UC-NRLF fll 155 VECTORIAL MECHANICS MACMILLAN AND CO., LIMITED LONDON BOMBAY CALCUTTA MELBOURNE THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO DALLAS SAN FRANCISCO THE MACMILLAN CO. OF CANADA, LTD TORONTO VECTORIAL MECHANICS BY L. SILBERSTEIN PH.D. (BERLIN) LECTURER IN NATURAL PHILOSOPHY AT THE UNIVERSITY OF ROME MACMILLAN AND CO., LIMITED ST. MARTIN'S STREET, LONDON 1913 COPYRIGHT PREFACE THE main object of this little volume is to present the chief principles and theorems of theoretical mechanics in the language of vectors, and thereby to contribute to the diffusion of the use of vectorial methods. No space has been devoted therefore to any discussion of the philosophical aspect or of the origin of such concepts as ' mass,' 'force,' 'work,' and so on, or to any description of the long and laborious route which, in the historical development of mechanics, has led to the fundamental principles of this branch of science, and especially to d'Alembert's Principle, nor of those routes which, in modern treatises, are supposed to lead to them. I emphasize d'Alembert's Principle rather than any other equivalent to it, because it is from this that a start is here made. If our object were simply and solely a translation of mechanics into the language of vectors, we could indeed begin anywhere. But this book is not intended merely to present a loose juxtaposition of mechanical theorems and of their vectorial formulae. On the contrary, after the enunciation, in the first section of Chapter II., of the principle mentioned already we shall be able to proceed by a continuous, deductive road, so that those readers who are acquainted with little more than d'Alembert's Principle will find here an almost systematic exposition of the chief parts of mechanics. Again, other readers who know the subject thoroughly, but only in its Cartesian form, which is based on the consideration of the scalar components of displacements, velocities, forces and so on, may perhaps wish to . see their knowledge translated into vectorial language, which to say the least is considerably shorter and more satisfactory to the imagination than the scalar language. In this work, only such problems will be treated as are not beyond the region proper to the vector language ; and consequently 28-4286 vi PREFACE everything which has no space-directional character will, with few exceptions, be omitted. Those readers, who are already acquainted with the elementary parts of vector algebra and analysis, may proceed at once to Chapter II. ; to those who are not, may be recommended ' The elements of vectorial algebra and analysis ' in Chapter III., Vol. I., of Oliver Heaviside's excellent book : Electromagnetic Theory, London, 1893, or E. B. Wilson's Vector Analysis, etc., founded upon lectures of J. W. Gibbs, New York and London, 1902, or the two sections of our Chapter I., in the first of which I have endeavoured to develop the fundamental concepts of Vector Algebra, and in the second the most important of the Differential and Integral Properties of Vectors. These sections taken together will contain, I hope, all that is needed for our mechanics, and possibly also for general Mathematical Physics. Chapters II. -VI., of which a certain part formed the subject of a series of articles by the author, published a few years ago in a Warsaw weekly (Przeglad Techniczny, i.e. 'Technical Review'), contain the General Principles of mechanics and their consequences, e.g. the three Special Principles and the essential part of the dynamics of Rigid and of Deformable Bodies, closing with Hydrodynamics. The not very numerous collection of Problems and Exercises may be useful in connexion with the various chapters, and the Appendix containing a kind of Vectorial-Cartesian dictionary to the whole volume may be helpful to freshmen in vectorial language. I gladly take the opportunity of expressing my sincere thanks to my friend Prof. A. M. Worthington, C.B., F.R.S., to Mr. J. F. M'Kean of the Royal Naval College, Dartmouth, to Profs. I. J. Schwatt and George H. Hallett of the University of Pennsylvania, Philadelphia, and to Profs. Alfred W. Porter, F.R.S., and R. A. Gregory, London, for their kindness in reading the MS. and the proofs and to the Publishers for the care they have bestowed on my work. L. S. LONDON, Jtine, 1913. CONTENTS CHAPTER I ELEMENTS OF VECTOR ALGEBRA AND ANALYSIS PAGE VECTOR ALGEBRA - i DIFFERENTIAL AND INTEGRAL PROPERTIES OF VECTORS 22 CHAPTER II GENERAL PRINCIPLES D'ALEMBERT'S PRINCIPLE - - 50 LAGRANGE'S EQUATIONS 54 HAMILTON'S PRINCIPLE - 58 CHAPTER III SPECIAL PRINCIPLES 1. THE PRINCIPLE OF VIS-VIVA 60 2. THE PRINCIPLE OF CENTRE OF GRAVITY - 62 3. THE PRINCIPLE OF AREAS 64 CHAPTER IV RIGID DYNAMICS GENERAL REMARKS 69 DIFFERENTIAL EQUATIONS OF PURELY ROTATIONAL MOTION - - 70 KINETIC ENERGY. PRINCIPAL AXES. EULER'S EQUATIONS OF MOTION 72 MOTION UNDER NO FORCES (Motion a la Poinsot) 78 ROTATION OF A RIGID SYSTEM UNDER THE ACTION OF GRAVITY - 82 KlNEMATICAL RELATIONS - - - - ' 87 THE LINE-INTEGRAL AND THE CURL OF A VECTOR ILLUSTRATED BY THE EXAMPLE OF A MOVING RIGID BODY 89 viii CONTENTS CHAPTER V GENERAL MECHANICS OF DEFORMABLE BODIES GENERAL REMARKS FUNDAMENTAL PROPERTIES OF THE LINEAR VECTOR OPERATOR STRAINS Infinitesimal Strain The Equation of Continuity Classification of Strains Longitudinal Strain Transversal Strain Surfaces of Discontinuity KINEMATICS OF A DEFORMABLE BODY 123 Kinematics of Surfaces of Discontinuity STRESS. DIFFERENTIAL EQUATIONS OF MOTION - - 132 CHAPTER VI HYDRODYNAMICS FUNDAMENTAL NOTIONS AND EQUATIONS ... 139 Conservation of Energy Clebsch's Transformation FLUID IN EQUILIBRIUM - 148 IRROTATIONAL MOTION - 150 PROPERTIES OF VORTEX MOTION 156 Kinematical Properties Kinetic Energy Steady Motion ; Ber- noulli's Theorem Sir William Thomson's Theorem Helmholtz's Theorems Creation of Vortices along the Intersections of Surfaces of constant density and constant pressure THE WAVE OF ACCELERATION ; HUGONIOT'S THEOREM - - 167 PROBLEMS AND EXERCISES ... - 170 APPENDIX. CARTESIAN EQUIVALENTS OF VECTOR FORMULAE 184 INDEX 194 CHAPTER I. ELEMENTS OF VECTOR ALGEBRA AND ANALYSIS. Vector Algebra. ANY magnitude which has size, in the ordinary algebraic sense of the word, as well as direction in space, is termed a vector,* whereas the common algebraic magnitudes, which have nothing to do with direction in space, which have no directional properties, but are each determined completely by a single (real) number, are called scalars. The typical case of a vector and, in fact, the intuitional representative of any vector, is a segment of a straight line of some definite length and of some definite direction! in space, the size of the vector being represented by the length, and its direction by the direction of the straight line. Thus, the displacement of a particle from some initial to some other final position is a vector, and is represented by the segment of the straight line joining the two positions and directed from the first to the second. Other examples of vectors are the instantaneous velocity of a particle, its acceleration, its momentum, the force acting on a particle, also the instantaneous rotational velocity of, say, a rigid body round a given axis, and so on. On the other hand, mass (in classical mechanics j), temperature, vis viva, energy in general ; gravitational, electric or magnetic potential (of fixed * As to the clause : ' if it obeys a certain rule of operation,' which some modern writers add to the definition of a vector, see footnote on page 4. The term ' magnitude ' is used above in the same sense as the more usual term 'quantity'; thus 'directed magnitude' stands for 'directed quantity.' t Its sense being included. 4: This reservation is necessary, as the electromagnetic mass of a moving electron, for example, has directional properties, the ' transversal mass ' being generally different from the 'longitudinal mass.' V.M. A 2 .