LIBRARY 
 
 University of California. 
 
 Class 
 
 \ 
 
QUATERNIONS 
 
 AS THE 
 
 RESULT OF ALGEBHAIC OPERATIONS 
 
 BY 
 
 ARTHUR LATHAM BAKER, Ph.D. 
 
 Head of Department of Mathematics, Manual Training High School, 
 Brooklyn, N. Y. 
 
 NEW YORK 
 
 D. VAN NOSTRAND COMPANY 
 
 23 MURRAY AND 27 WARREN STREETS 
 1911 
 
T53 
 
 Copyright, 1911 
 
 BY 
 
 D. VAN NOSTRAND COMPANY 
 
 THE SCIENTIFIC PRESS 
 
 ROBERT DRUMMOND AND COMPANY 
 
 BROOKLYN, N. Y. 
 
PREFACE 
 
 Beginners in the subject of Quaternions are generally 
 bewildered by the arbitrary manner in which the subject 
 is developed. They are forcibly introduced into a new 
 domain where the familiar rules of combination of symbols 
 are not valid. New magnitudes are arbitrarily assumed, 
 subject to arbitrary laws. The reader finds the logic con- 
 sistent and the results concordant with those of his previous 
 courses, but he hardly knows why. He finds himself in a 
 new country, but thoroughly and bewilderingly uncertain 
 as to how he got there. 
 
 It is in the attempt to avoid this uncertain journey, to 
 lead the student from the known to the unknown by 
 familiar steps, by steps which require no arbitrary limita- 
 tions of former laws, but merely their adaptation to new 
 circumstances, that these class notes have grown into their 
 present shape. 
 
 The backbone of the method of presentation is the use 
 of a one-to-one correspondence between the mathematical 
 concept and what I have ventured to call its idiographic 
 symbol, that is, a symbol wh^e spatial properties are the 
 same as the mathematical properties of the concept it 
 symbolizes. From this similarity of properties there exists 
 a one-to-one correspondence between the results of spatial 
 operations upon the symbols and the corresponding mathe- 
 matical operations upon the concept. 
 
 iii 
 
 236365 
 
iv PREFACE 
 
 These idiographic symbols are strokes, spherical shells, 
 and vectors, corresponding respectively to magnitudes 
 having size, and correlated sense of opposition, scalars, and 
 magnitudes having size, sense, and direction. 
 
 Spatial operations upon these symbols are used as sug- 
 gestions for a one-tc-one corresponding interpretation for 
 for the mathematical concept. 
 
 These spatial operations are rational and logical and require 
 no " standing loose for a time to logical accuracy." * As 
 they are rational and logical, so their interpretations are 
 rational and logical, and the reader does not lose his sense 
 of logical sequence. There is no " removal of barriers, of 
 limitations, of conditions." * Multiplication is the same 
 from beginning to end, whether applied to scalars, vectors, or 
 quaternions. Commutivity of factors may be permissible 
 in some cases and not in others, but this is a mere incident 
 and not an essential element of the operation. 
 
 The reader is not mystified by arbitrarily defining multi- 
 plication of one vector into another as the turning through 
 a right angle, etc., and left to wonder how one line can do 
 anything to another. In fact, the operations are not 
 defined a priori at all, but taking the properties of discrete 
 quantities as symbols of operations which the reader is to 
 perform, we find six possible operations, addition, subtrac- 
 tion, multiplication, division, reversion, and mean reversion. 
 These operations are defined accordingly a posteriori as 
 results of causes, not arbitrarily as assumptions. 
 
 The performance of these operations upon scalars leads to 
 or evolves successively vectors and quaternions. Thus qua- 
 ternions are evolved from discrete magnitudes, not arbitra- 
 rily, but of necessity, and along certain fixed and preordained 
 
 * Kelland and Tait, and others. 
 
PREFACE V 
 
 lines, by rules which the properties of discrete magnitudes 
 necessitate, and which cannot be altered or varied, and 
 with which the reader is already familiar. The reader is 
 not disturbed by the thought. Suppose we had made some 
 other assumption, what then? No assumptions are made. 
 He simply follows the road suggested by the properties of 
 discrete magnitudes, and can arrive at but one result. 
 
 We make no laws, lay down no rules, make no modifica- 
 tions or limitations. The only way in which we exercise 
 any choice is in the rational application of the laws we 
 discover to the proper operands and in a proper and logical 
 manner. 
 
 The '' interpretation of our results " is not made to 
 " depend upon the definition " as a foundation. The foun- 
 dation is the properties of discrete magnitudes, and the 
 definitions are merely rational statements of the results of 
 these properties being used as suggestions for operations to 
 be performed by the reader. 
 
 Considerable stress has been laid upon the avoidance of 
 the sole use of mere typographical symbols and upon the 
 auxiliary use of idiographic symbols upon which spatial 
 operations can be performed; as in the use of strokes for 
 merely reversible magnitudes, a spherical shell for scalar 
 magnitudes, arc strokes for quaternion multiplication, two 
 vector factors for the corresponding quaternion, etc., thus 
 making the treatment concrete and avoiding the difficulties 
 of abstractness. 
 
 The original features of the book are those specified above, 
 coupled with the general heuristic method by which the 
 student hews out his own concepts as he goes along. The 
 results, the examples, the applications, and the terms used 
 are those found in every treatise on the subject, of which 
 I have made free use and to whom should be accredited 
 
vi PREFACE 
 
 these features: particularly Hamilton, Kelland and Tait, 
 Tait, Laisant, Molenbrock, Hathaway. 
 
 As these notes are only intended as an introduction, not 
 an overabundance of examples or formula? has been pro- 
 vided, nor have any applications been made to problems in 
 Geometry and Physics. These will be found in the works 
 cited. Nor has the subject of differentiation been touched 
 upon. 
 
 The author's own experience with this method of presenta- 
 tion of the subject to beginners has been encouraging. It 
 is hoped others will have the same experience. 
 
CONTENTS 
 
 CHAPTER I 
 
 PAGE 
 
 Mathematical Operations upon Discrete Magnitudes 1 
 
 Discrete magnitudes, 1; Addition, 4; reversion, 5; sub- 
 traction, 6; multiplication, 7; division, 8; mean reversion, 9. 
 
 CHAPTER II 
 
 Idiographs 6 
 
 Strokes, 11; Argand diagram, 13; addition of strokes, 15; 
 algebraic operations, 17; scalars, 18. 
 
 CHAPTER III 
 
 Space Idiographs 10 
 
 Space idiographs, 19; mean reversion, 20; vector, 22; 
 reversal, 26; addition and subtraction of vectors, 27; 
 decomposition of vectors, 29; notation, 30; exercises in 
 combination, 31; examples, 37. 
 
 CHAPTER IV 
 
 Multiplication of Unit Vectors 17 
 
 Parallel vectors, 40; perpendicular vectors, 41; inclined 
 vectors, 42; exercises in i, j, k, 44. 
 
 vii 
 
vm CONTENTS 
 
 CHAPTER V 
 
 PAGE 
 
 Quaternions 21 
 
 Quaternion, 45; meaning of — , 47; examples, 48; commu- 
 
 tivity of vectors, 49; types of quaternions, 50; notation, 53; 
 exercises, 54; geometric meaning of TVafi and Sa^, 58; 
 meaning of Sa^Y, 59; sum of scalar and vector, 61; definitions 
 of algebra, etc., 63. 
 
 CHAPTER VT 
 
 Kinds of Quaternions 32 
 
 Reciprocal of a quaternion, 64; opposite quaternions, 65; 
 conjugate quaternions, 66; table of quaternions, 69; equality 
 of quaternions, 71; diplanar quaternions, 72. 
 
 CHAPTER VII 
 
 Quaternion Operators 37 
 
 Quaternion as a multiplier, 73; angle of a quaternion, 76; 
 arc strokes, 77; multiplication and division by arc strokes, 80; 
 table showing /g, 82; arc stroke of a vector, 84; table, 85; 
 square root by strokes, 86; examples, 88. 
 
 CHAPTER VIII 
 
 Products of Quaternions 45 
 
 Productsin general, 90; conjugateof a product, 91; coplanar 
 quaternions, 92; examples, 93. 
 
 CHAPTER IX 
 
 Versors 48 
 
 Versors, 94; multiplication of versors, 97; examples, 99; 
 conversion of qr into rq, 101; meaning of qrq~~^, 102; examples, 
 103. 
 
CONTENTS ix 
 
 CHAPTER X 
 
 PAGE 
 
 Interpretation of Vector Equations 54 
 
 a^y = H, 107; exercises and formulae, 111; applications, 
 114; reference formulae, 122. 
 
 CHAPTER XI 
 
 Quaternion Equation of the First Degree 62 
 
 General form, 127; linear scalar equation, 129; decomposi- 
 tion of vector part, 130; linear vector function, 131; properties 
 of 0, 132; conjugate strain function, 135; application of <p 
 to vector, 141; properties of (J), 143; nonion, 145. 
 
 CHAPTER XII 
 
 Applications of ^ 71 
 
 Change of volume, 146; special values of modulus, 148; 
 special applications of cfi, 153; properties of (0' — g), 154; latent 
 lines, latent roots, latent planes, 162; two latent roots equal, 
 164; three latent roots equal, 165; pure strain, 172; examples, 
 175. 
 
 Appendix 87 
 
 Functional Symbols. 
 
QUATERNIONS 
 
 CHAPTER I 
 
 MATHEMATICAL OPERATIONS UPON DISCRETE 
 MAGNITUDES 
 
 1. Mathematics is for school purposes the science of 
 magnitude. 
 
 Magnitude has been defined as that which can be increased 
 or diminished, and of two kinds, continuous magnitude 
 (continuum), which is usually called quantity and answers 
 to the question, How much? and discrete magnitude 
 (discreta), which is usually called number and answers to 
 the question. How many? 
 
 Quantity is differentness from nothingness, or quantita- 
 tively expressed, difference from nothing. 
 
 2. When we attempt to express quantity in symbols or 
 terms, or to conceptualize it, we find ourselves compelled 
 to express it in terms of some arbitrary unit adopted as a 
 standard. The symbol or concept for the measure of quan- 
 tity is called number, expressing its differentness from 
 oneness, the oneness of the standard. 
 
 In the case of natural units, soul, personality, inhabitant, 
 etc., the differentness from oneness which distinguishes a 
 group from an individual is number, giving rise to the 
 so-called primary numbers, 1, 2, 3, . . . 
 
 Number is differentness from oneness. The whole 
 
2 QtrA-TERNIONS 
 
 essence of number is to differ from another number. It 
 belongs to the primary categories of time, space, matter, 
 etc. No one of these can exist without the presence of 
 number, that is without a oneness and a difference from 
 that oneness. The very existence of time, space, matter 
 imphes multipHcity , three-f oldness. 
 
 The Hmitations of the mind require symbols to represent 
 the different concepts so that the mind can posit its judg- 
 ments and conclusions to await its return to make use of 
 them. Otherwise the mind soon becomes clogged with the 
 impedimenta of its own creation. 
 
 Upon the fortunate choice of these symbols depends very 
 greatly the progress in their use. A striking illustration of 
 this is seen in the enormous advantage of the Hindu position 
 system over the clumsy notation of the Greeks and Romans. 
 
 3. In the infancy of the subject pebbles (calculi) and 
 other concrete objects were the symbols used. And as a 
 negative pebble was impossible of imagination, negative 
 numbers were imaginary. Even as late as the sixteenth 
 century they were called by Cardan in his Ars Magna, 
 numeri ficti, imaginary numbers. After they were dignified 
 with the title of real numbers, expressions containing \/^^ 
 would intrude themselves, under the operations of mathe- 
 matical analysis, and as there was no way of writing a 
 numerical symbol whose square would be a negative 
 number, these were in their turn imaginaries, even to the 
 present day. So late a writer as De Morgan (1831) speaks 
 of them as void of meaning, self-contradictory, and absurd, 
 though of great utility in the formal mechanism of Algebra. 
 
 But it is not the number that is imaginary, it is the appli- 
 cation of it to certain data or symbols that produces the 
 imaginary features. Fractions are imaginary when count- 
 ing souls, inhabitants, events, phenomena, etc. Negative 
 
MATHEMATICS AND DISCRETE MAGNITUDES 3 
 
 numbers are imaginary when applied to length, bulk, 
 intensity, etc. The imaginaries of to-day become real 
 when applied to the proper symbol. 
 
 Before searching for a symbol we must inquire about the 
 properties of primary number and the operations perform- 
 able upon it. What are these performable operations, 
 guiding ourselves by the properties of number itself? This 
 guidance is necessary, the properties of a thing, if we thus 
 speak of number, being the determining factors as to what 
 purposes it can be put. 
 
 4. Number is differentness from unity. We can increase 
 this differentness quantitatively by combining the different- 
 ness of one number with the differentness of another 
 number. This operation we call addition, and by means 
 of it we can evolve all the primary numbers from unity. 
 In this two numbers are taken accumulatively. 
 
 5. We can imagine the differentness from oneness to have 
 a correlated sense of opposition, like debit and credit, 
 affirmation and negation, an antagonistic differentness. 
 One is the reversal of the other, the negative of the other; 
 and the process of converting one into the other is called 
 reversion. 
 
 Numbers having this correlated sense of opposition are 
 said to have size and sense. An example could be the 
 accumulative collection of material into the form of a mound 
 or tumulus. The workmen could proceed decumulatively 
 (this word is not in the lexicons, but its usefulness is apparent) 
 by undoing their former work, and might proceed below 
 the bed rock. We might not call this a negative tumulus, 
 but it would be the negative of the tumulus concept and 
 the number concept would be a negative number. 
 
 6. We can diminish the differentness from oneness which 
 is characteristic of a number by the characteristic different- 
 
4 QUATERNIONS 
 
 ness of some other number, giving rise to the operation called 
 subtraction. The continued application of this process to 
 numbers having size and sense leads to negative numbers. 
 Here the numbers are taken decumulatively. As addition 
 is accumulative combination so subtraction is decumula- 
 tive combination of numbers without regard to the size of 
 the subtrahend. 
 
 We can consider the characteristic differentness of one 
 number as a mandate for an operation to be performed 
 upon another number. Now this characteristic different' 
 ness from oneness or unity may be evolutory or involutory, 
 differentness from unity, .or differentness toward unity, a 
 differentness which evolves the number from unity, or a 
 differentness which converts a number into unity. 
 
 7. If we take the evolutory differentness as the mandate, 
 then the operation is called multiplication, the doing to 
 the operand (multiplicand) what was done to unity to 
 produce the operator (multiplier). If the first operand and 
 all the operators in a series of operations are the same, the 
 operation is called involution. Involution works from unity 
 with the first operand and all the operators alike. 
 
 A curious error is liable to creep in here unless care be 
 exercised. For example, in V2-3, one is liable to say that 
 \/2 is derived from 1 by doubling 1 and taking the \/~ of 
 the result. This applied to 3 gives the result \ 2T3, which 
 is, of course, wrong. The error consists in taking the 
 square root of the result, whereas multiplication is the 
 doing what was done to unity, and not what was done to 
 the results of operations upon unity. To get \/2 from 
 unity, we really operate upon unity with the tensor or 
 stretcher 1.4142 .. . =\/2. 
 
 8. If we take the differentness toward unity as the man- 
 date, then the operation is called division, the doing to the 
 
MATHEMATICS AND DISCRETE MAGNITUDES 5 
 
 operand (dividend) what was done to the operator (divisor) 
 to produce unity. 
 
 If all the operators and the last operand in a series of 
 operations are the same, the operation is called evolution, 
 and gives rise to irrational and so-called imaginary numbers, 
 e.g., V2, v-T, etc. 
 
 Evolution works towards unity with all the operators and 
 the last operand alike. 
 
 9. One of the most important phases of evolution is \/a^ 
 the breaking up of the operation of passing from unity to 
 a into two equal steps, so that the repetition of the first 
 step, \/a, shall produce a. When a = — 1, we get V^^. 
 
 Since — 1 is reversion, \/^^ is called mean reversion. 
 
