley. 45
Central axis, 35
Coaxial Quaternions, 16

Addition of. 18

Product of, 19

Quotient of, 19
Complete product

of three vectors. 28

of two vectors, 9, 24
Components

of quaternion, IT

of reciprocal of quaternion, IT

of versor, 16
Composition

of any number of simultaneous com-
ponents, G

of coaxial quaternions, 18

of finite rotations, 44

of located vectors, 34

of mass- vectors. 32

of simultaneous vectors in space,
21

of successive components, 7

of two simultaneous components,
4

Coplanar vectors, 9
Couple of forces, 33

condition for couple vanishing, 33

Cyclical and natural order. 15

Determinant

for scalar product of three vectors,

30
for second partial product of three

vectors, 28. 30
for vector product of two vectors,
25
Distributive rule. 27
Dynamo rule, 24

Electric motor rule, 24

Exponential theorem in spherical trigonom-
etry, 41

Hamilton's view, 43

Hamilton's

analysis of vector, 3

idea of quaternion, 16

view of exponential theorem in spher-
ical analysis, 43
Hayward, 19
Hospitaller system. 3

Imaginary algebraic. IT

Kennelly's notation. 3

Located vectors, 34

Mass-vector, 32

composition of, 32
Maxwell, 32
Meaning

48

INDEX

49

of I, 3

of ^TT as index. 37

of 7, 21

of dot, 3

of S. 10

of V. 11

of vinculum over two axes, 11

Natural order, 15
Notation for vector, 3

Opposite vector, 12

Parallelogram of simultaneous compo-
nents, 4

Partial products, 9, 25
of three vectors. 29
resolution of second partial prod-
uct, 29

Polygon of simultaneous components,
G

Product

complete, 9. 24
of coaxial quaternions, 19
of three coplanar vectors, 13
of three spherical versors, 40
of two coplanar vectors, 9
of two quadrantal versors, 38
of two spherical versors. 38
of two sums of simultaneous vec-
tors, 20
of two A'ectors in space, 25
partial. 9, 25
scalar, 10, 25

Quadrantal versor, 12
Quaternion

definition of, IG

etymology of. 16

reciprocal of, 17
Quaternions

Coaxial. 16

definion of, 1

relation to vector analysis, 1
Quotient of two coaxial quaternions,
19

Rayleigh, 18
Reciprocal

of a quaternion, 17

of a vector, 12
Relation of right-handed screw, 24
Resolution

of a vector, 5

of second partial product of three
vectors, 29

Rotations, finite. 44
Rules

for dynamo, 24

for expansion of product of two
quadrantal versors, 38

for vectors, 9, 23

for versors, 37

Scalar product. 10

geometrical meaning, 10

of two coplanar vectors, 9
Screw, relation of right-handed, 24
Simultaneous components, 3

composition of, 4

parallelogram of, 4

polygon of, 6

product of two sums of, 26

resolution of. 5
Space-analysis, 1

advantage over Cartesian analysis,
1

foundation of, T
Spherical trigonometry, 37

binomial theorem, 41

fundamental theorem of, 39
Spherical versor, 38

product of three^ 40

product of two, 38

quotient of two, 39
Square

of a vector, 10

of three successive components, 14

of two simultaneous components,
14

of two successive components, 14
Stringham. 19
Successive components, 4

INDEX 50

composition of, 7

Tait's analysis of vector, 3
Torque, 35

Total vector product of three vectors,
31

Unit-vector, 3

Vector

co-planar. 9

definition of, 3

dimensions of, 3

in space. 21

notation for, 3

opposite of, 12

reciprocal of, 12

simultaneous, 3

successive, 4
Vector analysis

definition of, 1

relation to Quaternions, 1
Vector product, 11

of three vectors. 31

of two vectors, 11
Versor

components of. 16. 38

product of three general spherical,
40

product of two general spherical,
38

product of two quadrantal, 38

rules for. 37

ADVERTISEMENT 51

SHORT-TITLE CATALOGUE

OF THE

PUBLICATIONS

OF

JOHN WILEY & SONS, Inc.