\ VECTORIAL MECHANICS Charges ;' or. ntagnets), mechanical or any other kind of work are all scalars. The size of a vector, or magnitude (absolute value) apart from direction, is called its tensor, or sometimes 'intensity.' Thus, the tensor of a vector is an essentially positive scalar. Every vector can be determined completely by three scalar quantities, for instance, by its projections on any three fixed axes, orthogonal or oblique, but not coplanar, these projections being commonly called the vector's ' components ' ; for example, the components of a force or the components of a velocity. We also may use polar coordinates, that is to say, we may define the tensor of a vector by the scalar r, and the direction by two other scalars, i.e. by two angles 0, e, say the geographical latitude and longitude. In this way we get again three mutually independent scalars determining a single vector. Obviously, such a decomposition of a vector into its three com- ponents or, more generally, into three mutually independent scalars, will in the majority of cases bring in some artificial elements, especially if the system of reference (axes, etc.) or the scaffolding constructed round ' the natural entities or phenomena be chosen quite at random without having anything in common with the essential characters of these entities or phenomena. Very often such a procedure gives rise to a hopeless complication of the resulting scalar formulae, a complication which does not arise from the intrinsic peculiarities of the phenomena in question, but is wholly artificial, a complication not due to Nature but to the (mathematizing) naturalist. Now, Nature is of herself won- derfully complicated ; so that supplementary complication is not wanted. This remark alone may suggest that to operate with vectors, each taken as a whole, without decomposing them into scalar components, may be more convenient and more simple, especially in those regions of research in which we are concerned mainly ivith vectors or directed magnitudes, as in Electromagnetism and in General Mechanics. But a true appreciation of the advantage of the vector method over the Cartesian (or scalar component) procedure is possible only when we see it actually at work, and the main object of each of the following chapters is to exhibit this working in Mechanics. Still more conspicuous is the service done by the vector method in Electromagnetism, especially in the hands of Oliver Heaviside, to VECTOR ALGEBRA AND ANALYSIS 3 whom also is due that simplified form of this mathematical method, which in its main features we shall now develop. Definition I. By saying that two vectors are equal to one another we mean that their tensors are equal and that they have the same direction, or, what is the same thing, that their representative straight line-segments have the same lengths and are parallel to one another and similarly (not oppositely) directed ; but the equality is independent of their position in space. According to this definition, the shifting of a given vector parallel to itself is quite immaterial, or does not change the vector. Thus, all the vectors represented on Fig. i are to be considered as equal to one another. The parallel shifting of a vector, which by convention leaves it the same, is, of course, not confined to one plane. FIG. i. Following the example of Heaviside and Gibbs vectors will be printed in heavy type (Clarendon), and their tensors will be denoted by the same letters printed in ordinary type (or simple italics). Thus A, B, C, etc., will be the tensors of the vectors A, B, C, etc., respectively. If the tensor of a vector, say a, be equal to unity (in a given scale), i.e. if a =i, then the vector a is called a unit-vector. By the definition, every tensor is an absolute or positive number. It has, of course, the same denomination as the physical, or geo- metrical, quantity represented by the vector, i.e. if A be a velocity, then A signifies so many centimetres per second, and similarly in all other cases. We pass now to the fundamental operations of vector algebra. These are : the addition of two vectors, and its inverse, the 4 VECTORIAL MECHANICS subtraction of one vector from another, and two different kinds of multiplication, the scalar and the vector multiplication of two vectors. (The division, i.e. the quotient of two vectors, belongs to the Calculus of Quaternions, due to Hamilton, and has nothing to do with Heaviside's and Gibbs' vector method to be developed here, notwithstanding that the latter has grown out of the former, historically.) Let us begin with the operation of addition and its result, the sum of two vectors. Definition II. If the end of the .vector A coincides with the beginning of another vector B, then we call sum of A and B and denote by A + B a third vector R which runs from the beginning of A to the end of B (Fig. 2).* FIG. 2. This definition of sum seems at first too narrow, as far as it appeals to the chain-arrangement of the two vectors ; but in fact it embraces the concept of the sum of any two vectors. For, if B or its representative line be originally given in a quite arbitrary manner *What we have called above a vector, without any further reservation, is by some modern authors called generally a directed qtiantity, whereas the term -vector is reserved for a subclass of directed quantities. Thus, Prof. Love in his Theoretical Mechanics (2nd edition, Cambridge Univ. Press, 1906 ; pp. 8, 9) gives the following definition of a vector : ' A vector may be defined as a directed quantity which obeys a certain rule of operation. . . . This rule may be divided into two parts and stated as follows : (l) Vectors represented by equal and parallel lines drawn from different points in like senses are equivalent. (2) The vector represented by a line AC is equivalent to the vectors represented by the lines AB, BC, the points A, , C being any points whatever.' Thus, the 'rule of operation,' implied in, and being 'an essential part of the definition' of a vector, comprises our Definition I. (of equality of two vectors) and Definition II. (of vector-sum). The rule of operation is embodied in the quoted definition of a vector itself, obviously, for the sake of exclusion of such 'directed quantities,' as the (finite) rotation about an axis, which do not obey that rule (and which, therefore, according to that definition, are not vectors). Of course, there is no serious objection to such a definition and nomenclature. But, for didactic and other reasons, it seemed to me preferable to adopt for the VECTOR ALGEBRA AND ANALYSIS 5 relatively to A, we can always shift it parallel to itself (which is allowed by Definition I.) till its beginning is brought into coincidence with the end of A. For the same reason we see that the sum of two vectors A, B starting from the same origin O is given by the diagonal OP of the parallelogram constructed on the addends A, B (Fig. 3). For, by Def. L, B' = B, since & = and B'ffB (i.e. B' parallel to and concurrent with B), in Euclidean space, of course. FIG. 3. Again, in the same parallelogram, A' = A (since A' = A and A' f t A), and therefore B + A = B + A' = R = A + B' = A + B ; hence, for any two vectors, present volume the usual definition (vector = directed quantity or directed magni- tude), leaving it an open question, to be decided in each particular case of application, whether the parallel shifting of a vector is or is not physically indifferent and whether the sum of two vectors A + B, taken according to Def. II., does or does not represent the physical resultant or the joint effect of the two ageYits or ' quantities ' represented separately by A and B. Thus, if A, B represent two rotations of a rigid body, namely through the angles A, B and about the axes a, b, respectively, we