 Unlike addition and subtraction, which merely adjoin the 
 elements without change, multiplication and division are 
 transformational operations which change the operand into 
 an entirely new and different number. They are purely 
 algebraic, being limited, unlike addition and subtraction, 
 to operations upon discrete number. A not entirely satis- 
 factory illustration of the difference between the two 
 operations of addition and multiplication is the putting of 
 two canes together for addition; for multiplication we 
 would be compelled because the first was triangular, black, 
 smooth, and uniform in size to transform the second cane 
 which was round, white, knotted, tapering, and with a 
 knob on the end, into a triangular, black, smooth, prismatic 
 stick. It is transformed into an entirely new and distinct 
 form, and its original features have disappeared. So in 
 numbers, addition and subtraction are merely adjunction 
 without change of the original elements. Multiplication 
 and division are destructive transformations, the destruction 
 of one element and the production of an entirely new and 
 different one, like the growth of a plant from a seed. 
 
CHAPTER II 
 IDIOGRAPHS 
 
 10. Mathematical operations are conducted by means of 
 symbols, and upon our choice of symbols depends Largely 
 the success of our operations. 
 
 In the dawn of mathematics, the fingers or other concrete 
 objects were the symbols used, and a negative quantity was 
 purely imaginary. With the introduction of written 
 symbols of quantity and the concept of debit and credit, 
 the negative quantity lost its imaginary quality and became 
 real. In these symbols, however, V^^ remained imagi- 
 nary, because the spatial and typographical properties of 
 the symbol and the mathematical properties of the thing 
 it represented did not agree. 
 
 11. With the introduction of the symbol called a stroke, 
 a straight line upon the surface of the paper, to represent a 
 magnitude of a given size, the space properties of the 
 symbol and the mathematical properties of the thing 
 symbolized became the same, and spatial operations upon 
 the symbols corresponded to mathematical operations upon 
 the thing symbohzed, and could be used to interpret and 
 control the mathematical operations. 
 
 Symbols whose spatial properties are the same as the 
 mathematical properties of the things represented might 
 be called idiographs {iBios, proper, peculiar) . 
 
 12. Representing a magnitude having sense .of generation, 
 sense of correlated opposition, by the idiograph >, and 
 
 6 
 
i/T 
 
 IDIOGRAPHS 7 
 
 of course its reversal by < , how can we convert > into 
 
 ^ ? 
 
 Obviously, only in one way, so long as we retain its 
 stroke characteristics, namely, by swinging it through an 
 angle of 180° in the surface of the paper. To attempt to 
 swing it out of the surface of the paper is to lose its stroke 
 characteristics, to give it absolute direction in space and 
 render it no longer an idiograph. 
 
 The revolution of 180° can be broken into two equal steps 
 of 90° each. Hence | or | must be the idiographic 
 equivalent of \/~- 
 
 Combining these into one diagram and i/^ 
 
 assuming the normal revolution as counter- _j 
 clockwise, we get the idiographic diagram 
 with the corresponding typographical sym- 
 bols. 
 
 13. This is the well-known Argand diagram, affording a 
 simple method of representing relatively directed quantities, 
 or as they are generally called, complex quantities, the general 
 type of which is x+a/^^?/, where x represents the normal 
 o'r the reversed portion and ^y — ly the mean reversed 
 
 portion. It is generally written 
 x-\-iy, i standing for \/^^. The 
 rectangular co-ordinates x and iij 
 determine a point, whose distance 
 from the origin, r, is called the mod- 
 ulus, and whose angular distance 
 
 from the axis of x, the angle (j), is called the amplitude 
 
 of the point or complex quantity x+iy. 
 
 14. The idiographic symbol for a magnitude, >, show- 
 ing its size and sense, we have already designated as a 
 stroke, a stroke forward or a stroke backward. Two 
 forward strokes need not be represented by the same 
 
8 QUATERNIONS 
 
 symbol on the paper. The forwardness is in reference to 
 its own backwardness, and has no reference to the forward- 
 ness or backwardness of other strokes. A stroke is a 
 straight Kne in a plane, symbolizing a given magnitude in 
 size and in relation to its sense of normalcy or of reversion, 
 or a condition betw^een these. Two strokes are equal when 
 they have the same lengths and the same direction in a 
 plane as regards a standard normal direction in the 
 plane. 
 
 15. If several strokes be taken in succession the sum 
 (result) of them is the same as the stroke from the beginning 
 of the first, to the end of the last, when they are arranged 
 end to end so as to be successive. Thus 
 
 Strokes will for the present be 
 represented typographically by lower- 
 carse Greek letters, as above. 
 
 The reader must notice carefully 
 that it is not the lengths of the stroke 
 which are added, but the results of 
 the strokes, including both length and direction on the 
 paper. 
 
 It is easily seen that the order of the strokes is immaterial, 
 and that any number of consecutive strokes can be replaced 
 by their sum. The addition of strokes is a commutative and 
 associative operation, that is, the order and mode of grouping 
 has no effect on the result. 
 
 A stroke is subtracted by reversing its direction and 
 adding. 
 
 16. If we attempt to break up the operation of rever- 
 sal into three or more equal and similar operations, 
 for example, three, a^ shown in the diagram, we find 
 
IDIOGRAPHS 9 
 1 -^/q" 
 
 that ^' — 1=— -f-t; , that is, the first of the three equal 
 
 operations is expressible in terms of the operation of 
 mean reversion, and of course the second 
 operation likewise. 
 
 Similarly for a greater number of 
 
 steps. Hence mean reversion is evi- ' ^— ^ 
 
 dently the unique operation, a multiple of which is rever- 
 sion, all the other partial equal operations whose con- 
 tinued application results in reversion being expressible 
 in this one. 
 
 17. Hence there are six unique and fundamental opera- 
 tions which can be performed upon a discrete magnitude: 
 addition, subtraction, reversal, multiplication, division, and 
 mean reversion, and no others. 
 
 By fundamental operations is meant operations based 
 upon the properties of discrete magnitudes, size, and sense 
 of correlated opposition. 
 
 18. Mere discrete magnitudes, considering size only, are 
 scalars, that is, they can be scaled off on a scale either in 
 a normal or in a reversed direction. 
 
CHAPTER III 
 
 SPACE IDIOGRAPHS 
 
 19. Space idiographs. In space we cannot idiograph- 
 ically represent a scalar by a line, for that would be assign- 
 ing to the symbol a characteristic direction, which the 
 magnitude it represents does not possess. If we are to 
 represent a scalar magnitude in space by any idiographic 
 symbol, the only one which seems available as possessing 
 perfect symmetry and therefore devoid of direction is a 
 spherical shell. 
 
 Just as we can assume our + unit of heat, pressure, etc., 
 anywhere on the scale, so we can posit our + spherical shell 
 anywhere in space. 
 
 Likewise as the — unit of heat, etc., would naturally 
 adjoin the + unit, along the scale, so naturally we should 
 expect the — unit shell to adjoin the + unit shell in some 
 position determined by previous assignment of direction. 
 According to this previously determined direction we shall 
 have units of a direction, of /? direction, etc., where a and 
 /? denote direction, not magnitudes, unless we say unit 
 magnitude, just as previously we had units of heat sense, 
 credit sense, etc. 
 
 20. How can we break up the operation of transforming 
 the + shell into the — shell into two similar and equal 
 operations. The + shell can be changed into the — shell 
 by the repetition of two different operations: 
 
 10 
 
SPACE IDIOGRAPHS 11 
 
 A. By revolving the — shell about its point of contact 
 with the + shell through an angle of 90° twice; 
 
 B. By moving the elements of the — shell perpendicularly 
 to the common line of centers in the proportion sin 6 (where 
 ^=cos~^ harmonic displacement* of the shell element) and 
 moving the resulting configuration, a directed line, one-half 
 its dimensions toward the correlatively reversed position 
 of the original operand, i.e., toward the position of the 
 + shell. A repetition of this operation would produce the 
 + shell. 
 
 21. Operation A is excluded by reason of its lack of 
 definiteness, leaving operation B as the operation producing 
 the mean reversed state. 
 
 Since we are dealing with iodiographs these spatial 
 operations must have a one-to-one correspondence with 
 mathematical operations performed upon the things they 
 symbolize. 
 
 22. Hence a mean reversed scalar is represented in all its 
 properties by a directed magnitude in space, a vector, as it 
 is called, which is definitely directed as soon as the -|- and 
 — shells are posited. 
 
 23. A vector is any magnitude having direction of exten- 
 sion in space, a directed line, plane, etc., such as velocity, 
 impulse, force, etc. 
 
 Vectors are equal when they possess the same quantitative 
 
 * If P represents an element of the shell, 
 OA, the projection of the radius on a given 
 diameter is its harmonic displacement, 
 
 = cos~^ OA. 
 
12 QUATERNIONS 
 
 and qualitative properties, viz., magnitude, sense, and 
 direction of extension. Direction of extension is that 
 property which prevents the vectors from coinciding (in 
 whole or in part) when brought together. 
 
 Parallel vectors of the same length are equal. Vectors 
 can be made coinitial without altering their properties. 
 
 24. It is customary to indicate the unit vector in a given 
 direction by Greek letters a, p, y, . . . Generally three unit 
 vectors at right angles to each other are assumed as reference 
 units. These are designated by i, /, k. 
 
 25. So far we have recognized two kinds of magnitudes, 
 scalars and vectors, and six operations, addition, subtrac- 
 tion, reversion, multiplication, division, and mean reversion. 
 
 Applying these operations to scalars we find that they 
 all produce scalars again, except in the case of mean rever- 
 sion, and that produces a vector. This gave us the second 
 kind of magnitude, to which we will now proceed to apply 
 the six fundamental operations. 
 
 26. Reversal is merely the turning of the vector into the 
 opposite direction, as the word implies. The result is some- 
 times called a revector. 
 
 27. Addition and subtraction of vectors. Subtraction of 
 vectors is merely addition with the minuend reverted. 
 Vectors are of the nature of strokes, with the property of 
 absolute direction added. The laws governing the addition 
 
 of strokes evidently hold 
 here also. Thus, vector ad- 
 dition is commutative and 
 associative, and this whether 
 the vectors are coplanar or 
 not. Thus 
 
 = /-'' +«"+/?'', etc., 
 
SPACE IDIOGRAPHS 13 
 
 where a, /?, y are the three edges of a parallelepiped. The 
 same reasoning would apply to additional vectors. 
 
 28. The following equations are self evident : 
 
 a -\-a +« ... to m terms =ma, 
 — « + ( — «)+... to m terms=m( — a) = —ma, 
 
 29. If a, /?, ;- be three coinitial vectors, then any fourth 
 coinitial vector, ^, can be expressed as 
 
 ^ being the diagonal of the parallelepiped whose edges are 
 xa, yP, and zy. 
 
 30. If a is a unit vector, and ma=A, then m indicated 
 generally by the symbol TA, which expresses the length of 
 the vector A, is called the tensor {tendere, to stretch) of the 
 vector A. a, denoted by UA is called the unit vector of 
 A. Therefore 
 
 A^TA'UA. 
 
 Vectors will be denoted by capital Greek letters when the 
 tensor and unit part are to be emphasized; by lower-case 
 Greek letters when the question of length is not important; 
 and by the corresponding lower-case English and Greek 
 letters when speaking of the tensor and unit part sepa- 
 rately. 
 
 Thus the same vector may be indicated hy A, aa, a. 
 
 The tensor is signless, just as any length, the height of 
 a steeple, for instance, is signless; or the height of a man. 
 A man cannot be —5 feet tall. 
 
14 QUATERNIONS 
 
 Exercises in Vector Combinations 
 
 31. If Za:+m/3=0, then 1=0, m=0, for in no other way 
 can two strokes in different directions cancel each other 
 so as to leave the pen at the point of beginning, unless 
 a =x^, i.e., unless a is parallel to /?. 
 
 32. If Za+m/?=Zi^.+mi^, 
 that is, (Z — Zi)a: + (m — mi)/3=0, 
 then, l=h, m=Wi. 
 
 33. If loL-^m^-{-nj=0, and Z, m, n are not zero, then a, 
 /?, and ;- are coplanar, for la and m/? determine a plane 
 which contains the ends of iiy, and therefore ny itself. 
 
 34. If Za+m/?+n;'=0, and Z, m, n are not zero but 
 Z+m+7i=0, then (Z+m+n)a:=0, and subtracting the first 
 
 equation, we get 
 
 m{a—p) -\-n{a — Y) =0, 
 
 whence «— /? and a — y are 
 parallel (§31). But a — y con- 
 nects the ends of a and ;', and 
 
 a—p the ends of a and /?, hence i/ Z+m+n =0, «, ^, anc? y 
 
 terminate in the same line. 
 
 35. Conversely, if a, /?, and y terminate collinearly and 
 la +m/3 -\-ny =0, ^/ien Z +m +n =0. 
 
 For by condition , a—p=x{a — y), 
 
 or {l — x)a—^-\-xy = 0, 
 
 in which 1— a;— l+a;=0. q.e.d. 
 
SPACE IDIOGRAPHS 15 
 
 36. If aa+h^-\-cy-\- . . . =dd, then evidently a-\-h-\-c 
 + . . ,>d,oT TA+TB + TC-\- . . . >TD, 
 
 or, Sum of the tensors > tensor of the sum, 21 T > Til , 
 or the distance a man travels > his distance from home. 
 
 Ti the vectors are parallel, I1T = T^. 
 
 37. The diagonals of a parallelogram mutually bisect each 
 other. 
 
 a=d+yy = j'-\-xd. 
 
 .-. §31, 
 
 d=xdj T=yTy 
 
 Q.E.D. 
 
 7- and d being parts of the diagonals to the point of inter- 
 section, and yy and xd the remaining portions respectively. 
 
 38. The lines joining the middle points of the opposite sides 
 of any quadrilateral, whether plane or gauche, mutually bisect 
 
 each other. 
 
 " — ^ . 1 1 
 
 One bisector is a=—X-\-ii +-^v. 
 
 The other is /?=| + i^+'^. 
 
 Find the vectors from any 
 assumed point to the middle points of these bisectors and 
 compare the results. 
 
 Thus the vector from the beginning of X to the middle 
 point of a is 
 
 „ ; « >^\n,^ ^v\ 3.1 ^1 
 
16 QUATERNIONS 
 
 The vector from the same point to the middle point of 
 
 ^-^+f+§ 
 
 ^+f4(f+'^+f) 
 
 .3 ^1 1 k + fi + v 3., 1 ,1 
 
 .*. '^i = s2, and the middle points coincide. q.e.d. 
 
 39. If the ends of two parallel vectors be connected by 
 straight lines, the join (connecting line) of the middle 
 points of the straight lines is half the sum or difference of 
 
 -ma met 
 
 the tensors of the parallel vectors: i.e., the median line of a 
 trapezoid is half the algebraic sum of the bases. 
 
 8=-^±a—^, taken ^long a ; 
 
 ma±a 
 
 P T 
 
 = — — + ma H-^, taken along ma. 
 
 Whence by addition, b = 
 
 2 
 
 „ ^ , ma±a 
 
 :. §36, d = ~-^, Q.E.D. 
 
CHAPTER IV 
 MULTIPLICATION OF UNIT VECTORS 
 
 40. Parallel vectors. Remembering that multiplication 
 is. the performing by the reader on the multiplicand of an 
 operation which is symbolized by the multiplier, viz., the 
 operation which produced the multiplier from unity, we 
 must in the product ii * ask what operation is the first i 
 the symbol of. The answer is, of course, of one of two equal 
 operations whose successive applications shall produce 
 reversal. Now i can be reversed by the repetition of each 
 of two methods. The one we have designated as operation 
 A (§ 20). The other we have designated as operation B. 
 Since the multiplier is exactly the same as the multiplicand 
 in all its properties, we must if possible use exactly the 
 same operation not only in kind, but also in detail, that 
 produced the multiplier, that is, operation B (§ 20). 
 
 This amounts to the repetition upon the multiplicand i 
 of the same operation which produced it from unity, and 
 of course results in —1, see § 20. That is, 
 
 ii = — \. 
 
 Hence, Multiplication {'performance of an operation sym- 
 bolized by the multiplier) of one unit vector into another 
 parallel to it produces reversion. 
 
 * i=some directed v — l^some directed mean reversed scalar. 
 
 17 
 
18 
 
 QUATERNIONS 
 
 41. Perpendicular vectors. Since the multiplier is now 
 perpendicular to the multiplicand, we must take it as the 
 symbol of an operation to be performed on the multipli- 
 cand, the same in kind but as far removed in detail from 
 that which would have been used had the multiplier been 
 parallel to the multiplicand as perpendicularity is removed 
 from parallelism. 
 
