c^S GIFT OF 1 J'. INTEODUCTION TO QUATEENIONS ^' INTRODUCTION TO QUATERNIONS BY THE LATE PROFESSORS PHILIP KELLAND, M.A., F.R.S. AND PETER GUTHRIE TAIT, M.A. THIRD EDITION. PREPARED BY C. G. KNOTT, D.Sc. LECTtlRER IN APPLIED MATHEMATICS IN THE UNIVERSITY OF EDINBURGH FORMERLY PROFESSOR OF PHYSICS IN THE IMPERIAL UNIVERSITY OF JAPAN MACMILLAN AND CO., Limited NEW YORK : THE MACMILLAN COMPANY 1904 All rights reseroed First Edition, 1873; Second Edition, 1882; Third Edition, 1904. oLAsr.ow : printed at the univrrsity press BY UOBKKT MACLKH08K AND CO. PREFACE TO THE THIRD EDITION. In preparing the third edition of Kelland and Tait's Introduction to Qnaternions I have been guided mainly by two considerations. In the first place, the average mathe- matical student of to-day attains either at school or in his early college courses a much higher standard than was possible in 1873 when Kelland wrote, or even in 1881, the date of the second edition. It seemed, therefore, desirable to delete many of the very simple geometrical illustrations which formed a large part of the text, indicating their nature by a word, or transferring them as exercises to the end of the appropriate chapter. In this way valuable space has been gained for the discussion of problems more fitted to bring out the power and beauty of the quaternion calculus. It is right to mention, however, that Chapter I. has been left exactly as Kelland wrote it ; and the greater part of Chapter II. is simply reproduced. The second consideration was the necessity for presenting the main features of Hamilton's great calculus in a brief but yet logically complete form. This has led to the recasting of Chapters III. and IX. In the new Chapters III. and IV. the calculus in its essential features is developed systematically from the definition of a quaternion as the complex number which measures the ratio of two vectors, with the further assumption that the associative law holds in product combina- O / O O i o vi PREFACE TO THE THIRD EDITION. tions. From these two root principles the whole of Hamilton's powerful vector algebra evolves itself simply and naturally. It is hoped that the mode of presentation will remove the difficulty which some have experienced in accepting Hamilton's identification of vector and quadrantal versor. O'Brien, Hamilton's brilliant contemporary, confessed that the difficulty was to him insurmountable. But the difficulty is really created by the sceptic himself, who fails to see that, so far as the mathematical definition goes, a vector quantity in quaternions has a much wider significance than the step or displacement or velocity by means of which the simple summation principles are first illustrated. The law of vector addition, which is common to all kinds of vectors, including the Hamiltonian, determines nothing as to the laws of product combinations. These may be anything we please among vectors, so long as the law of vector addition is satisfied. Now it is proved in Chapter HI., § 18, that quadrantal versors obey the vector law of addition. They are therefore true vectors ; and hence follows, from the geo- metrical point of view, the analytical identification of vector and quadrantal versor. The identification, no doubt, requires every vector (whatever physical quantity it may symbolise) to be subject analytically to the quadrantal versor laws in product combinations ; but this, as Hamilton himself proved, is tantamount to requiring that three or more vectors in product combinations obey the associative law. There is thus perfect consistency throughout. From the point of view of pure analysis the difficulty mentioned above cannot, of course, present itself. The quaternion is then a quantity involving four units, which are defined as reproducing themselves in product combinations and as satisfying certain general laws. The mathematical properties of the quaternion being thus established, the utility of the calculus will depend simply upon the mode of interpretation. Thus Professor C. J. Joly, by a new inter- PREFACE TO THE THIRD EDITION. vii pretation of the quaternion, has recently developed an interesting treatment of projective geometry. In Chapter IX. a completely new section has been intro- duced on dynamical applications. This seemed to be specially called for, inasmuch as vector ideas and notations are now a familiar feature of some of our best modern books on mathematical physics. It is to be hoped that they- will become so more and more, and that the powerful Hamiltonian method which develops the ideas and under- lies the notation will become equally familiar. The last four articles of Chapter IX. have to do with the chief properties of the remarkable diflPerential operator y. Differentiation in the ordinary sense was excluded from the earlier editions, although the method was implicitly used in the treatment of tangents. It was impossible, however, to give any true idea of the power of quaternions in dynamics without the explicit introduction of differentiation ; and this consideration seemed to me to outweigh all considera- tions based on artificial distinctions as to what is or is not suitable in an elementary book. The mathematical student who is able to appreciate the exquisite beauties of the linear vector function as expounded in Chapter X. will have no difficulty in appreciating the significance of Nabla. Tait's very instructive Chapter X. has been left practically untouched. It is the work of a recognised master, and has been a source of inspiration to many students of the subject. As a pupil of both Kelland and Tait, and as a colleague and friend of the latter, I have had peculiar pleasure in preparing this third edition of their joint work, and trust that it may draw the mathematical student into an attractive and largely unexplored field of mathematics. Analytically the quaternion is now known to take its place in the general theory of complex numbers and continuous groups ; it is remarkable that it should have viii PREFACE TO THE THIRD EDITION. provided for the geometry and dynamics of our visible universe a calculus of great power and simplicity. My thanks are due to Mr. Peter Ross, M.A., for his careful proof-reading of all but the very earliest Chapters. C. G. KNOTT. Edinburgh University, October, 1903. PREFACE TO SECOND EDITION. In preparing this second edition for press I have altered as slightly as possible those portions of the work which were written entirely by Prof Kelland. The mode of presenta- tion M'hich he employed must always be of great interest, if only from the fact that he was an exceptionally able teacher ; but the success of the work, as an introduction to a method which is now rapidly advancing in general estimation, would of itself have been a sufRcient motive for my refraining from any serious alteration. A third reason, had such been necessary, would have presented itself in the fact that I have never considered with the necessary care those metaphysical questions con- nected with the growth and development of mathematical ideas, to which my late venerated teacher paid such particular attention. My own part of the book (including mainly Chapter X. and worked out Examples 10 — 24 in Chapter IX.) was written hurriedly, and while I was deeply engaged with work of a very different kind ; so that 1 had no hesitation in determining to re-cast it where I fancied I could improve it. P. G. TAIT. University of Edinburgh, November, 1881. PREFACE TO THE FIRST EDITION. The present Treatise is, as the title-page indicates, the joint production of Prof. Tait and myself. The preface I write in the first person, as this enables me to offer some personal explanations. For many years past I have been accustomed, no doubt very imperfectly, to introduce to my class the subject of Quaternions as part of elementary Algebra, more with the view of establishing principles than of applying processes. Experience has taught me that to induce a student to think for himself there is nothing so effectual as to lay before him the different stages of the development of a science in some- thing like the historical order. And justice alike to the student and the subject forbade that I should stop short at that point where, more simply and more effectually than at any other, the intimate connexion between principles and processes is made manifest. Moreover, in lecturing on the groundwork on which the mathematical sciences are based, I could not but bring before my class the names of great men who spoke in other tongues and belonged to other nationalities than their own — Diophantus, Des Cartes, La- grange, for instance — and it was not just to omit the name of one as great as any of them. Sir William Rowan Hamilton, who spoke their own tongue and claimed their own nation- ality. It is true the name of Hamilton has not had the impress of time to stamp it with the seal of immortality. And it must be admitted that a cautious policy which forbids xii PREFACE TO THE FIRST EDITION. to wander from the beaten paths, and encourages converse with the past rather than interference with the present, is the true policy of a teacher. But in the case before us, quite irrespective of the nationality of the inventor, there is ample ground for introducing this subject of Quaternions into an elementary course of mathematics. It belongs to first principles and is their crowning and completion. It brings those principles face to face with operations, and thus not only satisfies the student of the mutual dependence of the two, but tends to carry him back to a clear apprehension of what he had probably failed to appreciate in the sub- ordinate sciences. Besides, there is no branch of mathematics in which results of such wide variety are deduced, by one uniform process ; there is no territory like this to be attacked and subjugated by a single weapon. And what is of the utmost impoi-tance in an educational point of view, the reader of this subject does not require to encumber his memory with a host of conclusions already arrived at in order to advance. Every problem is more or less self-contained. This is my apology for the present treatise. The work is, as I have said, the joint production of Prof. Tait and myself. The preface I have written without consulting my colleague, as 1 am thus enabled to say what could not otherwise have been said, that mathematicians owe a lasting debt of gratitude to Prof. Tait for the singleness of purpose and the self denying zeal with which he has worked out the designs of his friend Sir Wm. Hamilton, preferring always the claims of the science and of its founder to the assertion of his own power and originality in its development. For my own part I must confess that my knowledge of Quaternions is due exclusively to him. The first work of Sir Wm. Hamilton, Ledurci^ on Quaternions, was very dimly and imperfectly understood by me and I dare say by others, until I^rof. Tait published his papers on the subject in the PREFACE TO THE FIRST EDITION. xiii Messenger of Matheniaiics. Then, and not till then, did the science in all its simplicity develop itself to me. Subsequently Prof. Tait has published a work of great value and originality, An Elementary Treatise on Quaternions. The literature of the subject is completed in all but what relates to its physical applications, when 1 mention in addition Hamilton's second great work. Elements of Quaternions, a posthumous work so far as publication is concerned, but one of which the sheets had been corrected ])y the author, and which bears all the impress of his genius. But it is far from elementary, whatever its title may seem to imply ; nor is the work of Prof. Tait altogether free from difficulties. Hamilton and Tait write for mathematicians, and they do well, but the time has come when it behoves some one to write for those who desire to become mathematicians. Friends and pupils have urged me to undertake this duty, and after consultation with Prof. Tait, who from being my pupil in youth is my teacher in riper years, I have, in conjunction with him, and drawing unreservedly from his writings, endeavoured in the first nine chapters of this treatise to illustrate and enforce the principles of this beautiful science. The last chapter, which may be regarded as an introduction to the application of Quaternions to the region beyond that of pure geometry, is due to Prof. Tait alone. Sir W. Hamilton, on nearly the last completed page of his last work, indicated Prof. Tait as eminently fitted to carry on happily and usefully the applications, mathematical and physical, of Quaternions, and as likely to become in the science one of the chief successors of its inventor. With how great justice, the reader of this chapter and of Prof. Tait's other writings on the subject will judge. PHILIP KELLAND. University of Edinburgh, October, 1873. CONTENTS. CHAPTER I. PAGE Introductory, 1-6 CHAPTER II. Vector Addition and Subtraction. §§ 1-14. Direction a Fundamental Geometrical Conception, - - 7 Vectors and Scalars, -------- g.g Parallel and Non-parallel Vectors, ----- JQ Vector Addition, ........ n Unit Vector and Tensor, -..-... iQ Coplanar and Non-coplanar Vectors, - - - . . 17-19 Mean Point and Centre of Mass, 22-25 Velocity, 25-27 CHAPTER III. Quaternions and Versors or Quotients and Products OF Vectors. §§ 15-24. The Quaternion as a Geometrical Operator, ... 29 Tensor and Versor, 31 The Conjugate of a Quaternion, ..... 30 Quadrantal Quaternions and Versors, .... 33.35 Multiplication of Quadrantal Quaternions, . . . 35-37 Scalar and Vector Parts of a Quaternion, - . - - 37-38 Perpendicular Unit Vectors, 39 40 Cartesian Equivalents, --_--.. 40-41 Geometrical and Dynamical Interpretations, . - - 41-43 xvi CONTENTS. CHAPTER IV. Quaternion Products and Related Developments. §§ 25-34. PAGE The Versor as the Power of a Vector, _ . . . 51-52 Versors represented by Arcs on a Sphere, - - - - 54-55 The Rotational Operator, ------- 55-58 Finite Rotations, 58-59 Infinitely Small Rotations, ------ 59-60 Quaternion Products, ------- 60-61 Products of Vectors, -------- 61-64 Important Transformations, - . - - - . - - 64-68 CHAPTER V. Simple Geometrical Applications. §§ 35-42. Straight Line and Plane, ------- 71-78 Sphere and Circle ; Tangent Planes and Tangents, - - 78-81 Poles and Polar Planes, - - 81-82 Inversion, ---------- 83-84 Cones of Contact and Cylinders, . . . . - 85-87 CHAPTER VI. Cones and their Sections. §§ 43-56. Cone and Cylinder of Second Order, - - • - - - 90-91 Sphero-conics, --------- 91-93 Tangent Plane, --------- 93 Linear Vector Function, ------- 94-96 The Ellipse, 96-98 Cartesian Equivalents, ------- 99 Powers of Linear Vector Function, ----- 100 Conjugate Diameters of Ellipse, ----- 101 Poles and Polars, etc., 102-106 Parabola and Hyperbola, 107-116 CHAPTER VII. Central Surfaces of the Second Order. §§ 57-65. General Equation of Quadric Surface, - - - - 119 Tangent Plane ; Polar Plane, 119-121 Conjugate Diameters, 121-122 Square Root of Linear Vector Function, - - - - 122-123 Cartesian Equivalents, etc. 123-132 CONTENTS. xvii CHAPTER VIII. ^ PAGE Miscellaneous Geometrical Applications. §§ 66-70. Pascal's Hexagram, - - - - - ' - - - 135-137 Confocal Surfaces of the Second Order, ... - 137-143 Versor Equation of Ellipse, ..--.- 143-144 Various Applications, ....... 144-156 CHAPTER IX. Dynamical Applications. §§ 71-79. Differentiation of Quaternions, 158-159 Dynamics of Centre of Mass, ------ 160-161 Rigid Body with one Point Fixed, ----- 161-162 Spinning Top, - . . . 163-166 Mutual Action of Magnets, ..---- 166-168 Field of Force and Potential ; Nabla (V), - - - 169-172 Distributions of Matter, ------- 172-174 Convergence and Curl, -...--- 174 Electrical Distributions, .-.---- 175-176 CHAPTER X. Vector Equ.ations of the First Degree. Homogeneous Strain defined, - 178 Linear Vector Function, -..--.- 179 The Conjugate Function, - - - - - - - 181 The Symbolic Cubic, -------- 182 Volume Change one of the Coefficients, . . - . 184 The Invariant Coefficients of the Cubic, - - - . 185 Special Properties of Homogeneous Strain, - - - 187 Pure Strains, _ . - - I88 Rotation as a Strain, - - 190 Separation of Pure and Rotational Part, - - - - 194 The Square Root of a Strain, ------ 197 vSimple Shear, _ . . 199 Imaginary Vectors and Quaternions, ----- 202 raTRODUCTION^ TO QUATERNIONS. CHAPTEE I. INTRODUCTORY. The science named Quaternions by its illustrious founder, Sir William Eowan Hamilton, is the last and the most beautiful example of extension by the removal of limitations. The Algebraic sciences are based on ordinary arithmetic, starting at first with all its restrictions, but gradually freeing themselves from one and another, until the parent science scarce recognises itself in its offspring. A student will best get an idea of the thing by considering one case of extension within the science of Arithmetic itself. There are two distinct bases of operation in that science — addition and multiplication. In the infancy of the science the latter was a mere repetition of the former. Multiplication was, in fact, an abbreviated form of equal additions. It is in this form that it occurs in the earliest writer on arithmetic whose works have come down to us — Euclid. Within the limits to which his principles extended, the reasonings and conclusions of Euclid in his seventh and following Books are absolutely perfect. The demonstration of the rule for finding the greatest common measure of two numbers in Prop. 2, Book VIII., is identically the same as that which is given in all modern treatises. But Euclid dares not venture on fractions. Their properties were probably all but unknown 2 QUATERNIONS. to' liiru. Accordingly ,ve look in vain for any demonstration of the properties of fractions in the ^vritings of the Greek arithmeticians. For that we must come lower down. On the revival of science in the West, we are presented with cate- gorical treatises on arithmetic. The first printed treatise is that of Lucas de Burgo in 1494. The author considers a fraction to be a quotient, and thus, as he expressly states, the order of operations becomes the reverse of that for whole numbers — multiplication precedes addition, etc. In our own country we have a tolerably early writer on arithmetic, Robert Record, who dedicated his work to King Edward the Sixth. The ingenious author exhibits his treatise in the form of a dialogue between master and scholar. The scholar battles long with this difficulty — that multiplying a thing should make it less. At first, the master attempts to explain the anomaly by reference to proportion, thus : that the product by a fraction bears the same proportion to the thing multiplied that the multiplying fraction does to unity. The scholar is not satisfied ; and accordingly the master goes on to say : " If I multiply by more than one, the thing is increased ; if I take it but once, it is not changed ; and if I take it less than once, it cannot be so much as it was before. Then, seeing that a fraction is less than one, if I multiply by a fraction, it follows that I do take it less than once," etc. The scholar thereupon replies, " Sir, I do thank you much for this reason ; and I trust that I do perceive the thing." Need we add that the same difficulty which the scholar in the time of King Edward experienced, is experienced by every thinking boy of our own times ; and the explanation afforded him is precisely the same admixture of multiplication, propor- tion, and division which suggested itself to old Robert Record. Every schoolboy feels that to multiply by a fraction is not to multiply at all in the sense in which multiplication was originally presented to him, viz. as an abbre\aation of equal additions, or of repetitions of the thing multiplied. A totally INTRODUCTORY. 3 new view of the process of multiplication has insensibly crept in by the advance from whole numbers to fractions. So new, so different is it, that we are satisfied Euclid in his logical and unbending march could never have attained to it. It is only by standing loose for a time to logical accuracy that exten- sions in the abstract sciences — extensions at any rate which stretch from one science to another — are effected. Thus Diophantus in his Treatise on Arithmetic (i.e. Arithmetic extended to Algebra) boldly lays it down as a definition or first principle of his science that 'minus into minus makes plus.' The science he is founding is subject to this condition, and the results must be interpreted consistently with it. So far as this condition does not belong to ordinary arithmetic, so far the science extends beyond ordinary arithmetic : and this is the distance to which it extends — It makes subtraction to stand by itself, apart from addition ; or, at any rate, not dependent on it. We trust, then, it begins to be seen that sciences are ex- tended by the removal of barriers, of limitations, of conditions, on which sometimes their very existence appears to depend. Fractional arithmetic was an impossiliility so long as multipli- cation was regarded as abbreviated addition : the moment an extended idea was entertained, ever so illogically, that moment fractional arithmetic started into existence. Algebra, except as mere symbolized arithmetic, was an impossibility so long as the thought of subtraction was chained to the requirement of something adequate to subtract from. The moment Diophantus gave it a separate existence — boldly and logically as it happened — by exhibiting the law of minus in the fore- front as the primary definition of his science, that moment algebra in its highest form became a possibility ; and indeed the foundation-stone was no sooner laid than a goodly building arose on it. The examples we have given, perhaps from their very simplicity, escape notice, but they are not less really 4 QUATERNIONS. examples of extension from science to science by the removal of a restriction. We have selected them in preference to the more familiar one of the extension of the meaning of an index, whereby it becomes a logarithm, because they prepare the way for a further extension in the same direction to which we are presently to advance. Observe, then, that in fractions and in the rule of signs, addition (or subtraction) is very slenderly connected with multiplication (or division). Arithmetic as Euclid left it stands on one support, addition only, inasmuch as with him multiplication is but abbreviated addition. Arithmetic in its extended form rests on two supports, addition and multiplication, the one different from the other. This is the first idea we want our reader to get a firm hold of ; that multiplication is not necessarily addition, but an operation self-contained, self-interpretable — springing originally out of addition ; but, when full-grown, existing apart from its parent. The second idea we want our reader to fix his mind on is this, that when a science has been extended into a new form, certain limitations, which appeared to be of the nature of essential truths in the old science, are found to be utterly untenable ; that it is, in fact, by throwing these limitations aside that room is made for the growth of the new science. We have instanced Algebra as a growth out of Arithmetic by the removal of the restriction that subtraction shall require something to subtract from. The word ' subtraction ' may indeed be inappropriate, as the word multiplication appeared to be to Eecord's scholar, who failed to see how the multipli- cation of a thing could make it less. In the advance of the sciences the old terminology often becomes inappropriate; but if the mind can extract the right idea from the sound or sight of a word, it is the part of wisdom to retain it. And so all the old words have been retained in the science of Quaternions to which we are now to advance. The fundamental idea on which the science is based is that INTRODUCTORY. 