NEW YORK

London: CHAPMAN & HALL, Limited

Montreal, Can.: RENOUF PUB. CO.

ARRANGED UNDER SUBJECTS

De3cripti\"e ciicuIqih sent on application. Books marked with an asterisk (*) are sold at net
prices only. All books are bound in cloth unless otherwise stated.

AGRICULTURE — HORTICULTURE — FORESTRY.

ArMSBY — Principles of Animal Nutrition 3vo,

BOWNFAN — Forest Physiography 3vOj

Bpyawt — Hand Book of Logging [Ready. Fall 1913)

BUDD and HaWSEW — American Horticultural Manuak

Part I. Propagation, Culture, and Improvement 12mOj

Part IL Systematic Pomology 12mo,

Elliott — Engineering for Land Drainage 12nio,

Practical Farm Drainage. (Second Edition^ Rewritten) 12mo,

Fuller. — Domestic Water Supplies for the Farm 3vo,

G EtAH AM — Text-book on Poultry (/ti Preparation.)

Manual on Poultry (Loose Leaf Lab. Manual) {In Preparation.)

Graves — Forest Mensuration 3vo,

Principles of Handling Woodlands Small 3vo,

Green — Principles of American Forestry 12mOj

GrOTEWFELT — Principles of Modern Dairy Practice. (WOLL.) 12mOj

HaWLEY and HaWES — Forestry in New England 3vo,

HeR.R]CK — Denatured or Industrial Alcohol 3vo,

Howe — Agricultural Drafting oblong quarto^

Reference and Problem Sheets to accompany Agricultural Drafting each

KeITT — Agricultural Chemistry Test-book {In Preparation.)

Laboratory and Field Exercises in Agricultural Chemistry . {In Preparation.)
Kemp and WaUGH — Landscape Gardening. (New Edition, Rewritten) . . . 12mOj
L.ARSEW — Exercises in Dairying (Loose Leaf Field Manual) 4tO| paper,

Single Exercises each

and White — Dairy Technology Small 3vo,

LeVESON — Studies of Trees, Loose Leaf Field Manual, 4to, pamphlet form.

Price from 5—10 cents net, each, according to nuinber of pages.
McC.ALL — Crops and Soils (Loose Leaf Field Manual) {In Preparation.)

Soils (Text-book) {In Preparation .)

McKay and LaRSEX — Principles and Practice of Butter-making 3vOj '1 50

M.AYWARD — Landscape Gardening as Applied to Home Decoration 12mOj 1 50

Moon and Brown — Elements of Forestry {In Preparation.)

Record — Identification of the Economic Woods of the United States 3vOj '1 25

Recknagel — Theory and Practice of Working Plans (Forest Organization) 3vo, '2 00

S4 00

'5 00

1 50

1 50

2 00

1 50

'1 50

4 00

'1 50

1 50

2 00

•3 50

•4 00

•1 25

'0 20

•1 50

•1 00

•0 02

•1 50

Article 11

PROJECT GUTENBERG
"SMALL PRINT"

End of the Project Gutenberg EBook of Vector Analysis and Quaternions ,
by Alexander Maofarlane

*** END OF THIS PROJECT GUTENBERG EBOOK VECTOR ANALYSIS AND QUATERNIONS **"

««**^ This file should be named 13609-pdf .pdf or 13609-pdf .zip *•*<■**■■*
This and all associated files of various formats will be found in:
http;//Huw.gutenberg.net/l/3/6/0/13609/

Produced by David Starner, Joshua Hutchinson, John Hagerson, and the
Project Gutenberg On-line Distributed Proofreaders.

Updated editions will replace the previous one --the old editions
will be renamed.