 This we must do in order to take into account the per- 
 pendicularity of direction as opposed to parallelism. The 
 
 operation of mean reversion 
 which produced the multiplier 
 was operation B (§ 20). There- 
 fore we must use operation A. 
 Let i and / be operator and ope- 
 rand respectively. The only 
 position into which / can be re- 
 volved such that the reversal 
 of the signs of the two factors will give the same result is kj 
 one of perpendicularity to both factors. Thus 
 
 ij=k and 
 
 ■j = k, 
 
 since —i bears exactly the same relation to — / that i 
 does to k, and must therefore have the same effect. Any 
 other position than k for the product of ij would not do 
 this. 
 
 Hence, The multiplication {'performance of an operation 
 symbolized by the multiplier) of one unit vector into another 
 perpendicular to it residts in the turning of the multiplicand 
 through a. right angle in (to) a plane perpendicular to the operator. 
 
 '42. Inclined vectors. Naturally the result will be a com- 
 bination of those of §§40, 41, that is partly scalar and 
 partly vector, or 
 
 ap=— cos ^ + £ sin 0, 
 
MULTIPLICATION OF UNIT VECTORS 
 
 19 
 
 where a and /? are the two unit vectors incKned at an 
 angle d, and e. a unit vector perpendicular to a and /?, 
 since this formula satisfies both the limiting cases (§§ 40, 
 41). 
 
 Hence, The multiplication {performance of an operation 
 symbolized by the operator) of one unit vector into another 
 inclined to it at an angle 0, thus produci7ig the mean reversed 
 
 state induced by the operator symbol, turns the operand through 
 a right angle into a plane perpendicular to the multiplier, 
 makes its length sin and adds a sccdar, — cos 0. 
 43. We can get the same result as follows: 
 i^, the mean reversed state of /?, must be as to direction 
 some vector perpendicular to the plane of i and ^, since 
 — !•— /? must produce the same result as i^. Hence, 
 tentatively, 
 
 ip=sk, 
 
 where s is some scalar. Operating again with i to see if 
 the second application produces reversal, we get 
 
 i'i^=i-sk^S'ik=s- —j, 
 
 which is not reversal, but which would be, except as to 
 
20 QUATERNIONS 
 
 length, perhaps, by the addition of —ci, where c is some 
 scalar. But this would require 
 
 ifi=—c+sk, since i-ip=—ic-\-i-sk 
 
 = —ic — sj. 
 Now if — ic — s/=— /?, then c = cos^, s=sin^^ 
 
 and we have as before i^ = — cos d -\-k sin d. 
 44. Exercises in unit reference vectors. 
 
 ij = k, but ji = —k ; jk =i, but kj =—i. 
 
 Hence the factors are not commutative, 
 
 i-jk=i'i = — l, ij'k=k'k = — l. 
 
 Hence i-jk=ij-kj or the factors are associative. 
 ki=j, i-jk=ii=i^ = — l, 
 
 ji = —k, j-ki ^jj = — 1, 
 
 i- —j =—k, ijk =jki =kij = — 1, 
 
 jjlii =jH^ =i, k'ji=k- —k = —k^ = l. 
 
 ii-k=kk=k^ = — l. 
 
CHAPTER V 
 QUATERNIONS 
 
 45. Having ascertained that the product of two vectors 
 
 a, /? is 
 
 ap=— cos ^ -f- £ sin /9, 
 we can, § 29, express £ sin in terms of i, j, k, viz.: 
 e^m d=xi-\-yj+zk, or a^^—co& 6 -{-xi-\-yj-\-zky 
 
 which, being composed of four terms, a scalar and three 
 vectors, is called a quaternion, and will be symbolized 
 by 5. 
 
 A quaternion is evidently composed of a scalar plus a 
 vector. Later (§ 61) we shall find that, conversely, a scalar 
 plus a vector is a quaternion. 
 
 46. The plane of the factors (multiplier and multipli- 
 cand) of a quaternion is called the plane of the quaternion. 
 
 The plane of a vector is the plane perpendicular to it. 
 
 £ is called the axis of a/?. The most convenient rnethod 
 of defining it seems to be: The unit vector toward the 
 north pole when the multiplicand is to the east of the 
 multiplier, the equator being the plane of the quaternion; 
 toward the south pole when the multiplicand is to the 
 west. 
 
 47. Meaning of — . By the rule for division we must first 
 
 21 
 
22 
 
 QUATERNIONS 
 
 ascertain what must be done to a to produce 1. Return- 
 ing to our idoigraphic shell; to convert a into +1, we 
 must, repeating the operation which produced it, apply 
 operation B (§ 20), and then direct it one-half its dimen- 
 *j sions towards its correlatively reversed 
 
 position. Performing these operations 
 
 on the 
 
 numerator 
 
 we get by application of B, 
 and then by directing it, 
 
 whence — 
 
 This is verified by the fact that a- —a = —a^ = — • — 1 =1, 
 
 hence — = —a, since a— = 1, being a functional* operation 
 a a ' 
 
 followed by the inverse operation and therefore resulting in 
 
 the original operand. 
 
 Hence, the reciprocal of a unit vector is the unit vector 
 
 reversed. 
 
 48. Meaning off 4-, ^, etc. In a similar manner -^=k, 
 
 * See Appendix. 
 
 / 1 1 .a/?lll 
 
 t — means ? • — and not — • / . Thus we can write — — = a—/?— = a— 
 
 i i I P r ^ r r 
 
 =— . But we cannot write —• —=—, for ^— «— = /?( — 7-) a ( — .5) does 
 
 r r ^ r r ^ 
 
 not allow the 5's to cancel each other, the vector factors not being 
 commutative, § 44. 
 
QUATERNIONS 
 
 23 
 
 a 
 
 and ■3-=cos ^— £ sin ^, since this satisfies both the limiting 
 
 cases — = 1 and -^ = +k, being the angle from a to /?. 
 % 1 
 
 Examples for Practice 
 
 — k_. ik _ 
 
 j k k' 
 
 — =k. —i--~j = k 
 
 
 -i 
 
 ^=i. v^pk;^ = -m'- Yj = -^'-^ = --J-j^l 
 
 -7=1. ~]==l. 
 
 ■k y 
 
 k . .i . 
 
 1 1 
 
 j k 
 
 ■I- —1=1^ 
 
 ikj = kji == jik = —i'^ = ~p = l. 
 
 49. Since aa — =a- —a=l= — aa = —a-a, 
 aa aa * ^ 
 
 therefore a vector is commutative with its reciprocal. 
 60. Since ^= — /?, we can write 
 
 1 
 
 _/?=-^./? = l. 
 
 ^=aj = a(-^) 
 
 Hence, § 42, 
 
 = -cos {n^6) +£sin {jz^O) 
 = cos ^— £ sin d, 
 where, as before, is the angle from a to ^, 
 
 ■H 
 
24 QUATERNIONS 
 
 Similarly, 
 
 pa = —cos { — 0) +£ sin { — 0) = —cos — £ sin ^, 
 
 —a = — cos (tt — ^) + £ sin {n — 0)= cos 6^ + £ sin ^. 
 
 Hence -^a 
 
 r'^(?=4)' 
 
 —^ = — cos (tt + /9) + £ sin (tt + ^) = cos ^ — £ sin ^^ 
 
 — = — cos (jz — d) +£ sin (n—O) =cos 6^ +£ sin ^. 
 a 
 
 61. Introducing the tensors of a and /?, and collecting the 
 results, we have, 
 
 a/?=a6( — cos 6 + £ sin ^), ■^=^(cos — e sin ^), 
 
 Ba=ab ( — cos 6^ — £ sin ^), — =— (cos 6' + £ sin ^). 
 
 ^ ' a a 
 
 52. Distrihutivity of the vector multiplier. Let a, /?, ;- be 
 three unit vectors, making with each other the angles (j), 0, 
 2a, as shown; (^ is not a unit vector. 
 
 £ is the axis of a/?, -q oi ay^ X^ oi ad', coplanar, since the 
 the planes of the three quaternions have the common 
 edge a. 
 
 P + r=^, 1^1 =2 cos a. 
 
 [|^l means length of d. cos a=^ diag. of the parallelogram 
 
 on /?, r-] 
 
 a/? = — cos d + £ sin 0, ay = — cos + >? sin 0- 
 
 .*. a^ +«;-=- cos ^ — cos ^+£ sin ^ + )^ sin j>. 
 
QUATERNIONS 
 
 25 
 
 But by trigonometry, since the angle between the axes is 
 the same as the angle between the planes of the quaternions, 
 and since a=a' makes the sines of these angles propor- 
 tional to the sines of the adjacent sides, that is, sin 0, sin ^, 
 therefore £ sin ^ + >? sin will lie along the ^ axis and 
 
 (1) 
 
 «/? + «;- = — cos ^ — cos (l)-\-xX^, 
 
 where x is some unknown tensor. 
 
 (2) But «(/? + /-) =a^ =2 cos a( — cos + 1^ sin 0), and we 
 now have to show that this agrees with a^-^ay. 
 
 By trigonometry, 
 
 cos = 
 
 cos (j) =cos i[} cos a +sin ^ sin a cos 
 
 cos ^ = cos ^ cos a— sin ^ sin a cos 
 
 cos <j) — cos cos a _ cos (/» cos a — cos 8 
 sin (j) sin a sin 9^ sin a ' 
 
 cos<A+cos^ , 
 
 ,*. — ?r- =cos 0. 
 
 2 cos a 
 
26 QUATERNIONS 
 
 (3) .*. —2 cos a cos ^ = — cos 9^ — cos ^. 
 
 By trigonometry again, 
 x^ =sin2 -\-sm^ cf) +2 sin sin cj) cos A, 
 cos 2a = cos (j) cos ^ +sin d> sin 6 cos A. 
 .-. z^=sm^ ^+sin2 0+2(cos 2a — cos cj) cos 0) 
 
 = sin^ ^ + sin^ ^ + 2 (cos^ a — sin^ a — cos (f) cos 6^) 
 = 1 — cos^ ^4-1 — cos^ (j)+2 cos^ a — 2+2 cos^ a 
 
 — 2 cos <^ cos d 
 =4 cos^ a — cos^ — 2 cos ^ cos <p — cos^ <^ 
 =4 cos^ a— (cos ^S +COS /9)2. 
 But 
 
 , ^ /-. /cos oS + cos /^X 2 V 4cos2a— (cosQ^+cos^ (9)2 
 
 sin^ = \l- — ^r^- = . 
 
 ^ ^ \ 2 cos a / ^ cos a 
 
 (4) .'. x=2 cos a sin ^. 
 
 .-. «/?+«;-=— 2 cos a cos 0+2 cos a sin ^- 1^ (1).(3),(4) 
 
 =2 cos a( — cos 9^+ sin (Jf-'Q 
 
 =a(J^ + r)^ Q.E.D. (2) 
 
 Hence in vector multiplication, the multiplier is dis- 
 tributive over the operand. 
 
QUATERNIONS 
 
 27 
 
 53. li al^ ^ab(- cos -\-£ sin 0) =q. 
 
 ah is called the tensor of q and is symbolized by 
 ab = Tq. 
 
 — ah cos 
 
 ah sin 6 • e 
 
 ah sin d 
 
 — cos ^4-£ sin ^ 
 
 cos d 
 
 sin ^ • e 
 
 sin d 
 
 scalar part of q and is symbol- 
 ized by Sq. 
 
 vector part of q and is symbol- 
 ized by Vq. 
 
 tensor of the vector part and is 
 
 symbolized by TVq. 
 
 unit part and is symbolized by 
 Uq. 
 
 unit vector of the vector part 
 
 and is symbolized by UVq. 
 
 scalar of the unit part and is 
 
 symbolized by SVq. 
 
 vector part of the unit part and 
 is symbolized by VUq. 
 
 tensor of the vector part of the 
 unit part and is symbolized 
 by TVUq. 
 
 Exercises 
 
 54. Show that 
 {TVq)^ = -{Vq)^, 
 
 {TqY = {Sq)^-{VqY, 
 
 UVq = 
 
 TVq' 
 
 TVUq^^. 
 
28 QUATERNIONS 
 
 If q=w+xi+yj-\-zk, § 45, show that Sq=w. 
 
 Vq=xi+ijj-\-zk^ (1) Tq=Vw^+x^+y^+z^, 
 
 TVq =Vx^ +7/2 +z2, f/5 = 2-r^ 
 
 V'M;^+a:^+?/^+z^ 
 
 \/a;2 +2/2+^2' 
 
 If g and r are quaternions 
 
 C/g 
 
 (2) T'qr = TqTr, U-qr=UqUr, U^ 
 
 If q and r degenerate to a, T-a^ = —a^. [a^ 
 
 55. By § 54, Eqs. (1), (2), 
 
 T^r = V W^ +X2 + F2 +Z2, 
 where 
 
 r9=\/'w;i2+a;i2+?/i2+^i2^ Tr=\/^i^?T^2M^^^?+^2, 
 
 .-. Tf2+X2 + F2+Z2 
 
 = (Wi^ -\-Xi^ +yi^ +^l2) (W2^ +X2^ +2/2^ +2;22) , 
 
 or (Euler's Theorem) the sum of four squares may be 
 resolved into two factors, each of which is the sum of four 
 squares. 
 56. Show that 
 
 VaP = - Vpa, ap-pa = {Sap)'^-(Vap)^ 
 
 aP^Pa=2SaP, . (1) ^{TaQ)^. 
 
 (2) Sx^x, Sij=0, Vij=k. 
 
QUATERNIONS 
 
 29 
 
 67. Example. a:+/? = 7-. 
 
 (a ■VpY = f =a^ +al3 ^^a +/?2, 
 
 or § 56, eq. (1), -a^ +2Sa^-h'^ = -c^. 
 
 If this is a right triangle, at C, then 
 § 56, eq. (2), ^a/?=0 and a?-Vh'^=c^. 
 If not a right triangle, this becomes 
 
 ^2^n2 
 
 a? -\-h'^ — 2ah cos c. 
 
 (Law of Cosines. Trig.) 
 
 58. Geometric meaning of TVap and Sap. 
 
 Va^=ab sin d-e, 
 f^ f TVap=ab sin 
 
 = parallelogram on a/?. 
 Sa[^ = —ah cos 
 
 = — (one tensor -projection of the other upon it). 
 
 59. Meaning of Sap;-. Suppose a, p, y unit vectors, 
 ^= angle between a, 8, and 0= angle between ;- and plane 
 of a0. 
 
 ap = — cos /9 + £ sin 6, 
 
 SaPr=S-{-cosd ^-esmd)r 
 
 =Sey sin d. 
 
 But S£y = —sin 0, 
 
 .'. Saj^j- = — sin ^ sin 6. 
 
 If a, /?, ;- are not unit vectors, but have the lengths 
 a h, c, respectively, then 
 
 Sa^f = — ahc sin d sin <j) 
 
 = —volume of the parallelopiped on a, /9, ;-, as edges. 
 
30 QUATERNIONS 
 
 60. If Sal^-)'=0, «, /?, ;- are coplanar, and vice versa. 
 
 61. The sum of a scalar and a vector is a quaternion. Let 
 
 I/; = some scalar; A=aa =some vector. 
 
 Then w -\- afv =\^ rn^ -{- a'-^ I ^ - +<t ^ | 
 
 Ww^+a^ Vw^+a^/ 
 
 =a tensor (cos (j)+a sin 0) 
 
 = a quaternion. (§ 51.) q.e.d. 
 
 62. Idiographic proof. We can always construct in a 
 plane perpendicular to the vector a right-angle triangle, 
 one of whose legs shall equal the diameter of the sphere 
 which is the idiograph of the scalar, and the other leg 
 the tensor of the given vector. If we denote the hypoth- 
 enuse of this triangle by a6, then the leg corresponding to 
 the scalar length will be ah cos 6, and the other leg will be 
 ab sin 6, 6 being the angle adjacent to the first leg con- 
 structed. 
 
 On the sides including 6, lay off respectively the distances 
 a and 6 as vectors. Then the product of these two vectors 
 will be a quaternion, 
 
 — a& cos ^+a& sin ^-e, 
 
 which will be the given sum, and at the same time the 
 product of two vectors. 
 