5 of motion — of transference. Real motion is indeed not needed, any more than real superposition is needed in Euclid's Geometry. An appeal is made to mental trans- ference in the one science, to mental superposition in the other. We are then to consider how it is possible to frame a new science which shall spring out of Arithmetic, Algebra, and Geometry, and shall add to them the idea of motion — of transference. It must be confessed the project we entertain is not a project due to the nineteenth century. The Geometry of Des Cartes was based on something very much resembling the idea of motion, and so far the mere introduc- tion of the idea of transference was not of much value. The real advance was due to the thought of severing multiplication from addition, so that the one might be the representative of a kind of motion absolutely different from that which was represented by the other, yet capable of being combined with it. What the nineteenth century has done, then, is to divorce addition from multiplication in the new form in which the two are presented, and to cause the one, in this new character, to signify motion forwards and backwards, the other motion round and round. We do not purpose to give a history of the science, and shall accordingly content ourselves with saying, that the notion of separating addition from multiplication — attributing to the one, motion from a point, to the other motion about a point — had been floating in the minds of mathematicians for half a century, without producing many results worth recording, when the subject fell into the hands of a giant. Sir William Rowan Hamilton, who early found that his road was obstructed — he knew not by what obstacle — so that many points which seemed within his reach were really inacces- sible. He had done a considerable amount of good work, obstructed as he was, when, about the year 1843, he per- ceived clearly the obstruction to his progress in the shape 6 QUATERNIONS. of an old law which prior to that time, had appeared like a law of common sense. The law in question is known as the commutative law of multiplication. Presented in its simplest form it is nothing more than this, 'five times three is the same as three times five'; more generally, it appears under the form of ' ah = ha whatever a and h may represent.' When it came distinctly into the mind of Hamilton that this law is not a necessity, with the extended signification of multiplication, he saw his way clear, and gave up the law. The barrier being removed, he entered on the new science as a warrior enters a besieged city through a practicable breach. The reader will find it easy to enter after him. CHAPTER II. VECTOR ADDITION AND SUBTRACTION. 1. Direction as a Fundamental Geometrical Con- ception. The explicit recognition of direction as a funda- mental geometrical conception is the distinguishing mark of quaternionic and other vectorial methods. A little consideration will soon convince us that the comparison of directions is more intuitive than the comparison of lengths. The eye has no difficulty in judging as to the parallelism of two lines, but has considerable difficulty in judging as to the equality of their lengths especially if the lines are not parallel. The similarity of two triangles of different size when set with their corresponding sides parallel is apparent at a glance ; not so the equality of two triangles equal in all respects when they are set with their corresponding sides not parallel. These and other like illustrations show that the conception of direction is of a fundamental character. When we wish to determine the position of one point ^vith regard to another we must know not only their distance apart, but also the direction of the line joining them. In like manner the displacement of a point cannot be completely known unless both the direction and the amount of the displacement are given. We may obviously fully represent this relative position or this displacement by drawing from 8 QUATERNIONS. [Chap. n. any starting point or origin a line having the required direc- tion and having a length (drawn to a convenient scale) numerically equal to the required distance apart or to the required amount. Such a representative line is called a Vector. 2. Vectors and Scalars. The simplest example of a vector quantity is a directed line, the whole conception involving both length and direction. But it should be clearly understood from the outset that the word is descriptive of any quantity which may be represented by means of a directed line. Thus a vector quantity is one which possesses both direction and magnitude. As examples we may mention position, displacement, velocity, acceleration, momentum, force, moment of force, rotation, and so on. On the other hand, there are quantities which possess no direction but only magnitude. Such for example are time, temperature, volume, mass, work, energy ; and these are distinguished as Scalar quantities. The magnitude of a vector is evidently a scalar quantity, and may be assigned quite independently of the direction. In ordinary algebra and 'analytical geometry the symbols used are all scalars, being indeed essentially numbers or ratios. In co-ordinate geometry certain fixed directions are assigned once for all, and the co-ordinates of a point referred to these directions are simply the number of units contained in the distances of the point measured parallel to these directions. They are essentially ratios of parallel vectors. The importance of distinguishing clearly between vectors and scalars will appear as we proceed. We shall first take the simplest conception of a vector as represented by transference through a given distance in a given direction. Thus if AB be a straight line, the idea to be attached to 'vector AB' is that of transference or " step " from A to B. Art. 2.] VECTOR ADDITION AND SUBTRACTION. 9 For the sake of definiteness we shall almost invariably represent vectors by Greek letters, retaining in the mean- time the English letters to denote ordinary numerical or scalar quantities. If we now start from B and advance to C in the same direction, BC being equal to AB, we may, as in ordinary geometry, designate ' vector BC ' by the same symbol, which we adopted to designate ' vector AB.' Further, if we start from any other point in space, and advance from that point by the distance OX equal to and in the same direction as AB, we are at liberty to designate ' vector OX ' by the same symbol as that which represents AB. Other circumstances will determine the starting point, and individualize the line to which a specific vector corresponds. Our definition is therefore subject to the following condi- tion : — All lines which are equal and drawn in the same direction are represented by the same vector symbol. We have purposely employed the phrase 'drawn in the same direction ' instead of ' parallel,' because we wish to guard the student against confounding 'vector AB' with 'vector BA.' In order to apply algebra to geometry it is necessary to impose on geometry the condition that when a line measured in one direction is represented by a positive symbol, the same line measured in the opposite direction must be represented by the corresponding negative symbol. In the science before us the same condition is equally requisite, and indeed the reason for it is even more manifest. For if a transference from ^ to 5 be represented by -fa, the transference which neutralizes this, and brings us back again to A, cannot be conceived to be represented by anything but -a, provided the symbols H- and - are to retain any of their old algebraic meaning. The vector AB, then, being represented by -fa, the vector BA will be represented by -a. 10 QUATERNIONS. [Chap. ii. 3. Parallel Vectors. Further it is abundantly evident that so far as addition and subtraction of parallel vectors are concerned, all the laws of Algebra must be applicable. Thus in last paragraph AB + BC or a + a pro- duces the same result as AC which is t^vice as great as AB, and is therefore properly represented by 2a; and so on for all the rest. The distributive law of addition may then be assumed to hold in all its integrity so long at least as we deal with vectors which are parallel to one another. In fact there is no reason whatever, so far, why a should not be treated in every respect as if it were an ordinary algebraic quantity. It need scarcely be added that vectors in the same direction have the same proportion as the lines which corre- spond to them. We have then advanced to the following — Lemma. All lines draicn in the same direction are, as vectors, to he represented by manerical multiples of one and the same symbol, to which the ordinary laws of Algebra, so far as their addition, subtraction, and numerical multiplication are concerned, may be unreservedly applied. The converse is of course true, that if lines as vectors are represented by numerical or scalar multiples of the same vector symbol, they are parallel. 4. Non-Parallel Vectors. It is only necessary to add to what has preceded, that if BC be a line not in the same direction with AB, then the vector BC cannot be repre- sented by a or by any scalar multiple of a. The vector symbol a must be limited to express transference in a certain direction, and cannot, at the same time, express transference in any other direction. To express 'vector BC then, another and quite independent symbol [3 must be intro- duced. This symbol, being united to a by the signs -f- and - . the laws of algebra will, of course, apply to the combination. Art. 6.] VECTOR ADDITION AND SUBTRACTION. 11 5. Vector Addition. If we now join AC, and thus form a triangle ABC, and if we denote vector AB by a, BC by /?, AC lay y, it is clear that we shall be presented with the equation a + (3 = y, or, strictly speaking, with the identity a + /3 = y. This equation appears at first sight to be a violation of Euclid I. 20 : " Any two sides of a triangle are together greater than the third side." But it is not really so. The anomalous appearance arises from the fact that whilst we have extended the meaning of the symbol + beyond its arithmetical signification, we have said nothing about that of a symbol =. It is clearly necessary that the signification of this symbol shall be extended along with that of the other. It must now be held to designate, as it does perpetually in algebra, ' equivalent to.' This being premised, the equation above is freed from its anomalous appearance, and is perfectly consistent with everything in ordinary geometry. Expressed in words it reads thus : ' A transference from A to B followed by a transference from ^ to C is equivalent to a transference from A to C 6. Axiom. If two vectors have not the same direction, it is impossible that the one can neutralize the other. This is quite obvious, for when a transference has been efi"ected from A to B, it is impossible to conceive that any amount of transference whatever along BC can bring the moving point back to A. It follows as a consequence of this axiom, that if a, p be different actual vectors, i.e. finite vectors not in the same direction, and if 7na^n(3 = 0, where m and n are numerical quantities ; then must 7n = and n = 0. Another form of this consequence may be thus stated. If [still with the above assumption as to a and jS'] ma + nf3=pa + qfi, then must m,==p, and n = q. 12 QUATERNIONS. [Chap. ii. 7. Elementaky Illustkations. The properties of vectors must be based on the fundamental principles of geometry, and it may be well to show by a few elementary illustrations how these are involved. Thus let AB, CD be equal and parallel lines and represented by the same vector a. Then, if /3 represents the vector CA, we have the vector equation vector DB = -a + /3 + a = /^. In words, "the straight lines which join the extremities ^ Q oi equal and parallel straight lines are themselves equal and parallel." Again, ii AB and CD are given parallel, and CA, DB given Fig. 2. " parallel, we may represent them by the vector a, ma, /3, nfi where m and n are scalar multiples. Then ^8 + a = vector CB = na + m/3, whence by the last article n=l, m=l ; and consequently the opposite sides of the parallelogram are equal. Again as vectors AB = AO+ OB, GD=CO + OD, where is the meeting point of the diagonals. But AB = CD ; hence AO + OB = OD + CO ; Aa= OD, CO = OB. A few simple examples will show with what directness vector methods may be applied to plane geometry. 1. I'he bisectors of the sides of a triangle meet in a point which trisects each of them. Let the sides of the triangle ABC be bisected in D, E, F ; and let AD, BE meet in G. F/Cl \£; Let vector BD or DC be a, CE or '^ryC.X. EA fS, then, as vectors, ^^ |/^^ BA = BC+CA = 2a + 2(3=2{a + /3), Fig.3. DE = DC+CE=a + p, hence (§3) BA is parallel to DE, and equal to 2DE. Art. 7.] VECTOR ADDITION AND SUBTRACTION. 13 Again, BG + GA = BA = WE = 2(DG + GE). Now vector BG is along GE, and vector GA along DG. .-. (§6) BG = 2GE, GA = 2DG, whence the same is true of the lines. Lastly, bg^Ibe=^{bc+ce) .-. CG^BG-BC = ^(2a + (3)-2a GF=BF-BG =\ba-bg, hence CG is in the same straight line with GF, and equal to 2GF. 2. When, instead of D, E, F being points taken within BC, CA, AB at distances equal to half those lines respectively, they Fig. 4. 14 QUATERNIONS. [Chap. ii. are points taken in BC, CA, AB produced, at the same distances respectively from C, A, and B ; to find the intersections. Let the points of intersection be respectively G^, G^, Gy Retaining the notation of the last example, we have BD = Sa, CE = Z(i; and.-. BGs^xBE = x{2a + 3j3) (1), and BG^ = BD + DG^ = 3a + yDA = 3a + y{CA-CD) = 3a + y(2/3-a); .•. 2x = 3-y, 3x = 2y, and x = =-j .-. linei;6*3 = ^^5 Similarly line i?'G^i = ^FC, line DG2 = ^DA, and from equation (1) BG^ = ^(2a + 3f:i). But BG^ = BA + A G^ = 2a + 2/3 + AG^, .-. ^G^3 = |(2^-a); 2 hence line ^G^3 = ^ line D^ and similarly of the others. = 2DG2, 3. If DBF he drawn cutting the sides of a triangle ; then will AD.BF.CE = AE.CF.BD. -^ Let BD = a, DA = pa, AB = 13, EC = qf3, then BC = BA + AC = (I +p) a + {l+q)/3 and CF is a multiple of BC. Let CF=xBC = x{(li-i>)a + (l +!?)/?}. Fig. 5 Art. 7.] VECTOR ADDITION AND SUBTRACTION. 15 But CF=CE + EF^-EC+EF=^ -qP + y(pa + p); .-. equating, we have a;(l +p) = yp, x{l+q)-= - q + y, whence x = (l+x)pq, CF _BF AD CE^, ^•^' BCBG' BD' AE' .'. AD.BF.CE = AE.CF.BD. 4. The points of bisection of the three diagonals of a complete quadrilateral are in a straight line. P, Q, R, the middle points of the diagonals of the complete quadrila- teral ABCD, are in a straight line. Let AB = a, AD = fS, AE = ma, AF=nf3; .'. BF=nP-a and BC = x(nl3-a), ED = f3-ma and CD = y((3- ma). -^ow BC+CD = BD = AD-AB "^ QL f5"6. gives x{nl3 - a)+y(f3 -ma) = (3 - a, whence xn + y=^l,x + nuj=l, m - 1 t * JO ^^ mn - 1 and AP=^AC=^ If m-l , r, \1 2 (^ mn - 1 ^ ' j mn I m(n - 1) a + n(m - I) f3 2 mn- 1 1 AQ = ^(a + f3), AR = ^(ma + n^), AQ-AP = ^. ,, {(m-l)a + (7i-l)^}, AR-AF = : mn .{(m-l)a + {n-l)f3}, 2{mn- 1) or vector PR is a multiple of vector PQ, and therefore they are in the same straight line. 16 QUATERNIONS. [Chap. ii. Cor. Line PQ\ PR wl : mn AB.AD-.AE.AF triangle ABD : triangle AEF. We shall presently exemplify a very elegant method due to Sir W. Hamilton of proving three points to be in the same straight line. 8. Unit Vector and Tensor. It is often convenient to take a vector of the length of the unit, and to express the vector under consideration as a numerical multiple of this unit. Of course it is not necessary that the unit should have any specified value ; all that is required is that when once assumed for any given problem, it must remain unchanged throughout the discussion of that problem. If the line AB he supposed to be a units in length, and the unit vector along AB be designated by a, then will vector AB be aa (§ 3). Sir William Hamilton has termed the length of the line in such cases, the Tensor of the vector; so that the vector AB is the product of the tensor AB and the unit vector along AB. Thus if, as in the examples worked under the last article, we designate the vector AB by a, we may write a = Ta Ua, where Ta is an abbreviation for ' Tensor of the vector a ' ; Ua, for ' unit vector along a '. Take the following example : The three bisectors of the angles of a triangle meet in a point. Let AD, BE bisect A, B and meet in G ; CG bisects C. Let unit vectors along BC, CA, AB be a, /?, y, and let a, b, c be the lengths of the corresponding sides. Then aa + 6/5 + cy = (see below, § 9), AG^x(y-/3), BG = y{a-y), CG = CA+AG = bf3 + xy-x/3, and also CG = CB + BG = -aa + ya- yy. Hence {y - a)a + {x-b)l3 = (x + y)y = -{x + y){aa + b(3)/c. Art. 10.] VECTOR ADDITION AND SUBTRACTION. 17 Equating coefficients of the a's and ^'s, and solving, we find x = bc/{a + b + c), y = ca/{a + b + c), and finally CG = f (jS - a), '' a + b + c^'^ ' and CG bisects the angle between the unit vectors /3 and - a. 9. COPLANAR Vectors. If a, /?, y are non-parallel vectors in the same plane, it is always possible to find numerical values of a, b, c so that aa + b(3 + cy shall = 0. For a triangle can be constructed whose sides shall be parallel respectively to a, (3, y. Now if the vectors corresponding to those sides taken in order be aa, b(3, cy respectively, we shall have, by going round the triangle, aa + b/3 + cy = 0. If we multiply this equation by any quantity the right hand still remains zero, and the left hand represents a triangle similar and similarly situated to the original triangle but with its sides increased (or diminished) in a given ratio. Thus, though there is an infinity of values assignable to a, b, c, any one set is simply a multiple of any other. This may be proved directly as follows : Let aa + b(3 + cy = 0, and also pa + qfi + ry = 0. By eliminating y we get (ar - cp)a + (br -cq)(3 = 0; .'. (§ 6) ar = cp, br = cq, or a : b : c :: p : q : r, so that the second equation is simply a multiple of the first. 10. CoLLiNEAR Points. If a, (3, y are coinitial, coplanar vectors terminating in a straight line, then the same values of a, b, c which render aa + bf3 + cy = will also render a + b + c = 0. B 18 QUATERNIONS. [Chap. ii. Let vector OA = a, OB = 13, OC=y, ABC being a straight line; then AB = l3-a, ^B AC=y-a. But ^C is a multiple of AB, 0' ^ 0Ty-a=p{/3-a), ^S-^- i.e. (p- l)a-p;8 + y = 0. But (p-l)-p+l = 0; and as j?-l, - p, +1 correspond to a, b, c and satisfy the condition required, the proposition is proved generally. Conversely, if «, /3, y are coinitial coplanar vectors, and if both aa + b/3 + cy = 0, and a + b + c = 0, then do a, /i, y terminate in a straight line. For ay + by + cy = 0; therefore by subtraction a{y-a) + b(y-(B) = 0, i.e. y-a is a multiple of y - (B, and therefore (§4) in the same straight line with it : i.e. ^C is in the same straight line with BC. This criterion for the collinearity of the extremities of three vectors drawn from the same origin has many elegant applications. If p be any vector drawn from the same origin as a and fS and terminating on the straight line passing through the extremities of a and (3 we may write a + riiB aa + bB p=- or= -7-, ^ l+m a+b where m^bja. For, clearing of fractions, we have a{p-a) = b{l3-p), so that p-a. and /3 - p are coUinear, being parallel with a point in common. Also the end of p divides (/3-a) in the ratio bfa ( = w). Art. 11.] VECTOR ADDITION AND SUBTRACTION. 19 11. NON-COPLANAR VECTORS. If a, /S, y are three vectors neither parallel nor in the same plane, it is impos- sible to find numerical values of a, b, c, not equal to zero, which shall render aa + b/3 + cy = 0. For (§5) aa + b/S can be represented by a third vector in the plane which contains two lines parallel respectively to a, p. Now cy is not in that plane, therefore (§ 6) their sum cannot equal 0. Or we may reason in this way. The equation may be written in the form -cy = aa + bfB, so that, if a, b, c have finite values and if a and /3 are different vectors, y must lie in the same plane with a and (3. Hence with y not coplanar Avith a and /3, the above equation can hold only if a, b, and c all vanish. Thus with a, b, c unrestricted, the equation aa + b/S + cy = 0, means that a, (3, y are either parallel to one another or are in the same plane. These theorems find illustration in the following examples : 1. If two triangles are so situated that the lines which join cm-responding angles meet in a -point, then pairs of coiresponding sides being produced will meet in a straight line. ABC, A'B'C are the triangles; the point in which A' A, B'B, CO meet; P, Q, E the points in which BO, B'O, etc., meet : FQE is a straight line. Let 0A = a, OB ==(3, 00 =y, 0A'=ma, 0B' = nf3, 00'=py, then BA = a- (3, and BR = x{a-f3)', B'A' = ma- n(3, and B'B = y (ma - n(3). Fig. 8. 20 QUATERNIONS. [Chap. n. Now BF = BR- B'R gives {n - l)/3 = x{a - /3) - y/(ma - w/3) ; .'. n-\= -x + ny, = z- my^ and »= - m{n- 1) m-n whence OE=OB + BB^B- ^^"" ^\ «-/3) _n(m- l)/3 - m(n- l)a Similarly, op^^(^- 1)7 "M^ - 1)^, p-m (7n - n) (;; - 1) 072 + (% -^) (m - 1) (9P + {p-m){n-l)OQ = (). And also identically {m-n){p- l) + (n-p){m- l) + (p-m)(ii- 1) = 0, whence (§ 10) P, $, it are in the same straight line. 2. If AD, BE, CF be drawn cutting one another at any point G within a triangle, then FD, DE, EF shall meet the third sides Fig. 9. of the triangle produced in points tvhich lie in a straight line. Also the pivduad sides of the triangle shall be cut harmonically. Art. 11.] VECTOR ADDITION AND SUBTRACTION. 21 Take G as origin, and let a, /?, y be the vectors to the angles A, B, C ; then the vectors to D, E, F may be represented by aa, hf^, cy. Let p, cr, t be the vectors from G to the points L, M, N, the meeting points referred to in the enunciation. The vector equations GL=GB + BL =GF+EL, GM=GC+CM =GF+ FM, GN= GA + AN= GD + DN take the form p = p + x {y - P) = hp-Vy {cy -hP)A o- = y + a;' (a - 7) = cy + ?/' (aa - cy), \ T = a + 3;" (f3 -a) = aa + y" (b/3 - aa), J whence we find from the first pair f3{l -x-b + by) + y{x-cy) = 0, so that X = cy, (c - b)y =1 -b, (c- b)x = c(l ~b), with similar expressions for y'y", x'x". Substituting we get p(c-b)=^b{c-l)ft-c{b-l)yA o-(rt - c) = c(a - l)y-- rt(c- l)a, ]- r{b-a) = a{b-l)a-b{a-l)f3.) Multiplying by (tt-1), {b - 1), (c-l) respectively, and adding we find {a-l){c-b)p + {h-l){a-c)ir + {c-l)(b-a)T = 0. But evidently (a - l)(c -b) + {b - l)(a - c) + {c- l){b - a) = 0, hence p, o-, t terminate on the same straight line. Now aa may be written in the form (f3 + my)/{l +m) (§ 10), where m is to be found ; and in like manner b(3 and cy may be expressed, using 71, p instead of m. Thus we have the three equations (1 + m)aa - (i - my = 0,^ - 7ia + (I + n)bl3 - y = 0, \- -a-pf3 + {l +p)cy = 0j and these (§ 9) must be simply multiples of one another. 22 QUATERNIONS. [Chap. ii. TT I +m 1 Hence a = - r-^ -r = m, n {l+n)b ' {l+n)b 1 71= - from which m, n, p may be found. Thus c-cb a-ac b~ba b-bc c- ca a-ao TVT c-cb . , b-bc JNow x = r and x-l= r- c-b c-b Hence m or x-l DC__BL_JBL BD''BL-BO~ Cn and BL is cut harmonically. Again, since mnp= 1, we find DC EA FB BD' CE' AF'' 1. 12. The Mean Point of a group of points is that point whose vector position referred to any chosen origin is equal to the sum of the vector positions of the individual points divided by the number of points. Thus for two points a, fi, the mean point is a + /5 or /u. - a = /3 - /x, so that the point at the extremity of /x bisects the line ^ - a. For three points forming the triangle ABC, /,= -(a + /3 + 7). This point is the point G of Example 1, § 7 ; for it is the meeting point of the lines, which pass through A, D, and B, E. Let p be the vector of this meeting point. Then ^"~ 1 +m ~ \+n ' where a, /?, y are vectors not necessarily in the same plane. Art. 13.] VECTOR ADDITION AND SUBTRACTION. 23 Hence rearranging we have and the factors must vanish (§11). Hence 1 Wi = 71 = 2> and p = i(to±^)„l(, + ;3 + v). For four points A, B, C, D, with vector positions a, ^, y, S, ^ = _(a + ^ + 7 + S). These four points are the corners of a quadrilateral not necessarily plane. The middle point of the line joining the middle points of any pair of opposite sides is Ifa + Z? y + 8") the mean point. Hence the lines joining the points of bisection of the opposite sides of a quadrilateral in space meet and bisect each other. Again the point of bisection of the line which joins the middle points of the diagonals is the same mean point ; for ^ = 2 Ifa + y ^ + S1 a + /3 + y + S {^-^} = and so on for any number of points. 13. Centre of Mass. The properties of the centre of mass or centre of inertia of a system of particles are con- veniently discussed here. If we have a number of equal masses placed in given positions, then the centre of mass is simply the mean point. 24 QUATERNIONS. [Chap, ii. To extend the discussion to the more general case, we shall first define what Maxwell has called the mass-vector. The mass-vector of the particle of mass m whose position is assigned by the vector p is the vector mp, that is the vector whose tensor is m times the tensor of the vector p. Let there be two masses m^, m^ in positions p^, p^; then the vector ^J^hPi+^hN m^ + Wg is (§ 10) the vector of the point in the line joining m^ and m^ and dividing it in the ratio of m^ to rriy For m^{a--p^) = m^{p^-^), so that the product of each mass and its distance from the centre of mass is the same. Add to the system a third mass m^ in position py We may suppose m^ + mr, to be condensed at their centre of mass. Hence the centre of mass of (m^ + m^ and Wg got by the same process will be nij + ill., + Wig m^ + wig + ??lg Generalizing we arrive at the definition of the centre of mass of a system of particles as that point whose vector position is the sum of the mass-vectors divided by the sum of the masses. Thus, generally (r(m^ + 711.