will still call them vectors and A + B their vector-sum, though this last vector does not represent the resultant rotation of the body due to the first rotation followed by the second, nor the second followed by the first (unless the axes are coincident or the angles of rotation infinitesimal). And similarly in other cases. This can hardly give rise to misunderstanding. Notice that if we were to introduce the more restricted definition (embodying the 'rule of operation') and if we wished to be rigorously consequent, we would find the same difficulties with regard to scalars or ordinary quantities. Thus, energy, say the electrical energy of a given field, would not deserve the name of a scalar ; for, if 7j , U^ be the energies of two electrical fields, /i + 7 2 does not represent the energy of the resultant field obtained by their superposition (unless the fields I, 2 are mutually perpendicular). The same remark could be applied to the mass, say, of an electron at rest relatively to the observer (when it is deprived of directional properties) ; for the sum of the masses of two electrons is not equal to the mass of the pair of electrons (unless they are very far apart). Still, calling these entities scalars or (denominated) numbers, and operating on them as such, has never led to any trouble. VECTORIAL MECHANICS Now, the sum of two vectors being again a vector, we can add to R any third vector, thus getting Again, arranging A, B, C in a chain, i.e. so that the end of A is the beginning of B, the end of B the beginning of C, we see at once (Fig. 4) that A the result being always the same, namely to get from the beginning of A to the end of C. The same thing is true for the sum of four, five and more vectors. Thus we get the following theorem : Theorem I. The addition of vectors is commutative and associative, i.e. neither the order nor the grouping of the addends has an influence on the sum of any number of vectors. Thus, the fundamental laws of ordinary algebraic summation of scalars hold good for vectors, without any reservation whatever. If, in a chain-like arrangement of any number of vectors, the end of the last coincides with the beginning of the first vector, then the sum of all these vectors is nil. Thus, in Fig. 5, B A FIG. 5. In A vector is nil or zero, R = o, if its tensor vanishes, ,/? = fact, this remark is scarcely necessary. The sum of any number of vectors having the same direction (i.e. of vectors parallel and of the same sense) is a vector of the VECTOR ALGEBRA AND ANALYSIS 7 same direction. In this particular case the tensor of the sum is equal to the sum of the tensors. Thus, the common sum is a particular case of the vector-sum. Now, let us take the case of two or more equal vectors ; then we see at once that A + A or 2 A is a vector of the same direction as A but of twice its tensor, i.e. 2A, and that analogous properties belong to 3A, 4A, and so on. Again, understanding by iA, ^A, etc., vectors which, repeated 2, 3, etc., times (as addends), give the vector A, and recurring to the generally known limit-reasoning, we obtain the meaning of *A, where n is any real positive* scalar number, whole, fractional or irrational. Thus, A will be a vector which has the same direction as A and the tensor of which is nA. In other terms, ;/A will be the vector A stretched in the ratio n : i. Thus, if a be a unit-vector having the direction of A, remembering the definition of tensor, we may write A = A&. Any vector A may be represented in this way. Now A is one scalar, and a implies two scalars, for instance the angles 6, ; thus we see again that any vector implies 1 + 2 = 3 scalars. The addition of two (or more) vectors may be illustrated most simply by regarding them as defining translations in space of, say, a material particle. The translation A carries the particle from p to/' (Fig. 6), the subsequent translation B carries it from/' to/". P The result of A followed by B, or of B followed by A, i.e. A + B or B + A, is to carry the particle from / to /". Similarly, if A, B be velocities of translation, A + B will be the resultant velocity. The same applies to angular velocities, to accelerations or forces. If A, B denote two forces acting simultaneously on a material * Negative scalar factors n will be considered later. 8 VECTORIAL MECHANICS particle, A + B will be the resultant force acting on that particle. The well-known * parallelogram law ' of forces appears only as an example of the parallelogram construction (Fig. 3) and the corre- sponding commutative property A '+ B = B + A. But it must always be kept in mind that experience only can teach us whether any given physical quantities will or will not behave like vectors, and whether their physical 'resultant' or joined effect will be given by the sum of the corresponding vectors. Now for the subtraction of vectors. This is the inverse of addi- tion, and therefore will be defined precisely as in common algebra. Definition III. We call difference of two vectors A, B and denote b y A-B such a vector 0, which added to B gives A. In other words, we say that Herefrom we see at once that if A, B are arranged to be co-initial, i.e. to have the same origin O (Fig. 7), the vector C = A-B runs FIG. 7. from the end of "B to the e?id of A. Remembering what has been said above, we see now that A + B and A-B represent the two diagonals of the parallelogram which is constructed upon A and B as sides, according to Fig. 8. VECTOR ALGEBRA AND ANALYSIS 9 Again, from the parallelogram construction (Fig. 7), we see that where B' has the same tensor as B but the opposite sense. This gives us another simple rule for constructing the difference of two vectors. Comparing the last equation with the above definition, we have A + B' = A-B, so that - B means the same thing as + B', i.e. the opposite * of B. The same thing follows also immediately from the above definition (Def. III.) written out for the particular case A = o; for then we have C = o-B=-B and B+C=o; hence B + ( - B) = o, that is to say, - B is the vector which runs from the end of B to its origin, and this is precisely what has been called the " opposite " of B. Thus, for any two vectors A, B, A-B means the same thing as A + ( - B), and the subtraction is therefore reduced to the addition of vectors. Hence, if the scalar n be a negative number, then nA means the vector A turned through 180, and then stretched in the ratio n \ : i, or, what is the same thing, first stretched in this ratio and then turned.! Thus the meaning of the product of a vector by any scalar n is fixed. The concept of such a product does not imply, in fact, any new concepts besides the vector sum or vector difference. But now we will pass to the conception of other products, namely, of products of a vector by a vector, which are not reducible to the conception of the vector sum. These products are, as already remarked, of two different kinds, the scalar- and the vector-product of two vectors. Both of them, susceptible of far-reaching appli- cations, belong, together with the vector sum (and difference), to -the most fundamental concepts of Vector Method. Let us begin with the former of these two products of a pair of vectors. *That is, B turned by 180, in any plane passing through B. \\n\ means, generally, the absolute value of n. io VECTORIAL MECHANICS Definition IV. The scalar product of a pair of vectors A and B, whose included angle is 6, is defined to be the scalar AB cos 6, and is denoted by AB.