 63. We are now able to distinguish between the different 
 branches of mathematics of discrete magnitudes. 
 
 First, the mathematics of numbers, having size only. 
 This is Arithmetic, the algebra of tensors. The operations 
 are four only, addition, subtraction, multiplication, and 
 division. The operands are tensors. 
 
QUATERNIONS 31 
 
 Secondly, the mathematics of numbers having size and 
 sense, using in addition to the previous operations, reversion. 
 Its operands are scalars, generally symbolized by letters 
 and numerals. This is Algebra, in its broad sense, including 
 Calculus and allied subjects. 
 
 The usual line of demarcation between arithmetic and 
 algebra, the use or non-use of literal characters, is unphilo- 
 sophical and erroneous in its significance. If the letters 
 represent tensors (e.g., number of girls, boys, children), 
 gf +6 =c is arithmetic purely, and 4 — 5 = — 1 is pure algebra. 
 
 Thirdly, the mathematics of numbers having size and 
 sense, and which adds to the previous operations that of 
 mean reversion. This is the algebra of Complex Functions. 
 Its operands are scalars, symboHzed by strokes as well as 
 by letters and numerals. 
 
 Fourthly, that branch of mathematics which deals with 
 numbers having size, sense, and direction. Its operands 
 are scalars and vectors and combinations of these. This is 
 the subject of Quaternions. 
 
CHAPTER VI 
 
 KINDS OF QUATERNIONS 
 
 64. The reciprocal of a quaternion, denoted by Rq, is 
 the factor into which the quaternion must be multiplied in 
 order to produce unity. 
 
 If q=aab^, 
 
 11 11* 1 
 
 then aabB-f-p, — =aQyr •— =a— = l. 
 
 ^bp aa ^ p a a 
 
 Hence the reciprocal of a quaternion is found by taking 
 the product, in reverse order, of the reciprocals of the 
 factors. 
 
 6/3 "^aa 
 
 1 
 
 a/? 
 
 1 1 
 
 — — — rt/^o H 
 
 -e sin 6. 
 
 q — COS +£ sm u 
 
 1 1 
 
 — COS 
 
 ^ + £sin^. .-. -^7^--^. 
 af:{ a /? 
 
 11111 
 j^'a a^^a'p' 
 
 
 I{ap=q, 
 
 
 ^^4r^"- 
 
 
 * The assumption made here that ^—•a = B-—a is justified later, 
 
 in §90, since all vectors are (§ 97) quaternions. 
 
 32 
 
KINDS OF QUATERNIONS 
 
 
 R(xl^=-(-cosO-£sm d), ^'- = 1, 
 ^ ab ' ' ^ a ' 
 
 <^1- 
 
 a 1 a 1 a 1 ^ ^ 1 1. 
 
 
 33 
 
 Hence, a quaternion and its reciprocal are commutative: 
 
 65. Opposite quaternions are those in which one factor is 
 reversed. 
 
 a^=ab{ — cos d+s sin 6) 
 
 =ab(cos {n-d) +£ sin {n-O)), y^^ 
 
 a{-p) =ab(cos d-ssin d). y a '* 
 
 Hence if a/?=g, a{ — ^)=—q '^ 
 
 and g + (-5)=0, 
 
 or the sum of two opposite quaternions is zero. 
 
 or the quotient of two opposite quaternions is — 1. 
 Tq = T{-(i)=ab. 
 
 Opposite quaternions have a common plane, equal tensors, 
 supplementary angles and opposite axes. 
 
 66. The Conjugate of a quaternion, denoted by Kq, has 
 one of its factors turned an equal angle across the other 
 factor, the order of the factors not being changed: it 
 
 reverses the angle of the original quater- 
 
 ^ nion. 
 
 Kq=Kap=aP' 
 
 =abi- cos (-^)+£sin (-0)) 
 =ab( — cos d — e sin 0). 
 
54 QUATERNIONS 
 
 Kq=Sq-Vq. Kx=x. 
 
 q+Kq=2Sq. ap=K^a. 
 
 q — Kq=2Vq. §55. Kq=w-xi-ij]'-kz. 
 
 Conjugate quaternions have a common plane, equal 
 angles between the factors, equal angles (see § 76), equal 
 tensors, and opposite axes. (§ 46.) 
 
 67. Since TKq = Tq and UKq=^^. 
 
 .-. Kq = TKq-UKq=^^ 
 Whence by multiplication and division, 
 
 qKq = {Tq)\ Tq^^^'^^'' 
 
 Kq=Ka[^ = Tq-^ =ah{-cos O-ssm 6) =pa. 
 
 a 1 
 .*. Kap=^a and K-r=-ra. 
 
 Evidently, Kxq = xab { — cosO — £smd)= xKq. 
 Making x = — l, we have 
 
 K{-q) = -Kq. 
 
 68. 
 
 1 
 
 kI.kI.tI ' . 
 
 a a a j-Ji 
 J a 
 
 1 -rh"- 
 
 
 "f 
 
 -^a 1 \x^r 
 
 ■• 9 Kq 
 
 (X 
 
 Confirm this by diagram, using a/? instead of -^. 
 
KINDS OF QUATERNIONS 
 
 35 
 
 69. Collecting the q, q^^, —q, and Kq into a table, we 
 have: 
 
 
 
 
 T 
 
 0. 
 
 U 
 
 a 
 
 q 
 
 — ( — cos (b-\- £ sin c6) 
 
 
 
 Tq 
 
 
 
 Uq 
 
 I 
 a 
 
 q~* 
 
 —( — cos 96— £ sin 0) 
 a 
 
 1 
 
 -0 
 
 1 
 
 — a 
 
 -q 
 
 —(cos 9S— £sin (56) 
 
 
 7^? 
 
 7r+«9^ 
 
 1 
 
 Uq 
 
 1 
 
 r 
 
 Kq 
 
 —( — cos ^ — £sin ^) 
 
 
 Tq- 
 
 -<i> 
 
 1 
 
 ^1 indicates the angle from the multiplier to the multi- 
 plicand, and not the angle between a and /?. 
 
 From this table it is evident that if ^=0, KS=S, i.e.. 
 The conjugate of a scalar is the scalar itself. 
 
 If (f)=—,KV = — V, The conjugate of a vector is its opposite. 
 
 70. RRq = (-) =q, The reciprocal of the 
 
 ^ ^ reciprocal 
 
 — ( — q)=q, The opposite of the 
 
 opposite 
 KKq =g. The conjugate of the 
 
 conjugate 
 
 71. Equality of quaternions. If a, /?,;-, <? are four 
 coplanar unit vectors with the same angle d between a and 
 /?, and ;- and d, then 
 
 a/? = — cos ^ + £ sin 0, yd = — cosd + £ sin ^, 
 
 or a/? = yd. 
 
 Similarly, -^=^=cos^ — esin^. 
 
 is the quater- 
 nion itself. 
 
36 QUATERNIONS 
 
 Hence, Revolving the factors of a quaternion in the plane 
 of the quaternion does not alter the quaternion. 
 
 Quaternions having the same S and V parts must be the 
 same. Diplanar quaternions, that is, quaternions not 
 having their factors in the same plane, cannot have the same 
 V part, and therefore cannot be equal. To be equal, two 
 quaternions must be coplanar. 
 
 72. Two quaternions can always be transformed so as to 
 have the same vector for the numerator of one and the 
 denominator of the other, or what is the same thing, the 
 multiplicand of one the multiplier of the other. 
 
 For moving the quaternions in their planes until the 
 denominator of one and the numerator of the other lie 
 
 along the line of intersection of the planes, a vector along 
 this line can be taken as the numerator of one quaternion 
 quotient and the denominator of the other. Thus 
 
 a u r XV V 
 
 p V - o p p 
 
 X 
 
CHAPTER VII 
 THE QUATERNION AS A MULTIPLIER 
 73. Into the reciprocal of its multiplicand factor. 
 
 /5 
 
 li=qp^a. 
 
 Result: The quaternion multiplier turns the multiplicand 
 through the angle —0,'m. the plane 
 of the quaternion. 
 
 74. Into its mulitplier factor. 
 
 cc 
 qa =-n(x =«: COS d—ea sin 
 
 =a^Q: =q: cos -\-ae sin 0. 
 
 [§51. 
 Result: Same as in § 73. 
 
 75. Into any coplanar factor , y. 
 
 = —xa{co^ 0+£ sin 0) +ya = —xa cos d — x sin d'a£-\-ya. 
 at , 
 
 {-xco8e-^v)cL 
 Result: Same as in § 73. 
 
 37 
 
38 
 
 QUATERNIONS 
 
 76. Into any quaternion. Let the quaternions be reduced, 
 § 72, to a common numerator and denominator respectively, 
 viz.: 
 
 
 Denote the arcs which de- 
 termine the planes of the qua- 
 ternions by c, a, h respec- 
 tively, as shown in the figure. 
 The effect of the quaternion 
 whose axis is ax-c operating 
 upon the quaternion whose 
 axis is ax -a is to give a new 
 quaternion whose axis is ax • h. 
 This can be analyzed as follows: Suppose we move the 
 arc a horizontally along c, being careful not to change its 
 inclination to c, until it takes the position a'. Theaa;-a 
 will move horizontally around ax • c (conical revolution) until 
 it takes the position ax -a'. Then as we turn a' down to 
 coincidence with h around a as an axis, ax • a' will move up 
 the arc passing through ax-c (the equator of a) until it 
 takes the position ax • b. 
 
 That is, the multiplication of ^ into — turns the axis of — 
 
 (X 
 
 horizontally, i.e., parallel to the plane of -^ (conical revolu- 
 tion) through an angle equal to that between a and ,/?, and 
 raises or lowers it through an angle which depends entirely 
 
 upon — . Whatever the position of f with reference to 5, 
 
 the horizontal revolution is always the same, dependent 
 upon a and 8. 
 
THE QUATERNION AS A MULTIPLIER 39 
 
 OL 
 
 Hence, the effect of -^ as a multiplier is generally to revolve 
 
 the axis of the multiplicand conically through a definite angle 
 around the axis of the multiplier. This angle { — here), 
 the angle through which the axis of the operand is moved, 
 is called the angle of the quaternion, and is designated by 
 / q, or in these notes by D. 
 
 77. Since the effect of a quaternion, g, as a multiplier is 
 to revolve the multiplicand (axis) through the angle /.q 
 around the ax-q, the arc of a great circle which measures 
 Zg on the equator of ax-q, can be taken as an indirect 
 measure of Uq. 
 
 Just as a quaternion can be revolved in its plane without 
 alteration of value, so the arc representing it can be moved 
 in its great circle without alteration of value. 
 
 Arcs in different great circles cannot be equal. If 
 they are semicircles, any semicircle will have the same 
 effect. 
 
 78. Representing the quaternions -r, — , — , by their arcs 
 
 (arc strokes) c, a, b respectively, we have, considering the 
 arcs as arc strokes on the sphere surface similar to the 
 plane strokes of § 12, 
 
 a 8 a , T 
 
 77'— =— or a-{-c=o, 
 
 P r r 
 
 and we have reduced the multiplication of 
 quaternions to the addition of arc strokes. 
 In the same way 
 
 a r r 
 
 6 + (-c)=a. 
 
40 
 
 QUATERNIONS 
 
 1^^" 
 
 79. Notice carefully that the arc strokes corresponding 
 to the quaternion factors read from the left, are added up 
 from the right. 
 
 80. Evidently from the figure qq' is quite a different arc 
 from q^q, and hence, The addition of arc strokes is not commu- 
 tative, unless they are coplanar; and correspondingly, the 
 7nultiplication of quaternions is not commutative, unless the 
 quaternions are coplanar. 
 
 Rule for multiplication by arc strokes. 
 To the arc stroke of the multiplicand 
 add the arc stroke of the multiplier. 
 
 Rule for division by arc strokes. 
 Reverse the arc stroke of the divisor 
 and add the arc stroke of the divi- 
 dend. 
 Show that 
 
 qKq^Kq-q. q{-q)=-q.q. 
 
 11 -1-1 
 
 q-=-q=qq ^ =q \ 
 
 81. From this we see that the product of a quaternion, is a 
 quaternion, the sum of two arc strokes being an arc stroke. 
 
 Take two quaternions in the form of a scalar plus a 
 vector, and show that their product is a scalar plus a vector, 
 and therefore, § 61, a quaternion. 
 
 82. By multiplying the various forms of a quaternion, 
 
 a B . 
 
 ap, /?«, ■^, — , etc., into some vector, whatever will give the 
 
 result most readily, we find the following angles for the 
 quaternions, d being the angle from a to /?, Oi the angle 
 from the multiplier to the multiplicand, and D the angle of 
 the quaternion. 
 
THE QUATERNION AS A MULTIPLIER 
 
 41 
 
 2 
 
 D 
 
 e. 
 
 q 
 
 D 
 
 0^ 
 
 a/? 
 
 Tz-e 
 
 d 
 
 1 
 
 d 
 
 TZ-d 
 
 /?« 
 
 TZ+d 
 
 -d 
 
 a 
 
 -0 
 
 K + O 
 
 I 
 a 
 
 -d 
 
 TZ-\-d 
 
 -«/? 
 
 -e 
 
 -TZ + d 
 
 d 
 
 iz-d 
 
 
 
 
 Inspection of the table shows that: The angle, D, of a 
 quaternion is the supplement of the angle between the 
 factors measured from the multiplier to the multiplicand. 
 
 83. Representing the angle measured from a to ^ by ^, 
 
 and the angle of the quaternion by D, the angle of a/? by 
 
 ex. 
 (Jf, the angle of -^ by (f), and abbreviating cos D+£ sin D by 
 
 C£S D, we get the following table: 
 
 a/? 
 
 CcS (p 
 
 C£S {71+ (l)) 
 
 C£S (tz-O) 
 
 C£SD 
 
 fia 
 
 C£S (271-^) 
 
 C£S(7Z-c})) 
 
 C£S (tZ + O) 
 
 CSS D 
 
 a 
 
 1 
 
 C£S (7r+ (/f) 
 
 CSS <p 
 
 ces{-0) 
 
 C£SD 
 
 I 
 a 
 
 CSS (tz— (/>) 
 
 CSS (27r-f/.) 
 
 cesO 
 
 C£SD 
 
 1 
 
 r 
 
 CSS (tz— (p) 
 
 C£S(-0) 
 
 cesO 
 
 cesD 
 
 a 
 
 C£S {tZ-\- 4>) 
 
 C£S </> 
 
 ces(-O) 
 
 cesD 
 
42 
 
 QUATERNIONS 
 
 84. Since a vector is the special case of a quaternion, one 
 whose angle is — , we can represent a vector by an arc stroke 
 
 of — on its equator. As an example 
 
 take the case -^y. 
 
 a 
 
 arc -A=c, 
 
 arc y=a, 
 
 :/'=a+c =6 =arc <?, 
 
 and y has been revolved through Z ^. 
 
 85. Collecting the results of these articles we have the 
 table on the following page, where 
 
 di=7t-D, D=lq, il^-^lap, 
 
 *=z| = /l., 
 
 d= L. — = /_ 77a. d\ =angle from first factor to the second. 
 a [i 
 
 = angle from a to /?. 
 
 86. Since quaternions are multiplied by adding their arc 
 strokes, the square, q^, of a quaternion will have for its 
 representative arc stroke twice the stroke of q. Similarly, 
 one-half the arc stroke will be representative of qh. But 
 since there are two arcs D and — {2n — D),q^ will have two 
 
 representative strokes, ^D and —71 -{-—-, either of which if 
 
 A 
 
 doubled will give the stroke of q. Hence, Rule for extraction 
 of square roots of unit quaternions: Halve the angles of the 
 quaternion. 
 
THE QUATERNION AS A MULTIPLIER 
 
 43 
 
 
 w 
 
 «> 
 
 «o 
 
 Ui 
 
 (0 
 
 <o 
 
 01 
 
 
 
 «C> 
 
 <as 
 
 
 
 
 • 
 
 t> 
 
 
 !? 
 