-, + ???3 -I- . . . + m„) = m^p^ + m^p^ + m^p^ +...+ 7n„p„, giving in every case a perfectly definite vector o-. This relation may be written briefly (rlm = '2{mp), where 2 means the summation of terms of the type indicated. Art. 14.] VECTOR ADDITION AND SUBTRACTION. 25 Rearranging we find, since o- may be put under the summa- tion symbol, or, the mass vectors referred to the centre of mass would if put end to end form a closed polygon, 14. Velocity a Vector Quantity. As already explained, the simplest conception of a vector is that of a transference or step in a definite direction, through a definite distance. When a moving particle changes position from, say, position p to position p, the displacement is measured by the vector (P'-P). In the very simplest case imagine the particle to be moving with constant speed along this vector line {p - p). This vector will represent by its direction the direction of motion and by its length the distance travelled through. Suppose the transference to be eff"ected in t seconds of time. Then the vector ^ — ^, which is the ^th part in length of the vector // - p, will clearly represent in direction and magnitude the velocity of the moving particle. Velocity is therefore completely represented by a vector whose direction gives the direction of motion and whose tensor measures the speed or rate at which space is being described. Velocity is defined quite generally as the rate of change of position, and is clearly a vector quantity. For when a particle moves it must move in a definite direction with a definite speed. The velocity is therefore fully symbolized by a vector line drawn in this direction and of a length measuring the speed on a convenient scale. The relative velocity of two moving bodies is obtained at once by taking the vector difference of the vectors representing their velocities. If a, /3 represent the velocities of two particles A, B, we get their relative velocity by superposing on both such a velocity as will reduce one to rest. Thus, 26 QUATERNIONS. [Chap. ii. Pig. 10. for example, let - a be superposed. This will annul the + a in the one case and produce with /S a resul- tant velocity represented by /5 - a in the other case. The vector differ- ence /3 - a is the velocity of B relatively to A. Any given velocity may be decomposed into any number of components, the sole condition being that the components drawn end to end form with the original velocity reversed a closed polygon. If two directions be assigned coplanar with the given velocity, the components along these directions have deter- minate values. Thus in the figure on page 11, let^^Cbethe given velocity and AB, BC parallel to the given directions. Then it is clear that when lines parallel to these directions are drawn through the extremities A and C, they will meet in a determinate point B, and the components AB, BC have determinate values. Again, if any three non-coplanar directions are assigned, there is one way only in which a given velocity can be decomposed into components parallel to these directions. Thus let OA, OB, OC be the required directions in space and OP the given velocity, and let OB, OC be in the plane of the paper. Through P draw a line parallel to OA till it meets the plane containing OB and OC. Let PM be this line. Through M draw a line parallel to OC till it meets OB in N. Then the velocity OP is decomposed into the components ON, NM, MP, which Fig. n. are all determinate in direction and magnitude. Long before the calculus of quaternions or any system Art. 14] VECTOR ADDITION AND SUBTRACTION. 27 of vector analysis was invented these vector properties of velocities were known ; and they still form some of the most effective illustrations of the method. EXAMPLES TO CHAPTER II 1. If P, Q, R, S be points taken in the sides AB, BC, CD, DA of a parallelogram, so that AP : AB :: BQ : BC, etc., PQRS will form a parallelogram. 2. If the points be taken so that AP=CR, BQ — D8, the same is true. 3. The mean point of PQRS is in both cases the same as that of A BCD. 4. Tlie quadrilateral formed by bisecting the sides of a quadri- lateral and joining the successive points of bisection is a parallelogram, with the same mean point. 5. If the same be true of any other equable division such as trisection, the original quadrilateral is a parallelogram. 6. If any line pass through the mean point of a number of points, the sum of the perpendiculars on this line from the different points, measured in the same direction, is zero. 7. From a point £J in the common base AB oi the two triangles ABC, ABD, straight lines are drawn parallel to AC, AD, meeting BC, BD SitF,G; show that FG is parallel to CD. 8. From any point in the base of a triangle, straight lines are drawn parallel to the sides : show that the intersections of the diagonals of every parallelogram so formed lie in a straight line. 9 If the sides of a triangle be produced, the bisectors of the external angles meet the opposite sides in three points which lie in a straight line. 10. If straight lines bisect the interior and exterior angles at A of the triangle ABC in D and E respectively ; prove that BD, BC, BE form an harmonical progression. 11. The mean point of a tetrahedron is the mean point of the tetrahedron formed by joining the mean points of the triangular faces ; and also that of the mean points of the edges. 28 QUATERNIONS. [Chap, ii.] 12. If through any point within the triangle A BO, three straight lines MN, PQ, BS be drawn respectively parallel to the sides AB, AC, BC; then will MN PQ ES AB^AG'^BG 13. ABGD is a parallelogram ; PQ any line parallel to CD; PD, QG meet in S, PA, QB in A' ; prove that ^D is parallel to RS. 14. If the vertical angle of a triangle be bisected by a straight line which cuts the base, the segments of the base shall have the same ratio that the other sides of the triangle have to one another. 15. Find the expression for the centre of mass of a uniform wire bent into tlie form of a triangle, the lengths of whose sides are a, b, c. 16. The mean point of a triangle trisects the line joining the point of intersection of the perpendiculars on the sides from the opposite angles, and the point of intersection of perpendiculars on the sides from their middle points. CHAPTER III. QUATERNIONS AND VERSORS OR QUOTIENTS AND PRODUCTS OF VECTORS. 15. The Quaternion as a Geometrical Operator In the preceding chapter the laws of addition and subtrac- tion of vectors have been discussed ; and broadly speaking these laws are common to all vectorial systems such as are met with in the Barycentrische Calcul of Mobius and the Ausdehnungslehre of Grassmann. We now pass on to the discussion of products and quotients of vectors ; and it is well at the outset to state distinctly what are the peculiar features of Hamilton's Quaternions as compared with other systems of vector analysis. It lies in this, that, whereas the commutative law in multiplication no longer holds, the distributive and associative laws are still retained. In symbols, a/3 is not the same as (Ba ; but a (/3 + y) is the same as a/3 + a-y, and a^y has the same value whether it is regarded as a multiplied by /3y or as a/3 multiplied by y. The whole system may be developed analytically from these fundamental restrictions. Here, however, we shall develop the system geometrically, bearing in mind that the distributive and associative laws are to hold, and adopting the usual notations familiar to us in ordinary algebra, in so far as these are not inconsistent with the restrictions laid down. 30 QUATERNIONS. [Chap. hi. Given any two vectors a, /3 there must be some multiplier or operator whicli changes /3 into a. Writing q for this operator we have symbolically q(3 = a. If (3 and a were two scalar quantities of the same kind, q would be their ratio. When they are vector quantities of the same kind, we may still by analogy regard q as a ratio, and we may express it in the form a/j3 or a/3~'^. The relation a^-i./5 = a is obviously self-consistent, if we assume the associative law to hold, and if /3~i^ = l. This involves the definition of the reciprocal of a vector. Thus if y/3=l, y is the reciprocal of (3 and simply undoes whatever effect may be produced by /?. Evidently their tensors must be reciprocal in the ordinary arithmetical sense. Moreover, since the effect of a vector must depend in some way upon its direction it is reasonable to expect that the reciprocal vector will undo this effect in virtue of its having the opposite direction. That is to say, the most obvious interpretation of the reciprocal of a vector is a vector whose tensor is the reciprocal of the tensor of the original vector and whose direction is the reverse of that of the original vector. This we shall find to be its meaning in quaternion vector analysis. Since a vector possesses both direction and magnitude, the process by which it is changed into another vector must involve the two distinct operations of change of direction and change of magnitude. When, for example, /3 is to be changed into a, it must first be rotated through a definite angle in a definite plane until it is parallel to a, and then its tensor or length must be altered in the proper ratio so as to make it equal to the tensor or length of a. Or, we may first effect the alteration in length, and Art. 16.] QUATERNIONS AND VERSORS. 31 then effect the rotation. The result is the same, so that these operations are commutative. A further consideration shows that this process of changing one vector into another involves four numbers. (1) There is the change of tensor — one number ; (2) there is the angle of rotation — one number ; (3) there is the aspect of the plane in which the rotation takes place or the direction of the axis about which rotation takes place, and this requires two numbers for its determination. In all, four numbers. For this reason Hamilton called the multiplier q or a/5~i a quaternion. 16. The Constituents of a Quaternion. As already- pointed out, the process of changing one vector into another consists in general of two separable operations — the one effecting the necessary change of length, the other the change of direction. These are distinguished as the Tensor- and Versor parts of the quaternion, and are written Tq and Uq respectively.^ Thus q(3 = TqUqlB=UqTq(3, or symbolically, q = TqUq= UqTq. The operation represented by Tq is simply that of multiplying by a numerical factor, and requires no further discussion. Tq is in fact a scalar multiplier. When the value of Tq is unity, the quaternion is reduced to the expression Uq and is called a Versor, since its effect is simply one of turning. A Versor involves an angle and a plane or direction of axis — three numbers. A Vector involves a length and a direction — also three numbers. Any two vectors, a and /3, involve six numbers. But the quaternion, a/3~^, as has been shown above, involves only four numbers. Hence, although q{ = a/5~i) is completely determined when a and ft are given, a and (i are not completely determined when ^ The use of the selective symbol U in two senses as the unit of a vector and the versor of a quaternion will be found to lead to no confusion. 32 QUATERNIONS, [Chap, hi q is given. All we know, in this case, about a and f3, are the aspect of the plane in which they lie, the angle between them, and the ratio of their tensors — in short, just the four constituents of the quaternion. Any other pair of vectors having these same relations will give the same quaternion. Thus a given quaternion can be expressed as the ratio of two vectors in an infinite number of ways, the conditions attaching to these vectors being, that they are perpendicular to a given direction known as the axis of the quaternion, that they contain an angle known as the angle of the quaternion, and that their tensors have a ratio equal to the tensor of the quaternion. It should be mentioned that the axis of a quaternion is drawn in that direction which bears to the versor eff"ect of the quaternion the same relation which the translational motion of a right-handed screw bears to its rotation. 17. The Quaternion and its Conjugate. To each quaternion q there corresponds another quaternion Kq, called the conjugate, whose effect upon a vector operand is the same except that the angle of rotation is taken in the opposite direction. In other words, the axes of a quaternion q and its conjugate Kq are oppositely directed, the tensors and angles being the same. Thus, if qP = a, - then Kq(3 = a, ''-f \ ^ where a has the same tensor as "/j- a but lies on the opposite side of ft, making the same angle A with it (see Figure 12). Evidently if we operate on a by Kq, or on a by q, we shall' obtain a vector /3' parallel to f3, such that TP'/Ta=Ta/TI3, or T(i'ITft={ralTpf = {Tqf. Art. 18.] QUATERNIONS AND VERSORS. 33 Hence, Kqq(3 == Kqa = (Tqf 13, and qKql3==qa' ={Tqfli, or symbolically, qKq = Kqq = {Tqf, a scalar quantity. From the fundamental equations given above we get by addition and subtraction (q-Kq)(B = a-a'. But a + a is a vector parallel to (3 and having a length equal to 2Ta cos J ; and a - a' is a vector perpendicular to (3, and having a length equal to '2Ta sin A. Let Uy be unit' vector perpendicular to f3 in the plane af3. Then we may write {q + Kq) (3 = 2Ta cos A. Uf3, (q - Kq) (3 = 2Ta sin A. Uy, or (q + Kq)U(3 = 2 —cos A . UI3 = 2Tq cos A . Uf3, (q - Kq) U(3 = 2Ssin A . Uy =2Tq sin A . Uy. Thus q + Kq is a scalar multiplier, while q-Kq is a quaternion which rotates (3 through a right angle about the axis of the quaternion q and changes its tensor in the ratio of 2Tq sin A to unity. A quaternion which rotates the vector operand through a right angle is called a qmdrantal quaternion; and a versor which does the same is called a quaclrantal versor. 18. QuADRANTAL QUATERNIONS AND Versors. From the last paragraph we learn that any quaternion may be expressed as the sum of a scalar quantity and a quadrantal quaternion ; and that when this is done the conjugate of the quaternion is then expressible as the difference of the same two quantities. In symbols q = S+Q, Kq = S-Q, where S is the appropriate scalar, and Q the appropriate quadrantal quaternion, whose meanings are given in the last paragraph. » 34 QUATERNIONS. [Chap. hi. If the scalar vanishes, q becomes the quadrantal quaternion Q, and we see that the conjugate of a quadrantal quaternion is simply the quadrantal quaternion with its sign changed. This is obvious from the geometry of the operation, for Q and KQ rotate the vector operand in the same plane through right angles measured in opposite directions. In this case also QKQ=Q{-Q)=-Q\ But QKQ = {TQf. Hence the square of a quadrantal quaternion is equal to minus the square of its tensor. The same conclusion may be readily established by direct consideration of the geometry of the operation, for UQ operating twice in succession simply reverses. For simplicity of discussion let TQ = \, so that Q becomes a quadrantal versor. The properties of the quadrantal versor being once established, those of the quadrantal quaternion are immediately obtained by introducing any scalar factor. Follow- ing Hamilton we shall symbolize quadrantal versors by one or other of the letters i, j, k ; and these we shall represent geometrically by their axes, distinguishing them meanwhile from vectors by using two arrow heads instead of one. Let i, i' be two given quadrantal versors as shown in the figure, and let each act on the unit vector f3 perpendicular to both. Then i/5 = a, a unit vector perpen- dicular to jS and to i, the direction of rotation Pig 13, being right-handed with ,.-Vl« + i Art. 19.] QUATERNIONS AND VERSORS. 35 reference to the direction of the axis of i. Similarly, i'(3 = a', a unit vector perpendicular to f3 and to i'. By the dis- tributive law a + a =1/3 + i'P = {i + 1)(3. Hence (i + i') is the operator which changes (3 into (a + a). This operator is evidently a quadrantal quaternion, turning (3 through a right angle and changing its length to the value T{a + a'). This quadrantal quaternion will have its axis along the diagonal of the parallelogram formed by i and i', and its tensor will be equal to T{i + i'). It may therefore be completely symbolized by (i + i'), in which the versors i and i' are added like vectors to produce the quadrantal quaternion {i + i'). In other words, quadrantal versors, and (it is easy to show) quadrantal quaternions also, are com- pounded like vectors. Now, so far as our definitions go, any quantities which obey the vector law of addition may be regarded as vectors ; and if no inconsistency results we may extend to vectors any analytical properties which these new quantities may possess. The explicit identification, so far as regards their properties in analytical combinations, of quadrantal quaternion and vector is one of the outstanding features of quaternions. It has been taken exception to by theorists ; but there is no practical system of vector analysis in use in which the versorial character of a vector in product combinations is not either implicitly involved or explicitly assumed. The identification of versor and vector leads to no confusion and greatly facilitates transformations, 19. Multiplication of Quadrantal Quaternions. And now let us consider the result of operating with two quadrantal versors in succession. Let ^, i' be these versors drawn from as in Figure 14. Through draw planes perpendicular to them and let y be the unit vector along 36 QUATERNIONS. [Chap. hi. the line of i / * >v / \ t-- u. :^-\;-- j " ~~- ^\? / Fig. 14. that is, intersection in the direction which is positive with regard to right-handed rotation from ^' to i. Take /3 perpendicular to y and i, so that if3 = y; and take a perpendicular to y and i' so that iy = a. Then i'ijS = i'y = a, or i'i = a/j3. Hence, i'i is the quaternion (in this case, versor) which changes (3 into a. This versor i'i has its axis perpen- dicular to the plane containing a and f3, that is, to the plane containing the axes of the quadrantal versors i' and i; and its angle is equal to the complement of the angle between i' and i. By introducing scalar multipliers we may pass from versors to quaternions ; and thus any quaternion can be represented as the product of two quadrantal quaternions, the tensor of the product being the product of the tensors of the factors, the axis being perpendicular to the axes of the constituents, and the angle the complement of the angle between these axes. By the original definition, q = a/ (3 — aP~^ = a(3', where /?' is the reciprocal of (3. Hence a quaternion may be expressed as the product of two vectors. Thus we find that in their multiplication as well as in their addition, quadrantal quaternions and vectors obey the same laws. The identification of vectors and quadrantal quaternions leads at once to the following conclusions. The conjugate of a vector is its inverse. Thus aKa= -a? = {Ta)\ or the square of a vector is minus the square of its tensor. Art. 20.] QUATERNIONS AND VERSORS. 37 If the vector is a unit vector, aKa = 1 = aa~^ j a~^ = Ka= — a, or the reciprocal of a unit vector is equal to its inverse. Consequently, for any vector f3 we have U(3 f3 (S-^ = {m-\u(^y T/3 cm TRM 20. The Scalar and Vector Parts of a Quaternion. It has been shown (§ 18) that any quaternion may be represented as the sum of an appropriate scalar and an appropriate quadrantal quaternion. For quadrantal qua- ternion we may now substitute the word vector, and write q = Sq+Fq, Kq = Sq-Fq, when S and F are selective symbols separating out the scalar and vector parts of the quaternion. These parts have definite meanings, which have already been given. When q is a versor {Tq = l), Sq is the cosine of the angle through which q turns a vector perpendicular to its axis, or it is minus the cosine of the angle between the axes of two quadrantal versors or unit vectors whose product gives q ; and Fq is the vector (or quadrantal quaternion) measured along the axis of q and of length equal to the sine of the same angle. The extension to quaternions is easily given. Let a and b be the lengths of the vectors a and /3. Then where in which A is the angle between a and /?, and e is the unit vector or quadrantal versor perpendicular to a and (3. a 9 = T COS A, "r a . . e T sm A, 38 QUATERNIONS. [Chap. hi. Again, when q = a/3, where a and /3 may be regarded as quadrantal quaternions or as vectors, we have a(i = ^a^ + Fa/5, K{aji) = ^a^ - Vaft, where Saji = -ab cos A, Va/S = i. .ah sin A, e being unit vector perpendicular to a and (^, and A the angle between a and j^. K(al3) . a/5= {r(a/3)}2= (ra)2 (7/3)2 = a2/52. Hence, K{afi) . a = a^fS = /8a2 = ^aa, because a^ is essentially a scalar quantity and fulfils the commutative law. Thus, finally, multiplying into a~\ K{af3)=l3a = SI3a+F/3a. But K{af3) ^SafB-Vap, from which we conclude that Sa(3 = Sf3a, but Vf3a= - Fa/3. Altering the order of the factors in the product a/3 reverses the sign of the vector part but does not afi'ect the scalar part. We also find 25'a/3 = a/3 + /3a, 2Fa^ = a^-/3a. When »S^a^ vanishes, the quaternion a/3 becomes reduced to its vector part ; and this occurs when ^ is a right angle. The equation Saf3 = means that a is perpendicular to ^. If we suppose a to be given, and p to be any vector satisfying the equation Sap = 0, we see at once that p may be any vector passing through the origin perpendicular to a. The equation therefore represents the plane passing through the origin and having its normal parallel to a. When Fa/3 vanishes, the quaternion becomes reduced to its scalar part ; and this occurs when A is zero or equal to two right angles. Hence, Fa^ = means that a and /3 are parallel. Conversely, when a and /3 are parallel, Va/S = 0. If we suppose a to be given, and /o to be a vector satisfying the equation Vap = 0, then the sole condition is that p || a. Art. 21.] QUATERNIONS AND VERSORS. 39 There is no limit to the length of p. Consequently, the equation Fap = represents a straight line through the origin parallel to a. 21. Unit Vectors perpendicular to one another. When a, (3 are two mutually perpendicular unit vectors, the product a/3 has no scalar part, but is wholly a vector. Hence we may write a/3 = y, where y is the unit vector perpendicular to a and /?. Taking the conjugates of both sides, we have /3a = - y. Multiplying by /3, we get ^2a=z -/3y, or a = f3y, since (3^= -1. Or, multiplying into a, we get fSa^ = — ya, ov (3 = ya. These relations a/3 = y, ^y = a, ya = /3, necessarily hold among three rectangular unit vectors. It has become customary to use for such a system of rect- angular unit vectors or quadrantal versors the letters i, j, k ; and, as Hamilton showed, from the properties of these space units the whole calculus may be analytically developed. The properties of i, j, k, as usually given, are V = k = -ji, jk = i = - kj, ki= j = - ik, It is instructive to see what relations among these quantities are necessary and sufficient for the purpose. Assume, to begin with, that y = k= -ji, jk=i= -kj ki=j = - ik, and assume in addition that the associative law is to hold. That is to say, any combination, such as iij, is to have the 40 QUATERNIONS. [Chap. hi. same value whether it is regarded as made up of (ii)j or i{)j). On this assumption we have i^j = iij = i , ij = iJc= -j ; .'. i^= -I. Similarly, p= -I, k^= -I. Again, ijk = ii= - 1, and similarly for the products jki, hij, in which the same cyclical order is preserved. But ikj = {-j)j= +1, so that a change in the cyclical order changes the sign of the product. It should be noted that the triple product ijk is a scalar quantity. Let us take vectors xi, yj, zk, where xyz are the tensors of the vectors parallel respectively to i, j, k. Then the product xiyjzk= -xyz. 22. Comparison with Cartesian Methods. Let ijk be unit vectors measured along a system of mutually perpendicular axes in space. Any other vector p may be expressed in the form P = ix +jy + H where xyz are the coordinates of the extremity of the vector p measured along the directions i, j, k. Any other vector o- will have the corresponding form o- = ix' +jy' + k;^. Applying the distributive and associative laws we find for the product the form po- = {ix +jy + kz) (ix' +jy' + kz) = i^xx' +pyy' + k^zz' + ijxy' +jiyx' + ... = -XX' - yy' - zz' + k{x'if - x'y) + i{yz' - y'z) +j{zx' - ^x). But per = Sip(T + Vpo-. Art. 23.] QUATERNIONS AND VERSORS. 