Creating the works from public domain print editions means that no
one owns a United States copyright in these works , so the Foundation
(and you' ) can copy and distribute it in the United States without
permission and without paying copyright royalties . Special rules,
set forth in the General Terms of Use part of this license . apply to
copying and distributing Project Gutenberg-tm electronic works to
protect the PROJECT GUTENBERG-tm concept and trademark. Project
Gutenberg is a registered trademark, and may not be used if you
charge for the eBooks, unless you receive specific permission. If
you do not charge anything for copies of this eBook, complying with
the rules is very easy r You may use this eBook for nearly any
purpose such as creation of derivative works , reports, performances
and research. They may be modified and printed and given away--you

ARTICLE 11. PROJECT GUTENBERG SMALL PRINT'''

may do practically ANYTHING with public domain eBooks.
Redistribution is subject to the trademark license, especially
commercial redistribution.

*** START: FULL LICENSE «*•

THE FULL PROJECT GUTENBERG LICENSE PLEASE READ THIS BEFORE YOU
DISTRIBUTE OR USE THIS WORK

To protect the Project Gutenberg-tm mission of promoting the free
distribution of electronic works , by using or distributing this work
(or any other work associated in any way with the phrase "Project

Gutenberg") , you agree to comply with all the terms of the Full
Project Gutenberg-tm License (available with this file or online at
http: //gutenberg. net /license) .

Section 1 . General Terms of Use and Redistributing Project
Gutenberg-tm electronic works

1 . A . By reading or using any part of this Project Gutenberg-tm
electronic work, you indicate that you have read, understand, agree
to and accept all the terms of this license and intellectual
property (trademark/ copyright) agreement . If you do not agree to
abide by all the terms of this agreement , you must cease using and
return or destroy all copies of Project Gutenberg-tm electronic
works in your possession. If you paid a fee for obtaining a copy of
or access to a Project Gutenberg-tm electronic work and you do not
agree to be bound by the terms of this agreement, you may obtain a
refund from the person or entity to whom you paid the fee as set
forth in paragraph 1 . E , 8.

1 . B . "Project Gutenberg*' is a registered trademark. It may only be
used on or associated in any way with an electronic work by people
who agree to be bound by the terms of this agreement . There are a
few things that you can do with most Project Gutenberg-tm electronic
works even without complying with the full terms of this agreement .
See paragraph 1 . C below. There are a lot of things you can do with
Project Gutenberg-tm electronic works if you follow the terms of
this agreement and help preserve free future access to Project
Gutenberg-tm electronic works . See paragraph 1 . E below.

I.e. The Project Gutenberg Literary Archive Foundation ("the
Foundation" or PGLAF) , owns a compilation copyright in the
collection of Project Gutenberg-tm electronic works . Nearly all the
individual works in the collection are in the public domain in the
United States. If an individual work is in the public domain in the
United States and you are located in the United States , we do not

ARTICLE 11. PROJECT GUTENBERG SMALL PRINT''

claim a right to prevent you from copying . distributing, perf oriDiiLg,

displaying or creating derivative works based oil the work as long as

all references to Project Gutenberg are removed. Of course, ve hope

that you will support the Project Gut enberg-tm mission of promoting

free access to electronic works by freely sharing Project

Gut euberg-tm works in compliance with the terms of this agreement

for keeping the Project Gut enberg-tm najoe associated with the work.

You can easily comply with the terms of this agreement by keeping

this work in the same format with its attached full Project

Gut enberg-tm License when you share it without charge with others.

1 .D . The copyright laws of the place where you are located also
govern what you can do with this work. Copyright laws in most
countries are in a constant state of change. If you are outside the
United States , check the laws of your country in addition to the
terms of this agreement before downloading, copying, displaying .
performing . distributing or creating derivative works based on this
work or any other Project Gutenberg-tm work. The Foundation makes
no representations concerning the copyright status of any work in
any country outside the United States .