 
 
 
 
 <ji 
 
 
 
 
 
 -§ 
 
 1 
 I 
 
 1 
 
 
 1 
 
 
 1 
 
 1 
 
 
 
 CO 
 
 <3i 
 
 
 
 
 
 ■^s 
 
 
 
 02 
 
 m 
 
 «15 
 
 <^ 
 
 <55 
 
 <IS 
 
 CO 
 
 
 t? 
 
 o 
 
 s 
 
 o 
 
 o 
 
 
 § 
 
 O 
 
 o 
 
 
 
 1 
 
 1 
 
 e l-o 
 
 ^1 c 
 
 el^ 
 
 -o 1 e 
 
 
 
 ^ 
 
 ^1 « 
 
 -1^ 
 
 ^1 « 
 
 «1<^ 
 
 
 
 1 ^ 
 
 
 
 -e- 
 
 -e- 
 
 
 -e- 
 
 
 -e- 
 
 
 
 
 + 
 
 1 
 
 -0- 
 
 1 
 
 -0- 
 
 1 
 
 -0- 
 
 
 w 
 
 K 
 
 ti 
 
 
 1 
 
 
 
 
 
 
 -^ 
 
 -^ 
 
 -^ 
 
 -^ 
 
 -^ 
 
 -^ 
 
 
 NJ 
 
 -S- 
 
 1 
 
 + 
 
 1 
 
 + 
 
 1 
 
 + 
 
 
 
 
 
 K 
 
 k 
 
 K 
 
 ti 
 
 ti 
 
 
 Qi 
 
 <5S 
 
 ■5^ 
 
 
 <& 
 
 
 <3i 
 
 
 
 1 
 
 + 
 
 1 
 
 <5i 
 
 1 
 
 CO 
 
 1 
 
 k |(N 
 
 
 t: 
 
 k 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 « 
 
 -o 
 
 e 
 
 c l-o 
 
 ro 1 e 
 
 ^1 e 
 
 e IrO 
 
 -§ 
 
 rH 
 
 ■3 "5 
 
 c& 
 
 <3i 
 
 <3S 
 
 
 <^ 
 
 
 <C> 
 
 
 1 
 
 + 
 
 1 
 
 <3i 
 
 1 
 
 <& 
 
 1 
 
 H l(N 
 
 Q s 
 
 ti 
 
 t; 
 
 1 
 
 
 
 
 1 
 
 
 
 
 
 /— N 
 
 
 
 
 
 
 
 
 
 <& 
 
 
 
 
 
 
 
 
 <3S 
 
 1 
 
 
 
 
 
 
 
 <5i 
 
 C3 
 
 
 ■55 
 
 <^ 
 
 <^ 
 
 <3i> 
 
 
 
 .5 
 
 'm 
 
 c 
 
 fl 
 
 c 
 
 ^ 
 
 • S 
 
 
 
 
 
 
 
 •1-1 
 
 
 
 
 
 
 *> 
 
 *M 
 
 'm 
 
 m 
 
 n 
 
 *m 
 
 
 K?^ 
 
 4- 
 
 1 
 
 Ui 
 
 U) 
 
 u> 
 
 x> 
 
 «« 
 
 <u 
 
 !::) 
 
 •5^ 
 
 <I5 
 
 + 
 
 + 
 
 1 
 
 + 
 
 1 
 
 
 
 tn 
 
 22 
 
 ^ — s 
 
 <& 
 
 ■^s 
 
 <I5 
 
 <5i 
 
 
 
 O 
 
 O 
 
 <IS 
 
 03 
 
 m 
 
 
 
 
 
 o 
 
 t» 
 
 I 
 
 O 
 
 O 
 
 Q 
 
 o 
 
 
 
 1 
 
 1 
 
 •>— / 
 
 o 
 
 « 
 
 « 
 
 « 
 
 
 
 
 
 02 
 
 
 
 
 
 
 
 
 
 o 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 ■Si 
 
 1 
 
 CO 
 
 <C. 
 
 qii 
 
 <^ 
 
 <C) 
 
 
 
 <3i 
 
 + 
 
 1 
 
 + 
 
 1 
 
 + 
 
 klc^ 
 
 
 
 1 
 
 ti 
 
 K 
 
 1=; 
 
 ti 
 
 tJ 
 
 
 <51 
 
 
 4 
 
 «|Q3. 
 
 Q3.I « 
 
 ^1 « 
 
 
 1 
 
 <» 
 
44 QUATERNIONS 
 
 87. Since the angle of the quaternion is the supplement 
 of the angles measured from the multiplier to the multi- 
 plicand, that is, 7t — D=6i, a better working rule would be: 
 
 For square root. Revert the multiplicand and halve the 
 angles between this and the multiplier, for the factors of qi. 
 
 Similarly, a working rule for squaring quaternions: 
 Double the angle between the factors and revert the multipli^ 
 cand in its new position for the factors of q^. 
 
 In ^2 =py, or - =a|^, ^ bisects the angle between /? and ;-. 
 
 88. Using the arcs of quaternions, show that 
 
 (1) arc (72 = arc g+arc g=2 arc 9±2n;r. 
 
 (2) arc g^ = } (arc q ± 2nn) . 
 
 (3) arc g^=arc y+arc q-{- . . . =n arc q±2n7t. 
 
 i 1 
 
 (4) arc g^ = — (arc q ± 2n7i) . (5) (pg) ^ ^^ p^^. 
 
 89. When the quaternion is a scalar, its arc is some 
 multiple of n. All coinitial arcs of the same length, n, for 
 instance, will represent will represent the same scalar, — 1, 
 however the arcs are situated; that is, whatever the 
 direction of their axes. In halving these arcs to extract 
 square roots we will get an infinite number of answers, 
 one for each arc, each corresponding to its own axis. Or 
 in other words, there are an infinite number of vectors the 
 square of which is a scalar. 
 
 Similar reasoning shows that there are an infinite number 
 of quaternion nth roots of a scalar. On this account the 
 roots of a scalar are limited to scalars. 
 
 From Eq. (2) of § 88 we get (pg)^ = cos — +£ sin — ; 
 
 repetition of which gives (/>g)° = cos = 1, 
 
 thus illustrating for the quaternion number the familiar 
 
 fact of algebra. 
 
CHAPTER VIII 
 PRODUCTS OF QUATERNIONS 
 
 90. Having three quaternions, let the first two be reduced, 
 § 72, to a common denominator and numerator respect- 
 ively, viz., q=-n, ^=— ; and the latter two also, viz., r=— , 
 
 s =— , where, evidently, since they represent the same quater- 
 nion, 
 
 r ^' 
 
 The product of the three quaternions is 
 
 ■qr8. 
 
 At ^}p 
 
 Associating the first two factors, we have, 
 a Q V a V 11 
 
 ?r p r p r p 
 
 Associating the second two, we have 
 
 a I ti v\ a a 1 1 
 
 /?\y p) P p r P 
 
 But if ^=^ then ^=A 
 
 45 
 
46 QUATERNIONS 
 
 Multiplying this by - and into //, we have 
 1 1 
 
 Whence =-t,-— or qr-s^q-rs, 
 
 r P ? P 
 
 or quaternions are associative in multiplication. 
 91. Since, § 67, 
 
 Kq = -^ = Tq{-cos O-esin 0) =Tq(-cos(-0) +£sin(-^)), 
 
 the arc representing Kq will be the reverse of that repre- 
 senting q, or arc Kq = — arc q. 
 
 ar 
 
 Inspection of the figures shows that 
 
 — arc gr = +arc KrKq. 
 Hence 
 
 K(qr) =KrKq, or 
 
 The conjugate of the product of two quaternions is the 
 'product of their conjugates in a reverse order. 
 
 K{aP) = -p- -a=da. Conf. § 67. 
 
 92. Coplanar quaternions. Using arc strokes for q and 
 r show that 
 
 qr=rq. Conf. § 80. 
 
PRODUCTS OF QUATERNIONS 47 
 
 Therefore, since a quaternion and its reciprocal, opposite, 
 conjugate, and any power are coplanar, 
 
 qKq=Kq-q. q~'^ • —q = —q-q~'^. Kq-q'^^q^Kq. 
 
 93. Using q=Sq + Vq, r=Sr + Vr, show that qr is a 
 quaternion, the sum cf a scalar and a vector. 
 Show that 
 
 Sqr =SqSr +8- VqVr. V • VqVr = -V- VrVq. 
 
 Vqr=SqVr-\-SrVq-\-V'VqVr. qr^^rq. unless coplanar. 
 
 Srq =SrSq +S ■ VrVq. T-qr^ TqTr. 
 
 Vrq^Sr^q +SqSr + V • VrVq. V-qr = VqVr. 
 
 Srq = Sqr. S-qrj^ SqSr^ 
 
 unless S'VqVr^O, i.e., unless the planes of the two qua- 
 ternions are perpendicular. 
 
 Vqr 7^ Vrq, 
 
 unless V • VqVr^O, i.e., unless the quaternions are coplanar. 
 Since 
 
 (p +^) ir +s) =V^ +?^ +ps +5^ =P^ +ps +9^ +?^ =etc., 
 
 therefore the distributive and associative law applies to 
 quaternions, verified by resolving the quaternions into their 
 vector factors and applying the results of § 52. 
 
CHAPTER IX 
 
 VERSORS 
 
 94. i[/a/? = — cos ^+£ sin ^=cos D+£ sin D, turns any 
 quaternion to which it is applied as a multiplier, horizontally 
 through the angle D. 
 
 li D=— then Ua^=£^ turns the multiplicand quaternion 
 through the angle — . Applied again it turns it through 
 
 Zi 
 
 another right angle, or in all through the angle n. That is 
 £ turns the multiplicand through one rt. Z , £^ turns it 
 through two rt. Z 's, and so on.* Hence, by the law of 
 mathematical continuity, e^ should be the symbol of turning 
 
 2 
 
 through J rt. Z , etc. If ^ is expressed in radians, z "^ turns 
 through the angle 0. 
 
 If £ is perpendicular to its operand, we recognize the 
 familiar equations, 
 
 ij =kj i^j = — /, i^j = —k, etc. 
 
 According to this notation then, 
 
 2 
 
 UaS = cos 7) + £ sin 7) = £ 'f . 
 
 ♦Remember that e^ does not mean here e multiplied by itself, but 
 e applied twice to some operand. See Appendix. 
 
 48 
 
VERSORS 49 
 
 .2-D 
 
 96. Since e and —e are opposed, £ "" , which turns clock- 
 wise from the upper side of the plane, must have the same 
 
 effect as ( — £) '^ , which turns counterclockwise from the 
 lower side of the plane. Hence 
 
 r"^'' = (-£)^'', or e-^ = (-£)^. 
 
 If D =-Jr, then £~^ = — £, or to put it in the more familiar 
 
 form, i~^ = —i. 
 
 96. Again, § 85, 
 
 8 -e 
 
 U- = c,os d-\-£ sin d^e"" , 
 a ' 
 
 and 
 
 C/|=cos (-0) +£ sin (-0) =£ 
 
 therefore, £~^=-t. 
 
 
 From these £- ^=cos ( — 6^) +£sin (- 
 
 -d)=(-s)^, 
 
 — (s^) = — cos d — £ sin d. 
 
 
 Hence in general. 
 
 
 {-e)' = e-<>^-U'), 
 
 
 unless ^ = 1, in which case, as before, e ^ = — e. 
 
 97. The unit portion of a quaternion (considered as the 
 symbol of an operation to be performed by the reader) 
 results in the turning of the operand, and is therefore aptly 
 termed a versor. 
 
60 QUATERNIONS 
 
 It must not be forgotten that all vectors (considered as 
 symbols of an operation to be performed by the reader) 
 are versors, as indeed all vectors are special cases of qua- 
 ternions. The evanescence of the scalar part does not 
 affect its versorial character. 
 
 i, j, and k are called quadrantal versors. 
 
 98. The method of expressing the unit part of a quater- 
 nion as the power of a vector gives a convenient method of 
 indicating multiplication and division of coplanar quater- 
 nions. 
 
 Uq- 
 
 = COS(j) +i 
 
 ■sm(j)=£'' . Uq' = 
 
 = COS 
 
 (p+£ sin (/f = 
 
 1^ 
 
 £"" . 
 
 •*• 
 
 UqUq' = 
 
 = (cos ^+£ sin (j))(cos 
 
 'P+ 
 
 £ sm ([>) =£^ 
 
 
 
 = cos {(f) +(p) +£ sin (9^+^) 
 
 
 2 
 
 
 Similarly, 
 
 e<f>.e-~'^=^ = £<l>- 
 
 .4, 
 
 t 
 
 
 
 id 
 
 
 (£«^)m=^m^^ 
 
 
 
 
 Hence, The algebraic law of indices holds good for versors. 
 Hence, The angle of the product of two coplanar quaternions 
 is the sum of the angles of the two factors. 
 99. According to this notation, 
 
 a^=4:{- cos 30° +£ sin 30°) 
 
 =4(cos 150° +s sin 150°) =4A 
 
 a/? =2 (cos 60° - £ sin 60°) 
 
 =2(cos (-60°) +£ sin (-60°)) =2£-* 
 
VERSORS 
 
 51 
 
 4i 
 
 Show that (f = \ Tqicos 9S +£ sin 0) {2 = Tq)^e ^ 
 = (Tq)^ (cos 2(l>+e sin 2cl>). 
 q"' = i Tq) " (cos n<j)-\-s. sin n<f)) . 
 Hence any quaternion can be written, where p is some 
 
 vector, 
 
 24> 
 
 p'=p-. 
 
 
 100. In the diagram to § 57, 
 
 
 2A 2B 
 
 2C 
 
 2(A + B + C) 
 
 ,\ £ "^ = — 1 or 
 
 A+B+C =7r. 
 
 101. What will convert rq into qrf Assume for the moment 
 that the tensors are unity. Let the strokes of q and r be 
 as shown. Then the strokes of qr 
 and rq will take the positions shown. 
 
 From the table of § 83, we find 
 
 9 = 
 
 qr 
 
 r^-^n 
 
 and the problem becomes: To con- 
 
 vert —. into — . Now 
 a y 
 
 a a ^ 
 
 
 a y' a' 
 
 _a r' /? 
 ^ a' a 
 
 §71. 
 §71. 
 
 Or substituting for these their equivalents, § 69, 
 qr=q{rq)q-^. 
 
52 
 
 QUATERNIONS 
 
 102. Meaning of q(r)q~'^. Indicate q and r by their arc 
 strokes, as shown in the diagram. The other strokes will 
 have the values indicated. 
 
 The two triangles have two sides and the included angle 
 
 respectively equal, and, as 
 shown in the figure, by the 
 angle 6, qrq~^ is r revolved 
 in the plane of q through 
 2 /_q. This amounts to re- 
 volving the axis of r conic- 
 ally around the axis of q 
 through 2 Z g. 
 
 Hence qrq~'^ differs from 
 
 r only in being rotated through a certain angle. Hence 
 
 q{ )q~^ may be aptly called a rotator, since it rotates 
 
 any quaternion inserted in the parenthesis. 
 
 This is a special case of a more general function, called a 
 
 nonion, which we shall meet with farther on. 
 
 103. If r is some multiple of q, say q^q, then 
 
 q{q'q)q-^=qq'qq-^=qq\ 
 
 Hence qq' is q'q rotated through the angle 2 Z ? in the 
 plane of q. 
 
 Compare this with the figure of § 80, and see how they 
 agree. 
 
 Similarly, q~hq rotates r negatively through 2 /_q. 
 
 Since, Tq-T-^l, therefore, T-qrq-^ 
 
 Tr, 
 
 so that in the discussion above only the unit or versor parts 
 needed to be considered. 
 
 104. qk-'=^', 
 
 revolves the plane of /?, or /?, conically around the axis of 
 q through 2 Zq- 
 
VERSORS 53 
 
 106. apa-^=p\ 
 
 revolves /? conically around a through the angle 2;r, that is, 
 turns /? in the plane of a/? to a corresponding position on 
 the other side of a. 
 
 106. Exercises. If oiy=q, show that aqa~^ ^Kq. 
 
 If a , ^, 7- are coplanar, show that o: • /?;- • a: "^ == a '^^ya = K^y. 
 
 rqBq~h~^=rq-B'{rq)~^. 
 