41 Thus the analytical expressions for the scalar and vector parts of the product of the two vectors are Spo- = - (xx + ijy' + zz), Vp, sin {A + B) = sin ^ cos 5 + cos A sin B. Aet. 26.] QUATERNION PRODUCTS. 53 If the actions of the versors be in opposite directions, /? lying beyond y, we obtain similarly the expressions for sin(A-B), cos(A-B). As another example let us find the cosine of the angle of a spherical triangle in terms of the sides. Let afSy be unit vectors, OA, OB, OC, not on the same plane. The identity ya~^ = y/3~^ . (3a~'^ is still true, and this may be expanded in the form p. ^g ^sl.s^+sl.yl^+sl^.Fl+vl.Fi papa a p pa Taking the scalar part of both sides, we find a pa \ p a/ In the usual notation S- = cosh, SL = cosa, S'=cosc, a p a and s(vl V^-] = tfI TV^SUVl UV^ \ p a/ pa pa = sin a sin c cos B, B being the angle between the planes OBA, OBC. Hence cos h = cos a cos c + sin a sin c cos B. Again, let e be the unit vector perpendicular to the plane of the triangle whose sides are parallel to the unit vectors afSy ; and let A, B, C he the angles opposite sides. Then £2.4/^^ = -y, e^-Bl^y = - a, e'^C^a = - /3, whence - /3 = e^-l^^a = - e^(^l^ . ^-^I^y = + ^^{C+B+A)l^j^^ and g2(.^ + S + C)/:r= _i^ or A+B + C = Tr. The angles of a triangle are together equal to two right angles. 54 QUATERNIONS. [Chap, iv. 27. Versors represented by Arcs on a Sphere. If a, /?, y, etc., be unit vectors drawn from a given origin, their extremities A, B, C, etc., lie on a sphere. The operation a/3~i which changes (3 into a may be represented by the arc of the great circle passing through BA. An equal arc taken anywhere on this great circle will represent the same versor. Let this ^^^' ^"^ versor be symbolized by p. Similarly, let q be the versor f3y~^, represented by the arc CB. Then From the pq = al3 1 . /3y~^ = ay 1 = CA = r, say The conjugates of p, q, r are p~^, q~'^, r~'^ (§ 17). equation pq — r, we get by successive multiplications pqr~'^ = l, qr~'^ =p~'^, so that Kr = K{pq) =- KqKp. It is easy to see that this relation holds for quaternions as well as for versors; and that generally the conjugate of the product of any number of quaternions, versors, or vectors, is equal to the product of the individual conjugates taken in the reverse order. Let us now find how the combination qp is to be represented. The versor arcs must be so arranged that the operation p is completed at the point where q begins. Hence p must end at B, and q must begin at B. p is therefore to be represented by A'B (Fig. 21), an arc equal to BA and on the same great circle. Similarly, q is to be represented by BC'{ = CB). Then qp is represented by the arc A'C, which is evidently equal in magnitude to the arc CA, but in general lies on quite a different great circle, that is in a different plane. It is therefore not the same versor. Let this versor qp Art. 27.] QUATERNION PRODUCTS. 55 be symbolized by s, which, it must be remembered, may be represented by any arc equal to A'C in the same great circle, such for example as CD. Since the associative law holds, we have s = qp = qpqq~'^ = qrq~'^. Thus the complex operator q( )q~'^ changes the versor r into the versor s, the great circle containing r being moved into the position of the great circle containing s. It is obvious also because of the equality of Fig. 21. the angles of the curvilinear triangles AGB, A'C'B that the great circles containing s and r cut the great circle containing q at the same angle. Hence the motion by which r is changed into s may be effected by a rotation, C moving into the position C", and AC moving into the position DC. Moreover any other great-circle arc drawn through C will be simultaneously rotated into a corresponding position with reference to C. The particular great circle which meets q at C orthogonally will remain perpendicular to it at 6" after the rotation ; and the meeting point Q of these two great circles will be the pole of q, and will be the extremity of the axis about which all rotations are effected. Any given network of great circles, and therefore the corresponding vector lines drawn from the centre 0, will, when operated on by q{ )q~''^, be rotated about the axis of q through an angle equal to twice the angle of q. Since q = Tq Uq, and Kq = Tq{Uq)-\ we find q{ )Kq = { Tqf Uq( ) ( Uq)-\ 56 QUATERNIONS. [Chap. iv. Hence the operator q{ )Kq acting on any vector or collocation of vectors will have the same rotational effect as q( )g-i, but will increase the length of every vector in the ratio (Tqf : 1. The operator q{ )Kq therefore repre- sents a simple rotation about the axis of q, accompanied by a uniform expansion (or contraction) of the system. It is a particular form of strain. 28. The Rotational Operator otherwise deduced. The operator q{ )q~'^ may be built up directly from the original definition of a quaternion. The problem is to find the quaternion opei'ator which will rotate any vector about a given axis through a definite angle. The versor Q, acting on a vector perpendicular to its axis, turns that vector through the appropriate angle. But we may represent any vector p as composed of two parts, CT parallel to the axis of Q and v perpendicular to it. Hence the vector p' = (TT + Qv is Avhat the vector p becomes when it is rotated conically about the axis of Q through the angle of Q. Taking con- jugates of both sides we get Kp=Kv5 + KvKQ, or - p' = - — - v()~i, or p' = uJ + I'Q-i. Now when p is rotated about the axis of Q through the angle of Q, it becomes p" = To -\- QvQ~'^ = Q{^ + v)Q-' = QpQ-\ because Q and CT having parallel axes are commutati\'e. Hence Q{ )Q~^ rotates p through twice the angle of Q. Another way of considering the effect oi q{ ) Kq as an operator is to write q = a + a, Kq = a- a, T^q = a^ _ ^2^ Art. 28.] QUATERNION PRODUCTS. ' 57 and expand the expression qi^Kq. It becomes p' = qpKq = {a + a)p(a-a) = a^p + a(ap - pa) - apa = a^p + 2a Vap — 2aSap + pa? = {a? + a?)p- 2aSap + 2aVap. Now in all cases we may write p = a~^ap = a~^Sap + a~^ Vap, giving the components of p parallel to and perpendicnlar to a. But evidently a~^Sap' = a~^ { (a- + a-)Sap — 2a"Sap} = a-1 (a2 _ a2) Sap = T\a-^Sap, SO that this component is increased in the ratio of T\ or qKq to unity. The other component is a-l Vap' = a-l { {(fi + a'-) Vap + 2aV . aVap] = (^2 + a-) a"l Vap + 2aaa~l Vap. Put a-Wap = /3 and a-^Vap=f3', then ft' = (a? + a2) /3 + 2aaf3, /3'/3-i = a2 + a2 + 2aa=:^, say. Hence Q = (a + «)'' = q^, and /3 is turned into direction f3' through twice the angle of q, and its tensor is increased in the ratio of T-q or qKq to 1. Thus the effect oi q{ )Kq on p is to change it into the vector p, whose projections along and perpendicular to the axis of q are greater than the projections of p in the ratio of qKq or T'^q to unity, so that the angle which p makes with the axis of q is unchanged, while at the same time the projection perpendicular to the axis of q is rotated through twice the angle of q. The same result may be obtained with ease by use of the versor in the form a"" or by use of the expanded binomial form for a versor, namely, cos A + asm A. All give interesting exercises in quaternion transformations. 58 ■ QUATERNIONS. [Chap. iv. The strain symbolized by q(^ )Kq was noticed by Gauss, who saw that it involved four numbers. These are given by the scalar coefficients in the expanded qua- ternion form, {w + xi + yj + zJc) ( ){w-xi- yj - zk), where i, j, k form a set of rectangular unit vectors. 29. Composition of Finite Rotations. Let a rigid body be acted upon by a rotation q( )q~'^ and then by a rotation P( )P~^j ^^^ resultant effect is M( h-^p-' = r{ )r-i(§27). But r is a definite versor represented by the great-circle arc drawn from the beginning of the representative arc q to the end of the representative arc p, as shown in Fig. 20. Hence, when a rigid body with one point fixed is subjected to a series of rotations about various axes, the final position can be arrived at by a single resultant rotation from the initial position about a definite axis through a definite angle. The consideration of the spherical triangle gives the position of the resultant rotation at a glance. The resultant angle of rotation and the direction of the axis of rotation are determinate, and may easily be calcu- lated. For example, let p = cos ^ + a sin ^, and q. Thus C is determined. Akt. 30.] QUATERNION PRODUCTS. 59 Then equating the vector parts of both sides, we get y sin C=a sin A cos B + /3 cos ^ sin ^ + VafS sin A sin B, so that the components of the vector y along the directions a, /3, and the directions perpendicular to these, are determined, and the position of y is known. Finite rotations are not in general commutative, for, as shown above, qp is not the same as pq. Hence the rotation qp( )p~\~^ is not in general the same as ])q{ )q~^P'~^- They are the same when the rotations are coaxial, so that^ and q may be represented by arcs along the same great circle ; also when each component rotates through four right angles. 30.. Composition of infinitely small Kotations. If we write q in the form a + a, Kq is a — a, and {Tqf = qKq = a'- + {Taf. We pass to the case of infinitely small rotations by taking Ta very small, so that its square may be neglected in comparison with a-, which in the present case may be taken as equal to unity. Hence, if we write q in the form l+|^ee, where e is a very small quantity and e is a unit vector, the rotation is symbolized by (l+iee)( ){\-Ue), e being unit vector along the axis of rotation, and e the measure of the (small) angle of rotation. Any vector p becomes p' = {l+U€)p{\-\ee) = p + ^e{ep- pi) - le-epe = p + eVep, the term in e^ being negligible. Thus p - p = eVep is the displacement of the extremity of p. It is in a direction perpendicular to both p and e, and its value is eTVtp = eTp sin A, where A is the angle between € and p. 60 QUATERNIONS. [Chap. iv. If there are two simultaneous small rotations ee about axes e e', the vector p becomes p' = (l+|ee)(l+ieV)p(l-ieV)(l-icc) = {1 +i(ee + ee')} p {1 -i(« + 60}. neglecting products of the small quantities e, e'. Hence p' = p+ F. (ee + e'e')p, so that the resultant rotation is obtained by the same process of vector addition as resultant displacements and velocities are obtained. Generally for any number of simultaneous small rotations e-^e^e^ . . . about axes e^e^e^ ... the displacement of any point p is p' - p= F . {e^e^ + e^e.2 + e^e^+ ...)p= F .^{ee)p. There is no displacement when p is parallel to ^(ee). This vector is therefore parallel to the axis of rotation, and the resultant angular displacement about this axis has the value The quantity pFep is evidently a vector, being the product of two perpendicular vectors ; and the summation {~7np Fep) of quantities of this kind in which e is any vector, and p is one of a number of given vectors, is an example of what is called a linear vector function of the vector e. In the present case its value depends upon the distribution of matter in the body. The linear vector function is one of the most beautiful of Hamilton's discoveries. Some of its properties are discussed in Chapters VI., VII., and X. 31. Quaternion Products. The product of any number of quaternions is a quaternion. This follows at once from the representation of versors on a sphere ; for the passage from versors to quaternions requires simply the introduction of the scalar factors known as the tensors. Thus, in the equation jyq = r, where p and q are given quaternions, r also is a quaternion whose tensor is equal to the product of the Art. 32.] QUATERNION PRODUCTS. 61 tensors of f and q, and whose versor is the resultant of the versors oif and q, as discussed in § 27. The product ifygs of three given quaternions is at once by the foregoing reduced to rs, a product of two quaternions, and this is a quaternion ; and so on for any number. Again, since T{pq) = TpTq, we have pq^^ipq) = T^ (pq) = TyT^q. Multiply by Kp, and then by Kq, and divide out the scalar factors. This gives Kq{TpfqK{pq) = KqKpT-pT^q, and finally J^iVl) = KqKp. And generally the conjugate of the product of any number of quaternions is the product of the conjugates of the con- stituents taken in the reverse order ; in symbols K{pqrst) = KtKsKrKqKp. 32. Products of Vectors. What is true of quaternion products in general will be true of particular types, such as quadrantal quaternions or vectors. Thus the continuous product of three or more vectors is in general a quaternion, degenerating in special cases to a vector or a scalar. Consider the quaternion q = a(iy, with its conjugate K{a(iy) = KyK/SKa = ( - 7) ( - /3) ( - a) = - yft-- From the general relations '2Vq = q- Kq, 2Sq = q + Kq, we have 2V.al3y = afty+yl3a, 2S . a(3y = o,f3y ~ yl3a. Suf3y and Va^y are the scalar and vector parts of the product a^y. The geometrical meaning of S . afSy is easily deduced. For S. apy = S. a(SI3y+V/3y) = S. aFj3y, because aSfSy being a vector can have no scalar part. But S. aVPy may be written in the form TF/3yS . aUF/3y. Now 62 QUATERNIONS. [Chap. rv. TF/3y is the area of the parallelogram contained by (3 and y, i.e. twice the area of the triangle OBC (fig. 22). Then - S . all V (By is the resolved part of OA perpendicular to OBC, i.e. the perpendicular from A upon the plane OBC. But the pro- duct of twice the area OBC and the height to A is evidently the volume of the parallelepiped whose base is the parallelo- gram contained by OB and OC and whose opposite face passes through A. In short, it is the volume of the parallelepiped whose edges are a, /3, y. Since S.afty = Sa F/3y = ,S'( Fpy) a = ^^ya = -S.aFyl3= -Say/S, and so on, we see that so long as the cyclical order is unchanged the scalar of the product has the same value ; but that if the order is changed the sign is changed. If we express a, (3, y in terms of a set i, j, k of rect- angular unit vectors, namely, a = ci-^i + a.-,j + ajc, f3=\i + hj + hjc, y = Cji -f- c^j -t- cjc, and form the scalar of the product a^y, we notice that all terms of the form iij or jjk, being vectors, must vanish. Hence, only terms in ijk can exist. But since ijk= -1, we find -S.afiy. «!, «2' 0^3. K b„ ^3. ^■v ^2, Cs, the well-known determinant expression for six times the volume of the tetrahedron whose corners have the coordinates 000, a-^^a^a^, b-J)^bg, CJC2C3. The vector F . af3y may, like any vector, be expressed Art. 32.] QUATERNION PRODUCTS. 63 linearly in terms of the non-coplanar vectors a, /?, y. This is most simply effected as follows : 2V. afty = a(iy + y(ia + ay/3 - ay/3 + r«/5 - r«/5, adding and subtracting the quaternions ay^ and ya/3. Combining in pairs, we get 2>a^y = a (^y + y/3) + y (/3a + a/3) - (ay + ya)^ = 2a5'^y + lySaji - 2^»Say, or V.af3y= aSpy - f3Sya + ySafS. From this form we see at once that a and y may be inter- changed without affecting the value of the quantity. Or F.al3y=V.yf^a. Again, since V . apy = aSfiy + V.a Ffiy, we obtain the further identity F.aF(3y = ySaf3-l3Sya, an extremely important formula of frequent use in trans- formations. When Saf^y = 0, the volume of the parallelepiped becomes zero, which means that a, /3, y cannot form a parallelepiped. If they have different directions they must be in one plane. In fact, any one, say a, must be perpendicular to the common perpendicular to the other two, namely, F/3y. In other words, all three are perpendicular to the same line, and must therefore be coplanar when drawn from one point. Under these circumstances the product a/3y must be a vector. Call it 8. Then S = a^y, or Sy~i = a^. Hence Sy~'^ and a^ represent equal quaternions, showing that the operation which changes y into 8 will also change ^"^ into a. In other words, a/3y8 when drawn from one point or continuously end to end are coplanar vectors, and the angle between 8 and y is the same as that between a and y3~^ (or - ^). Thus we can draw the direction of 8 at once, the vectors a/3y being given ; and then the tensor of 8 is equal to the product of the tensors of a, j8, y. 64 QUATERNIONS. [Chap. iv. If af3yS form the sides of a closed quadrilateral, then the interior angle between a and /3 is equal to the exterior angle between y and 8 ; and the quadrilateral is inscribable in a circle. If a/?y form the sides of a triangle, then S( = afdy) is drawn in the direction of the tangent at the point (a, y) to the circle circumscribing the triangle. 33. Transformations of Scalar and Vector Parts OF Products. The formulae of transformation for F. a/3y and »S' . a/3y are of great importance in applications to geometry and dynamics. We shall give a few of these. In expressions of the form *S' . af3y, it is evident that the vector part only of the product of any pair is of importance, for aSjSy is necessarily a vector, and can have no scalar part. Thus the expression S . Vafi Fj3y Fya may be written SFa/S F{ F/3y Fya) = S Fo.f3 ( - y,S'a^y + aSy/3y) = - SafSySafSy + ; :. S. Fa(3F/3yFya= - (SafSyf. This formula may be readily transformed into Cartesian coordinates ; and occasionally practice of this kind is useful, if only to show how much more concise and expressive the quaternion notation is. Thus, with i, j, k as rectangular unit vectors, we have a = rtji + rtoi + a^k, I3 = b-^i+ b.J + bjc, y = Cji + cj + cjc, Fafi = (^2^3 - a^K^i + (ag&i - a^h.^)] + {a^\ - aj)^)k, FfSy = etc., Fya = etc. By forming the products and taking the scalar parts, we readily find as the analytical equivalent of the formula given above the determinantal identity «2^3 - flg&o, a^\ - «l^3' (^A -a-zh = «1, «2' ag i./a - &3C2, hh - Kh^ ^^2 -k,q K K h C2«3 — CgCtgJ %«! - Ci"3. c^a.^ - C./l^ 'v C2, (^3 Art. 33.] QUATERNION PRODUCTS. 65 Fig. 23. When a, (i, y are unit vectors, the formula F. Fal3F(3y= -fSSaf^Y has an immediate application in spherical trigonometry. Let A, B, C be the extremities of the vectors a, f3, y on the unit sphere ; and let a, b, c be the arcs subtending the angles A, B, C i-espectively. Then throwing the above formula into the form TFafSTFISyF. UFa(3UFI3y= - pTFyaS . fSUFya, we deduce the relation sin c sin a sin .B = sin b sin ^2? where p^ is the perpendicular arc from B, upon the arc AC, being the supplement of the arc whose cosine is - Sf3 UFya. If, on the right hand, we write - (3TF/3yS . aUFfty, TF/3y divides out, and we get sincsini> = sinpp where p-^ is the perpendicular arc from A on CB. Similar expressions for p^, p^ may be written down at sight. The transformation S. Faf3FI3y = S.aFf3Fl3y ^ P'^So.y - Saf3SI3y, when interpreted in the same way with af3y as unit vectors, leads to the formula sin a sin c cos B= - cos a cos c + cos b. Again, P sini? TF. Faf3F/3y _ _ Saf3y *^" ^ ~ cos ^ ~ S. FafS Fpy ~ Say + Sa/SSf^y' giving tan B{- cos b + cos c cos a) = sin a sin p-^ = sin b sinp^ = sine sin py E 66 QUATERNIONS. [Chap. iv. These examples show with what peculiar readiness the calcu- lus of quaternions attacks problems of spherical tiigonometry. The most immediate geometric interpretation of the formula F. FapF(3y=^ -ftSafSy is that the line of intersection of two planes is perpendicular to the normals of these planes. For FafS is perpendicular to the plane containing a and /8, and F/Sy is perpendicular to the plane containing (3 and y. But these planes have the line fS in common, and (3 is by the above formula per- pendicular to its constituents Fa^, FfSy. 34. Kelation connecting Four Vectors. The expan- sion of Fa/Sy as a linear function of the three non-coplanar vectors a, (3, y (§ 32) is a particular case of the general truth that any vector may be so represented. Let p = xa + yl3 + zy, where x]/z are the coordinates of the extremity of p referred to axes parallel to a, [3, y. To express x in terms of the vectors, operate by S . Ffiy, that is, multiply by Ffiy and take the scalar part. Then since S/SF/Sy and SyF/3y both vanish, we get at once SfSyp = xSafSy. Similarly, Syap = ySafiy, Safip = zSajSy. Hence, generally, pSal3y = aSf3yp + l3Syap + ySaftp (1) Now the vectors Faf3, F/3y, Fya will be non-coplanar if af^y are, for each is perpendicular to the plane containing its constituents. Hence p must be expressible in the form p^x' FafS + y' F(Sy + z Fya. Operating by S . a, we find Sap = y'Safiy. Similarly, S(ip = z'Saf3y, Syp = x'SafSy ; and consequently, pSa/3y = Fa/iSyp + FfSySap + FyaS/3p (2) Art. 34.] QUATERNION PRODUCTS. 67 The vector whose components are Va(3, V/3y, Vya has an important property. Calling it S, we have, operating by S{a - /3) S = S(a - f3) F(a/3 + (3y + ya) = SafSy - SfSya = 0, the other products vanishing because they are of the form SpVpa-. Similarly, S(l3-y)S = 0, S(y-a)S = 0. Hence 8 is perpendicular to the plane passing through the extremities of a/3y. In like manner it may be shown that the vector a + (3 + y is perpendicular to the plane passing through the extremities of Fa(3, F(3y, Fya. The condition that these two planes should meet at right angles to each other is S(a + l3 + y)F(afi + l3y + ya) = 0, or 3Sa/3y = 0. Hence a/Sy are coplanar, and so are the vectors Fa ft, r(3y, Fya. The vector lines a, (3, y drawn from a point form in general three of the edges of a tetrahedron ; and the perpendiculars on the faces from the opposite angles are parallel to the vectors Faft, FfSy, Fya, F(af3 + /3y + ya). What is the condition that these perpendiculars meet in a point 1 Evidently the edge a and the perpendicular F(3y must lie in the same plane with F{af3 + fty + ya) ; and similarly for /3 and Fya, and for y and Faft. Hence, S . aFl3yF(a/3 + f3y + ya) = 0, or S(ySaf3-ftSay)F{a(3 + l3y + ya) = 0. 68 QUATERNIONS. [Chap. iv. This reduces at once to Sa/3y{SafS-Say) = 0, or Sal3ySa{l3 -y) = 0. Hence, since a, (3, y are not coplanar, Saf^y has a finite value, and the other factor Sa{f3-y) must vanish, or a is perpendicular to (/3 - y). Similarly, /3 is perpendicular to (y - a), and y perpendicular to (a - (S). Thus the six edges form three groups of perpendicular pairs. This also implies that the sum of the squares of any two opposite edges is the same for the three sets of pairs. For a2 + (/3 - y)2 = a2 + (3-^ + y2 - 2Sfty, and /3'^ + (y-a)2 = a2 + /32 + y2_2^ya, y2 + (a - ^)2 = a2 + ^2 + .^2 _ 2Sal3, and these have the same values because, as proved above, S/3y = Sya = Sal3. Having shown that 8 = V(af3 + fty + ya) is perpendicular to the plane passing through the extremities of a, /?, and y, let us next find the value of the perpendicular from the origin. It will be some scalar multiple, x, of 8, such that x8 — a, a- 13, (3 -y will all lie in one plane, or S{x8-a){a-(3){(3-y) = 0. This reduces to xS8 V{aj3 + I3y + ya) = Sa(3y, or x8^ = Sa/3y ; hence x8 = 8~'^Sal3y Thus the vector perpendicular from the vertex of the tetrahedron a, /j, y upon the opposite face is Saf3y V{a(3 + f3y + ya)' and its length is ~ "'' ^ -. ^ TV{a.\3 + I3y + ya) Art. 34.3 QUATERNION PRODUCTS. 69 This suggests one form of the equation of a plane passing through the extremities of the coinitial vectors a, f3, y, namely, if p is the vector to any point on the plane, S{p-a)(a-l3){(3-y) = 0, or Sp F{al3 + I3y + ya) = Sa.(3y. And generally, if e is a given vector, Spe = — 1 represents a plane perpendicular to e. Throwing it into the form we see at once that the plane must pass through the point EXAMPLES TO CHAPTER IV. 1. Prove that 5^. (a + 13) (13 + y)(y + a) ^2S . a^y. 2. S . Va^ Vpy rya=-( SajSy T: 3. S.V( Fa/3 F/Sy) V( F/Sy Fya) V( Vya. FajS) = - (S . apy)\ 4. S( F/37 Vya) = y^SaP - S^ySya. 5. a2^ V - ( Va^y)-' - (Sa^y)'- 6. = a2 (5/37)2 + pHSya)^ + y'^iSa^)^ - (Sa^y)^ - 2SapSpySya. 7. S{yV.apy) = y^Sap. 8. (a,37)2 =. a2;3V + 2apyS . a^y. 9. S( Fa;87 F/37a Vyap) = iSa^S^ySyaS . a/Sy. 10. The expression Fa/3 F75 + Fa7 F5/3 + I^aS F/37 denotes a vector. What vector ? (Tait's Quaternions. Miscellaneous Ex. 1.) 1 1 . SapS . I3y8 - S^pS . ydc. + SypS . 5a/3 - SdpS . a/37 = 0. 12. (a^7)2 = 2a2/3V + a- [pyf- + [3"- (a7)2 + 7^ (a^)^ - 4aySal3Sl3y. (Hamilton, Elements, p. 346.) 70 QUATERNIONS. [Chap, iv.] 13. When A, B, C, D are in the same plane, a.BGD-^. CDA+y.I>AB-5.ABC=0, where BCD, etc., are the areas of the triangles and a^yd the vectors to A BCD from any origin. 14. SV . cL^y + aV. ^yo + ^V . 75a + 7F. 5a/3 = 45. a/375. 15. Fa/3 Vyd + V^y V8a + Vyd Fa/S + F5a V^y is a scalar. What is its geometrical meaning ? 16. If P be any point within the tetrahedron A BCD, and if a, b, c, d be the points in which the produced lines AP, BP, CP, DP meet the opposite faces, then PalAa+BblBb + PclCc + PdlDd=l. 17. Expand .9 . 