I.E. Unless you have removed all references to Project Gutenberg:

1 . E . 1 r The following sentence, with active links to, or other

immediate access to, the full Project Gutenberg-tm License must
appear prominently whenever any copy of a Project Gutenberg-tm work
Cany work on which the phrase "Project Gutenberg" appears, or with
which the phrase "Project Gutenberg" is associated) is accessed,

displayed, performed, viewed, copied or distributed:

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever. You may copy it . give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www .gutenberg. net

1 . E . 2 . If an individual Project Gutenberg-tm electronic work is
derived from the public domain (does not contain a notice indicating
that it is posted with permission of the copyright holder) . the work
can be copied and distributed to anyone in the United States without
paying any fees or charges. If you are redistributing or providing
access to a work with the phrase "Project Gutenberg" associated with
or appearing on the work, you must comply either with the
req^uirements of paragraphs 1 . E. 1 through 1 . E. 7 or obtain permission
for the use of the work and the Project Gutenberg-tm trademark as
set forth in paragraphs l.E^S or I.E. 9.

1 . E. 3 r If an individual Project Gutenberg-tm electronic work is
posted with the permission of the copyright holder, your use and
distribution must comply with both paragraphs l.E.l through I.E^Y
and any additional terms imposed by the copyright holder.

ARTICLE 11. PROJECT GUTENBERG SMALL PRINT^

Additional terms will be linked to the Project Gutenberg-tm License
for all works posted with the permission of the copyright holder
found at the beginning of this work.

1 . E.4 r Do not unlink or detach or remove the full Project
Gut enberg-tm License terms from this work^ or any files containing a
part of this work or any other work associated with Project
Gutenberg-tm .

1 . E.5 . Do not copy, display, perform, distribute or redistribute
this electronic work, or any part of this electronic work, without
prominently displaying the sentence set forth in paragraph 1 . E. 1
with active links or immediate access to the full terms of the
Project Gutenberg-tm License.

1 . E . 6 . You may convert to and distribute this work in any binary,
compressed H marked up, nonproprietary or proprietary form, including
any word processing or hypertext form. However, if you provide
access to or distribute copies of a Project Gutenberg-tm work in a
format other than "Plain Vanilla ASCII" or other format used m the
official version posted on the official Project Gutenberg-tm web
site (www. gutenberg. net) , you must . at no additional cost , fee or
expense to the user, provide a copy, a means of exporting a copy, or
a means of obtaining a copy upon request , of the work in its
original ''Plain Vanilla ASCII" or other form. Any alternate format
must include the full Project Gutenberg-tm License as specified in
paragraph 1 . E . 1 .

1 . E. 7. Do not charge a fee for access to, viewing, displaying,
performing, copying or distributing any Project Gutenberg-tm works
unless you comply with paragraph I.E. 8 or I.E. 9.

1 . E. 8 r You may charge a reasonable fee for copies of or providing
access to or distributing Project Gutenberg-tm electronic works
provided that

- You pay a royalty fee of 20"i of the gross profits you derive from

the use of Project Gutenberg-tm works calculated using the method
you already use to calculate your applicable taxes. The fee is
owed to the owner of the Project Gutenberg-tm trademark, but he
has agreed to donate royalties under this paragraph to the
Project Gutenberg Literary Archive Foundation. Royalty payments
must be paid withm 60 days following each date on which you
prepare Cor are legally req^uired to prepare) your periodic tax
returns . Royalty payments should be clearly marked as such and
sent to the Project Gutenberg Literary Archive Foundation at the
address specified in Section 4 , " Information about donations to
the Project Gutenberg Literary Archive Foundation. "

- You provide a full refund of any money paid by a user who notifies

ARTICLE 11. PROJECT GUTENBERG SMALL PRINT''

you in writing (or by e-mail) within 30 days of receipt that s/he
does not agree to the terms of the full Project Gutenberg-tm

License . You must req^uire such a user to return or

destroy all copies of the works possessed in a physical medium

and discontinue all use of and all access to other copies of

Project Gutenberg-tm works.

- You provide, in accordance with paragraph 1 . F. 3 , a full refund of

any money paid for a work or a replacement copy, if a defect in
the electronic work is discovered and reported to you within 90
days of receipt of the work.