* Thus (see Fig. 9) A FIG. 9. AB = AB cos 0, A, B being the tensors of A, B, as explained above. We may say, equivalently, that AB is the component of A taken along B, multiplied by the tensor of B, or, what is the same thing, the component of B taken along A, multiplied by the tensor of A. First of all, we have, according to the definition itself, BA = AB, that is, the order of the factors in the scalar product is quite immaterial, or more concisely, the scalar product is commutative, precisely as an ordinary algebraic product. If 0<-, then AB is positive, and if 2L>0>-, then AB is 2 22 negative. If A, B are normal or perpendicular to one another ', i.e. if 6? = -, then A T, AB = o, independently of the tensors of the two vectors. Conversely, if AB = o, then we can infer only that AJ.B, and not that one of the factors vanishes. This is an important difference between the scalar product of a pair of vectors and the ordinary algebraic product. If A, B have opposite directions, then cos#= - i, and AB= -AB. If A, B have the same directions, then cos0=i, and In this particular case the scalar product of a pair of vectors degenerates into the algebraic product of their tensors. * This is Heaviside's notation ; Gibbs writes for the scalar product A . B. VECTOR ALGEBRA AND ANALYSIS n If B = A, then AB becomes what may be called the scalar square of the vector A, and may be written Hence, if a be any unit-vector, then Conversely, if a 2 =i, we can infer only that a is a unit-vector without ascertaining anything about its direction in space. If a, b be a pair of unit-vectors, then ab = cos = cos (a, b). Since AB is a scalar, its multiplication by another scalar n or by a vector C does not present any difficulty. Thus, AB or AEn means the same thing as (AB) or (AB);/, that is n times AB or AB times ; again (AB)C means AB times C, i.e. AB cos 6 . C. (The parentheses above are to be considered as separators.) But, of course, we must not confound (AB)C with A(BC) or B(CA), and therefore the brackets (or some other separators, as dots used by Heaviside) are quite indispensable. Observe that, A, B, C, D, etc., being any series of vectors, we have AB = scalar, (AB)C = vector, (AB) CD = scalar, etc., that is to say, by introducing fresh vector factors we get alternately scalars and vectors. (This will be contrasted afterwards with the other kind of multiplication, the vector multiplication of vectors by vectors.) In various branches of physics scalar products frequently have reference to energy, or work, or activity, i.e. work per unit time. Thus, if F be a mechanical force and s the displacement (say, infinitesimal) of its point of application, then F. cos (F, s) . s, i.e. the scalar product p is the work done by the force F. If, F having the same meaning as above, v is the velocity of its point of application, then Fv is the activity of the force. Again, if E be the electric force and D 12 VECTORIAL MECHANICS the dielectric displacement (which in an eolotropic medium is not generally identical in direction with E), then JED is the electric energy per unit volume. Similarly, half the scalar product of magnetic force and induction gives the magnetic energy per unit volume. In the special case of isotropy D is concurrent with E, namely D = A'E the scalar K being the dielectric constant or ' permittivity ' of the medium ; in this case the density of electrical energy becomes and similarly for the magnetic energy. Let us now pass to the scalar product of a vector A, and the sum of two other vectors B + C, which is fundamental in vector algebra. By Definition IV., we have A (B + C) = A x projection of (B + C) on A; but the projection of a sum of vectors on any direction or axis is seen immediately to be the same as the sum of the projections of the single vectors on that axis (see Fig. 10). Hence FIG. io. C), that is to say A(B + C) = AB + AC. Similarly, A(B + C + D + ...) = AB + AC + AD + ... . Thus we get the following theorem : Theorem II. The scalar multiplication of pairs of vectors and of vector-sums is commutative and distributive. Thus, the scalar product of a pair of vector-sums is developed precisely as in ordinary algebra: for example, VECTOR ALGEBRA AND ANALYSIS 13 Similarly, we have A(B - C) = AB - AC, since B - C = B + ( - C). Also (A-f-B)(A-B) = A 2 -B 2 = ^ 2 -^ 2 , which the reader may put easily into the form of a geometrical theorem, remembering that A + B and A-B are the diagonals of the parallelogram constructed on A, B. Another example is and, particularly, if AB, which is simply the theorem of Pythagoras. But as this book will be full of illustrations of the use of the scalar product and the remaining fundamental concepts of Vector Calculus, we have no need to look about for illustrations in this preparatory chapter. We come now to the second kind of product of a pair of vectors, namely the vector product. Definition V. The vector product of two vectors A, B is a third vector C, whose tensor is equal to the area of the parallelogram A, B, and whose direction is perpendicular to the plane of A, B,* the positive direction of C being such that a right-handed rotation about C through an angle less than 180 carries the vector A into B (Fig. n). The vector product thus defined is denoted by C = VAB. FIG. ii. Thus, if B be the angle included by A, B, we have C= A3 sm0, and as, by the definition, C_LA, B, we have AC = o, BC = o, or AVAB = o, BVAB = o. * Any two vectors can always be brought into one plane, by shifting one or the other of them parallel to itself, according to Def. I. 14 VECTORIAL MECHANICS Also, we see by Def. V. that on interchanging the factors, the direction of C is reversed, its tensor remaining the same, that is VBA= -VAB. Thus, the vector product is not commutative, and the order of its factors has always to be treated with care. Again, if A, B are parallel to one another, i.e. if = o or 6 = TT, then sin B = o, and consequently VAB-o. Conversely, from VAB = o it follows only that A, B, otherwise unknown, are parallel to one another. In particular, we have for any vector A, VAA = o, which may be put in words by saying that the vectorial a///0-product of a vector vanishes. 7T Again, if A_LB, or # = -, sin0= i, we get, as the tensor of VAB, C=AB; thus, caeteris paribus, the maximum tensor is reached for 6 = 90. If m, n be any pair of scalars, then, by Definition V., just as for the scalar product If, in particular, A = ^4a, B = ^b, where a, b are the corresponding unit-vectors, then If i j, k be a right-handed system of mutually perpendicular unit-vectors, then yjj = since sin(i, j) = sin9o=r, and similarly Vjk = i, Vki = j, so that the three equations follow from one another by cyclic permutation, the order being always kept in mind as the proper order. Inverting it, we have to change the sign, thus Vji= -k, etc. VECTOR ALGEBRA AND ANALYSIS 15 Remembering the definition of scalar product, we have for the same system of unit vectors, which throughout the whole of this book will be denoted by i, j, k, ij = o, jk = o, ki = o, $x~ u ~ and i2=j2 = k 2 =I> while, of course, Vii = Vjj = Vkk = o. The components of any vector A taken along i, j, k, i.e. the scalar products Ai> ^ ^ will be denoted by A lt A^, A z , respectively ; thus A = A$. + A. 2 j + A 3 ls.. Such a decomposition will be made use of incidentally, but as a rule we shall avoid any artificial decomposition of vectors, except in transitional work or in order to show the reader the equivalent Cartesian form of some fundamental vector formula. (See also the 'Appendix' containing a great number of such equivalents, com- posed especially for those readers who are not used to the Vector Method.) But, meanwhile, let us return to the vector product of any pair of vectors, say B, C. Since VBC is a vector, it may in its turn be multiplied by a third vector, say A, either scalarly or vectorially, giving rise respectively to i) A (VBC) or simply AVBC, and 2) VA(VBC) or simply VAVBC. The second of these products will be treated later. For the present let us consider the first of these triple products, i.e. AVBC, as it will enable us to prove the most important property of vectorial multiplication, namely its distributivity (Theorem IV. below), without any artificial decomposition of the vectors involved. Now, putting for the moment we have AVBC = AR. But the tensor of R, R = Csm (B, C), is the area of the parallelogram B, C, which may be considered as i6 VECTORIAL MECHANICS the basis of the parallelepiped A, B, C (Fig. 12), and the direction of R is perpendicular to this basis; hence, if the arrangement of A, B, C be right-handed (as in Fig. 12), then AYBC = A/fr = j?Ar = base x height = volume of the parallelepiped A, B, C. Fir,. 