0^75 and V . a/376 in terms of scalars and vectors of the products of afiyd in pairs. 18. Show that F. aFj37, F./3F7a, F. 7 Fa/3 are coplanar, and that their mutual perpendicular is Fa/3 F/37 Vya Sa^'^ S^y'^ Sya 19. Expand q^ and q^ in terms of the scalar and vector parts of q ; and thence find S . q\ V . q\ S . q^ V . qK Give trigonometrical interpretations of the identities established. 20. Find a solution of the equation Q^ = q'^ in the form Q=±'J^^(Sq. UVq-TVq). (Hamilton, Lectures, p. 673.) 21. Show that the equation p{ )Kp + q( )Kq + r( )Kr = 0, where p, q, r are quaternions, is impossible except under very limited conditions. Find these conditions. 22. Show that for any three vectors a, /3, 7, we have ( (7a/3)2 + ( C7/37)" + ( Uy is a linear operator. In the present case cm nn 9P = P+ o ^' and for another vector o-, aSfScT+BSna- (per = cr H . 2 By addition we find = is distributive. In particular, i>{xp) = x4,p. If we form the expressions o-^p and /3p=ScTp+^-^^^^^p±^M^ = S . p'l^a-, SO that in this case p and o- may be interchanged without aflfecting the value of the expression. AVhen this can be done for any linear vector function <^, ^ is said to be self-conjugate. That this self -con jugate character is not a necessary property of the linear vector fimction may be seen at once by con- sidering the very simple case /) = aSfSp, in which S(Tcf)p = So.irSf3p = Sp (/SScut) = Sp4>'(r, say. Evidently ^V = /ISW is not in general the same as ^cr = aS(3(T. (See Chapter X.) Art. 45.] CONES AND THEIR SECTIONS. 95 With and ^ are that they are linear vector functions of the variables and that they are self-conjugate. When we meet with any equation of the form Sp(f>p= -a-, which, being a scalar equation of the second degree in Tp, must represent a surface of the second order; and if we know 96 QUATERNIONS. [Chap. vi. that (f) is self-conjugate * — then we may at once write down the equation of the tangent plane at the point p by simply changing one p into CT, the vector to the tangent plane, namely, SzHifip = Sp4>'^ = - a^. Also p forn — ~ ^, where 4>P is a vector fr(l-e-) which coincides with p only in the cases in which either a coincides with p or when Sap=^0, i.e. in the cases of the principal axes ; the equation of the ellipse becomes Sp^P=-i. : (3) The same equation is, of course, applicable to the hyperbola, e being greater than 1. It is evident that ^ is of the same type of function as that already discussed in last section. It is distributive and self -con jugate ; and many of the properties of the ellipse and hyperbola can be deduced with ease by its means. The method is identical with that which will be used in the discussion of the more general properties of the ellipsoid and hyperboloid ; and a few examples will suffice to show its power. 47. Tangents and Normals. From demonstrations already given, we may at once write the equation of the tangent line to the ellipse (or hyperbola) in the form STJ, that is, it is parallel to the normal at the point p. Art. 48.] CONES AND THEIR SECTIONS. 99 The equation Vptf^p = means that p is parallel to p= —1. Again, if x, y be the coordinates of T, a point in the tangent, 7n = x'i + y'j, 100 QUATEKNIONS. [Chap. vi. and _A.„^,_5(.'i.,7)(5 + |) = ?J^.|:; . ^^' ,yy' _. the equation of the tangent. 49. Powers of the Linear Vector Function. The values of p and ^/a exhibited in the last article, viz. : P = xi + yj, cj,p= +f^^ + ^\ (1) ,a2 ' b enable us to wTite Applying the operation a second time, we have iSip , jSjp\ „. a^ ' b^ Also it is easily verified that the inverse function iSip - b'^'^jSjp = -iSip-jSjp = xi + yj. If, further, we write *-(^^-f). («) we shall have ,2 /, f^Sip jSjp\ = ^P l6) Thus the operator \p may be regarded as the square-root of the operator 4>- Also, i/'- V = - (aiSip + bjSjp), p = ^-^xPp = -{aiSi^pp + hjSjxPp) (7) It is evident that the properties of p (p. 95) are pos- sessed by all these functions. Aet. 50.] CONES AND THRIR SECIIO^S.; „ 101 Now Sp4>p = - 1 gives Spxp{^p)= -1. But since Sp\p(r = Scnpp, this becomes Sippipp = (ipp)- = -1, or Txpp = \; which shoAvs (1) that -^p is a unit vector; (2) that the equation of the ellipse may be expressed in the form of the equation of a circle, the vector which represents the radius being itself of variable length, deformed by the function xp. Lastly, when a ^ are such as to make we have Saxp-fB = Sxpaip = ; therefore \pa, xpfi are vectors at right angles to one another. This may be exhibited without use of the i/', thus S.a(f,f3 = S. a^acf>^l3 = 0. 50. Conjugate Diameters : Parallel Chords. The equation S . a(f)/3 = in the case of the ellipse or hyperbola means that a (or /3) is perpendicular to the normal at the point on the curve whose vector is parallel to (3 (or a). In short, a is parallel to the tangent line at the point where fB meets the curve ; and ft is parallel to the tangent line at the point where a meets the curve. Two diameters which have this property that either is parallel to the tangents at the extremities of the other are called conjugate diameters. Let a /3 he two conjugate radii, so that Sa<{,j3 = Sft(t^a = 0. Any vector to the ellipse (or hyperbola) may be represented by p = xa + yl3, hence p = x<^a + ?/0/3, and - 1 = Spffip = x'^Saa + fS/34>f3 --'''-f=-m^-^2- (Fig ^\ P- 103) 102 'QUATERNIONS. [Chap. ^^. For each value of x, there correspond two equal and opposite values of y; and for each value of y there correspond two equal and opposite values of x. Hence each diameter bisects chords parallel to its conjugate. The converse that the locus of the middle points of parallel chords is the diameter conjugate to the diameter parallel to these chords is easily proved. Let CP, CD be the conjugate semi-diameters a, /3 ; and let DC be produced to meet the ellipse again in D' ; then vector DP = a-(i, vector UP -=a + /3; and ^(a + ^) (a - /8) = 'S'(a + /?) (<^a - c^^) = Saa - Sf3(f>B - Sacpp + Sft^j^a = 0, because Sa(jia and Sfifftf^ are equal quantities. Therefore a + /3, a-^ are parallel to conjugate diameters. This is the property of Supplemental Choi'ds. 51. Poles and Polars. From any one point two tangents can be drawn to the curve. Let CT be the vector CT, and p^ p^ vectors to the points of contact Q, R of the two tangents. Then by the equation of the tangent Smcpp^= -1, STHcj^p.-, = - 1 . Hence »S'CT(^ (p^ - P2) = = >S^(Pi - P2) ^^^ or the chord of contact p^ - p^ is perpendicular to the normal at the point where CT meets the curve. In the figure CT, QR are parallel to conjugate diameters. The equation Sp4>T^= - 1, CT being constant, is (under present limitations) the equation of a straight line. Pj po are tAvo values of p satisfying the equation. Hence the equation is that of the straight line passing through the points of contact. Art. 52.] CONES AND THEIR SECTIONS. 103 The point N where this straight line cuts CT is found by putting p = xTH, whence xS^4>^= -I. If CP = yTT^, the equation of the curve gives whence or Pig. 31. Suppose now that any chord such as QE is drawn through a iixed point E, vector o-, and that in all positions of this chord the meeting point T of the tangents at Q and B is found. Then since Str(f)Tn= - 1, we have also S'U5(^(t= - 1. Hence with a- a fixed vector rn describes a straight line. The point o- and the straight line S7;s/3. Then by the equation of the tangent we have Sx(f>l3(f)B= - 1, or xcf>B= - -j-^. Consequently the length of the perpendicular on the tangent at the extremity of (3 is \jT(^f3. Hence J'^"/? = J^^ ^#' and since VV'xB= U-^ = - U$- all being in one plane, we have finally To show that Fa/3 is the same whatever pair of conjugate vector radii be chosen, let us express any other pair a, (3' in terms of a, (3. We have already shown (§50) that if a =xa + 1//3, we must have x^ + y^ = 1. Hence we may write a' = xa±JlTF^f3. Similarly, f3' = x^a ± Jl - x^% But 5/3>a' = 0, or xx^Sacf^a ± Jl -x\l- x^'^Sf3f3 = 0, or xx^ = ±s/l -x'^. I - x^", or = 1 - a;2 _ .r^2_ Hence x^= ±Jl- x^. Art. 52.] CONES AND THEIR SECTIONS. 105 Bearing in mind the positions of any two self-conjugate diameters relatively to any other pair, we see that a' = a;a + n/1 - x'^f^, \ represent a pair of self-conjugate radii. Hence Fa 13' = - ( 1 - a;2) F"/?a + a;2 Fa/3 = {\-X^)Vafi + xWaf3 = Fal3. CP, CD are conjugate semi-diameters of an ellipse, as also CF, CD' ; PF, DU are joined ; to prove that the area of the triangle PCP' equals that of the triangle BCD'. Let a, j3, a', (3' be the vectors CP, CD, CF, CD' ; k a unit vector perpendicular to the plane of the ellipse. Since a = i/'-i i/ri = - {aiSixpa + hjSjxpa), etc., etc., (§49.7) therefore Vau! = V{aiSiy^o. + bjSj\pa) {aiSi\pa' + hjSjrpa) = abk{Si^aSj\l^a' - SjipaSiiJya) ==abkSi{x}^aSj^a' - xpa'Sjxpa) = ahkS .ijFifa'xj/a = - abkS . kFixl^aifa'). Similarly, r/3/3' = - abkS . kF{xp(3^Pf3'). Now \pa, \p(3 are unit vectors at right angles to one another ; as are also ipa, xpj3'; therefore the angle between if/a and \pa' is the same as that between xp/S and \ff3'. Hence S . kFixJ^axpa') = S . kF(:^ (3x^(3'), and . Faa'=F(3f3', i.e. area of PCP' = that of triangle DCD'. We end this section with a few examples : 1. The product of the perpendiculars from the foci on the tangent is equal to the square of the semi-axis minm: 106 QUATERNIONS. [Chap. vi. We have SY the vector perpendicular ==x4>p, and as Z is a point in the tangent, and CY=CS + SY=a + xp= - 1, x{(f)p)" = - 1 - SaP 1 - S^acjip Now (§46) ay = _ 6f2ap - a* ( 1 - e% a-p + aSap ^ p= + a\l-e^) Oj .'. SY.HZ = a^{\-e^) = h\ 2. The perpendicular from the focus on the tangent intersects the tangent in the circumference of the circle described about the axis major. Retaining the notation of the last example, we have CY =a + x4>p _ 4>p{l-Sap) , and the square of this is easily found to be equal to - a^. 3. To find the locus of T when the perpendicular from the centre on the chord of contact is constant. Art. 53.] CONES AND THEIR SECTIONS. 107 If CT be TT, the equation of QR, the chord of contact, is S(r(fiTr= — 1, and the vector perpendicular is - -— ; (pTT .-. (c^7r)2=-c2, or S4>7r . (/)7r = - c\ or Sir(f)({>Tr = - c^ J the equation of an ellipse whose Cartesian equation is 53. The Parabola. Cer- tain properties of the parabola may be discussed by the method employed for the ellipse. Thus if S be the focus of a parabola, DQ the directrix, we have SP = PQ, SA =AD = a. If SP = p, SD = a, we have (§ 37, Ex. 2) Q V^ \ T D a( S N G Fig. 32. If aV=(a2-5'a/))2. , p — a~^Sap 'PP= 12 ' (1) •(2) to which the properties of 4>P in § 45 evidently apply, the equation becomes Sp(p is a vector perpendicular to the axis. 108 QUATERNIONS. [Chap. vi. From the same equation " - a -0"ap «^^p= ^2 a? =-a\pf (7) From (4) the normal vector is ^P + «~'; (8) therefore the equation of the normal is cr = p + a; ((/)/3 + a~i) (9) Equation (2) when exhibited as a?p, (10) SN=a-^Sap; i.e. =aSa-^p (11) For the subtangent AT, put xa for tt in (5), and there results by (6) X + Sa~^p— 1, whence ix-~\a = -a-aSa~'^p', i.e. vector AT= - vector AN (by 11); .'. line^2'=^J\^; By similar processes we may easily prove that ST=SP, NG = SD, vector 5'Z> = vector GP, vector AY=- vector NP. The following examples may also be worked out by means of Equation (3), namely : (a) Find the locus of the middle points of parallel chords. (b) Find the locus of the point which divides a system of parallel chords into segments whose product is constant. (c) Find the locus of the point in which the perpendicular from A on the tangent at P meets the line PQ produced. Art. 53.] CONES AND THEIR SECTIONS. 109 {d) Find the locus of the intersection with the tangent of the perpendicular on it from the vertex. To solve this last problem let it be the vector perpendicular on the tangent from A, then by (8) and the equation of the tangent gives, putting tt + - in place of 77 in (5) and multiplying by 2, 2STrcj>p + -iSa-^TT + 2Sa-^p = 1, we have also From these three equations we have to eliminate x and p. The first gives Sa-JT = X, which gives x, and STrp)'\ which substituted in the second gives 1x{^pf + 2/S'a-V + 2So.-^p = 1. Also, substituting or{(^pY for Sp4>p (equation 7), the third equation gives a2(c/)p)2 + 2p = ap + baSap + c (aSfSp + (3Sap) + ... we shall have Sp(})p = ap2 + hS^ap + 2cSapSI3p + ... = -1, the equation required. We easily see that ^ is a linear vector function, distributive and self -conjugate, and fulfilling all the conditions given in § 45. 58. The Tangent Plane. By reasoning identical with that employed several times already, it is easily shown that if T be a vector in the tangent plane, ST(bp = 0, 120 QUATERNIONS. [Chap. vii. or, if o7 is a vector to a point on the tangent plane, SV5<^p = Sp(f>p = - 1 , which is the equation of the tangent plane. The vector (fjp is perpendicular to the tangent plane at the extremity of the vector p. 59. Perpendicular on the Tangent. If 07 be the perpendicular from the centre on the tangent plane ; then, since (j>p is a vector perpendicular to that plane, OY=zcf)p and Sz{(fipy = - L, giving vector OY=^X(f)p= - -— = - ( of a plane perpendicular to (pa. Now 4>a is the normal vector of the point where OT cuts the ellipsoid ; consequently the curve of contact lies in a plane parallel to the tangent plane at the extremity of the diameter drawn to the given point. When a is vector to a point inside the ellipsoid, there can of course be no real tangent planes drawn ; but in all cases the equation Spa = 0, the equation of a plane through the origin perpendicular to ^a : that is, the curve of contact lies in a plane through the centre parallel to the tangent plane at the extremity of the diameter which is parallel to the given line. 62. Conjugate Diameters. Let us first find the locus of the middle points of parallel chords. Let each of the chords be parallel to a, cr the vector to the middle point of one of them; then TH + xa, in-xa are points in the ellipsoid. From the first, »S'(c7 + xa) (f)(~ + xa) = - 1 J ie. ^'CTc^CT + 2xSTna = - 1 ; .•. subtracting, »S'CT<^a = 0, i.e. the locus is a plane through the centre perpendicular to <^a, or parallel to the tangent plane at the extremity A of the diameter which is drawn parallel to a. Let fB be any vector in this plane So5a = 0, and therefore Sa(fif3 = 0, or a satisfies the equation Sto4>/3 = of the plane which bisects all chords parallel to /3. These two planes, bisecting chords parallel to a and [3 respectively, will intersect along a line through the centre. Let y represent this line ; then, since y is a vector in both the planes »S'u7(^a = 0, and »S'CJa. = 0, Sy/3==0. 122 QUATERNIONS. [Chap. vii. Hence Sa(f,y = 0, Sf34,y = 0, so that a and (3 are both vectors in the plane bisecting chords parallel to y. Let OA, OB, OC be the vector radii parallel to a, /3, y respectively ; then these lines are such that all chords parallel to any one of 'them are bisected by the diametral plane which passes through the other two. We may term these lines conjugate semi-diameters, and the corresponding diametral planes conjugate diametral planes. It is e"\ddent that the number of sets of conjugate diameters is unlimited. We have then the following equations : S(S^y = = Sy^fi, Sacf)y = = Syf^a. They show that y is perpendicular to both (^a and 4>/3, and is therefore a vector perpendicular to their plane ; hence y = xF<{ia(f)f3. In the same way, since f^y is perpendicular to both a and (3, we have 4>y = yVa(i ; or we have the following pairs of parallel vectors : y II F0ac/)^, /3 II r(^ac/)y, a \\ F(/./3(^y, P is a linear and vector function of p, its vector projections along the principal axes will be multiples of iSip, jSjp, kSkp ; we may therefore write fiSip jSjp kSkp\ .ON the particular multipliers having been chosen in order to make the equation Sp4>p = - 1 coincide with the Cartesian equation 3.2 y2 z"^ _ a^ 52 g2 124 QUATERNIONS. [Chap. vii. As 4>P=^Hp, (4) we require to take ifp so that ij/ operating twice in suc- cession on p shall give the same result as ^ operating once. Now a comparison of equations (2) and (3) will show that the latter operation introduces —„ etc., into p ; it is evident it therefore that the former operation (ip) is to introduce -, etc., i.p= _^^p+i|p+^j^^, (5) It may perhaps be worth while to verify this result. We have rp^P = - (^ J^ + ^-^) a\ a b c J '" .i"Sip Vfl^ 62 -*- c2 ) 'iSip jSjp kSl-p\ .^. ^"^1^ + ^^;' ^ ' 4>~^P produces p. ^-^p= -{aiSip + hjSjp + ckSkp), ■•••(?) p = \p-'^xl^p= - (aiSiijyp + bjSjx/yp + ckSkxpp) (9) It is evident that the properties of Art. 44 ai^ply to all these functions. 65. This section contains a series of examples, chiefly on the ellipsoid, chosen with a view to variety of treatment. Other examples will be found at the end of the chapter. The Art. 65.] CENTRAL QUADRIC SURFACES. 125 student will find it a very good exercise to work through by quaternionic method the theorems and problems given in any recognised treatise on solid geometry — such as Salmon's or Smith's. 1. Find the point on an ellipsoid, the tangent plane at which cuts off equal poiiions from each of a given set of conjugate axes. Let aa, bfS, cy be the set of conjugate vector radii, a, f3, y unit vectors. Then the vector to the required point may be written p = xa + yf3 + zy. Let p be the length cut off from each axis ; so that pa, pP, py will be vectors to the tangent plane. Hence Spa(f)p= - 1, or o. + y4)(i + Z({>y) = - 1, 1 fl^' 2 Similarly, py = b-, pz = c^. and p = -(a2a + 52^ + c^y), p and - 1 = Spf^ip = -Tj_S{aaa + hhfi + ccy) (rtc^aa + 5^5^ + C(^cy) a2 + 52 + c2 .-. p = JW+W+^'^. Let X, y, z be the coordinates of the point, p the portion cut otf, then p = xi + yj + zTc. Now pi, pj, pk are points on the tangent plane ; .*. Spi<:f)p=\, which gives or px_ a 2 126 QUATERNIONS. [Chap. vii. Similarly, QUATERNIONS. 62- -I, pz = 1, X y z 1 ~c^~p~ JcC' 1 T? 62- + h'^ + c^ 2. To find the locus of a jmnt ivhen the perpendicular from the centre on its polar plane is of constant length. Let CT be the vector to the point, then Sput)2= -(7-, by the question. But (4^^)^ == S . <^77(jiZJ = STocji^-Vj ; hence Suy'^rn = - C^, the equation of an ellipsoid whose Cartesian equation is a* 6^ c^ 3. To find the surface enveloped hy the polar planes with respect to one ellipsoid of points which lie on another ellipsoid. Let Sp4>p= - 1, Sp\pp=^ - 1 be the two ellipsoids. If a is a point on the latter, Saipa= - 1, and the polar plane to a has the equation — 1 = Sp(jia = Sp4>\p~^ipa = S^~^({)p'\fra, Hence a = \f/~^cf)p, and - 1 = Sp4>a — Sp(jiylr~'^(f)p, an ellipsoid. 4. The sum of the squares of three conjugate semi-diameters is constant. Let a, (i, y be the semi-diameters ; xpa, \J//3, \fy are rect- angular unit vectors (§ 63). Now a = - {aiSixpa + bjSj\pa + ckSkipa) ; (G4. 9) ... (ra)2 = - a2 = a^Si^ay + b^Sjifay + C^SkxJ^af, {T(if = a^Sim' + i- {Sjm' + C {Sk^pi3)\ {Tyf = a~ {Si^y)'' + 6-" (S}Yy)2 + c'-' {Slfyy^ : Art. 65.] CENTRAL QUADRIC SURFACES. 127 adding, and observing that (Siipaf + (SiiPfSY + {Si^Pyf = 1, we get (Taf + {T(3f + {Tyf = a2 + J2 + ^2^ i.e. a'2 + 6'2 + c'2 = ft2 + J2 + c2. 5. The mm of the squares of the p-ojedions of three conjugate diameters on any of the principal axes is equal to the square of that axis. Let a, /3, y be conjugate semi-diameters ; then, since a = - {aiSitpo. + hjSjxpa + ckSkxpa), (63. 9) Sia = aSii/ya. Similarly, Si/3 = aSi^fi, Siy = aSiipy ; .: {Siaf + (Si/3)^ + {Siy}' = a^ {(^Sixj^ay + (S^.^i3)2 + {Six/^yf } = a\ (31. Cor.) because \pa, \p(i, \py are at right angles to on another (62) But -Sia is the projection of Ta along the axis of x; and similarly of the others. Hence the proposition. 6. The sum of the reciprocals of the squares of the three per- peivliculars from the centre on tangent planes at the extremities of conjugate diameters is constant. Let Oy^, Oy2, Oy^ be the perpendiculars. ^,--(^«)' (58) {Siaf (Sjaf {Ska)\ \ = -^^+-^ + -15-' ^^^- ^^ 1 JSifif ISjPY {S,(i)\ Oy.-r a* ^ b^ c^ ' I JSiyf (SjyY (Sky)\ Oy^^ a^ "^ 6* -*- c4 ' -l-hl <^-'> 128 QUATERNIONS. [Chap. vii. 7. A, B, and C are three similar and similarly situated ellipsoids; A and B are concentric, and C has its centre on the surface of B To show that the tangent plane to B at this 'point is parallel to the plane of intersection of A and 0. Let a be the vector to the centre of C. Sp(f>p = - a the equation of A, Spcl^P= -b B, S{p-a)(f>(p-a)= -c 0. Now at the intersection of A and C, p is the same for both ; therefore the equation of the plane of intersection is to be found by subtracting the one from the other. It is therefore 2Sp4>a = *S'a<^a - a + c ; and the equation of the tangent plane to B at the centre of C is S7rcf)a = -b ; .'. both planes are perpendicular to ^a, and are con- sequently parallel. 8. Two similar and similarly situated ellipsoids are cut by a series of ellipsoids similar and similarly situated to the two given ones; and in such a manner that the planes of intersection are at right angles to one another. Shoiv that the centres of the cutting ellipsoids lie on another ellipsoid. Let Spcl^p^ -1, (1) S(p-a)4.{p-a)=-C, ....(2) be the given ellipsoids ; S{p-7r)cf.{p-7r)=~X, (3) one of the cutting ellipsoids. T^{Tr — (f)a) = 0, or (S(:r - a)(^"7r = 0, the equation of the locus of the centres of the cutting ellipsoids. It represents an ellipsoid of which the semi-axes are pro- portional to the squares of the semi-axes of (1). The Cartesian equation is 9. Find the equation of the curve described by a given point in a line of given length whose extremities move in fixed straight lines. This is a good example of the ease with which the quaternion method attacks the most general case. Suppose the lines not to meet, and take the origin half way' along the line perpendicular to both. Let y, - y be the vectors perpendicular to the lines, and let a, /S be unit vectors along the lines. Then the extremities of the line of given length are determined by the vectors y + xa, -y + y(3, and the vector of the point which divides this line in the ratio of m to 1 is _ (y + .ra)-fm(-y + y/3) P ~~ 1 — i ' 1 -1-m or {\ +m)p = xa + myP + (l -m)y. Also constancy of length gives T{2y + xa-yf3) = l. From the first of these two equations, we get so that the extremity of p lies in a plane perpendicular to y. I 130 QUATERNIONS. [Chap. vii. We also get, by operating by S . VfSy, S . Vya the values of X and y, namely xSafiy = {l+m)Sl^yp, ySafSy = (1 + m)Syap, and finally { 2ySaf3y + ( 1+ m) aSfSyp - ( 1 + m) /^^yap } 2 = - /2^a/8y, or, if 4y'- = - c-, S'-f3yp + S'-yap + 2Saf3Sf3ypSyap = -I^_ S^-af3y. Since y occurs in the second degree in every term, we may multiply it throughout by any number. We may therefore put in its place Vaf3, and so obtain an equation involving only a, (3, I, c, and 7n as constants. If we write 4>pS'-af3y = (SySpifSy + yaSa{3) + yaSp(ya + /3ySa.f3), the equation becomes Spa= — 1. The equation Sp4>p = {Sp4>a)", a;2 v2 ^2 /a^c' yy' zz'\^ a- b- c^ \a^ b^ c- ) represents a surface passing through the points of contact ; and is the cone required. [For it is homogeneous in Tpi\ 11. The equation p = fa + u"(3+(t + u)y is that of a cone of the second order touched by each of the three planes through OAB, OBC, OCA; and the section ABC through the extremities ^f «) A 7 ^5 an ellipse touched at their middle points by AB, BC, CA. (!) If the surface be referred to oblique co-ordinates parallel to a, 13, y respectively, we shall have p=^xa + y(3 + zy, therefore x = t^, y = u^, z={t + «/)-, or z = {Jx + Jyf = x + y + •2jxy, which gives (z-x-y)^ = ixy, a cone of the second order. (2) If t= -n, the equation becomes p = t%a + (3), the equation of a straight line bisecting the base AB, which, since it satisfies the equation relative to t, shows that this line coincides with the cone in all its length ; i.e. the cone is touched in this line by the plane OAB. Similarly, by putting / = 0, u = respectively, we can show that the cone is touched by the plane BOC, CO A in the lines which bisect AC, CA. (3) Restricting ourselves to the plane ABC, we have the section of a cone of the second order enclosed by the triangle 132 QUATERNIONS. [Chap. vir. ABC, which triangle is itself the section of three planes each of which touches the cone. 12. I'he equation p = aa + b/3 + cy with the condition ab + bc + ca = is a cone of the second order, and the lines OA, OB, OC coincide throughout their length with the surface. (1) It is evident that the equation gives xy + yz + zx = 0. (2) That a b = 0, c = 0, the question is satisfied by p = aa, whatever be a, therefore, etc. 13. The lines which divide proportionally the pairs of opposite sides of a gauche quadrilateral, are the generating lines of a hyperbolic paraboloid. Let A BCD be the quadrilateral. ^P AD, BC are divided proportionally in P and B. LetCA = a, CB = f3, CD = y; CE = mf3, DP = mDA; i.e. CP-y = m(a - y); therefore RP = CP - CR = y + m{a-y)- m/S, p = CQ = CR-¥pRP = m(i +p{y + m(a - y) - m/?