CHAPTER X 
 
 INTERPRETATION OF VECTOR EQUATIONS 
 
 107. a:/?;'=a/?2/?-Y 
 
 
 a, /?, ;- coplanar, i.e., 
 
 Sa^-jr=0. U - —-X acting on /? turns it through an angle 
 
 into —a. Acting on ;- it turns it through the same angle 
 
 into d. li a, p, Y represent in direction the successive sides 
 
 a. 
 of a polygon (which is always possible) then U- —-^ must 
 
 turn ;- into a direction coinciding with the fourth side of 
 a polygon inscribed in a circle drawn 
 through the intersection of a, /?, and 
 y. The diagram shows this. Hence 
 d is in direction the fourth side of 
 the inscribed polygon of which a, /?, 
 and ;- are the other three. 
 
 If the circle passes through three 
 
 intersections, that circumscribes a /?, ^ as a triangle, then 
 
 d becomes tangent to this circle. 
 108. Locus of ^ in [^=a$. 
 
 ^=a-^l^=Sa-'^^ + Va-^l^. 
 
 Sa-'^^=0, e = Fa-i/?, 
 
 and ^ is a constant vector perpendicular to a and to ^3, and 
 therefore locates a point. 
 
 54 
 
INTERPRETATION OF VECTOR EQUATIONS 55 
 
 109. Locus of e in 7ae=/?. 
 
 /?±a. a^ =Sa^ + Va$ =x +/?. 
 
 .*. ^=a~^x-\-a~^^=za-\-j'. [since a _L/?, 
 
 and the locus of l^ is a vector through the point determined 
 by ;'(_L«, _L/?), and parallel to a. 
 
 Notice the difference in the reading and interpretation 
 of the equations of this and the previous section. In one 
 /? is the whole of a^, in the other only the vector part. 
 
 110. Locus of e in Sa$=0, Sal^=0, Sj^^^-c. 
 
 Sa^=0 limits $ to the plane of a. 
 Sal^=0 puts /? in the same plane. 
 S^^ = —c or 6a;cos^=c, 
 
 where b and x are the tensors of /?, $ respectively, makes the 
 
 c 
 projection of I" on /? a constant, viz., x cos 0=-r. Therefore 
 
 the locus of ^ is a line J_ to /? in the plane of a and through 
 
 c 
 the point of ^ distant from the origin -r- 
 
 111. From the table of § 85, we find 
 
 a/? = cos ^ +£ sin (f). (a +^) (a +/?) = (a +/?)2 
 
 /?« =cos — £ sin ^. =a^+a^+^a+^^. 
 
 a^-^a =2Vap = -2F/?a. 
 
 112. By §§ 53, 54, the last equation of § 111 becomes the 
 well-known formula for triangle, 
 
 c2=a2 4-62-2a6cos^. 
 
56 
 
 QUATERNIONS 
 
 113. As in § 111, (a:-/?)2=a2_2^Q;^+^2^ 
 
 =a2_2Fa/?-/?2. 
 
 a^.pa = {Sap + 7a:/?) (Sa/S - 7a:/?) 
 = (>S'a/?)2-(7a/?)2 
 =a262 cos2 D - £2 sin2 D • a262 
 =a262(cos2 7)+sin2 2)) 
 = {Tap)^. 
 
 ap'^a^aP'^a by the associative law. 
 ==a:2/?2 since /?2 is a scalar. 
 7^(a/?)2. Conf. § 88, (5). 
 ap-ap = {aP)^ a:^-a:/?=cos2 Z) +£ 2sin D cos i)-sin2 Z) 
 
 = (*S + 7)2. =cos 2/) +£ sin 22). Conf. § 98. 
 
 114. Applications. // the diagonals of a parallelegram are 
 perpendicular to each other, the parallelogram is a rhombus. 
 
 By hypoth., 
 
 '^"^^ ^(a: +/?)(/?- a) =0 §56. 
 
 =,Q^-a^. § 113. 
 
 .-. {TpY = (TaY. Q.E.D. 
 
 115. The joins * of the mid-points of the sides of a rhombus 
 are at right angles. 
 
 = }(27a:/?+/?2-o:2). § 113. 
 
 .*. Sr^=\{p^-a^) =0. [since Ta=Tp. 
 
 • *. T-L^- Q.E.D. 
 
 * Join = the line joining. 
 
INTERPRETATION OF VECTOR EQUATIONS 57 
 
 116. In any plane triangle to find a side in terms of the 
 other two sides and the opposite angles. 
 
 Multiplying by (or into) a, (or any 
 vector, in order to get a quaternion) 
 
 Taking the scalar parts, we have 
 -a^+Sal^=Sar. -a^-ab cos (180°- C) =-ac cos B. 
 
 .'. a=b cosC + c cos B. (Conf. Trigonom.) q.e.i. 
 
 117. Had we taken the vector parts in § 116, we would 
 have found 
 
 Va^ = Vay. ab sin C •£=ac sin B- e, 
 
 or 6 sin C =c sin B. (Conf. Trigonom.) 
 
 118. In § 116, had we divided by some vector in order 
 to get a quaternion, say y, then we would have had the 
 same result, c=a cos B-\-b cos A. 
 
 119. Had we divided by a or /?, say a, then if the triangle 
 
 were right-angled at C, since *S— =0, we would get 
 
 l=S—=— cos B, or cosB=—. 
 a a c 
 
 120. Had we taken the vector parts in § 118, we should 
 have had a sin B^b sin A. (Law of Sines, Trig.) 
 
 Had we taken vector parts in § 119, then since sin C = l, 
 
 sin 5=-^. 
 c 
 
 121. V T is representable by a prolate spheroidal shell 
 with its major axis i and its two minor axes ^j and ^k. 
 
58 QUATERNIONS 
 
 Repetitions of this process tend toward the unit shell and 
 i° = l. 
 
 In the same way \/^=\/a\^T is represent able by a 
 prolate shell whose major axis is i and whose other axes 
 a-re ij, ik, leading similarly to (a/?)° = l. 
 
 From § 113, since, 
 
 \/(a/?)2 = \/cos 2D+£ sin 2D =cos D+e sin D. 
 '.'. (a/?)°=cosO° = l, 
 the same as above, and agreeing with § 89. 
 122. Formulae for reference and practice. 
 
 = angle from a to ^. 
 
 i)= angle of the quaternion. [§ 76. 
 
 p, q, r . . . = quaternions. 
 
 1. q = Tq(- cos d + e sin 0) =a^. 
 
 2. =Tq{cosD + esinD). 
 
 2 
 
 3. = Tq-Uq^Tqe-^. 4:. q=Sq + Vq. 
 
 2 
 
 5. Kq=Sq-Vq = Tq(cos D-e sin D) =Tq£~^^. 
 
 6. Sq = Tq cos D = Tq(- cos 0). 
 
 7. TVq = Tq sin d ^Tq sin D. 
 
 8. Vq - TVq- UVq = Tq sin D • s. 
 
 9. £^+^'. 10. Sq=i(q+Kq). 
 
 11. Vq=i(q-Kq). 
 
 12. {Tq)^=qKq=Kq'q = {SqY-{VqY. 
 
 13. (rg)2 = (^g)2 + (r7g)2. 20. ra2 = _a2. 
 
 »«--<^v 
 
 21. 5q:=0. 
 
 22. 7a=a:. 
 
 15. KKq=q, 
 
 23. KaP=^a. 
 
 lG.Kx=^x. 
 
 24. Sal^=Sda. 
 
 17. Rq=q-^. 
 
 25. 7a/? = -7/3a:. 
 
 IS, K(-q)==-Kq. 
 
 26. ap+^a=2Sa^. 
 
 19. Xa = -aa:. 
 
 27. ap-pa=2Vap, 
 
INTERPRETATION OF VECTOR EQUATIONS 5^ 
 
 28. (a±/?)2=a2_t25a:/?+/?2. 
 
 29. Tipqr...)=Tp'Tq-Tr.., 
 
 30. U(pqr . . .) =Up-Uq-Ur , . . 
 
 31. S{pqr . . . ) =S{qr . . . p) =S(r . . . pg) =. . . 
 
 32. Kipqr ...)=... Kr-Kq-Kp. 
 
 33. ;S(a:p+2/g + . . . )'' = xSp+ySq+. . . 
 
 34. V{xp-{-yq-\-. . . )*=a:y7)+2/Fg + . . . 
 
 35. AS(g+r+s + . . . ) ^Sq+Sr-\-Ss^. . . 
 
 36. V{q^-r+s + . . . ) =Vq-\-Vr + Vs+. . . 
 
 37. 7g'g=>S5'7g4-^gFg' + 7(F5'7^). 
 
 38. S{a'^r) =Sa{Spr-^ypr) =SaVpr' 
 
 39. SaVM^=SaMI^, [M=m-\-pL. 
 
 40. =ASfa: (m +yO/? =mSa^ ^S-aii^. 
 
 41. =mS^a-SP/ia. [§ 51. 
 
 42. =Spma-Sp}ia=S'pKMa. [§122(5). 
 
 43. ^(a4-«)(6+/?)=.Sf(6+/?)(a+a:). 
 
 44. Spq=Sqp. 
 
 45. Kal^r=^^i^l^-r) =Kr-Kap==-r-Pa. 
 
 46. aP-r-y-^a=2Sa^r = -2Srpa. 
 
 47. Sa^r=^ioL^-r) S(P''d) =Siq'P). 
 
 48. =AS(;'a5) =>S;'«/? =aS/?7'«. 
 
 49. ^(«./?.r)=^(>Sa:/? + Fa/?)r. 
 
 50. =>S.;'7a^ = ->S.r7/?a. 
 61. =-,S-r(7/9a:+iS/?a). 
 
 52. =-SrPa. 
 
 53. ==-S.3ar. 
 
 54. iS(a:ia2. • . a)n = (- l)«*S:(a'„ . . . «2ai). 
 
 55. Sal3r=SaVl3r. 
 
 56. =>S-aF/?;'=^./?Fra:=ASfrFa/?. 
 
 57. a/?;' + ;'/?a=27a:/?;-=27;'^a. 
 
 58. 2VaPr =aPr -\-{ar^- arp-rap + raP) +rl^a. [57. 
 
 * See Appendix. 
 
60 QUATERNIONS 
 
 59. 2Vai^r=(^(l^r+rl^)-(^r+r(^)l^+r(^^-^M- 
 
 60. =2(aSpr-I^S(^r+rS(^l^)' [26. 
 
 61. Vai^r-(^^^r=Va[^r-V'^^Pr' 
 
 62. =Va(^r-^M- 
 
 63. =V'aVpy. 
 
 64. =-pSra^-rSap. [60. 
 
 65. =-F.7(/?r)«. [62,25. 
 
 66. .-. 7-7(/?r)«=/5>Sf;'«-r>S^«/?. [64. 
 
 67. 7.(7a/?)7r^ = -r^^7a:/?+^>S.(7a,/?)r. [63,64. 
 
 68. = - r^apd +dSa^r- [^6, 24. 
 
 69. V'(Val^)Vrd=+aS-I^Vrd-l^S-{Vrd)a, [66. 
 
 70. =aSl^r^-^Sard. [56. 
 
 71. dSa^r =(^S^r^ +^Srad + rSa^^. [68, 70. 
 
 72. 7-7a:/?7/?;' = ;'>S-(7a:^)/?-^>S:.r^«^. [67. 
 
 73. =r^pVa^-pSra^. [24,56. 
 
 74. =^Spar. [^/?7a:^=0, 60. 
 
 75. a^a:/?;' = 7-7/?a7Q:r. [74. 
 
 76. ^Sapr=y-yrpyp^- [48, 75. 
 
 77. rSa^r=vvarVr[^. 
 
 78. (5 = - (tiSi^ +/>Sy^ +A;aS/c^). [71. 
 
 123. In (63), (64), VaV^y is perpendicular to a and 
 coplanar with /?, ;-. That it is perpendicular to a can be 
 shown by multiplying by a and taking scalars thus, 
 
 SaiV-aVpr) =SarSa^-Sa^Sar=0. [63, 64. 
 
 It is in the plane of /?;- since 
 
 v-aV^r=r^<^?-^^(^r- [63, 64. 
 
 124. In (58), (60) Va^y is of the form xa +yl^-\-zy and is 
 therefore the intermediate diagonal of the parallelepiped 
 of which the edges are aSl^y, —^Say, ySa^. 
 
INTERPRETATION OF VECTOR EQUATIONS 61 
 
 125. In (67), since S-rSal^=S'dSa[^=0, 
 
 and it is evident that V- Vaj^Vyd is coplanar with d and y. 
 Moreover, since 
 
 y • Va^Vyd =^V'VdrVap =l^Sdra-aSpdr, [25, 52, 67. 
 
 it is also coplanar with a and /?, and therefore must be along 
 the intersection of the planes of a, /? and y, d. 
 
 126. In (71), dSapy is the intermediate diagonal of the 
 parallelepiped of which the three edges are aS^yd, j^Syad, 
 
 rSapd. 
 
CHAPTER XI 
 QUATERNION EQUATIONS OF THE FIRST DEGREE 
 
 127. An equation of the first degree with respect to an 
 unknown quaternion X, is that which contains the quater- 
 nion to the first power together with known quaternions, 
 either isolated or under the symbols S or V. 
 
 The general equation will then have the form 
 
 i:axb+i^csaxb+i.D'VaxB'E=^f, 
 
 The third term assumes the form of the first two if we 
 replace VAXB by AXB — SAXB, so that the general 
 equation reduces to the form 
 
 i:axb+^csaxb=f. 
 
 128. To resolve this equation we decompose the quater- 
 nions into their scalar and vector parts. Thus, putting 
 A=a+a, B=h-\-^, etc., we have 
 
 2(a+a)(:r + e)(6+/?)+S(c + ;')*S.(a+a)(a:+0(^+/?)=^ + ^. 
 
 The sum of the scalar parts of this will be found to be 
 embraced in the general term Sa^, therefore the scalar part 
 of the equation is /Sa^=c?. 
 
 129. Solution of the linear scalar equation Sa^=(i. 
 This may be written, § 122, (33), 
 
 Sa{^-da-^)=0, [ada "i = d 
 
 62 
 
QUATERNION EQUATIONS OF THE FIRST DEGREE 63 
 
 where, § 56, (2), evidently {$—da~^) is some vector /?, JLa, 
 or p = VaY, where ;- is an arbi- 
 trary vector. Therefore, 
 
 The geometrical interpreta- 
 tion of this is shown in the 
 diagram, where since y is arbi- 
 trary, the locus of the extremity "^ 
 of ^ must be the plane _L a and 
 through the point da~^. 
 
 130. The vector part of the second term in § 128, is, 
 neglecting the S, 
 
 Multiplying out the first term we find the vector part to be 
 
 axp-\-aM -\-a^p-\-hxa -\-xa^-\-ha^ +a^^. 
 Of these forms 
 
 =Sa^l3+aS^l3-^Sap+^Sae. [§ 122, (61), (64). 
 
 Of the final forms, abx-y, aS^^-y, xSa^-y, h-Sa^-y, 
 Sa$l^-r, ohxy, ax^, hxa, aS$^, xVad, bSa^* aS$^, ^Sa^, 
 are comprehended under the general form, 
 
 ah^, $Sa^ are comprehended under the general form, 
 Vm$ = V(ab+Sa^ + . ..); 
 
 * Or neglected as purely scalar. 
 
64 QUATERNIONS 
 
 aF^/?, bVa^ are comprehended under the form, 
 
 Hence the vector part of the general equation becomes 
 
 or 2a,S/?e + y(m + //)e=^. [§ 122, (34). 
 
 or I,aS^e-\-V'Me=d. [§61. 
 
 This is generally abbreviated under the functional sign, 
 
 131. Hamilton called this a linear vector function of the 
 vector ^. Considered as an operator its application to ^ 
 has many interesting results, some of which will be investi- 
 gated in the following pages. 
 
 We will consider first the properties of (j) itself, and then 
 the result of its application to the vector $. 
 
 For reasons given later this function is called a strain 
 function. 
 
 132. Properties of ^. Since § 122 (35), 
 
 and § 122, (36), 
 
 V'M($ + ri+...)=V'Me + V-Mr)+.,. ; 
 /. I,l3Sa(e + 7)+. . . ) +V-M($ + 7)+. . . ) 
 
 = (i:^Sa$ + V-M$) +(^pSa7) + V'M7)) -f . . . , 
 or (t>(^ + r)+. . .)=^e+^>?+. .., 
 
 that is, is distributive over a sum. 
 
QUATERNION EQUATIONS OF THE FIRST DEGREE 65 
 
 133. li $ = r}=, , , to n terms, 
 
 then (fm^ =n(f)^ , 
 
 that is, (j) is commutative with a scalar factor. 
 