12. Thus the triple (scalarly-vectorial) product AVBC has a very simple geometrical, or say stereometrical, meaning; it represents the volume of the parallelepiped constructed upon A, B, C as edges, if the order A, B, C be a right-handed one. But, if it be left-handed, AVCB represents the same thing. Again, considering C, A as the basis, we see that BVCA represents the same volume, and similarly CVAB the same volume of the parallelepiped A, B, C. Hence the following theorem : Theorem III. The triple, scalarly-vectorial product of any triad of vectors retains its value when its factors are cyclically permuted, AVBC = BVCA = CVAB. This theorem is of fundamental importance, in mechanical, electromagnetic, and other applications. As the reader knows already that VCB = - VBC, it is unnecessary to emphasize that AVCB = -AVBC, i.e. that any one of the above three products changes its sign if the cyclic order A, B, C be reversed. VECTOR ALGEBRA AND ANALYSIS 17 Coming now to the capital point, let us consider the vectorial product of a vector by a vector sum or, more generally, the vector product of a pair of such sums. We then have the following fundamental theorem : Theorem IV. Vector multiplication is distributive, i.e. and also V(A + B)(C + D) - VAC + VAD + VBC + VBD, as for scalar multiplication, with the only difference that vector products are not commutative. To prove this important property, let us put VA(B + C) - VAB - VAC = Q. Then we have only to show that the vector Q vanishes. Multiply th sides of the last equation scalarly by A, or B, or C \ then 1 ) AQ = AVA (B + C) - AVAB - AVAC = o, ice VA(B + C), VAB, VAC are perpendicular to A; again 2) BQ = BVA(B + C)-BVAC -BVAC + (B + C)VBA, by Theor. III., = - BVAC + BVBA + C VBA = -BVAC + BVAC = o, ind similarly 3 ) CQ=...=BVCA-CVAB = o. Thus, by i), 2), 3), AQ = o, BQ = o, CQ = o. Hence Q either vanishes or is normal simultaneously to all the three vectors A, B, C ; but if these are not coplanar, the last alternative is obviously impossible, so that Q must vanish, i.e. VA (B + C) = VAB + VAC. Thus, for A, B, C not coplanar the theorem is proved. Again, if A, B, C are coplanar, we can always add, say to C, a fourth vector D including any angle with the plane of A, B, C; then A, B, C + D will not be coplanar, and therefore . VA [B + (C + D)] = VAB + VA (C + D) ; but making D approach zero, we get again VA(B + C) = VAB + VAC, so that the first part of the Theorem IV. is proved for any triad of vectors. V.M. B i8 VECTORIAL MECHANICS To prove the second part of this theorem, i.e. that V (A + B) (0 + D) = VAC + YAD + VBC + VBD, put C + D = R, and observe that V(A + B)R = -VR(A + B) = -VRA-VRB then the second part 'of Theor. IV. is reduced to its first part ; thus the whole theorem is proved. The importance and practical applicability of vector as well as scalar products of vectors are based mainly on their distributivity, in which property they both resemble the common algebraic product. The scalar product is, besides, commutative (i.e. BA = AB), the vector product has not the benefit of this property in as far as VBA= -VAB; but the trouble is, in fact, not very considerable ; when inverting the order of the factors we have to do but a very little, namely to change the sign of the product, and this is very easily remembered. In Physics, and especially in Mechanics and Electromagnetism, including Optics, the vector products are as useful as, and perhaps more than, the scalar products of vectors. The following mechanical chapters will be full of their applications. Here therefore a pair of electromagnetic examples will be sufficient. Thus, if E, M be the electric and magnetic force in any point of a field, the product <:VEM (c velocity of light in vacuo, an ordinary scalar) gives the flux of electromagnetic energy, per unit time and per unit area, at that point. This product is generally known as the Poynting vector. Thus, the electromagnetic energy flows normally to the plane containing both the forces E, M. Again, 'the electromagnetic force,' according to Maxwell's ter- minology, i.e. the pondero-motive force, per unit volume, acting upon a conductor supporting an electric current, placed in a magnetic field, equals the vector product of the current C (per unit area) and the magnetic induction B : F = VCB. In fact, the force F is normal to both C and B, its intensity Fv$> given by the area of the parallelogram C, B, and its direction is determined by saying that F, C, B is a right-handed system. VECTOR ALGEBRA AND ANALYSIS 19 Now, all these rules are certainly remembered more easily if condensed 'into the short formula F = VCB. But let us go on with our vector algebra. Once in possession of Theorem IV., we can immediately develop the product VAB into its Cartesian form, i.e. represent it by the components of its factors. In fact, if A = Aj. + A 2 j + AJs. and B = ^ 1 i + ^ 2 j+^ 3 k, then VAB = A^Vii +...+ A^Vij + . . . ; but, as has been shown above, Vjk = i, Vki = j, and thus we get or, in determinant form, VAB = i J k A l A, AS \ (I) (l) by taking a pair of unit vectors a, b, whose components or direction-cosines are a^, ^etc., ^, etc., respectively, and whose included angle is 6, we have the known trigonometrical formula, Similarly, using the distributivity of the scalar product, we have AB = iM^! + . . . + ij (A^B Z + A%B^ + . . . ; hence AB = A l ^ 1 + A 2 ^ 2 + A^ 3 , (2) and in particular, for a pair of unit-vectors a, b, ab = cos 6 = a l b l 4- aji^ + z are scalars. The process of vector multiplication may be repeated any number of times, giving always a vector. In this regard vectorial multiplica- tion differs characteristically from scalar multiplication (see the corresponding remark above). Thus VD( VAVBC) will be a vector, namely normal to D and to VAVBC, and so on. But, in practice, we shall have to do with hardly more than triple vectors ; thus, it will be sufficient here to consider only VAVBC. To develop this triple product, we have only, in the determinant (i), to replace B^ B^, B z by ^2^3 ~ BtPli B*C\ ~ B\ *3 B\ Q -B\\ hence the first component (i.e. the i-component) of VAVBC will be = (AC)^-(AB)C 15 by (2); similarly, the second and the third components of VAVBC are respectively; hence, compounding the three components, i.e. adding them, after the first has been multiplied by i, the second by j, the third by k, VAVBC = (AC) B - (AB) C, (4) which is, in fact, the form (4'), foreseen at the beginning, with _y = AC and z= -AB.