} = a;a + ?//3 + sy, say; therefore x=pm, y = m-pm, z=p(l -m) ; therefore m = x + y, p = , ^ -^ x + y X Z = X, x + y or {x + z){x-\-y)==x, the equation referred to ol^lique co-ordinates parallel to a, fi, y. Fig. 34. Chap. VII.] CENTRAL QUADRIC SURFACES. 133 EXAMPLES TO CHAPTER Vn. L Find the locus of a point, the ratio of whose distances from two given straight lines is constant. 2. Find the locus of a point the square of whose distance from a given line is proportional to its distance from a given plane. 3. Prove that the locus of the foot of the perpendicular from the centre on the tangent plane of an ellipsoid is (aa:)2+(6i/)- + (c2)2 = (a;2 + 2/2 + 22)2. 4. The sum of the squares of the reciprocals of any three radii at right angles to one another is constant. 5. If Oy^^, Oyo, Oy^ be perpendiculars from the centre on tangent planes at the extremities of conjugate diameters, and if Q-^, Q^, Qz be the points where they meet the ellipsoid ; then 1 1 , 1 _1 , 1 , 1 ! + /Tir 2 7T7V-1 + /I \^ 2 r\r\ 2 ~ „4 "*" ^.4 "^^ ^i' 6. If tangent planes to an ellipsoid be drawn from points in a plane parallel to that of xy, the curves which contain all the points of contact will lie in planes which all cut the axis of z in the same point. 7. Two similar and similarly situated ellipsoids intersect in a plane curve whose plane is conjugate to the line which joins the centres of the ellipsoids. 8. If points be taken in conjugate semi-diameters produced, at distances from the centre equal to p times those semi-diameters respec- tively ; the sum of the squares of the reciprocals of the perpendiculars from the centre on their polar planes is equal to p- times the sum of the squares of the perpendiculars from the centre on tangent planes at the extremities of those diameters. 9. If P be a point on the surface of an ellipsoid, PA, PB, PG any three chords at right angles to each other, the plane ABC will pass through a fixed point, which is the normal to the ellipsoid at P ; and distant from P by 2//J_ 1 ^\ where p is the perpendicular from the centre on the tangent plane at P. 134 QUATERNIONS. [Chap, vii.] 10. Find the equation of the cone which has its vertex in a given point, and which touches and envelopes a given ellipsoid. 11. Find the locus of the points of contact of tangent planes to an ellipsoid, when the tangent planes make a given angle with one of the principal axes. 12. The sum of the squares of the three perpendiculars from the centre on three tangent planes at right angles to one another is constant. 13. If through a fixed point within an ellipsoid three chords be drawn mutually at right angles, the sum of the reciprocals of the products of their segments will be constant. 14. Establish for the central surface of the second order the theorems of Poles and Polars corresponding to those established for the sphere (§ 39). 15. If through a given point chords be drawn to an ellipsoid, the intersection of pairs of tangent planes at their extremities all lie in a plane parallel to the tangent plane at the extremity of the diameter which passes through the point. 16. If a tangent plane be drawn to the inner of two similar con- centric and similarly situated ellipsoids, the point of contact is the centre of the elliptic section of the outer ellipsoid. 17. If two of a system of three rectangular vectors are confined to given planes, show that the third lies in a cone of the second order whose circular sections are parallel to the given planes. 18. Find the locus of a point, the sum of the squares of whose distances from a number of given planes is constant. CHAPTER VIII. MISCELLANEOUS GEOMETRICAL APPLICATIONS. 66. Pascal's Hexagram. Let be the origin, OA, OB, OC, OD, OE five given vectors lying on the surface of a cone, and terminated in a plane section of the cone ABCDEF, not passing through ; OX any vector lying on the same surface. Let(9^ = a, OB = (S, OC=y, 0D = 8, OE^e, OX=p. The equation S. F{Fal3V8€)F{F/3yFep)V{Fy8Vpa)^0 (1) is the equation of a cone of the second order whose vertex is and vector p along the surface. For 1. It is a cone whose vertex is because it is not altered by writing xp for p. Also it is of the second order in p, since p occurs in it twice and twice only. 2. All the vectors OA, OB, UC, OD, OE lie on its surface. This we shall prove by showing that if p coincide with any one of them the equation ( 1 ) is satisfied. If p coincide with a, the last term of the left-hand side of the equation viz. Fpa, becomes Faa = Fa- — 0, and the equation is satisfied. If p coincide with ft, the left-hand side of the equation DGC0ni6S S . F{ Fa/S FSe) F{ Ff3y F€f3) F{ Fy8 F/3a) (2) 136 QUATERNIONS. [Chap. viii. Now F(Ff3yVe/3) = - FiFef^Fpy), (§ 30), is a vector parallel to fS (§ 33), call it m/S ■ and F.{F{ Fa/3 FSe) F{ FyS F/3a) } = F.{F( Fa/3 FS^) F{ Fa/3 FyS) } = a multiple of Fa/3 = nFa/3, say. P. Fig. 35. Hence the product of the first and third vectors in ex- pression (2) becomes scalar + n Fa/3, and the second is m/3 ; therefore expression (2) becomes (§32) S.{sca\iXT + nFa/3)m/3 =^mnS/3Fa/3 = 0, because Fa/3 is a vector perpendicular to /3. Equation (1) is therefore satisfied when p coincides with y8. If p coincide with y both the second and third vectors are Art. 67.] GEOMETRICAL APPLICATIONS. 137 parallel to /3 (§ 33) ; therefore their product is a scalar, and equation (1) is satisfied. The other cases are but repetitions of these. Hence equation (1) is satisfied if p coincide with any one of the five vectors a, /?, y, 5, e ; i.e. OA, OB, OC, OD, OE are vectors on the surface of the cone. 3. Let F be the point in which OX cuts the plane ABODE ; then ABODEF are the angular points of a hexagon inscribed in a conic section. 4. Let the planes OAB, ODE intersect in OP ; OBC, OEF in OQ ; OCD, OF A in OR ; then F.Fa/3F8€ = mOP, F.Ff3yFep = nOQ, F.FyWpa^pOR; therefore S . F{ Faj3 FSe) F{ F/3y Fep) F{ Fy8 Fpa) = mnpS{OP . OQ . OR) ; hence the equation (1) gives S{OP.OQ.OR)=^0, or (§ 32) OP, OQ, OR are in the same plane. Hence PQR, the intersection of this plane with the plane ABODE F, is a straight line. But P is the point of intersection of AB, ED, etc. Therefore, the opposite sides (1st and 4th, 2nd and 5th, 3rd and 6th) of a hexagon inscribed in a conic section being produced meet in the same straight line. Cor. It is evident that the demonstration applies to any six points in the conic, whether the lines which join them form a hexagon or not. 67. CoNFOCAL Surfaces of the Second Order. Two surfaces are said to be confocal when their principal axes have the same directions, and when the squares of the lengths of the corresponding axes difiPer by the same quantity. Thus, 138 QUATERNIONS. [Chap. viii. if a^, b^, (? are the squares of the semi-axes of one surface, then a^-i-A, V^-vh^ c^ + h are the squares of the semi-axes of another confocal with it. Here h may be positive or negative. Let (o, rn be the appropriate linear vector functions, so that iSip jSjp kSkp iSip jSjp kSkp a^ + h~ W+h ~ (? + K "^=-^--^ c^ ^p= - We have m = + \, oyj = + 1 u>k = + 1 X CTi= +-2 — r, etc., etc., or {>i~H = d\ it}~y = by, o)~^k = c"k, Tn-H = (a- + h}i, rn-^j = {b- + h)j, u5-^k = {c- + h)k; and, by subtraction, (CT-i-co-i)e = /iz, (u7-i-(o-i)y = /y; (CT-i-w-i)/!; = /iyt, and generally (^"^ - ^~^)p = hp. Let 0)"^ = ~'^p= - 1 and Sp{(f) + h)~'^p= -1 represent two confocal quadric surfaces. Or generally >Sp{ci> + h)-^P= -1 represents a series of confocal quadric surfaces ■with para- meter h. From the expanded semi-Cartesian form or from the theory of Chapter X., we see that, if a point p be chosen, there are three distinct surfaces passing through it corresponding to the three roots of the cubic in h. Certain other properties of confocal surfaces may be established with great ease by means of the quaternion form of the equation. Art. 67.] GEOMETRICAL APPLICATIONS. 139 1. Let Spcl>-'^p + 1=0, Sp{cfy + h)-^p+l=0, be two confocals having a common tangent joining the point p on the one to the point cr on the other. The vector p-a- is perpendicular to the two normals 4>~^Pj {4> + h)~''-a-. Hence S{p--^P = 0, S(p-(T){ct, + h)-^p = 0, or . Sp + h)~^(T, and consequently o=sp{l-^y=hSprH^^hr. Hence the normals are perpendicular to one another, and form with the common tangent a rectangular system of vectors. The theorem will hold however close the two points are, and will still hold when the points coalesce. Consequently any two confocals which meet at a point intersect at right angles ; and generally the three confocals which meet at a point form a set of orthogonal surfaces. This theorem also follows at once from the identity 1 c /_! i_i ''K-h''y4, + h 4, + h'l'' 2. Let Sp'f)~^p + 1=0 be the equation to one of three confocals meeting at a point a. The central section by a plane parallel to the tangent plane at a is given by the two equations -'^7r>. Hence T^jf^-'^TH and ^"^a lie in one plane, and we may write CI + A^ - ICT = ?/^ - la, (2) where h and y are arbitrary constants. This gives at once (fiTH + Act = 7ja, To find h, operate on (2) by aS . CT. The result is 7n^= +h, or h^ -{Tv5f. Now {(f) + h)-'^a is the normal to the confocal passing through a. Hence the principal axis cr of the central section conjugate to a in the surface (/)-Ms parallel to the normal to the confocal {(f>- {Tz7)^-^ which meets the surface -! at the point a. The other principal axis of the same central section will be parallel in like manner to the normal to the other confocal. Let p be the vector to any point common to the surfaces ^-1, (<^ + /i)-i, and let o- be a vector radius to the surface ^~i, parallel to (^ + /t)-i/3, and therefore perpendicular to (f>~^p. Hence, operating by »S' . cr, we find 0-2 + /i6'cr<^- 1(7 = 0, OT h= +(r2. In words, a diameter (2o-) in the quadric 0-1 drawn parallel to the normal to the confocal (<^ + A)-i at any common point of the two is of constant length. If (cfi + h')-'^ represent the other confocal, then -h and -h' are the squares of the semi-axes of the section of <^-i conjugate to the vector to the point common to the three confocals. Hence if p is the perpendicular on the tangent plane to <^-i at the point p, pjhh' = - area of circumscribing parallelepiped, or = abc. That is ((j^ + h)(T\ip, or o- + /i + h)~'^z:j+l=0 is the equation of the polar plane with reference to the point CT. Let this plane be given in position, and let its equation be given in the form Spa +1=0. Then (cfi + hy'-TT) = a or u] = {cji + h)a. Hence FaCT = Vacf>a = y, a constant vector. That is, the poles with reference to a series of confocals of a given plane lie on a straight line parallel to a. Thus, when a particular plane is regarded as the polar plane with reference to a series of confocals, the corresponding poles lie on a straight line which is perpendicular to the plane, and which is therefore the normal at the point of contact to the particular surface which touches the plane. 4. The equation Sp4>~^p= - 1 niay be written in the form Sp~^'ci,-^-p = Sp~^~^P = {~^p)^= - 1, so that T~^p==l. For a confocal surface, T(cj, + hy^P=i. Let us consider the relation between the points on these two surfaces for which a constant unit vector. 142 QUATERNIONS. [Chap. viii. When this relation is satisfied, the points p, pj on the two confocal surfaces are called corresponding points. Let 0-, o-j be another pair of corresponding points on the same two surfaces, so that another constant unit vector. These give whence Spa-^ = Scrp^ = Sacfi^cfj + h)^/3. Also P'-p^^ = Sa(f>a-Sa( + h)a = h, and (T- - 0-^2 = ^ Consequently, p- + o-j- = p^^ + a-'^. Adding or subtracting from each side the quantity 2Sp(r^ or 2Sp^a; we find T(p±a,) = T{p,±cr). If we take the negative sign we get Ivory's theorem that the distance between two points, one on each of two confocal ellipsoids, is equal to the distance between the two corre- sponding points. If we take the positive sign we have the relation that the other diagonal of the parallelogram formed by the vectors to two points, one on each of two confocal ellipsoids, is equal to the similar diagonal of the parallelogram formed by the vectors to the corresponding points. We may regard the equation /3 = ((^ + A)2a, with Ta=l, as representing a system of confocal surfaces. If a is a constant unit vector, then, as h varies, p traces out the locus of a set of corresponding points. Let h change l^y a small quantity e, while p changes by a small vector increment r, Art. 68.] GEOMETRICAL APPLICATIONS. 143 which in the limit will give the direction of the tangent to the curve traced out by p. We have neglecting higher powers of e. Hence r = he((fi + hy^a = ^e{ + h)-^y= -1. In other words the locus of corresponding points on a series of confocal ellipsoids cuts the ellipsoids orthogonally, and is therefore the line of intersection of two of the confocal hyperboloids. 68. Vkrsor Equation of the Ellipse. We shall indicate briefly another quaternion mode of discussing the ellipse, leaving the .student to fill in the steps. As shown in Chapter III., § 25, the operator a", where Ta=l, is a versor which acts on any vector perpendicular to a so as to turn it through the angle -ttx. When it operates on any vector /?, the product a^/3 is a quaternion whose vector part, p=V . a^fS, will trace out an ellipse. Since a^ is of the form ^'a^ + Fa"", p is of the form m/3 + nFal3, and lies in the plane whose normal is parallel to FI3Fa(3 = f3Sal3-al3l Expanding a" in the form TTX . TTX cos ^ + a sin -^, we find p = cos -^ . ^ + sin — Fa/3, It li an ellipse referred to conjugate vector radii. 144 QUATERNIONS. [Chap. viii. From last section we may at once write down — or we may deduce from the equation of the tangent — the expression for the vector radius conjugate to p, namely o- = - sin '^ . /3 + cos ^ra/3 = F. a'+'p. The vectors /3 and VafS are perpendicular to each other ; no value of Tp can be greater than Tf3 ; hence /3 and FafS are the major and minor semi-axes. Evidently Fa^\\a=pa, say. Then SaFa^= -p=^Sa'+\ and Fa^ = - aSci^+\ By use of this equality it is easily shown that Fp(r=F.f3ra/3, as in § 52. 69. 1. The sum of the squares of ths areas of the faces of all parallelepipeds, constructed on the semi-conjugate diameters of an ellipsoid, is constant. By § 64. 9, a = - (aiSiipa + hjSjxLa + ckSk^a) f3= - {aiSixf'/3 + bjSji^fS + ckSkxpP) ■ therefore Fafi = abk {Si\j^aSjxp(i - Si xjyf3Sj\pa) -t- acjiSirPaSkxP(3 - Sii/^fSSkxI^a) + bci{SjxPaSkiP(3 - SjxppSkxjya). Now SixpaSjxjyjB - SiiPfSSji^a - ;S' Fij Fipjixpa = -Sk^y; (§63) therefore Fa/3= - {abkSk\j/y -\-acjSj\}^y +hciSi\py\ Fya = - (abkSklp/S + acjSjx^fi + bciSilp/3), Ff3y = - (abkSkipa + acjSj\pa + bciSixpa). If now we square and add these expressions, observing that because ^a, xpfS, \fy are unit vectors at right angles to one another, (Si^PaY + {SiiP/3)'' + {SiiPyf - 1, we shall have {Fa/3y^ + (Fayf + {F/3yf= - {{ab)^ + (a^)^ + {bcf}, which is the proposition to be proved. Art. 69.] GEOMETRICAL APPLICATIONS. 145 2. To find the locus of the intersections of tangent planes at the extremities of conjugate diameters of an ellipsoid. Let TT be the vector to the point of intersection of tangent planes at the extremities of a, /3, y : then TT = a + /? + y, <^7r = (^a + <^/5 + (fij, and consequently STTcfiTT = S(a + f3 + y) (cjia + <^/? + ^y) = Say = -3, for all terms of the form Sacf)^ vanish. This is an ellipsoid similar to the given ellipsoid. 3. IJ 0, A, B, C, D, E are any six points in space, OX any given direction, OA', OB', OC, OD', OE' the projections of OA, OB, OC, OD, OE on OX; BCDE, CDEA, DEAB, EABC, ABCD the volumes of the pyramids ichose vertices are B, C, D, E, A, with a positive or negative sign according as the m'der of the letters naming the angles at the base is right-handed or left-handed as seen from the vertex ; then OA' . BCDE + OB' . CDEA + OC . DEAB + OD' . EABC -\-OE'.ABCD = 0. Let OA, OB, OC, OD, OE be a, /?, y, 8, e respectively. Write for aS{y - /3)(8 - /3){€ - j3) its value a{S.y8e-S.8€p + S. c/3y - S . (3y8), and similar expressions for [3S{a-y){8-y){i-y), etc., and there will result, by addition, aS{y-(3){8-f^){,-/3)+(3S{a-y){8-y){e-y) + yS{a -8)i/3- 8) (e - 5) + 8S{a - e) {/? - e) (y - e) + eS{l3 - a){y- a) {8- a) = 0, or, using the notation explained above, OA . BCDE + OB . CDEA + OC . DEAB + OD . EABC -\-OE.ABCD = 0. Now let TT be a vector along OX ; then the operation by iS . 77 on the above expression gives the result required. E 146 QUATERNIONS. [Chap. viii. 70. In some of the examples which follow, Ave will en deavour to show how a problem should not, as well as how it should, be attacked. 1. Given, any three planes, and the direction of the vector perpendicular to a fourth, to find its length so that they may meet in one point. Let Sap = a, S/Sp = b, Syp = c be the three planes, and let 8 be the vector perpendicular to the new plane. Then, if its equation be Sop = d, we must find the value of d that these four equations may all be satisfied by one value of p. Formula (2), § 34, gives pS . a/Sy = Va/3Syp + FftySap + FyaS/3p = cFaP + aVf3y + bFya, by the equations of the first three. Operate by S .8, and use the fourth equation, and we have the required value dS . a^y = aS . fSyS + bS . yaS + cS . a/38. 2. The sum of the {vector) areas of the faces of any tetra- hedron, and therefore of any polyhedron, is zero. Take one corner as origin, and let a, /?, y jje the vectors of the other three. Then the vector areas of the three faces meeting in the origin are ^FafS, TiFjiy, ^Fya, respectively. w •^ -^ That of the fourth may be expressed in any of the forms lF{y-a)(/3-a), lF(a-(3)(y-P), \F{fi - y){a- y). ^ -J -J But all of these have the common value \F{yli + Pa + ay), which is obviously the sum of the three other vector-areas taken negatively. Hence the proposition, which is an elementary one in Hydrostatics. Art. 70.] GEOMETRICAL APPLICATIONS. 147 Now any polyhedron may be cut up by planes into tetra- hedra, and the faces exposed by such treatment have vector- areas equal and opposite in sign. Hence the extension. 3. If the pressure he uniform throughout a fluid mass, an immersed tetrahedron {and therefore any polyhedron) experiences no couple tending to make it rotate. This is supplementary to the last example. The pressures on the faces are fully expressed by the vector-areas above given, and their points of application are the centres of inertia of the areas of the faces. The co-ordinates of these points are l(a + /3), l(/i-hy), ^^(7 + a), i(a + ^ + y), and the sum of the couples is If. { Vaji.{a -f (i) + r/3y.(/i + y) + Tya. (y + a) + r(y^ + /ia+ay).(a-F/3 + y)} = -^F{Fal3.y+F(3y.a+Fya.f3) = 0. 4. What are the conditions that the three planes Sap = a, SPp = h, Syp = c, shall intersect in a straight line ? There are many ways of attacking such a question, so we will give a few for practice. (a) pS . al3y = Fa/3Syp + F/3ySap + FyaS/3p = cFafi + aFPy + bFya' by the given equations. But this gives a single definite value of p unless both sides vanish, so that the conditions are ^.a^y = 0, and c Fa/3 + a Ff3y + h Fya = 0, which includes the preceding. (b) S(la - mft)p = al - bm is the equation of any plane passing through the intersection 148 QUATERNIONS. [Chap. viii. of the first two given planes. Hence, if the three intersect in a straight line there must be values of /, m such that la - m(3 = 7, la - mh = c. The first of these gives, as before, S.a/3y = 0, and it also gives Fya = m Fa/3, Ff3y= -I Fa/3, SO that if we multiply the second by Fa/?, laFa/3-mbFa/3 = cFa/3 becomes - a F/3y - h Fya = c Fa/3 ; the second condition of (a). ((•) Again, suppose p to be given by the first two in the form p=pa + q/3 + x Fa/3, we find a =po? + qSa/3, because 5a Fa/3 = 0, h=pSa/3-\-q/3'^; therefore {p-xFa/3) a2, Sa/3 = a a, Sa/3 + /? a-, a Sa/3, /3^ h, /3'- Sa/3, h so that the third equation gives, operating by ♦S' . y. {c - xSa/3y) a% Sa/3 = Say a, Sa/3 + SPy a^, a Sa/3 , 1^' b, /32 Sa/3,b Now a determinate value of x would mean intersection in one point only ; so, as before, AS'.tt^y::=0, c{a'/3^ - S'-ap) = a {/3'^Say - Sa/3Spy) - h{Sa/3Say - a:^S/3y). The latter may be written S . a[c(a/32 - /3Sa/3) - a{y/3'^ - /3S/3y) - b(aS/3y - ySa/3)] = 0. Now S.a{a/3'^- /3Sa/3) = Sa{/3 . /3a - /3S/3a) = S.a{/3F/3a) = -S. a{/3Fa/3) = - S(a/3Fa/3) Art. 70.] GEOMETRICAL APPLICATIONS. 149 Similarly, S . a(y/32 - (SSfSy) = S{af3Ff3y), and S . a{aSf3y - ySafS) = S.a{F.f3 Fya) = S{apVya). The equation now becomes S. a(3{cFa(3 + aVlSy + hVya) = 0. Now since S . afiy = 0, a, /3, y are vectors in the same plane ; therefore y may be wi-itten ma + nfi, and c Fa/3 + a Vfty + b Fya assumes the form eVa/S, which, unless e = 0, gives S{al3Fal3) = 0, or FajS is in the same plane with a, f3 ; but it is also perpen- dicular to the plane, which is absurd ; therefore e = 0, or cFal3 + aFl3y + bFya = 0; thus the third and prolix method leads to the same conclusion as the first. 5. Find the surface traced out by a straight line ivhich remains always perpendicular to a given line while intersecting each of two Hxed lines. Let the equations of the fixed lines be cn — a + xf3, CTj = ttj + a;j/3j. Then if p be the vector of the new line in any position, p = CJ + !/(CT^-CT) = {l -y){a + o:f3) + y(a^+x,l3{). This is not, as yet, the equation required. For it involves essentially three independent constants, x, .t^ y ; and may therefore in general be made to represent any point whatever of infinite space. The reader may easily see this if he reflects that two lines which are not parallel must appear, from every point of space, to intersect one another. We have still to introduce the condition that the new line is perpendicular to a fixed vector, y suppose, which gives S.y{7;j^-rn) = = S.y [(a^ - a) + x^fS^ - xft]. 150 QUATERNIONS. [Chap. viii. This gives .Tj in terms of o:, so that there are now but two indeterminates in the equation for p, which therefore represents a surface, which, it is not difficult to see, is one of the second order. 6. Find the condition that the equation S . p<^p = - 1 may represent a surface of revolution. The expression 4>P here stands for something more general than that employed in Chap. VII. above,, in fact it may be written 4>p = aSa-^p + (3S/3^p + ySy-^p, where a, a^, /3, ft^, y, y^ are any six vectors whatever. This will be more carefully examined in the next chapter. If the surface be one of revolution then, since it is central and of the second degree, it is obvious that any sphere whose centre is at the origin will cut it in two equal circles in planes perpendicular to the axis, and that these will be equidistant from the origin. Hence, if /• be the radius of one of these circles, € the vector to its centre, p the vector to any point in its circumference, it is evident that we have the following equation, 8pp + 1 - C{p' + r^) = {S^pY - e\ where C and e are constants. This, being an identity, gives Sp^p — Cp^ = {Sep)^j The form of these equations shows that C is an absolute constant, while r and e are related to one another by the first ; and the second gives 4>p = Cp + eSep. This shows simply that S . (p4>p = 0, i.e. 6, p, and 4>p are coplanar, i.e. all the normals pass through a given straight line ; or that the expression Vpcf)p, whatever be p, expresses always a vector parallel to a par- ticular plane. Art. 70.] GEOMETRICAL APPLICATIONS. 151 7. If three mutually perpendicular vectors be drawn from a point to a plane, the sum of the recipvcals of the squares of their lengths is independent of their directions. Let Sip = - 1 be the equation of the plane, and let «., ft, y be any set of mutually perpendicular unit-vectors. Then, if xa, yft, zy be points in the plane, we have xSae= - 1, ySfte= - 1, zSye= - 1, whence e = - {aSa€ + ftSfte + ySyi) (64. 2) = - + '- + ^. z y z Taking the tensor, we have x^ y^ z^ 8. Find the equation of the straight line which meets, at right angles, two given straight lines. Let o7 = a + xft, ST = a^ + X^ft-^, be the two lines ; then the equation of the required line must be of the form V5 = a^ + X^Vftft-^, where a.-, only needs to be determined. Since the first and third equations denote lines having one point in common, we have S.ft Fftft^{a -a2) = 0. Similarly S . ft^ Vftft^ (a^ - a.^) = 0. Let tto = yft + y-^ft-^ (it is obviously superfluous to add a term in Vftft^), then S.aftVftft,=y,TWftft„ S.a,ft,Vftft,^-yTWftft„ and, finally, u5 = ^^^ {ft,S . aft Vftft, - ftS . a,ft, Vftft,) + X, Vftft,. 