 134. If we define (j)~^ by the equation, ^"^0=1,* then 
 
 and from § 132, 
 
 or e + >? + ... =(?^-i(0^+0^ + . ..)• 
 
 But <j>^=d, etc., and (l)~^d = $, etc., and therefore, 
 
 Hence <f)~^ has the same properties as (j). 
 By repeated appHcations we can easily get, for all integral 
 values of k, 
 
 135. Conjugate strain functions. Operating on ^f = 
 
 llaSp^ + V'M^ with >S-7?, we get 
 
 = ^S$l^Sarj+Srj(m + pL)^ 
 
 = ^S^l^Sa7)+S$im + n)r) [§ 122, (24), (38). 
 
 = j:S$^Sa7)+S$(m-fi)7j [§ 122 (52). 
 
 =S${^l3Sar) + VKMt)) [§ 122 (39) , (5) . 
 
 4> and (j)' are called conjugate strain functions. They 
 evidently differ in the interchange of the known vector a, 
 |9, and of the quaternion M and its conjugate KM. 
 
 * See Appendix. 
 
66 QUATERNIONS 
 
 136. When (j>=(j)\ that is, the function conjugates into 
 itself, the functions are called self-conjugate strain func- 
 tions, in which case St)^^ =S^(j)r). 
 
 137. Types of self-conjugate functions. Since 
 
 and writing * (cj) +^0^ for (l)^-\-^^^, 
 
 which shows that the operator </> 4- 9^' is always self con- 
 jugate. 
 
 138. Furthermore, S^^^ =Se(l>'^, 
 
 whence /Se(^-^0^=O, 
 
 and therefore 
 
 the vector (^ — (j)') ^ is perpendicular to I" 
 
 or (cl>-cj)0^==Ve$, 
 
 where e is some unknown vector. 
 
 Consequently, ^^ = i (^ + ^0 ^ + i (9^ - 00 ^ 
 
 which shows that a linear vector function of ^ differs from a 
 self-conjugate function only by a term of the form Vs^. If it 
 is already self conjugate the vector £=0. 
 
 139. Since 
 
 S'^cl>cl>^r)=S$ct>{ct>'7)) S-^cj>cj>'ri=S'-q<j>(^'^) [§ 135. 
 
 =S'CJ>'T]ci>'^ [§135. =S7i4>cj>'^, 
 
 =S-<t>'^4>'ri\ [§122(24). 
 therefore ^^' is a self-conjugate strain function. 
 
 * This is allowable because the functional symbol <^ has the same 
 properties as an algebraic factor (distributive over a sum and commu- 
 tative with scalar factors) and can be treated like one. See Appendix. 
 
QUATERNION EQUATIONS OF THE FIRST DEGREE 67 
 
 140. If by ((p-^g)^ we understand ^f +g6, where g is a 
 scalar, ^ +g is also a linear vector function, since it has all 
 the properties of as the reader can easily demonstrate for 
 himself. Hence 
 
 S'^(cf>+g)r)=S-$(<j>7)-{-grj) =S-^cl>7j-\-S'$g7), 
 
 =S-i^ct>'^+9S'r)^ -S'rj(ct>^+g)e, 
 
 or <f>' -\-g is a conjugate function to cj) +g. 
 
 141. Application of <;6 to a vector $, If an elastic soHd, 
 that is, the vector connecting the several elements, be 
 subjected to the operation ^, then 
 all its particles, for instance, those / \pa 
 determined by the vectors a, p, y, are 
 displaced to positions determined by / J^ 
 the vectors (j)a, 0/?, 0;'. In general, 
 any particle whose vector is ^ occupies ^ 
 after the operation the position whose vector is ^^. 
 
 Also any vector a is displaced to the position ^a, for since 
 
 ^a =(j)Y — <j)^. 
 
 Hence, Any straight line of particles parallel to a, ^ay xa, 
 
 is homogeneously stretched and turned by the operation cj) into 
 
 a straight line of particles parallel to ^a, and the ratio of 
 
 . X(ba (f)a 
 extension is —^~- = — . 
 xa a 
 
 The operation (j> is called a strain and the property that 
 
 parallel lengths are strained into parallel lengths and 
 
 stretched proportionally is the physical definition of linear 
 
 homogeneous strain. Portions of the body originally equal, 
 
68 QUATERNIONS 
 
 similar and similarly placed remain after the strain equal, 
 similar and similarly placed. 
 
 142. If in the general equation of the first degree with 
 respect to an unknown quaternion, § 127, instead of sepa- 
 rating the scalar and vector parts, we had merely indicated 
 the vector part, thus 
 
 I,VQXR = VF, 
 
 this operator VQ( )R must of course be the undeveloped 
 form of ^. 
 
 If R=Q~^, we have the rotator of § 102 as a special case 
 of 0. 
 
 If Q and R degrade into scalars, we have SFmX=nX, 
 or dilatation merely. 
 
 A combination of rotation and dilatation makes the strain 
 just defined. 
 
 143. Properties of ^. To get a slightly different view, 
 let us suppose «, /?, 7- to be unit vectors at right angles to each 
 other. Hence, § 122 (78), 
 
 By the definition of homogeneous strain this is changed into 
 
 where a\ /?', f are three vectors upon which the same 
 proportional distances Sa$, S^^, Sy^ are laid off. This is 
 necessary in order to preserve the similarity required by 
 homogeneous strain. By substituting a, p, f for ^, we find 
 
 «'=0«, /?'=#, r'=#. 
 
QUATERNION EQUATIONS OF THE FIRST DEGREE 69 
 
 But 
 
 ^a =«' = - {aSaa' +^S^a' +rSra') =aA „ -\-l^Ba + rCa, 
 
 ^^ =^' = - {aSap' +/?^/?/?' + r^rP') =aA ^ -V^Bp + yC ^, 
 
 ^r = f = - (<^Saf +l^S^f+ySrf) =aAr +^B, -{-jCr, 
 
 which equations contain nine arbitrary constants, Saa\ 
 S^a', etc., since a', /?', f are entirely independent of a, 
 
 .8, r- 
 
 Operating on (f)^ with S-t) we have 
 
 5. ,^e = - (Sa'fjSa^ +Sl^'7)S^^ -}-SfrjSr^) 
 
 = -S-^(aSa'rj+^dSl^'r)+rSfr)), [§122(24). 
 
 where the expression in the parenthesis is a linear vector 
 function of tj, which will be shown (§ 144) to depend upon 
 the same nine scalars il«, B^, etc., as those in </>, and which 
 we may therefore appropriately designate by ^', thus 
 
 whence obviously S-i^^$ =*S- <^^')^, 
 as before. 
 
 144. Substituting «, /?, ;- for tj in the expression for (^'t^, 
 we find 
 
 4>'r=^-{aSa^r'rps?'r-\-rSrr), 
 
 or using the notation of § 143, 
 
 <^'a: = a A„ +/?^ p + T-^r i 
 
70 QUATERNIONS 
 
 which shows that ^' depends upon the same nine scalars 
 as (^. 
 
 145. If a, /?, Y are given non-coplanar vectors, then § 29, 
 
 whence (j)^ =X(j)a +y<f>[^ +z(l>]r. 
 
 The vector <^6 is known when the three vectors (jya, ^/?, 
 (py, are known. Each of these vectors involves three scalar 
 constants as in the case of $, multiples of the reference 
 vectors a, /?, y. Therefore the value of depends upon 
 nine scalar constants. It has therefore been called a 
 nonion. 
 
CHAPTER XII 
 
 APPLICATIONS OF ^ 
 
 146. Changes of volume due to <^. p, c/S^, cfy^p are, in 
 general, not in one plane, and hence, § 29, 
 
 <p^p=m2(l)^p — mi^p+mp, . . . . (1) 
 
 where m2, mi, m are scalars independent of p. The inde- 
 pendence is obvious, since we may put «, /?, ^ in succession 
 for p and thus obtain three equations from which they can 
 be obtained. 
 
 From § 59 the volume of the parallelopiped whose three 
 conterminous edges are p, <j)p, <pp is 
 
 — S'p<j)p(jy^p. 
 
 After the strain this volume is 
 
 — S • (j)pc[)^p<j)^p. 
 
 Multiplying Eq. (1) by (j)p<j)^p and dividing by pcjypcjy^p, we 
 have 
 
 S'(j)p^'^p4>^p 
 S'pcj>pct>^p ""^ ^^^ 
 
 This ratio of dilatation m due to ^, the ratio between the 
 volumes after and before the strain, is called the modulus 
 of the strain <j). 
 
 71 
 
72 QUATERNIONS 
 
 147. By referring to the equations of § 143 we see that 
 the scalar part of the product <f)a(j>P(j)y is confined to those 
 terms in which all three of the vectors a, p, y appear, that 
 is, taking S-a^y = l, 
 
 '^~ s-apr 
 
 ^CMpBr-ArB^). 
 In the same way from § 144, 
 
 =m. 
 
 Hence, Conjugate strains produce equal changes of volume. 
 
 148. Special values of m. If m = 1 there is no change of 
 volume caused by the strain. 
 
 If m=0, there results a zero volume due to the solid 
 becoming strained into a plane, a line, or a point, and in 
 this case cf) is called a null function, singly, doubly, or triply 
 null according as the strain results in a plane, a line or a 
 point. The plane, line, or point into which the solid is 
 strained is called the strain plane of (j), strain line, etc. 
 
 When m=0, then S(j)a(j)p^y , § 147, becomes zero and, 
 § 60, (j)a, <j}pj (jyy are coplanar vectors, and we can have a 
 relation, 
 
 or § 132, cl){xa ^y^-^zf) =0. 
 
 The vector, xa+yl^-\-zj-, whose strain, i.e., the result of 
 the application of the strain function (p, is zero is called a 
 null direction of (f>. In this case ^ is singly null, unless 
 (f>a, (j)^, (j)Y becomes collinear. 
 
APPLICATIONS OF cj) 73 
 
 149. If there is only one vector, say a, whose strain is 
 zero, i.e., (fta =0, then 
 
 and therefore m=0, and ^/? cannot be parallel to (fyy 
 (i.e., ^^f^xcjyy), for otherwise (j)(^ — xf) would be equal to 
 zero and there would be a second vector, ^ — xy, with a 
 strain of zero, which is contrary to the hypothesis. Hence, 
 if there is only one null vector and p=xa •\-yp-\-Z'f, then 
 
 <j)p=ycl>l^-{-z<l)r, 
 
 and (jyp is a vector In a plane parallel to ^,/?, <^;', that is, <^ 
 is singly null. 
 
 150. If there are two (only) vectors whose strain is zero, 
 say ^a:=0, (f>l^ =0, then (j)p=Z(f>y, which confines <j)p, the 
 strain of the general vector p, to the line parallel to (j))-, 
 and in this case ^ is doubly null. 
 
 151. Similarly, if <;6a =0, #=0, cjyj-^^O, then ^^=0 for 
 all values of p and (^ is a triply-null function. 
 
 152. When ^ is singly null, say (j)a =0, then the general 
 vector of any point in the strain plane of (j) is, § 149, 
 
 <j)p=ycl)^+Z(j)r, 
 
 and all particles that strain into this plane have the vectors, 
 
 p=xa+yl^+z-jr, 
 
 where x is arbitrary, since <j)a =0, and therefore the locus 
 of |0 is a line parallel to a. Hence, 
 
 A singly null function strains each of its null lines into a 
 definite point of its strain plane. Similarly, 
 
 A doubly-null funcion strains each of its null planes into 
 a definite point of its strain line. 
 
74 QUATERNIONS 
 
 153. Special applications of ^. What vectors, if any, are 
 unchanged in position by the strain (f>f 
 If p is unchanged by the strain, then 
 
 (jyp 
 
 ^S'iO, 
 
 (j)2p^g2p^ 
 
 cj^^p: 
 
 =9\ 
 
 and the equation 
 
 of § 146 becomes 
 
 
 
 
 (.7^- 
 
 -m2g^-\-mig-'i 
 
 m)p=\ 
 
 [), . 
 
 . . (1) 
 
 which must have one real root gi and may have three. In 
 other words, the curious fact that however a body may be 
 homogeneously strained, there is always at least one vector 
 whose direction remains unchanged. Hence 
 
 and S- /i(j)p — giS/xp =0, [/i=any vector, 
 
 and S-p{cl>'ii-g,/i)=0=S'p{cl>'-g)/L [§ 135. 
 
 Hence, § 56 (2), (^' — gfi)/z is perpendicular to ^o. 
 
 From this it appears that the operator {(j)' — g) applied to 
 any vector /z throws it into a plane perpendicular to p, or, 
 in other words, cuts off the component of the strained fx 
 which is parallel to p. 
 
 Also, p\\y-{<t>'-Qi)P-^^'-gi)^, [/I, V any vectors, 
 
 or multiplying out 
 
 ^||7(^V(/)'y-^i(r/>>.i; + /i0^) +sriV) 
 
 154. Properties of i^' — g). Like 0, this operator pro- 
 duces a homogeneous strain, as can be shown in a manner 
 similar to that in the case of (j). It has the same general 
 properties and has a modulus nig. Omitting the primes, 
 
APPLICATIONS OF ^ 75 
 
 which will not affect the discussion, we have, after expand- 
 ing, 
 
 155. In § 153 we found the expression, 
 V-{ct>^-gi)fx{cl>'-gi)v, 
 
 the vector part of the product of two strained vectors. To 
 investigate this we return to the last equation of § 146, 
 which can be written, 
 
 _ s-4,a4,p4>r s-{4>?<j>Ma _ 8a<t,'V{<i >^<i>r) . . ,„, 
 
 or since a is any vector whatever, 
 
 ^'7(#(jSr) =m7/?r, (1) 
 
 or since, § 147, cj) and ^' have the same modulus, 
 
 <l>v(<i>Wr) =mv^r, 
 
 Using (<j) — g) in place of <f>, we have 
 
 (4.-g)V(4>' -g)p{^' -g)r=m„v^r 
 
 ={<p-g) V[<l>'^4>'r -9(4''^ ■ r +P<P'r) WM 
 -{4>- g)[v<j>'^4,'r-g{v<i>'^ ■ r + vp<i>'r) +g^vpr\ 
 ^{ci>-g)[m4>-^v^r-9(.v<j>'^-r+v^'t>'r) +9^vpr] [d) 
 
 =={m-in'ig+m'2f-g^)VPr- 
 
76 QUATERNIONS 
 
 156. In this equation and the last expression of § 153, we 
 have the four vectors Vj^y, V-^'P-y, V-^<j)'x, (jiVPy, and 
 we can write 
 
 Operate successively with S-a, S-^, S-y and we get 
 
 spr^y^zs-rpct>y. 
 
 But § 122 (53), y = -l,z = -l,and 
 
 X =—^^n^ ^L-^ 1^:^^!. =.rn'2. [§ 154. 
 
 157. Hence, § 156, 
 
 or substituting in the equation of § 155 and using $ for 
 VPr, we get 
 
 or expanding, 
 {m+gcj)^+m'2g<l>+g^(t>-g'Jn^~'^+g^m'2-g^-g^)^ 
 
 = {m-m\g+m'2g^-g^)^, 
 or {ct>'^-m'2(j>-m^-^)^^-m\^, . . . (1) 
 
 or {(j)^ — m'2(l>^+rn'i^ — m)$=0, 
 
 where ^ is any vector whatever. 
 
APPLICATIONS OF ^ 77 
 
 158. Comparing this with Eq. (1), § 146, we have 
 
 The expression of § 153 now becomes, by §§ 155, 157, 
 and the equation above, 
 
 ^||(m0-i-gfi(m2-^)+gfi2)e 
 
 [dividing by gi and operating with ^ or, multiplying and 
 dividing by ((f> — gi) and then substituting the value of 
 
 9i 
 
 from (1) of § 153, 
 
 p\\ ^^ 6. [6 = any vector. 
 
 159. Since (1) of § 153 is true for all vectors, and since 
 is commutative with scalars, it can be written, 
 
 and, §158, ^||(^_^2)(<5^-^3)e, 
 
 that is, the operator (^— S'2)(^ — S^s) when applied to any 
 vector whatever results in a vector parallel to p, where 
 
 {cj>-g{)p==0, 
 
 or <j>p=g\p, 
 
 p being unchanged in direction by the strain (j>. 
 