- You comply with all other terms of this agreement for free

distribution of Proj ect Gutenberg-tm works.

1 . E. 9 . If you wish to charge a fee or distribute a Project
Gutenberg-tm electronic work or group of works on different terms

than are set forth in this agreement , you must obtain permission m
writing from both the Project Gutenberg Literary Archive Foundation
and Michael Hart , the owner of the Project Gutenberg-tm trademark.
Contact the Foundation as set forth m Section 3 below.

l.F.

1 . F . 1 . Project Gutenberg volunteers and employees expend
considerable effort to identify, do copyright research on,
transcribe and proofread public domain works in creating the Project
Gutenberg-tm collection. Despite these efforts. Project
Gutenberg-tm electronic works, and the medium on which they may be
stored, may contain "Defects, " such as, but not limited to ,
incomplete, inaccurate or corrupt data, transcription errors, a
copyright or other intellectual property infringement . a defective
or damaged disk or other medium, a computer virus , or computer codes
that damage or cannot be read by your equipment .

l.F. 2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the
"Right of Replacement or Refund" described in paragraph 1 . F. 3 , the
Project Gutenberg Literary Archive Foundation, the owner of the
Project Gutenberg-tm trademark, and any other party distributing a
Project Gutenberg-tm electronic work under this agreement , disclaim
all liability to you for dajoages , costs and expenses, including
legal fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE,
STRICT LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT
THOSE PROVIDED IN PARAGRAPH F3. YOU AGREE THAT THE FOUNDATION, THE
TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT
BE LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL,
PUNITIVE OR INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE
POSSIBILITY OF SUCH DAMAGE.

l.F. 3, LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a

ARTICLE 11. PROJECT GUTENBERG SMALL PRINT^

defect in this electronic work within 90 days of receiving it , you
can receive a refund of the money (if any) you paid for it by
sending a written explanation to the person you received the work
from. If you received the work on a physical medium, you must
return the medium with your written explanation. The person or
entity that provided you with the defective work may elect to
provide a replacement copy in lieu of a refund , If you received the
work electronically, the person or entity providing it to you may
choose to give you a second opportunity to receive tke work
electronically in lieu of a refund , If the second copy is also
defective, you may demand a refund in writing without further
opportunities to fix the problem.

1 . F.4 . Except for the limited right of replacement or refund set
forth, in paragraph 1 . F . 3, this work is provided to you ' AS-I3 ' , WITH

ND OTHER WARRANTIES DF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT
NOT LIMITED TO WARRANTIES OF HERCHANTIBILITY OR FITNESS FOR ANY
PURPOSE.

1 . F .5 . Some states do not allow disclaimers of certain implied
warranties or the exclusion or limitation of certain types of
damages. If any disclaimer or limitation set forth in this agreement
violates the law of the state applicable to this agreement , the
agreement shall be interpreted to make the maximum disclaimer or
limitation permitted by the applicable state law. The invalidity or
unenforceability of any provision of this agreement shall not void
the remaining provisions.

1.F.6, INDEMNITY - You agree to indemnify and hold the Foundation,
the trademark owner, any agent or employee of the Foundation, anyone
providing copies of Project Gutenberg-tm electronic works in
accordance with this agreement , and any volunteers assoc iated with
the production, promotion and distribution of Project Gutenberg-tm
electronic works, harmless from all liability, costs and expenses,
including legal fees , that arise directly or indirectly from any of
the following which you do or cause to occur : (a) distribution of
this or any Project Gutenberg-tm work, (b) alterat ion, modification,
or additions or deletions to any Project Gutenberg-tm work, and (c)
any Defect you cause .

Section 2 . Information about the Mission of Project Gutenberg-tm

Project Gutenberg-tm is synonymous with the free distribution of
electronic works in formats readable by the widest variety of
computers including obsolete, old, middle-aged and new computers.
It exists because of the efforts of hundreds of volunteers and
donations from people in all walks of life.