152 QUATERNIONS. [Chap. viii. 9. If Tp=Ta = Tf3=^l, and S. aftp = 0, show that S.U{p-a)U{p-(3) = ^l{l-Sa(3). Inter2Jret (his theorem geometrically. We have, from the given equations, the following, which are equivalent to them, p = xa + 7jf3 J Hence -x- - y" + 2xySaP = - 1, {z-l)a + yf3 U{p -a) = U(p - /3) s/(x-iy-2(xy-y)Sal3 + y^ xa + {y- l)/3 \/x^-2(xy-x)Sal3 + {y-lf S.U(p-a)U{p-(3) -x(:x-l) + [xy + {x-l){y-\)]SaP-y(y-l) Jx^ + y^-2x+\--2{xy- y)Sa/3 >Jx^+y^ -2y+\-2 {xy - x)Sa$ z + y-{x + y- l)Sa/3 - 1 ' V2 - 2x + 2ySaf3 j2-2y + 2xSafB (x + y-l){l-Sa/3) "2j{l-x-y){l- SajS) +xy{l- {Sa(3y } x + y- I I 1 - SafS '■ 2~Vl 1 -x-y + xy{l+ SafS) x + v-l I I - Saf3 x-y+h(2xy + x:--{-y-- 1) x + y-l r 1-Saf3 42 \ l-2(x + y)+x^ + : = ±^^(l->S'a/5). y^ + 2xy Of course there are far simpler solutions. Thus, for instance, the given equations show that p, a, fH are radii of some unit Art. 70.] GEOMETRICAL APPLICATIONS. 153 circle. Hence the expression is the cosine of the supplement of the ansfle between two chords of a circle drawn from the same point in the circumference. This is obviously half the angle subtended at the centre by radii drawn to the other ends of the chords. The cosine of this angle is - Sa/3, and therefore the cosine of its half is VI Ul-Sa/3). 10. Find the relative position, at any instant, of two points, which are moving unifo^-mly in straight lines. If a', /3' be their velocities, t the time elapsed since their vector positions were a, /3, their relative vector is p = a + ta' -(i-t(i' = (a-ft) + t(a'-p'), so that relatively to one another the motion is rectilinear, and the relative velocity is a - f3'. To find the time at which the mutual distance is least. Here we may write P = 7 + ^S, Tp^= - y"^ - 2tSy8 - t'-B'^ As the last term is positive, this expression is least when it vanishes, i.e. when t= -S.yS-\ This gives p = y - SSyS ~ i the vector perpendicular drawn to the relative path ; as is, of course, self-evident. 154 QUATERNIONS. [Chap. viii. 11. Find the locus of a given point in a line of given length, when the extremities of the line move in circlei- in me plane. (Watt's Parallel Motion.) Let o- and t be the vectors of the ends of the line, drawn from the centres a, j3 of the circles. Then if p be the vector of the required point /5=(a-f o-)(l -e) + c(/i + T), subject to the conditions {a + cr-(/3 + T)P=-Z2, 570- = 0, »v=o> 0-2^ _a\ t2= -5-'. From these equations o- and t must be eliminated. We leave the work to the reader. There is obviously an equation of condition S.y{(i-a) = 0. 12. Classify the curves rep/resented hy an equation of the form _a + xf3 + x^y a + bx + cx^^ where a, /3, y are given vectors, and a, b, c given scalars. In the first place we remark that x- in the numerator merely adds a constant vector to the value of p, unless c = 0. Thus, if c do not vanish, the equation may be written, with a change of a and /3 and in general a change of origin, _ a + x(3 a + bx + cx^ ' and this again, by change of x and of a and ft, as _a + xft a + cx^' It is obvious that this represents a plane curve. 4 , Sap _a^+ xSaft Sft'p^Saft + xft^' Art. 70.] GEOMETRICAL APPLICATIONS. 155 Hence both numerator and denominator of x are of the first degree in 6ap, 5'/?p ; and therefore -, a- + xSaR gives an equation of the third degree in p by the elimination of X. When we have Saf3 = 0, Sap = a + cx"' SPp~ ""^ a + cx^ o:^spp I J- oap whence and aiSapf + c"^^ (S/Spf = a^Sap, a conic section. If c = 0, then with a change of x, a, fS, y, the equation may be written a hyperbola — so long at least as b does not also vanish. If b and c both vanish, the equation is obviously that of a parabola. If a and b both vanish, whilst c has a real value, we have again a parabola. If a vanish while b and c have real values, we have again a hyperbola. 13. Find the locus of a point at which a given finite straight line subtends a given angle. Take the middle point of the line as origin, and let ± u be the vectors of its ends. At p it subtends an angle whose cosine is -SU{p-a)U{p + a). 156 QUATERNIONS. [Chap. viii. This, equated to a constant, gives the locus required. We may write the equation a? - p~ = cT(p - a) T(p + a). This is, obviously, a surface of the fourth order ; a ring or tore formed by the rotation of a circle about a chord. When c = 0, i.e. when the angle is a right angle, the two sheets of this surface close up into the sphere A plane section (in the plane a, (3 (suppose) where 1/3= Ta and SafS — 0) gives p = xa + yf3, or {l-(x2 + ^2)p^c2{(x2 + ?y2+ 1)2-4x2}, * 2cv or, finally, 1 - (x2 + ^2) = + -j^^, which, of course, denotes two equal circles intersecting at the ends of the fixed line. 14. A ray of light falls on a thin reflecting cylinder, show that it is spread over a right cone. Let a be the ray, t a normal to the cylinder, p a reflected ray, /3 the axis of the cylinder. Then t is perpendicular to /?, or ,S'^T = (1) Again p and a make equal angles with r, on opposite sides of it, in one plane ; therefore p II TttT or r.TaTp=.0 (2) Eliminating t between (1) and (2) we have p2_/^\2 a2 \Sal3j ' the equation of the right cone of which (3 is the axis, and a a side. Chap, viii.] GEOMETRICAL APPLICATIONS. 157 EXAMPLES TO CHAPTER VIII. 1. Find the equation of the surface described by a straight line which rotates about a fixed axis, the axis and straight line not being in the same plane. 2. Find the locus of a point whose shortest distances from two straight lines have a constant ratio. 3. Find the equation of a sphere circumscribing a given tetrahedron. 4. A straight line intersects a fixed line at right angles and turns uniformly about it while it slides uniformly along it. Find the equation of the surface described (1) when the fixed line is straight, (2) when it is circular. 5. Find the equation of the surface described by a circle which is made to rotate about any chosen axis in its plane. 6. Show that the equation Sp(pp= -\ may be expressed in the following forms : Sp{gp+V'KpfM)= -1, X and /J. being normals to the circular sections, and g a scalar constant ; a(rap)2 + ?<(.V/3p)2=-l, where a and h are constant scalars, and a, /3 constant vectors ; and THcp + pK) = (K^-Lr% t, K being two vector constants, which are real only when the equation is that of the ellipsoid. 7. Show that the equation of the surface generated by lines drawn through the origin parallel to the normals to Sp(l>''^p= -I along its lines of intersection with the confocal surface ((p + h)''^ is uT^-hST;j{

p= - 1. 10. Find the locus of all points on Sp(t>p= -1 where the normals meet the normal at the point a. CHAPTER IX. DYNAMICAL APPLICATIONS. 71. Differentiation of Quaternions. In the follow- ing dynamical applications we shall assume the simpler processes of differentiation and integration as in ordinary- analysis. In general, time will be the independent variable flowing continuously ; and in terms of it the rates of change of other varying quantities are expressed. "When a scalar quantity, such as x, is varying continuously its rate of variation at any instant will have a definite value, and this we shall, following Newton, represent by the notation x. The more usual notation dx will also be used when necessary. There is no difficulty in extending the methods of the Differential Calculus to quaternions and functions of quater- nions if we bear in mind the rion- commutative character of quaternion products. For example, ii q = af3, then the rate of change is q = 6L(3 + a$= F(af3 + a^) + S(kf3 + a$). Hence it follows that the symbol of differentiation is com- mutative with the selective symbols Faud *S'. Thus dVpui = ^m{FiopY = 3/n ( FUwpYT^w. Now '2.m(TF'U(Dp)- is the moment of inertia about the axis 0) and Tw is the angular speed. Hence - Saxpio represents twice the kinetic energy of rotation. If no couples act on the body, the kinetic energy and moment of momentum are each constant ; hence Sw(f)(j)= — €, ^ /.). c ^'> ■> P ^""' and the intersection of these two ellipsoids gives the cone in space described by the axis of spin. Its equation is The second equation of (2) may also be written, Tp= T(f)(o ; so that the perpendicular on the tangent plane at the extremity of w to the ellipsoid (f) is constant. Hence this ellipsoid rolls on a fixed plane perpendicular to p. Art. 74] DYNAMICAL APPLICATIONS. 163 74. The Spinning Top. In the case of the ordinary spinning top the couple acting on the body may be written Vf3a, where /3 is a unit vector drawn along the axis which passes through the centre of mass, and a the vector drawn vertically downwards with tensor equal to the product of the weight of the body and the distance of the centre of gravity from the origin. Since (3 is the vector position of a point in the body $-F->P (1) The dynamic equation is F(3a = fL = j-2m Vp'p = Swi Vpp. Hence, since p = Viop, p = Vuip + Fio'p, we find, after a slight transformation, Ff3a = ^mpVwp + ^mV . ojFpViDp = i^ = Ai>y-{C-A)kSkio = Au>-(C-A)ftSf3w, if we take k = p. Equation (2) becomes, by differentiation of c^w, and by use of (1), F/3a = A6,-{C- A)f3Sf36, - (C - A)Fu>(3S/3w (2) Operate by = 0, hence (jLI> and (5/30), which measures the angular speed about the axis of figure, is constant. Equation (2) takes the simplified form, F(3a = Aui-{C-A) ViofiSfii^ = Adj + {C-A)$.c, (4) where c (= - S/3w) is the constant angular speed about the axis of figure. Byuseof w/3 = FojfS + SiofS = $ - c, (5) we may eliminate either [3 or oj, and find the equation satisfied by w or (3. The w equation is somewhat complex (see Tait's Scientific Papers, Vol. I., p. 126); but that in /3 is comparatively simple and is easily obtained. For, multiply- ing into fS and taking the vector part, we find 0,= -FI3ft + (ic, and w=-Ff3(3 + $c. Hence substituting in (4), we get AF(3'fi + Ccfi=F(3a (6) Art. 74.] DYNAMICAL APPLICATIONS. > 165 If the second term be omitted, the equation becomes identical in form with the equation of motion of the conical pendulum. Operate on (6) in succession by S . V/Sp, S . a, and S. Vaft, and integrate the first two. There result ^Ar'f3/3 = Sal3 + H\ (7) ASaf3$ = H'-CcSap, (8) ASa ( - /? - pSftfi) + CcSafSfS + F'-(3a = (9) Eliminating Sa^ between (7) and (8), we find that the vector F/3/3 describes a curve on a spherical surface whose centre lies in the vertical line a. Equation (7) is one form of the energy equation (3), and may be written + hA$-^ = Saf3 + H\ since 5/3/i = 0. Also, differentiating SfSp = 0, we find Hence substituting in (9) from (7) and (8), and making a few transformations, we get ASafi = ^S-'aP + iSafSfH-' - ^j + ^ + a^, and finally, multiplying by Sa0 and integrating, hA^'a^ = S^aft + S-^af3 (^ - ^) + SafS (^ + a^ \ + K. If we put «S'a/3 = 0, we get the usual cubic for determining the limiting positions of the top. Returning to the fundamental equation (6), let us study the simple case in which the precessional motion is steady, the axis /? describing a right cone about a. Evidently fS\\F/3a; and if a is the precessional angular speed, T$ = aTFf3Ua=~TF(3a. la 166 QUATERNIONS. [Chap. ix. Hence ^=^F/3a, Ta a -r-rr. CC^ la I -a a2 ^^JaSaft-M, a2 and Ff3f3 = - ^ F^aSafi. Hence equation (6) becomes in this case J -a i a or - Aa^cos 6 + Cat - Ta = 0, where is the inclination between fS and - a. Writing Ta = nigh, we get the usual quadratic equation expressing the processional angular speed a in terms of the rate of rotation c about the principal axis, namely, Aa^cos 6 - Cca + mgh = 0. 75. Mutual Action of Magnets. When a magnet with pole-strength m and vector length [jl/vi is placed in any position in a uniform field of force /3, the couple acting on it is mFB^=FBfx, m tending to bring jjl parallel to (3. The quantity yu, the product of the pole strength into the distance between the poles, is called the magnetic moment of the magnet. The work done in moving the magnet so that the positive pole moves against /3, and the negative pole with f3, is -mSI3'^= -SfSdfx. ' m But the work done in bringing from infinity the positive and negative poles to such positions that the vector yu, lies perpendicular to /3 is evidently zero. Hence the integral of S/3diJ., namely, ''^f^f.t., measures the potential energy of Art. 75.] DYNAMICAL APPLICATIONS. 167 the magnet /x in the field /3, being equal to the minimum value - TfiTiJi when /x is co-directional with (i, to the maxi- mum value + T/ST/ji^, when /x is turned the other way, and to zero when /x is perpendicular to /3. Let the field /3 be due to a second magnet A with its centre at the origin ; and let p be the vector position of the centre /x. Both magnets are supposed to be short compared to their distance apart. Considering the action of the individual positive and negative poles at the extremities of the short vector X/n, n being the strength of the positiA^e pole, and assuming the law of the inverse square, we have for the force at the point p, ^ iU( p-XI-2n ) _ U {p + \,2 n) \ ' \T\p-).l2n) r'{p + \/27i)j ..^r p-^/2n ^ p + XI2n ] \(-p2 + SpX/nf (p-' + SpX/n)^} neglecting {X/2n)- in comparison with p'. Expanding each denominator by the binomial theorem, we find ''=iS;;{(^-4)04'*''-)-(''4.)('-i*'^^-')} = jL(„A. + 3/&V->) (1) Hence the couple acting on /x because of A is V/3f,= ~{FiiX+3VppSXp-^) (2) Similarly, the couple acting on A because of /x is -L(FXf. + srpxSf.p-^) (3) The mutual potential energy of the two magnets is S(3p, = ~{- SXp. + SSixpSXp - 1) i ^p = ^( - aS'A/x - 3*S> UpSX Up) SXp, SSfipSXp ^ "Wp W^' 168 QUATERNIONS. [Chap. ix. To find the translation force acting on either magnet, calculate the increment of this expression when p becomes p + dp. This is equivalent to differentiating the expression, p being the only variable. We find for the work done in effecting this displacement against the force, the ex- pression MTpSXp. 1 bSjipSkp . dip SSpdpSXp + 3SfMpS\dp "'' f^^ Wp Wp _^ f liUpSXfji l5UpSp.UpSXUp -.ifxSXUp 3XSp.Up ] ~ ^\ T^p T^p T*i~ T^p ] = Sdp{ 3Up(~ SXp. - bS/ji UpSX Up) _ SixSXUp _ 3XSp.Up ] The part in the brackets represents the total force against which work is done during the small displacement dp. It consists of three parts, one parallel to p, and the others parallel to the axes of the two magnets — all varying inversely as the fourth power of the distance. The couples acting on the magnets vary inversely as the cube of the distance. As a particular case, let X be set parallel to p, and /x perpendicular to p, and the two magnets to be in the same plane. Then the couples are on /x, p-(/xA - .3/xA) = - Ys~, , 1 . pX on A, j^Xp, = 'Js~p' so that the couple acting on the one is twice the value of the couple acting on the other. If we suppose the two rigidly fixed together, the system seems to be acted upon by a couple equal to - 3p.X/T'-^p. But then the translational force acting on either is equal to +— ^rj — ; and these two equal and opposite forces give rise to a couple +3nX/ T'^^p acting on the system. Thus the system is held in equilibrium. Art. 76.] DYNAMICAL APPLICATIONS. 169 The general proposition may be easily established that, if the magnets are fixed relatively to each other by a rigid framework, the sum of the couples acting on the two magnets is balanced by the moment of the translational forces acting on them. 76. Field of Force and Potential ; Properties of Nabla (y). In a field of force, gravitational, electric, or magnetic, the force /? at any point has a definite magnitude and direction. The work done by the force ^ acting through distance dp is - Sfidp ; and the integral of this along any path connecting two points is defined as measuring the difference of potential between the points. If dp is perpen- dicular to /S, this expression vanishes and no work is done, ■dp is then an element in an equipotential surface. The field may be imagined as mapped out by a series of surfaces h = c, where u is a scalar function of the position p and c a parameter which is constant for any one surface and varies as we pass from one surface to another. Let c be chosen so as to measure the work done against the forces in bringing up from infinity unit mass of the matter acted upon. Then if we pass from one surface to another near it the change du = dc will measure the difference of potential between the two surfaces, that is, the work done in passing from one to the other. Since u is a scalar function of p, its differential will consist of terms, each of which contains dp once. We may write du in the form Svdp, where v is a vector function of p. If dp lies in the equipotential surface, du = 0, and therefore S\dp = 0, so that V is a vector parallel to the normal to the equipotential surface at the point p. Hence we may write du--=dc= -Sf3dp (1) where /3 ( = - v) is the force associated with the equipotential system of surfaces u = c. 170 QUATERNIONS. [Chap. ix. It is clear that fi the force is derived from u the potential by a definite analytical process involving differentiation. Let V be the operator which derives ^ from u, so that /3 = ^u. It is defined by the equation du= -SdpVu (2) The vector quantity Vw is such that when resolved in direction dp and multiplied l)y the length of dp it gives a quantity which measures the work done in passing from the one to the other equipotential surface passing through the extremities of dp. It is in fact the force due to the potential u and acts in the direction of u diminishing. Let dp be written in the form idx where i is unit vector parallel to dp and dx is the tensor of dp. Then (2) may be written du= - SidxVu, du ^„ or y = - SiVu, giving the rate of change of n per unit distance in any assigned direction i. For three perpendicular directions i, j, k, we have '^=-SiVu, '^^-SjVu, ^=-SkVu. dx dy ■' dz But the vector Vw = - iSiT^u -jSjVu - kSk\/U 'dx^^dy^'^d^'"' ^^ which assigns the analytical expression for V in terras of the rates of change along any three perpendicular directions. It was in this form that Hamilton first defined the operator V. From (.3) we can at once verify that 5iV?< = — du/dx, and so on ; and we see that we may form the operator SiV first and then operate on the scalar function u, or we may form Vii first and then operate by S .i; symbolically SiVu^SN.u (4) Akt. 76.] DYNAMICAL APPLICATIONS. 171 Since any vector o- may be written in the form o- = ui + vj + wk where u, v, w, are scalar functions of p, we find d(T = du. i + dv .j + dtv . k = - SdpV . {ui + vj + wk) = -SdpV .(T, (5) a vector quantity which is necessarily diiferent from the scalar quantity - Sdp'va: Thus the identity (4) does not hold when a vector is the operand. Eeturning now to the discussion of the potential let us take the equipotential surfaces to be parallel planes, or u= - Sap = c, then du= - Saxlp — - SVudp, or VSdp = - a, giving a a constant force perpendicular to the planes. Let M be a function of the distance (7p = r) from the origin, say, u=f{Tp). Then du=f'{Tp) . dTp= -f{Tp) SUpdp= - SVudp. Hence Vi'r)- Upf'{r). For example let f{Tr) = ar = aTp. We find f'{r) = a, and f=aUp, so that the force is radial and constant in magni- tude throughout all space. Again let u=f{r) = ar~'^, f {r)=- - ar~'\ so that V,,--"-^^ (6) the important law of the inverse square, including the dynamic theories of gravitation, electricity, and magnetism. Again let u depend upon the distance from a given axis. If a is unit vector' along this axis, TVap is the distance of 172 QUATERNIONS. [Chap. ix. any point from the axis, and U=f{TVap), du= -SUFapVadp.f'(TVap) = - Sdpa-^ UVap .f'(TFap). The force will have the value aa'^UVaplTVap if f{TVap) = alogpTFap, the potential for cylindrical distributions. 77. Potential due to Distributions of Matter. The potential of a continuous distribution of matter may be written in the form * mdv u = Tp' where m is the mass in unit volume, dv is the element of volume, and the integration is taken through the region occupied by the attracting matter. Tp is the distance between the element dv and the point P at which u is the potential. The force acting at P is Vu = ^ ' mdv r J 'Tp^W •J mdvV 1 since V may be taken inside the integral or summation symbol and act on each term separately. Hence mdv Y^^= - Then Vm = nyiv^^. Apply V a second time V%= - T^6- jp.^Tp.p "^^^(-y^ps ■iUp. mdv i.-AV (7) ly Tpy This vanishes for all finite values of /j. Hence if Pis wholly outside the attracting matter V-m = 0. If P is a point occupied Art. 77.] DYNAMICAL APPLICATIONS. 173 by attracting matter V-m may have a value, which must depend solely upon the matter at F. For we may draw a small closed surface round P and consider the potential at P to consist of two parts u^ and w^, the former being due to matter within the small closed surface, the latter to matter without it. But, since u = u-^ + u^ we have V% = VH-^ + V^ti.,, of which the latter necessarily vanishes. Hence V% = V%j. The value of V-u^ is most easily found by considering the value of VyS, where ^ is any vector function of p. Let (i be the value at p the centre of the small parallelepiped, whose edges are idx, jdi/, kdz. At the face p + ^ idx, the value of f3 changes to /3 - ^ dxSiV . ft, and multiplying by + idydz, the vector area, we get the whole value over the surface element. Similarly, on the opposite end, looking the other way, the value of the corresponding quantity is {f3 + h dxSi'y . f3) X -idydz. Hence adding we obtain - iSi^ . /Sdxdydz. Similar expres- sions are obtained for the surface integrals on the other faces ; and adding all three together we find for the surface integral over the parallelepiped the value (3dv = - (iSiV +jSjV + kSk\7) . fSdxdydz = + Vf^dv, where dv is the vector area of the surface looking outwards and dv is the enclosed volume. This may be at once extended to finite volumes and enclosing surfaces in the form ^/3dv fSdv (8) Now if there is matter at the point /' the force P is outwards over the surface of any small enclosing sphere of radius Tdp ; and so far as it depends on the matter at P its value is ^t^, ^=-{Tdpf'^'^'- 174 QUATERNIONS. [Chap. ix. Let the area be divided into n equal parts, where ri is very large ; then V/3fZy; = \idpY n But Hence = iTrmdv. ^^u = 4:Trm •(9) This includes (7), for when there is no attracting matter at P, m vanishes and V-m = 0. 78. Convergence and Curl. From (8) we may derive very simply the important physical meanings of S^fi and FV^. Take first the scalar part of (8), namely : SVI3dr= Sf3dv. Let /? be the flow of fluid. Then the surface integral Sf3dv represents the amount of fluid which has entered the region ; and thus SVfS represents the convergence or increase of density of the flowing fluid. If the fluid be incompressible Secondly, take the vector part of (8). This gives V'^(3dv = Vfidv. Draw from any origin the vector areas dv for all points of the surface, and from their extremities draw the corre- sponding /3's Then if we consider ^ to be a force the surface integral will represent a couple or moment of force. Hence Wfi is the measure of this moment per unit volume. Maxwell has called it the curl of the vector (3. If FV/5 = 0, there is no curl, there is no molecular couple, or there is no vorticity in fluid of which (3 is the displacement. Art. 79.] DYNAMICAL APPLICATIONS. 175 When /3 is a force derived from a potential and Vf3 = V% essentially a scalar quantity. Hence FV/3 = 0, or there is no curl when the vector quantity can be derived by differ- entiation from a scalar function. 79. Electrical Distributions. An electrically charged conductor is at the same potential throughout. There is no electric force within it, and the charge is wholly on the surface. Let us apply theorem (8) to a region enclosing a small part of the charged surface and bounded in the field outside the conductor laterally by lines of force which of course spring normally from the charged surface, and terminally by a small area parallel and very close to the element of the surface. Since in this case F\^/3 = 0, equation (8), becomes Sj3du = Vhidv. On the sides of the region considered S(idv vanishes because ^ is perpendicular to dv. Within the conductor /S has no value. On the end of the region f3 is parallel to dv, which in this case is ultimately equal to the vector area element on the surface. But V^u = 47r X volume density. Hence V^udv = 4:? X volume density x area of element X thickness of electrified layer = 47r x surface density x ai-ea of element. Hence we obtain at once for the electric force just outside the surface the expression Tji = 47r X surface density. This in fact is the dynamical definition of the surface density of the charge. 