* The formula (4) is very important in practical applications. *The formula (4) can also be proved without splitting the factors A, B, C into their rectangular components. See ' Problems and Exercises,' where some necessary hints are given. VECTOR ALGEBRA AND ANALYSIS 21 Observe that VAVBC does not retain its value on cyclic per- mutation of its factors (as AVBC did) ; we have, indeed, according to W' VBVCA = (BA)C-(BC)A, (40) VC VAB = (CB) A - (CA) B ; (4^) now these are certainly three totally different vectors, since (4) lies in the plane B, C, whereas (40), (46) lie in the planes C, A and A, B respectively. But it is interesting to remark that the sum of these three vectors vanishes ; in fact the sum of the right sides of (4), (4^), (4^) vanishes identically, since CA = AC, etc ; hence VAVBC + VBVCA + VCVAB = o. The particular case of (4), in which C = A, is most often met with ; we have then VAVBA = ^ 2 B-(AB)A, (5) and especially, if A be a unit-vector, say = n, then VnVBn = B-(Bn)n. (50) Now Bn is the (scalar) component of B taken along n, and therefore (Bn)n is the part* of B along n, as regards both intensity and direction ; subtracting it from the whole vector B we get its part normal to n. Thus we see that VnVBn gives the part of B normal to n, as regards both size and direction. If, for instance, n be the normal of a surface, at a given point, then VnVBn gives immediately the part of B tangent to the surface. In what has now been given we have all that is required for easily following the vector-algebra of the subsequent chapters on Mechanics. There would remain the so-called ' linear vector-operator,' which might be treated in this preparatory survey of vector-algebra. But this operator is considered in the text itself, as the 'symmetrical' operator in the chapter on Rigid Dynamics, and as the ' non- symmetrical' or general linear vector-operator in the chapter devoted to Non-rigid Bodies. All that is still required are a few notions of elementary Vector- Analysis, to which we shall now pass. * The ' component ' is a scalar, the ' part ' -of a vector is a vector. 22 VECTORIAL MECHANICS Differential and Integral Properties of Vectors. Let us consider a vector A as a function of some independent variable scalar; to fix the ideas, let this independent variable be the time /, reckoned from some 'initial' instant. Then, in general, both the direction and the tensor of A will vary with /. To illustrate, let A = r be the vector drawn from a fixed point O to a material particle P moving about in space along any curved path (Fig. 1 3) ; then T will vary with the time or r will be a function of /. 1, FIG. 13. Again, in an electromagnetic field the electric and magnetic forces E, M are generally variable in time, as regards both their intensities and their directions, i.e. the vectors E, M are functions of /. If these functions be periodic, we have the special case of electro- magnetic oscillations, the end-points of the corresponding vectors describing closed paths. A is said to be a continuous vector function of the time /, if both its direction and its tensor vary with / in a continuous manner. Thus, any vector being the incarnation of three scalars, the con- tinuity of a vector implies the continuity of three scalar functions as considered in common analysis. But here again we shall avoid any artificial decomposition, and treat the vector function A as a whole. The differentiation of A with respect to / is quite as simple as the differentiation of an ordinary or scalar function. Hence, on this point, only a few remarks are needed. To obtain an instructive picture of such differentiation we may proceed as follows. VECTOR ALGEBRA AND ANALYSIS 23 First of all, any parallel shifting of the whole vector being, by Duf. I., an indifferent matter, let us suppose that the origin of the changing vector A is fixed, say in O. Then the statement that A is a function of the time /means (as above for r) exactly the same thing as saying that the end-point of the vector A moves about in space ; and the continuity of the vector function A implies the continuity of this point's motion. Let P (Fig. 14) be the ^ .^H- P' AA FIG. 14. position of this point at the instant /, and P' its position at any later instant / + A/, where A/ is a finite increment of /. Then OP= A is the vector- value at the instant t and* at the instant / + AA The vector running from P to P' or will be the increment of the vector under consideration during the time A/. The ratio ^A ~A7 will be the average rate of change of the vector A or the mean velocity of its end-point, in the time-interval A/. Now, if this ratio tends to a definite limiting vector, when A/ is infinitely reduced, this limiting vector is called the differential coefficient of A with respect to / or the flux of A, and is denoted by or by A ; thus dt dt ds or w = vl + v 2 - ds 26 VECTORIAL MECHANICS Now, 1 being a unit-vector, or I 2 = i = const., we have .d\ l ds= Q > hence, the vector d\fds is perpendicular to 1, i.e. normal to the path ; besides, it is contained in the plane of the two tangents 1 and 1 + d\, that is in the osculating plane of the path. Thus the vector d\jds points from the moving particle towards the centre of curvature ; moreover, its tensor is easily seen to be equal to the curvature or to the inverse radius (ft) of curvature at the given point of the path. Hence, denoting by n a unit-vector pointing from the particle towards the centre of curvature, the last equation may be written ^2 w = i>\ + -y, n, /i showing that the resultant acceleration (for any path, plane or tortuous) consists of the tangential component and of the normal component zfyfc which is towards the centre of curvature, the well-known result of kinematics. Here is another kinematical example, affording a beautiful illustration of the use of vector products, of their distributive property and the consequent formula (7) of their differentiation. Let a material particle P (Fig. 15) move along a plane path so that FIG. 15. the areas swept out by the radius vector r, drawn towards it from a fixed centre O in the plane of the path, are proportional to the time, i.e. so that the area swept out per unit time is constant, which, for P= planet and (9 = sun, is Kepler's second law. Let P be the position of the particle at the instant /, and P' at the instant t + dt. VECTOR ALGEBRA AND ANALYSIS 27 Then, by the definition of vector product, the area POP' swept out during dt is given by the tensor of the vector <#.F = Vr(r + v <#) = <#. Vrv, since Vrr = o, or per unit time, F = JVrv. Now, by the supposition, the tensor of F is constant, and as F is normal to r, v and as the whole path is supposed to be in one plane, the direction of F is also constant, i.e. the whole vector F is invariable in time ; it is an invariant of the system. Hence dTjdt=Q, or by (7), Vrv-f Vrv = o. But r = v*and Vw = o; thus, v = w being the acceleration, Vrw = o, that is to say : the acceleration is always towards, or from, the centre, or the motion is central. Vice versa, if the motion is supposed to be central and plane, we have Vrw = Vrv = o, and, as Vrv vanishes identically, also Vrv *t- Vrv = o or Vrv = const., i.e. Kepler's second law. Thus, the reciprocal equivalency of central motion and of Kepler's second law is seen by the aid of the vector language almost immediately. If a vector, A, be a function of two or more independent scalar variables r, s, ..., instead of d the symbol 3 of partial differentiation will be used; thus 3A , 3A 3 a A. = -^ dr + -x as + . . . . ?>r 3s Of particular importance, especially for physical applications, is the case of a vector, say R, depending on three scalar variables denning the position of a point in three-dimensional space, that is to say, the case in which the vector R is to be considered with regard to its distribution in space. It is particularly in this branch of vector analysis that some special concepts have been created, remarkably adapted to the very nature of vectors, concepts that are of great theoretical interest and capable of numerous applications, and which therefore require, and deserve, a special treatment. 