Volunteers and financial support to provide volunteers with the

ARTICLE 11. PROJECT GUTENBERG SMALL PRINT^

assi stance they need, is critical to reaching Project Gutenberg-tm's
goals and ensuring that the Project Gutenberg-tm collection will
remain freely available for generations to come. In 2001 , the
Project Gutenberg Literary Archive Foundation was created to provide
a secure and permanent future for Project Gutenberg-tm and future
generations. To learn more about the Project Gutenberg Literary
Archive Foundat ion and how your efforts and donations can help, see
Sections 3 and 4 and the Foundat ion web page at
http: / /vwv . pglaf . org .

Section 3. Inf ormat ion about the Project Gutenberg Literary Archive
Foundation

The Pro J ect Gutenberg Literary Archive Foundation is a non profit
BOlCc) (3) educational corporation organized under the laws of the
state of Mississippi and granted tax exempt status by the Internal
Revenue Service . The Foundation's EIM or federal tax identification
number is 64-6221541. Its 501(c)C3) letter is posted at
http: //pglaf . org/f undraising. Contributions to the Project
Gutenberg Literary Archive Foundation are tax deductible to the full
extent permitted by U.S. federal laws and your state's laws .

The Foundation' s principal office is located at 4567 Mel an Dr. S.
Fairbanks, AK , 99712 . , but its volunteers and employees are
scattered throughout numerous locations . Its business office is
located at 809 North 1500 West, Salt Lake City, UT 84116, (801)
596-1887 , email businessQpglaf . org , Email contact links and up to
date contact information can be found at the Foundation ' s web site
and official page at http : //pglaf . org

For additional contact information:
Dr. Gregory B. Newby
Chief Executive and Director
gbnewby^pglaf . org

Section 4. Inf ormat ion about Donations to the Project Gutenberg

Literary Archive Foundation

Project Gutenberg-tm depends upon and cannot survive without wide
spread public support and donations to carry out its mission of
increasing the number of public domain and licensed works that can
be freely distributed in machine readable form accessible by the
widest array of equipment including outdated equipment . Many small
donations (Si to $5,000) are particularly important to maintaining
tax exempt status with the IRS.

The Foundation is committed to complying with the laws regulating
charities and charitable donations in all 50 states of the United
States . Compliance requirements are not uniform and it takes a

ARTICLE 11. PROJECT GUTENBERG SMALL PRINT''

considerable effort, much paperwork and many fees to meet and keep

up vifh. these requirements. We do not solicit donations in
locations where we have not received written confirmation of
compliance . To SEND DONATIONS or determine the status of compliance
for any particular state visit http: //pglaf . org

While we cannot and do not solicit contributions from states where
we have not met the solicitation requirements, we know of no
prohibition against accepting unsolic ited donations from donors in
such states who approach us with offers to donate .

International donations are gratefully accepted, but we cannot make
any statements concerning tax treatment of donations received from
outside the United States . U.S. laws alone swamp our small staff .

Please check the Project Gutenberg Web pages for current donation
methods and addresses. Donations are accepted in a number of other
ways including including checks, online payments and credit card
donations. To donate, please visit : http ; //pglaf . org /donate

Section 5. General Information About Project Gut enberg-tm

electronic works.

Professor Michael S. Hart is the originator of the Project
Gut enberg-tm concept of a library of electronic works that could be
freely shared with anyone . For thirty years , he produced and
distributed Project Gut enberg-tm eEooks with only a loose network of
volunteer support .

Project Gut enberg-tm eBooks are often created from several printed
editions , all of which are confirmed as Public Domain in the U.S.
unless a copyright notice is included. Thus , we do not necessarily
keep eBooks in compliance with any particular paper edit ion.

Host people start at our Web site which has the main PG search
facility :

http: //www .gutenberg . net

This Web site includes information about Project Gut enberg-tm,
including how to make donations to the Project Gutenberg Literary
Archive Foundation, how to help produce our new eBooks , and how to
subscribe to our email newsletter to hear about new eBooks .