78 QUATERNIONS 
 
 160. We can fortify this by another line of attack. As 
 before, § 153, under the condition, ^^ =g^, we have 
 
 Supposing for the moment that the three roots are real, 
 the solution of the problem will be given by one of the 
 directions a, p, y which satisfies the conditions, 
 
 {cj>-g{)a=0, (0-^2)/?=O, (0-^3)/9=O. . (1) 
 
 Now ^=aa-{-hp-\-cj. 
 
 Operating with {(j) — gi), we get 
 
 (ct^-gi)^=b{ct>-gi)^-\-c(cf>-g,)r 
 
 = {b(j)-hgi+bg2-b(t))^-\-{ccl)-cgi-\-cgs-ccf))f 
 
 [subtracting h{^ — g2)^=0 and c{(j) — gs)}'=0 
 
 =Hg2-gi)P +c(g3-gi)r' 
 
 161. Thus we see that the operator {4>—g\) cuts off from 
 the general vector ^ the component parallel to a. 
 
 Operating again, with (^— §^2), this becomes 
 
 ^<f>-gi){4>-g2)^ =c{g3-gi){g3-gi)r' 
 
 In the same way, 
 
 ((t>-gi)((l>-g3)^=b(g2-gi)(g2-g3)P, 
 ((l>-g2)i(l>-g3)^=aigi-g2)(gi-g3)a. 
 That is, the operator, 
 
 ((l>-g2){(j>-g3), 
 
APPLICATIONS OF (f> 7^ 
 
 operating on a, since $ is any vector, leaves its direction 
 unchanged, and 
 
 «ll(<5^-9'2)(^-g'3)^. 
 
 If the three roots gi,g2, 93 are unequal the three unchanged 
 directions are given by the operators above. 
 
 162. Those lines that are unchanged in direction by the 
 strain <f> have been suggestively called latent lines of ^, and 
 evidently from the relation {(t> — gi)p=^, the latent lines of 
 are the null directions of (<f)—gi). 
 
 Hence ^ — gri is a null strain, or § 154, 
 
 mod. (^ — gi)=mg=m — mig+m2g^—g^=0. 
 
 The roots of this cubic, gi,g2, gs are the ratios of dilatation 
 or extension of the latent lines of (j) and are called the 
 latent roots of <^. Planes whose vectors are not strained out 
 of the plane by are called the latent planes of <j). 
 
 163. If a, ^, y are the latent directions of cj), with the latent 
 roots gi, g2, gz, then (/?, ;-), (;-, a), (a, /?) determine the latent 
 planes of (p, which are enlarged by the strain (j> in the ratios 
 
 g2gz, g^gi, gm- 
 
 For if x^ +2/7' be a vector in the plane /?;-, then 
 
 cj> {xp +yr)= a:# +y(j>r'- ^g2P + ygz r, 
 
 and evidently the strained vector remains in the plane, 
 though its direction has been changed. q.e.d. 
 
 Also for the two vectors in the plane /?, ;', 
 
 T7^(x/3 +yr) (xip +yir) = TV{xg2^ +ygsr) (xm^+yigsr) 
 
 =g2g3TV(xp +yr)(xip +yir.) q.e.d. 
 
 Conf. § 58.] 
 
80 QUATERNIONS 
 
 164. If 92=93, and we operate upon upon ^ (§ 160) with 
 (j> — 92, remembering that {^ — 92)P = {^ — 92)T'^^} we get 
 
 {4>-92)^=a{gi-g2)a, 
 
 and the latent direction is given by 
 
 Operating upon ^ by {(j>—gi) we get 
 
 {<t>-9i)^ = i92-9i){hP+cr), 
 
 Since (96-9^2) W+cr) =(#-^2/?) +c(^-^2)r, 
 
 =0 [(^-9r2)/?=0, {cjy-92)r=0. 
 
 therefore every line of the plane (^ly is unchanged in direc- 
 tion, as well as kept in the plane. 
 
 165. gi =02 =93- 
 
 Operating on ^=aa +6/?+cgr with 4^ — 91 we obtain 
 
 (0-Sri)e=O, 
 
 that is, when the latent roots are all equal, all the vectors are 
 latent vectors. 
 
 166. There cannot be two latent directions for the same 
 root 9i. For if ((j)—9i)d=0 as well as {(f) — gi)a=0, we 
 should get by the method of § 161, 
 
 {(l>-92){<j>-93)^=d(gi-g2){gi-g3)^, 
 or a{gi-g2) (gi -93)0^ =d{9i -92) {91-93)0, 
 
 which cannot be unless 
 
 9^1 =92, 9i =93r 
 that is, unless the roots are all equal. 
 
APPLtCATlONS OF ^ 81 
 
 167. Since the latent planes are strained in different 
 ratios (the latent roots being unequal) they cannot coincide 
 and their intersections, the latent vectors, a, /?, 7-, cannot be 
 coplanar. 
 
 ^ — g\ strains all vectors into the plane ^, y, that is, into 
 the plane determined by the other latent vectors. 
 
 For if ^ =xa+y^+zy = general vector, 
 
 then {^-g\)^=x{<j)-gi)a+y{<h-gi)P+z{cj)-g{)y 
 
 =yig2-gi)P+z(g3-gi)y, [(0-gfi)o:=O. 
 
 =vector in plane p, y. q.e.d. 
 
 Repeating this operation we get 
 
 (^-^1) (0-^2) (^-6^3)6=0, 
 
 as already shown in § 159. 
 
 The strain plane of (j) — gi is the latent plane of cj). 
 
 168. Two conjugate strains have the same latent roots. For 
 
 since {(j) — gi)a=0. 
 
 .'. 0=S^{cl,-g,)a=Sa(cl>'-g^)$, [§135. 
 
 and evidently (t>' — gi strains all vectors into a plane per- 
 pendicular to a, i.e., (f>' —gi is a null function and gi is a 
 latent root of ^' (§ 162). 
 
 169. The latent plane of one strain is perpendicular to the 
 corresponding latent line of the conjugate strain. 
 
 The strain plane of (j>^ — gi is the latent plane of ^', § 167, 
 and therefore the latent plane of 0' is perpendicular to a, 
 the latent line of <j). 
 
82 QtTATERNIONg 
 
 170. // a strain is self conjugate its three latent roots are 
 real. 
 
 Suppose Qi+ti^ — I to be one of the roots, and let the 
 application of the other factors of the cubic be denoted by 
 
 Then, by § 160(1), 
 
 ^(« +aiV~l) = {gi -{-tiV^) (a +ai\/^, 
 or equating the reals and imaginaries, 
 
 (pa^gia — tiai, (l)ai=giai+tia. 
 Whence Saicpa^giSaia — tiSaiai, 
 
 /Sa^rti =giSaai -\-tiSaa, 
 or, since Sai(f)a=Sacj)ai, 
 
 0=t,{a^+ai^), 
 whence ti =0. q.e.d. 
 
 171. If a strain is self conjugate, that is, if ^=0' or 
 Srj^^=S$(l)7), then by § 170 it must have three mutually 
 perpendicular latent directions. 
 
 Conversely: If ^ have three mutually perpendicular 
 latent directions i, j, k, with the corresponding latent roots, 
 a, b, c, then cj) is self conjugate. For, § 122 (78), 
 
 $ = -iiSie+jSj^+kSm, 
 
 (j)^ = — (aiSi^ -\-bjSj^ -{-ckSk^ ; [(j>i =ai, etc. 
 
 S^(l>$ = - aSrjiSi$ - bSrjjSj^ - cSfjkSk^; 
 
APPLICATIONS OF (fy 83 
 
 7) = — (iSi7)-\-]'Sjrji-kSk7j); 
 ^T} = — aiSiTj — bjSJT) — ckSkr) ; 
 S$(l)7) = - aSi^SiT} - bSj^Sjrj - cSk^Skrj 
 
 and (j) is self conjugate. q.e.d. 
 
 172. When the strain is self conjugate (0=^') and the 
 latent roots equal, the strain is non-rotational and is called 
 pure. 
 
 Since the strain is self conjugate, § 168, the latent lines 
 are i, /, k, and 
 
 $ = -iSi^-jSj^-kSk^, 
 (f)^ = — aiSi^ — ajSj^ — akSk$ 
 =a(-iSi^-jSj^-kSk^) 
 ==a$, [a = latent root. 
 
 and ^ is not rotated. 
 
 That is, all vectors are latent lines. q.e.d. 
 
 When ^ is self conjugate ((j) — (j)'), by reference to §§ 143, 
 144, if the strain is given by 
 
 (j)i=xi-{-yj+zk; 
 
 (j)j=x'i-\-y'j-\-z'k', 
 
 (j>k=x"i^-y''i+z''k\ 
 
 then when the strain is pure, 
 
 x'=y, x"=z, y"=z', 
 
 and a pure strain depends upon six instead of upon nine 
 scalar constants. 
 
84 QUATERNIONS 
 
 173. // the drain is self conjugate (<f) = 0') and the three 
 latent roots a, b, c are not equal, only the vectors i, j,k are non- 
 rotated. 
 
 As in the previous section, 
 
 ^e = - aiSi^ - bjSje - ckSk^ 
 
 7^X^. Q.E.D. 
 
 <f)i = — aiSii — bjSij — ckSik 
 =ai. 
 Similarly, ^/= 6/, ^k=ck. q.e.d. 
 
 174. Example. If <j) have three latent directions a, p, 
 y with the three latent roots a, b, c, then 
 
 ^^= — ^^^r~' 
 
 and <f>=<i>', or <}> is self conjugate. Where is the error? 
 
 Ans. (j)'a7^aa. 
 
 176. Examples. Solve Vae^ = r = <t>^ for e. By § 146 
 (2), 
 
 O • A/IV 
 
APPLICATIONS OF 85 
 
 where X, n^ v are any three non-coplanar vectors. Putting 
 for these a, ^, y, if they are not coplanar, we get 
 
 [VarP =aSrP-rSa^ +^Sar. [§ 122 (64). 
 By § 158, 
 
 m2 = — Sa^. 
 By § 157(1), we get 
 
 [§ 122 (64). 
 whence ^ V = "^= — ' SB 
 
APPENDIX 
 
 FUNCTIONAL SYMBOLS 
 
 As every discrete magnitude is of necessity derived from 
 unity by some algebraic operation, so the symbol repre- 
 senting a discrete magnitude can be considered as the 
 symbol of some operation whose operand is unity. 
 
 Symbolizing the operation of converting unity into the 
 magnitude x by the symbol x-1, we have the functional 
 equation {x operating upon 1), 
 
 X'l =x. 
 
 A second application gives 
 
 x{x-l) =x'^-\=x'^, 
 
 where the 2 of x'^-l shows the number of operations, and 
 the 2 of x^ the result. 
 
 The inverse of the operation must be symbolized by 
 
 X 
 
 In the expression x-1 occur two concepts, the operation^ 
 X'l and the effect of the operation, x. 
 
 87 
 
88 QUATERNIONS 
 
 The inverse of the operation is x~^-l= — 1, the inverse 
 
 of the effect is (x)~^=— , and in this case the two results 
 
 coincide. 
 
 If we take a different operand, say y, then x~^-y and 
 {^y)~^i the inverse of the operation and the inverse of the 
 effect are not the same. 
 
 Every algebraic expression can be considered as a func- 
 tional operation. Thus adding 1 to x can be considered as 
 an operation symbolized thus, 
 
 /(a;)=x + l. 
 
 The inverse operation would be whatever operation is 
 necessary to change back from x + 1 to a; to original operand. 
 Thus 
 
 Here again /"^t^t, where, of course, f~^ symbolizes the 
 
 inverse of the operation and (f)~^ ^j, the inverse of the 
 effect. 
 
 Similarly, log~i x ^ (log x) ~i, 
 
 sin~i X7^(sin x)"^. 
 
 Examples. If/(x)=x + l, f{x)^x-\, p{x)=x^-2. 
 
 If /(x)4+., /-Kx)=--=^, /.(.)=-^+i±^. 
 
APPENDIX 89 
 
 If f(x) =x2 +2a; +6, /"i {x)=-l± V^T, 
 
 P(x) = (x2 +2x +6)2 +2(a;2 +2x +6) +6. 
 In these examples, omitting the operand, show that 
 
 In the same way prove 
 
 sin sin~i x =x^ log log~^ x =x, 
 
 log~^ log x=x, sin~i sin2a; =sin x. 
 
 If fix) =x^+3, F{x) =2-V^, then 
 
 fF(x)=(2-\ x)^+'S, F'f{x)=2-Vx^-^S. 
 2x—\ 
 
 The algebraic symbol is distributive over and commuta- 
 tive with its operand, that is, xy+xz + . . .=x{y+z+, , . ), 
 xy=yx. 
 
 Other symbols of operation which like these are dis- 
 tributive over and commutative with the operand will be 
 subject to the same algebraic laws. Hence we can treat 
 these symbols of operation just as we treat algebraic symbols 
 of operation. 
 
 Hence we can write § 122 (33), (34), § 132, § 135, 
 
 Sp + Vp = (S + V)p; ct>^-cj>'^ = {^-cf>')^, 
 
90 QUATERNIONS 
 
 We recall the similar results in Calculus, where, omitting 
 the operand, we have 
 
 (D2-a2)=(Z)+a)(Z)-a) [0=^ 
 
 dx 
 
 (D2-D-2)=(D + l)(D-2) 
 
 xW2 =x^D-D= x^D- [xD = 0. 
 
 X 
 
 = 0.0-6 = 6(0-1), 
 
 where is commutative with constants but not with either 
 X or D. 
 
INDEX 
 
 Addition, 3 
 
 Algebra, 31 
 
 Amplitude, 7 
 
 Angle of a quaternion, 39 
 
 Argand diagram, 7 
 
 Arithmetic, 30 
 
 Axis of quaternion, 21 
 
 Calculi, 2 
 
 Complex functions, 31 
 
 Complex quantity, 7 
 
 Conjugate of a quaternion, 33 
 
 Conjugate-strain function, 65 
 
 Continuum, 1 
 
 Coplanar quaternion, 46 
 
 Diplanar quaternions, 36 
 
 Discreta, 1 
 
 Division, 4 
 
 Euler*s Theorem, 28 
 
 Evolution, 5 
 
 Functional symbols, 87 
 
 Harmonic displacement, 11 
 
 Idiograph, 6 
 
 Idiograph, space, 10 
 
 Involution, 4 
 
 Latent lines, 79 
 
 Latent planes, 79 
 
 Linear homogeneous strain, 67 
 
 Linear scalar equation, 62 
 
 Mean reversion, 5 
 
 Modulus, 7, 71 
 Multiplication, 4 
 
 * * of vectors, 17 
 
 Nonion, 70 
 Null direction, 72 
 Null function, 72 
 Operation A, B, 11 
 Opposite quaternions, 33 
 Parallel vectors, 17 
 Plane of quaternion, 21 
 Plane of a vector, 21 
 Quadrantal versors, 50 
 Quaternion, 21, 31 
 
 " , angle of, 39 
 
 '* , axis of, 21 
 
 ** , plane of, 21 
 
 ** , reciprocal of, 32 
 Quaftemions, equality of, 35 
 
 " , diplanar, 36 
 
 Revector, 12 
 Reversion, 3 
 Rotator, 52, 68 
 Scalar, 9 
 " part, 27 
 * * of the unit part, 27 
 Self-conjugate strain function, 66 
 Singly null, 72 
 Space idiograph, 10 
 Stroke, 6 
 
 91 
 
92 
 
 INDEX 
 
 Strain, 67 
 
 Strain, linear homogeneous, 67 
 
 Strain function, 64 
 
 " " conjugate, 65 
 
 Strain plane, 72 
 Subtraction, 4 
 Tensor, 13, 27 
 
 ** of the vector part, 27 
 Unit part, 27 
 
 " vector, 13 
 
 " " of the vector part, 27 
 
 Vector, U 
 
 " , addition, etc., 12 
 unit, 13 
 
 ' * , inclined, 18 
 
 " , parallel, 17 
 
 ' ' , perpendicular, 18 
 
 " part, 27 
 
 ' ' part of the unit part, 27 
 
 " , plane of, 21 
 Versors, 48, 49 
 
 " , quadrantal, 50 
 
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