28 VECTORIAL MECHANICS None the less I shall not pretend to develop here fully this branch of vector analysis, but only such parts of it as may be needed for the subsequent work, and which,* in fact, will prove to be sufficient for almost any physico-mathematical purpose. T every point of space, or of a certain portion of space, let there correspond a given vector R, of definite direction and tensor, generally varying from point to point. Then the space, or its portion in question, considered as the seat of these different R's, is called a vector field or the field of R. Thus, if R be the electric force or the magnetic force, we have an electric or magnetic field or a field of electric or magnetic force, respectively. Again, a portion of space occupied by a -moving fluid will also be a vector field, namely the field of a vector representing at each point the absolute value of the velocity and the direction of the motion of the fluid. If R, as regards both its tensor and direction, be constant every- where, we have a homogeneous field, otherwise a heterogeneous field. Thus, the electrostatic field between the plates of a condenser (plane and parallel) is approximately homogeneous, provided that we do not approach the edges of the plates. But the vector fields met with in nature are generally heterogeneous. A vector field is said to be continuous, if the corresponding vector R, both in direction and absolute value, varies in a continuous manner from point to point. We shall suppose, as a rule, that the field under consideration is not only continuous but also that the vector R admits everywhere definite differential coefficients, at least of the first and second orders, with respect to space, i.e. that, if / be the (scalar) length 3R 3 2 R measured in any direction, both and -^ exist as certain definite vectors. Nearly all that is needed for the investigation of the most characteristic differential properties of a vector field is concentrated in a certain differential operator, called the Hamiltonian (or sometimes 'Nabla' or 'Atled') and denoted by V. The best and most natural way of arriving at this differential operator, and of grasping its true meaning, is to consider, at the * Especially if combined with the supplementary notions of vector method developed occasionally, in the subsequent chapters themselves. VECTOR ALGEBRA AND ANALYSIS 29 starting point, certain integral, not differential, concepts, namely the so-called line-integral and the surface-integral of the vector R. To the definition of these fundamental concepts of vector analysis we therefore now pass. Let s (Fig. 1 6) be any continuous line joining any two points FIG. 16. i, 2 of the vector field R; call i -> 2 the positive direction of the line s. Let the infinitesimal vector ds represent an element of the line s, both in length and direction, the direction of ds being coincident with the positive direction of s at the place where the element lies. Take the projection of R upon ds and multiply it by the (scalar) length ds of ds, that is to say, take the scalar product lids of any element ds of the path s and of the corresponding R, and sum up or integrate from i to 2. Then the integral supposing that it exists as a definite limit of a sum, is called the line-integral of R taken along s. The line-integral, thus defined, is a scalar, of course. If the line of integration ^ be dosed, or what is called a circuit, then we shall denote the line integral by -fc -Rds, e positive sense of the circuit s being fixed in a definite manner. Again, let o- be a continuous surface drawn in the vector field R ; let us call one of its sides the positive (or + ) side and the other the negative (or - ) side. Let the unit vector n represent 30 VECTORIAL MECHANICS the normal of any (scalar) element dv of the surface a-, crossing it from the - to the + side (see Fig. 17). The scalar product Rn is the normal component of R, at any point of the surface. Multiply this product by the infinitesimal area dv and sum up or integrate over the whole surface tr; then if it exists, is called the surface-integral of R extended over v. This integral is, of course, again a scalar. In considering its value, we must always keep in mind how the positive sense of the normal n has been fixed. Especially, if the surface o- be a dosed surface, it is usual to consider its outer side as positive, i.e. to draw n outwards. In mathematical physics both the line-integral and the surface- integral of a vector are of frequent occurrence ; this is the reason that these concepts have been constructed at all and are studied in vector analysis. Thus, referring always to Fig. 16, if F is a mechanical force and s the path of the point of its application, then the line-integral (Ws is the mechanical work done by this force. Again, the so-called ' electromotive force ' along any line s joining a pair of arbitrary points of an electric field is the line-integral of the electric force E, ULF.-JI An analogous magnitude M.M.F., also of frequent use, is obtained by taking the magnetic force M instead of the electric E. Of course the E.M.F. and M.M.F. depend, generally, not only on the position of the terminals i, 2, but also on the choice of the path s leading from the first to the second. Especially useful for the description of electromagnetic laws is VECTOR ALGEBRA AND ANALYSIS 31 the consideration of E.M.F. and M.M.F. for closed paths or circuits, /.(-. according to our notation : f Eds and I M^/s. Jw Again, let v be the velocity of fluid motion, at any point of the space occupied by a fluid, at the instant / of time ; then it is often useful to consider the line-integral Jd) called the circulation round the curve s. It is closely related to vortex motion, and also to the most characteristic properties of irrotational motion, especially if the fluid occupies a cyclic space, as will be seen in the chapter devoted to Hydrodynamics (Chap. VI.). Fluid motion affords also the best illustration for the surface- integral. Thus, let p be the density of the fluid, i.e. its mass per unit volume, which generally may vary with time and in space, and let v have the above meaning. Then pv will be the current, i.e. the amount (mass) of fluid crossing unit area of a surface perpendicular to v, per unit time. Hence pv cos (v, n) do- or /ovn da- will be the normal component of the current crossing any surface- element do; and the surface-integral So- = I /ovn da- will be the total amount of fluid crossing the surface a- from its negative to its positive side, or the total current through a-. In particular, if or be a closed surface, then S& will be the amount of fluid leaving* the space limited by o-, supposing of course that fluid (mass) is neither created nor annihilated on its path. Keeping in mind the above definitions of the line- and surface- integrals, the reader will see almost immediately the truth of the following propositions. First of all, on inverting the sense of integration, we change only the sign of the line-integral, i.e. * If S(T>O, and entering into, if S where - s denotes the opposite of s. Secondly, if acb is any continuous line joining a pair of points a, b of a circuit s = adbea (Fig. 18), then ^adbca = ^adb + ^hca i -*acbea = * bea ~^~ -*acb 5 and as, by the above, f bca = - f acb , it follows: fadb + -^bea = *(*} = ^adbca + ^acbea 5 or, denoting the circuit adbca by s l and the circuit acbea by ^ 2 , ^(s) = ^i) + ^)- Similarly, decomposing the given circuit s, by the introduction of an appropriate network of lines, into three or more circuits s lt s. 2 , ^ 3 , etc. (Fig. 19 \ we get / ( . s) = /(,,) 4- /(.,) + /(.,,) + . . . . ( i o) FIG. ig. Now, consider any continuous surface o- bounded entirely by the given circuit s, and the whole network of lines drawn on this surface, as above. Take one of the sides of or as the + side, and the other as the - side, and draw everywhere its normal n crossing o- VECTOR ALGEBRA AND ANALYSIS 33 from the - side to the + side. Let, by convention, i\\Q positive sense of any circuit drawn on -^"~ \L' Arr tends to a definite limiting value \t, when Ao- is infinitely reduced, we may conclude from the above that the line-integral / (s) must be reducible to an integral taken over the surface a- bounded by the circuit s : f f