176 QUATERNIONS. [Chap. ix. As a final example consider the distribution on an ellip- soidal conductor. Let u= - Sp(j)p = c represent the ellipsoidal equipotential surface. Then du= -2Sdp4>p= -Sdp'^u. Hence the force at the point p of the surface is Vu = 2<^/3. But (§ 59) the perpendicular (p) on the tangent at the point p is equal to cjT4>p; hence and the surface density is c/27rp, that is inversely as the perpendicular from the centre on the tangent plane at the point. These are some of the simple applications of the important differential operator V, the theory of which was developed by Tait. For further discussion the reader is referred to the works of Tait and M'Aulay, and to Joly's Appendix to the second edition of Hamilton's Elements. EXAMPLES TO CHAPTER IX. 1. A particle is moving under the action of a constant force. Prove that the hodograph is a straight line and that the path is a parabola. 2. Two equal and opposite magnetic poles are placed at A and A' (vector AA' = 2a). Show^ that the equation giving the direction of the line of force at any point P (vector distance p from the middle point of A A') leads to the result Sa{U(p + a)^ f(p-a)} = const. 3. Show that in uniplanar motion, the motion of any rigid figure may in general be represented by a rotation about a determinate point ; and that if the motion is continuous, the velocity of any point is given by p-ci{p-(T), where P= - {a Sap + /S'S/Sp + y'Syp), where c^ itself depends upon nine independent constants in- volved in the three equations <^a = a " <^/? = /3' For a, IS', y' may of course be expressed in terms of a, /3, y : and, as they are quite independent of one another, the nine coefficients in the following equations may have absolutely any values whatever ; ^a = a' = Aa + cfi + b'y \ cl>ft=.(3' = c'a +Bl3 + ay^ (a) ^y=y'=ba +a'/3+Cy} 180 QUATERNIONS. [Chap. x. In discussing the particular form of <^ whicli occurs in the treatment of surfaces of the second order we found, § 45, that it possessed the property S . crcf)p = S . p^fxr, (b) whatever vectors are represented by p and cr. Remembering that a, f3, y form a rectangular unit system, we find from (a) S.a(l>fS= -c'y with other similar pairs ; so that our new value of (^ satisfies {h) if, and only if, we have in (a) {<■) The physical meaning of this condition, as will be seen immediately, is that the distortion expressed by ^ takes place without rotation. In this case the nine constants are reduced to six. But, although {h) is not generally true, Ave have S . (rcf)p = - (Sa'aSap + Sj3'(rS/3p + Sy'crSyp) = -S. p{aSa'a- + f3S/3'(r + ySy'ir), where the expression in brackets is a linear and vector function of cr, depending upon the same nine scalars as those in ; and which we may therefore express by <^', so that 'o-= -{aSa'a- + (3SP'cr + ySy'cr} (d) And with this we have obviously S . (rcf>p = S . p(fi'(r, (e) which is the general relation, of which (b) is a mere particular case. By putting a, ft, y in succession for o- in (d) and referring to (a), we have (f)'a = Aa + c'ft + by ~| cf>'ft= ca +Bft + ayy (/) 'y=b'a Araft + Cy) Chap, x.] VECTOR EQUATIONS OF FIRST DEGREE. 181 Comparing (/) with {a), we see that whatever be p, provided the conditions (c) be fulfilled. This agrees with the result already obtained. Either of the functions ^ and 4)', thus defined together, is called the Conjugate of the other : and when they are equal (i.e. when (c) is satisfied) 4> is called a Self-Conjugate function. As we employed it in Chap. VI., is self -con jugate by writing e = 0, the physical interpretation of which equation is of the highest importance, as will soon appear. If we form by means of (a) the value of e as in (/i), we get 2e = (cy - b'p) + {aa - c'y) + (bf3 - a a) = {a-a)a + {h-h')f3 + {c-c)y, which obviously cannot vanish unless (as before) the three conditions (c) are satisfied. 182 QUATERNIONS. [Chap. x. By adding the values of p + 4>'p = - {aSa'p + a Sap + ftSft'p + ft' S ftp + ySy'p + y'Syp). Operating by S .a- we see at once that this new function of p is self-conjugate. Hence we may write { + 4>')p = 2^P, (J) where the bar over ST signifies that it is self-conjugate, and the factor 2 is introduced for convenience. From {g) and (i), we have 4>'p = '^P - ^^pj If instead of ^p in any of the above investigations we write (4> + g)p, it is obvious that c^'p becomes {(f)' + g)p: and the only change in the coefficients in (a) and (/) is the addition of g to each of the main series A, B, C. We now come to Hamilton's grand proposition with regard to linear and vector functions. If ^ be such that, in general, the vectors p^ <^p^ ,^2^ (where ^p (a contraction for ^(((/)p))) can be expressed in terms of them as in 31. 5. Thus (fi^p = m.^ffi^p - 7n^(f)p + mp, (k) where m, m^, ni^ are scalars whose values will be found immediately. That they are independent of p is obvious, for we may put a, ft, y in succession for p, and thus obtain three equations of the form a + ma, (I) from which their values can be found. For hy repeated applications of (a) we can express (Z) in the form This gives ^=0, ^ = 0, € = 0. Chap, x.] VECTOR EQUATIONS OP FIRST DEGREE. 183 These are three equations connecting m, m-^, m„, with the nine coefficients in (a). The other two groups of three equa- tions, furnished by the other two equations of the form (/), are merely consistent with these ; and involve no farther limitations. This method, however, is very inferior to one which will shortly be given. Conversely, if quantities m, m^, m.2 can be found which satisfy (/), we may reproduce (k) by putting p = xa + '//f3 + ~y and adding together the three expressions (?) multiplied by X, y, z respectively. For it is obvious from the expression ^ Z(^p == cf) (xp), Xfji^p = 4>'^ (xp), etc., whatever scalar be represented by x. If p, 4>p, and (f>'^p are in the same plane, then applying the strain (fi again, we find (fip, (fi'^p, <^^p in one plane ; and thus equation (k) holds for this case also. And it of course holds if <^/3 is parallel to p, for then ^^p and <^^p are also parallel to p. We will prove that scalars can be found which satisfy the three equations (/) (equivalent to nine scalar equations, of which, however, as we have seen, six depend upon the other three) by actually determining their values. The volume of the parallelepiped whose three conterminous edges are A, /i,, v is (§ 32) — S . Xp.v. After the strain its volume is — *S' . . ^ = P, \ H- = ^P, ( 184 QUATERNIONS. [Chap. x. 711, ,(m) and by inspection of (h), we find S . )^fji.v S . p<^p<^'^p which gives the physical meaning of this constant in {k). As we may put if we please we see by (a) that m = S.afSy A, c, V c, B, a b, a, C which is the expression for the ratio in which the volume of each portion has been increased. This is unchanged by putting (fi' for ^, for it becomes, by (/), m= A, c', b c, B, a' b', a, C Hence conjugate strains produce equal changes of volume. Recurring to (in), we may write it by (e) as from which, as X is absolutely any vector, we have or (ji VFij.v. Let (t)F{jiv = xFixi' + yF(f)'ixv + zFfjLcfi'v. Operate by aS' . A, S . /x, S . v successively, then S . /j.v(f>'X = xS . A/xv + yS . vXcfi'fx + zS . Xficfi'v, S . fxv(fi'fi — yS . i^/x(^'/i, S . fiv4>'v =ZS . Vfl'v. •(?) 186 QUATERNIONS. [Chap. x. The two last equations give (§ 32) y= -1, ^'= -1, and therefore the first gives S . fXV' /jLu - Vjx(f)'v (r) Substituting this in {q), and putting o- for F/xv, which is any vector whatever', we have («^ + g) [m<^-i + g (fi, -cf>) + f] a- = {m + fi^g + fi^g'- + g-") a; or, multiplying out, (m - g(f>- + ix,g(j> - g^ + gnifl^-^ + g^ + g'/j., + g^)a- = {m + fx^g + fi,g' + g'^)(T; that is (-(/)"- + /X., (/) + ?«X(j)fjL I ' ] ■is) "'i = /'i = 5.A/.V and thus the determination is complete. We may write (k), if we please, in the form m(f) ~^p — m-^p- 'Hh4>p + 4'^ Pi (^') which gives another, and more direct, solution of the equation (above mentioned) (/> — ^ Physically, the result we have arrived at is the solution of the problem, "By adding together scalar multiples of any vector of a body, of the corresponding vector of the same strained homogeneously, and of that of the same twice over strained, to represent the state of the body which Chap, x.] VECTOR EQUATIONS OF FIRST DEGREE. 187 would be produced by supposing the strain to be reversed or inverted." These properties of the function <^ are sufficient for many- applications, of which we proceed to give a few. (I.) Homogeneous strain converts an originally spherical portion of a body into an ellipsoid. For if p be a radius of the sphere, cr the vector into which it is changed by the strain, we have and Tp = C, from which we obtain or 5'.<^-ic7c/)-io-= -6'^ or, finally, 5 . (7<^' " V" ^o- = - C"-. This is the equation of a central surface of the second degree ; and, therefore, of course, from the nature of the problem, an ellipsoid. (il.) To find the vectors whose direction is unchanged by the strain. Here (^p must be parallel to p or # = gp- This gives <^-p = g'p, etc., so that by {k), we have g^ - m.^g''- + rti-^g - rii = 0. This must have one real root, and may have three. Suppose g-^^ to be a root, then i'P - 9\P = 0. and therefore, whatever be X, ,S'A<^p - g^SXp = 0, or S.p{cf>'X-g^X) = 0. Thus it appears that the operator ' - g^ cuts off from any 188 QUATERNIONS. [Chap. x. vector A. the part which is parallel to the required value of p, and therefore that we have p\\MF.(^ + m,-^4> -m)p = Q, being true for all vectors whatever, may be written (<^-5'i)(^-.'72)('^-5'3)P = 0, and it is obvious that each of these factors deprives p of the portion corresponding to it : i.e. - g^ applied to p cuts off the part parallel to the root of {4>- gi)cr = 0, etc., etc., so that the operator ( - ^2)('^~ S's) when applied to a vector leaves only that part of it which is parallel to o- where (ill.) Thus it appears that there is always one vector, and that there may be three vectors, whose direction is unchanged by the strain. Def. Pure, or non-rotational, strain consists in altering the lengths of three lines at right angles to one another, without altering their directions. Hence, if 'kPi = 9iPv 4>P2 = 9iPi-> Chap, x.] VECTOR EQUATIONS OF FIRST DEGREE. 189 the strain 4> is pure if, and not unless, p^, p.^, p^ form a rect- angular system. [There is a qualification if two or more of ffi 92 ffs be equal] Hence, for a pure strain, we have Sp2'i'P\ = 9lSp2Pl = 0' and Spi4'P2 = 9'2^PiP2 = ^' or Sp-^({>p2 = Sp2i<^p2 = ^P-z't^'Pr As we have two other pairs of equations like these, we see that 4> = 4>' when the strain is pure. Conversely, if 'P = i^\ the three unchanging directions p^, p.j, p^ are perpendicular to one another. For, in this case, the roots of are real. Let them be such that (-f/i)pi = 0^ ('^- ^2)^2 = j' then 9i92^PiP2 = ^'i'Pi^P2 = Sp^ffict'po (because, by hypothesis, the strain is pure) = 92'^PiP2y for p=Tp \ ^^^ Spa- = «S' . (fipcfxTj Chap, x.] VECTOR EQUATIONS OF FIRST DEGREE. 191 i.e. the lengths of vectors, and their inclinations to one another, are unaltered. In this case, therefore, the strain can be nothing but a rotation. It is easy to see that the second of these equations includes the first ; so that if, for variety, we take as represented in equations (a), and write p = xa + i//3 + zy, we have, for all values of the six scalars x, y, z, ^, 7], C, the following identity : - {x^ + pj + zO = S . (xa' + >/f3' + zy') {^o! + ,,/3' + ii) = a-^X^ + f3'-^yr^ + y2zC + (■'■'; + I/O S<^'/^' + (yC + .--'?) Sf3'y' + (z^ + xO Sy'a. This necessitates '^ ' , (m) Sa(B' = S(3'y' = Sy'a=0} i.e. the vectors a', f3', y' form, like a, /i, y, a rectangular unit system. And it is evident that any and every such system satisfies the given conditions. But the system a', (3', y must be similar to a, ^, y, i.e. if a quadrant of 'positive rotation round a changes /3 to y, etc., a quadrant of positive rotation about a must change /i' to y', etc. When this is not the case, the system a, /i', y' is the perversion of a, /3, y, i.e. its image in a plane mirror; and the strain is impossible from a physical point of view. This is easily seen from another point of view. The volume of the parallelepiped whose edges are rectangular unit vectors a, (3, y is - S . af3y if a positive quadrant of rotation round a brings ^ to coincide with y, etc. But, in the perverted system, the volume has changed sign and is expressed by S . a/3y. 192 QUATERNIONS. [Chap. x. (VII.) It may be interesting to form, for this particular case, the equation giving the values of g. We have ,S'.(<^ + ^)a(<^ + 5r)/3(<^ + r/)y ^S.{a: + ga){ft'i.g(S){y'+gy) S . ajiy = \-gS{al3'y' + a'fty' + a'j3'y) - g'S{a(3y' + a/3'y + a' (iy) + g\ Recollecting that a, ^, y; «', ft\ y' are systems of rect. angular unit vectors, we find that this may be written ^. = 1 - (^ + ^2)'S'(aa' + liji' + yy') + g^ = C9+l)[/-^{l+6'(aa' + ^/3' + yy')} + l]. Hence the roots of M^ = are in this case; first and always, ^1= -1, which refers to the axis about which the rotation takes place : secondly, the roots of g' - r/{ 1 + S{aa' + ftfi' + yy')} + 1=0. Now the roots of this equation are imaginary so long as the coefficient of the first power of g lies between the limits ±2. Also the values of the several quantities Saa, Sf3f3', Syy' can never exceed the limits ±1. When the system a, f3, y coincides with a, f3', y', the value of each of the scalars is - 1, and the coefficient of the first power of ^ is +2. When two of them are equal to + 1 and the third to - 1 we have the coeflicient of the first power of ^ = - 2. These are the only two cases in which the three values of g are all real. In the first, all three values of g are equal to - 1, i.e. for all values of p, and there is no rotation whatever. In the second case there is a rotation through two right angles about the axis of the - 1 value of g. Chap, x.] VECTOR EQUATIONS OF FIRST DEGREE. 193 VIII. It is an exceedingly remarkable fact that, however a body may be homogeneously strained, there is always at least one vector whose direction remains unchanged. The proof is simply based on the fact that the strain-function depends on a cubic equation (with real coefficients) which must have at least one real root. IX. As an illustration of what precedes (though one which must be approached cautiously), suppose a body to be strained so that three vectors, a", /3", y" (not coplanar, and not necessarily at right angles to one another), preserve their direction, becoming e^a", eM", e^y"- Then we have 4>pS . a"/3"y" = e,a"S . f3"y"p + e,(i"S . y"a"p + e,y"S . a"f3"p. By the formulae (m, s) we have S . 4>a"4,f3"cj,y" _ m — ^ a"8"y" ~ ^1^2^3' S(a"f3".l>y" + f3"y"a" + y'»/3") _ % = S' a"8"y" ^^ ^^ S{a"(3"^y"+(ry"cl.a"+y"a"f3") _^ , , , , . so that we have by (k) {cf,-e^){4>-e.^{cf>-e^)p = 0. Though the values of g are here all real, we must not rashly adopt the conclusions of (iv.), for we must remember that a", f3", y" do not, like a, f3, y, necessarily form a rect- angular system. In this case we have 'pS . a"^"y" = e, Ff3"y"Sa"p + e,Fy"a"SP"p + e^ ra"/3"Sy"p. So that, by (g) and (h), 2eS. a"(3"y" = F. {e^aF(i"y" + e^ft"Fy"a + e,y"Fa"ft") : (^2 - %a"5/3"y" + e^- e^(i"Sy"a +e^- e.^y'Sa'/S"). N 194 QUATERNIONS. [Chap. x. This vanishes, or the strain is pure, if either 1. Sa"/3" = Sft"y" = Sy"a"=.0, i.e. if a", /3", y" are rectangular, in which case e-^, e.^, e.^ may have any values ; or 2. ^^ = ^2 = ^3, in which case cfy'pS . a"f3"y" = e^{ Ff3"y"Sa"p + Fy"a"SI3"p + Va"ft"Sy"p] = e,pS.a"f3"y" by (§34.2), so that (f)' p = e^p = -''p = - ^2, and in this it is obvious that ^~- is self-conjugate, or at least is to be treated as such : for a non-conjugate term in (f)~'~p would be (g) of the form Fep, and would therefore not appear in the equation. For the proper treatment of rotations, the following simple but excessively important proposition, due to Hamilton, foi'ms the best starting-point. If q be any quaternion, the operator q{ )q~^ turns the vector, quaternion, or body operated on round an axis perpendicular to the plane of q and through an angle equal to double that of q. For the proof we refer the reader to Hamilton's Lectures, § 282, Elements, § 179 (1), or Tait, § 3-53. It is obvious that Chap, x.] VECTOR EQUATIONS OF FIRST DEGREE. 195 the tensor of q may be taken to be unity, i.e. q may be considered as a mere versor, because the value of its tensor does not affect that of the operator.* [A very simple but important example of this proposition is given by supposing q and r to be both vectors, a and ^8 let us say. Then aBa~^ is the result of turning (3 conically through two right angles about a, i.e. if u he the normal to a reflecting surface and /3 the incident ray, - a/ia^i is the reflected ray.] Now let the strain (f> be effected by (1), a pure strain CT (self -con jugate of course) followed by the rotation q{ )q~^. We have, for all values of p, (fyp = q(zijp)q--\ (i;) whence (ji'p = C7 {q'^pq). The interpretation is that, under the above definition, the conjugate to any strain consists of the reversed rotation, followed In/ the pure strain. We may of course put, as in Chap. VII., uJp = e^aSap + e^pspp -h e^ySyp, where a, f3, y form a rectangular system. Hence (/)p = e-^qaq-^Sap + e^qfiq~'^SIip + e-^qyq'^Syp. Here the axes are parallel to qaq-^, qPq-\ qyq-\ and we have S . qwf^qfiq-^ = S . qa(3q-^ = Saf3 = 0, etc. So far the matter is nearly self-evident, but we now come to the important question of the separation of the pure strain from the rotation. By the formulae above we see that 4>'4>p = T:5q~'^(f)pq = ^q-Hqmpq-'^)q *The proof is now given above, Chap. IV., §27. 196 QUATERNIONS. [Chap. x. so that we have in symbols, for the determination of S, the equation That is, as we see at once from the statements above, any strain, followed by its conjugate, gives a pure strain, which is the square {or the result of two applications) of the pure part of either. To solve this equation we employ expressions like (k). (f^'cf) being a known function, let us call it w, and form its equation as o>3 - Wigf"' + '"i^ - )/} = 0. Here the coefficients are perfectly determinate. Also suppose that the corresponding equation in v^ is m'^-gp'^ + g^C5-g = Q, where g, g^, g^ are unknown scalars. By the help of the given relation cj^ = oj, we may modify this last equation as follows : (77w - g.^ui + g^TH - g = 0^ _ g + gM whence tn = ; ^i + co i.e. CT is given definitely in terms of the known function, w, as soon as the quantities g are found. But our given equation 57^ = (^ may now be written or w3 _ (^g^2 _ 2g^) w2 + (^^2 _ 2gg_^ w - g'^ = 0. As this is an equation between w and constants it must be equivalent to that already given ; so that, comparing coefficients, we have 9'2'- -2g^ =m^. ffl' - - 2gg^ = mj. f = m; CuAP. X.] VECTOR EQUATIONS OF FIRST DEGREE. 197 from which, by elimination of g and g.^, we have The solution of the problem is therefore reduced to that of this biquadratic equation ; for, when g^ is found, g^ is given linearly in terms of it. [A neat way of arriving at the same result is to throw the equation in CT into the form square both sides and multiply up. The result is or (.3 - (g^ - 2g,)u.^^ + {g{^ - 2gg.^ io-g^^ = 0]. It is to be observed that in the operations above we have not been particular as to the arrangement of factors. This is due to the fact that any functions of the same operator are commutative in their application. Having thus found the pure part of the strain we have at once the rotation, for (v) gives 4)Z^-^p = qpq-\ or, as it may more expressively be written, J(f)'(f3 If instead of (v) we write <^p = 7n(rpr-i), (v) we assume that the rotation takes place first, and is succeeded by the pure strain. This form gives <^'p = r-i(trrp)r, and 4"f^'p = <^V) whence nr is found as above. And then (?/) gives TS-^ = r{ )r-\ 198 QUATERNIONS. [Chap. x. Thus, to recapitulate, a strain followed by the rotational strain <^— ==, or to the rotational strain -ttT' ^ followed by the pure strain sf^'. This leads us, as an example, to find the condition that a given strain is rotational only, i.e. that a quaternion q can be found such that Here we have ^' = q-H )q, or But or (w) 4>'=r' whose conjugate is m^) = m^ - m.24) + <^ and the elimination of <^' between these two equations gives I.e. - (m^ - mm^" + 'Im^m^ - m) 4> (m-'/rtj - mm-^m-2 + ni-^^-) - {m^ -mm^ + ^m-^i^'^ - {mnio - 2m^ - m.f)4>^ - 2m.24>^ + 4>^ by using the expression for c/)* from the cubic in (^. Now this last expression can be nothing else than the cubic in ^ - m.^ff)-^ + 'm.24> -1=0, or (^-i)[^-i + (i-,n.,)4> + l}=0, where m^ is left undetermined. By comparison with the result of (vii.) we see that in the notation there employed m.^= -S{aa' + ftP' + yy'). The student will perhaps here require to be reminded that in the section just referred to we employed the positive sign in operators such as 4^ + g. In the one case the coefficients in the cubic are all positive, in the other they are alternately positive and negative. The example we have given is a particularly valuable one, as it gives a glimpse of the extent to which the separation of symbols can be safely carried in dealing with these questions. Def. a simple shear is a homogeneous strain in Avhich all planes parallel to a fixed plane are displaced in the same direction parallel to that plane, and therefore through spaces proportional to their distances from that plane. Let a be normal to the plane, ^ the direction of displacement, the former being considered as an unit-vector, and the tensor of the latter being the displacement of points at unit distance from the plane. 200 QUATERNIONS. [Chap. x. We obviously have, by the definition, Sa/3 = 0. Now if p be the vector of any point, drawn from an origin in the fixed plane, the distance of the point from the plane is - Sap. Hence, if o- be the vector of the point after the shear, (T=<^p = p - (3Sap. This gives p = aSap + {fBcosd + y sin 8)SPp + {y cosd - f3 sin d)Syp = ( 1 - cos 6) aSap - p COS 6 - Fap sin 6. Forming the quantities m, ni-^, w., as usual, we have * <^3 _ (1 + 2 cos ^)-l)p = {l -cos6) {aSap + p) - sin 6 Vap, -(-(l+0}=0- We will conclude our treatment of strains by solving the following prol>lem : Find the conditions ivhich must be satisfied by a simjyle shear which is capable of reducing a given strain to a pure strain. Let (/> be the given strain, and let the shear be, as above, then the resultant strain is xpi^ = (fi + f3S . a(f> = cf> + (3S.i.. 204 QUATERNIONS. [Chap. x. Again, (4) 2e- = 6' . (/)'a/3e = S . acfi (f3e), or -ma='2V . fi~^4>e. Hence w-e may assume any vector perpendicular to e. for /?, and a is immediately determined. When two of the roots of the cubic in ?/> are imaginary let us suppose the three roots to be ep e.2±e^J - 1. Let P and y be such that Then it is obvious that, by changing throughout the sign of the imaginary quantit}^ we have These two equations, when expanded, unite in giving by equating the real and imaginary parts the values ^y = e.,y + e^l3l To find the values of a, (3, y we must, as before, operate on any vector by two of the factors of the cubic. As an example, take the very simple case 4>p = e Vip. Here it is easily seen by (in), (s), that m = 0, TOj = + e~, m^ = 0, so that* 4y' + e-4> = Q, that is {hy-jz + J -\p) II -jy-kz + J -\{ky-iz) \\jy + ]cz-J^^{iz-ky). AVith a change of sign in the imaginary part, this will represent so that (3=jy + kz, 7 =j- - ky- Thus, as the student will easily find by trial, /3 and y form with a a rectangular system. But for all that the system of principal vectors of ^, viz. a, /3±yV"l does not satisfy the conditions of rectangularity. In fact we see by the above values of /3 and y that It may be well to call the student's attention at this point to the fact that the tensors of these imaginary vectors vanish, for This gives a simple example of the new and very curious modifications which our results undergo when we pass to Bivedors ; or, more generally, to Biquaternions. 206 QUATERNIONS. [Chap. x. As a pendant to the last problem we may investigate the relation of two vector-functions whose successive appli- cation produces rotation merely. Here cf) = il^x^^ is such that by (w) i.e. x'"V = X'/'~^ or x'x = f'/' = ^-, since each of these functions is evidently self-conjugate. This shows that the pure parts of the sti'ains \p and x are the same, which is the sole condition. One solution is, obviously, i.e. each of the two is itself a rotation ; and a new proof that any number of successive rotations can l^e compounded into a single one may easily be given from this. But we may also suppose either of i/-, x, suppose the latter, to be self -con jugate, so that X' = X = X> or f^^ = x2, which leads to previous results. EXAiMPLES TO CHAPTER X. 1. If a, (3, 7 be a rectangular unit system S . VaaV^. * leiQUI JAN^ 74-4PII R^EXTD CD 7Fet)'64Zf LD 21-100m-12,'46(A2012sl6)4120 ■r 376U. K 4 . A . ^^ UMVERSriY OF CAUFORNl A UBRARY ^ W^f<