being a scalar, we may consider it as the projection of a certain vector C upon the normal n, i.e. as the normal component ofC: V = Cn. Then (u) will assume the form (i 2 ) f R d* = f Jw J i.e. the line-integral of R taken round s will be equal to the surface- integral of C taken over o-, the surface o- being bounded by the circuit s. The vector C which, as we shall see a little later more explicitly, 34 VECTORIAL MECHANICS depejids only on the local properties of the vector-field B, i.e. on the distribution of B at the given place in the field, is called the rotation* or the curl of B. We shall, following the example of most English writers, choose the latter name, thus writing C = curl B. Then equation (12) may be stated as follows : Theorem V. The line-integral of a vector B taken round the circuit s is equal to the surface-integral of its curl taken over any surface bounded by s : / f* = n curl B d 3^ 1 dy dz Similarly, by cyclic permutation, Hence the expansion of C or curl R : curlR = i(-^--^)4-j(^- 1 - ^^ J )+k(^-^-^- J ), (13) \oy dz J \dz dx J \dx dy J or in determinantal form, if V 1? V 2 , V 3 be written, after Heaviside, instead of --, , , respectively, dx dy dz curl R = i J k Vi V 2 V 3 * Since i is normal to Fig. aoth's plane, drawn away from the reader, i.e. vertically downwards, if he reads his book on a horizontal table. 36 VECTORIAL MECHANICS Now, introducing the symbol and comparing the structure of ( 1 3^ ) with that of the vector product (i#), we see that (130) may be written curl R = WE. (13$) As V l3 V 2 , V 3 , the 'components' of V, are not ordinary scalar magnitudes but scalar operators, namely differentiators, V is not a vector but an operator; none the less it has the character of a vector and may be applied, as above, to a real vector R, the result of such an operation having a perfectly definite meaning. Allegorically, then, curlR may be called (and in fact has been called by Heaviside) the vector product of V and R, the true meaning and the real sense of this allegory being that curlR is the result of the operation V, when applied vectorially to R. It is this operator V = i^ + j^- + k^ which is called the fix 3y ^>z Hamiltonian. It may be applied to a vector either vectorially, as above, or scalarly, as we shall see a little later. By the very definition of curl we are certain that curl R cannot depend on the particular choice of the system of coordinates such as x,y, z, but only on the distribution of the vector R in the given field. But it will, nevertheless, be an instructive exercise for the reader to introduce instead of #, y, z another rectangular system #',/, z', with the same origin (which is immaterial), but with different directions of axes, say i', j', k', and the corresponding operator and to prove then that curl' R or W'R is identically the same vector as curl R, i.e. WR. Using the formulae of transformation x = ax + by + cz, etc., with constant a, b, c, etc., and the well-known conditions of ortho- gonality, the reader will realise this in a moment. Observe that we arrived at the curl by the means of the line- integral /. Now let us repeat, mutatis mutandis, the whole reasoning with the surface-integral S, instead of /. VECTOR ALGEBRA AND ANALYSIS 37 As we started with / for a closed curve, so let us now take the surface-integral 5( a ) for a closed surface , when AT is infinitely reduced, weget - f " -=U. lRndr = \ The scalar p, thus defined, i.e. the limiting value of the above ratio or the limit of the surface-integral, per unit volume, is called the divergence of R, and is denoted shortly by p = div R. Substituting this symbol, we have, analogously to Theor. V., the Theorem VI. The surface-integral of a vector R taken over any closed surface a- is equal to the space-integral of the divergence of R taken throughout the volume T enclosed entirely by a- : Rn d, z be the coordinates of the centre of the parallelepiped, the pair of sides dy, dz normal to i contributes to the surface-integral the sum similarly, the contributions from the two other pairs of sides of the parallelepiped, normal to j and k, are, respectively, ^dydzdx, ^dzdxdy. dy ' dz Hence the surface-integral will be fdR, 97? 9 a&A , , , ( ^~ + -^ + ~~r dxd y dz - \ dx dy dz J But dx dy dz = dr is the volume of the parallelepiped ; hence the surface-integral per unit volume or, by the above definition, the divergence of R will be divB-^ + ^ + ^-V^ + V^ + V,*,. ( I5 ) Now, comparing this with the Cartesian expansion (2) of the scalar product of a pair of vectors, we may write shortly divR = VR. (150) Both the curl and the divergence are independent of the choice of the system of coordinates and, generally, of any system of reference, and depend only on the properties of the given vector- field R, and, as the reader will see from the subsequent chapters, both are specially characteristic of such a field. Now, it is remarkable that both of these important magnitudes, the first a vector and the second a scalar, are obtained by the application of one and the same operator, namely the Hamiltonian. In fact, we have (13^) and (150), which formulae tell us that the Hamiltonian V if applied to a vector vectorially gives its curl, and if applied scalarly gives its divergence. The most immediate kinematical illustration of curl is given at the end of Chap. IV., where it is shown explicitly that if v be the resultant velocity of any point of a rigid body, moving in the most general way, then curl v is twice the angular velocity of the body. A larger application of curl, namely to deformable bodies, in which the rotation generally varies from point to point, is developed fully in Chap. V. In the same chapter frequent use will be made 40 VECTORIAL MECHANICS also of div. Nevertheless we may illustrate here, also, the meaning of this operator in an immediate way, namely by our previous example of fluid motion, in which it has been shown that = I taken for any closed surface v).^r is the same thing as the surface-integral taken over the surface enclosing the elementary volume dr. Hence div (/>v) dr is the amount of fluid leaving the volume-element, and therefore div(pv) the amount leaving it, per unit volume (and per unit time, of course), whence also the name of 'divergence.' The meaning of the Theorem VI. and of its short formula (B) becomes now quite obvious ; in fact, if we put R = />v, it says simply that the amount crossing the surface cr is equal to the algebraic sum of all amounts leaving the elements of the volume r enclosed entirely by this surface. The fundamental Theorems A r . and VI. bring the curl and divergence of a vector into close connection with its line-integral and surface-integral, respectively. Combining both theorems, we may now prove an important general property of curlE, without recurring to any system of reference. Let o- 1? cr 2 (Fig. 23) be a pair of surfaces bounded by the same n=n. FIG. 23. circuit s, and themselves enclosing completely a portion of space r. VECTOR ALGEBRA AND ANALYSIS 41 Then, by Theor. V., (R ds = I HJ curl R dv l = I n 2 curl R dcr 2 , (.-) J 0-1 J *, etc. It is worth noticing that (16) may also be written VVVR = o. (i6a) Thus, also in this respect the Hamiltonian V behaves like a simple vector. Remember that, by the fundamental property of the vector product, AVAR = o identically. If divR = o throughout the whole field, then the vector-field R is said to be solenoidal or sourceless.* And if curlR = o everywhere, then the field R is called irrotational.f Thus, the identity (16) may be put in words by saying that the field derived from any field R by 'curling' it, is always a solenoidal field, or briefly that the curl of any vector is solenoidal. Instructive examples of solenoidal and of irrotational fields will be found in the subsequent chapters on Mechanics ; and as to * The reason of the last name is obvious by one of the above illustrations, and will be seen again in Chap. IV. t That is ;; on-rotational ; remember that curl is closely related to rotation. 42 VECTORIAL MECHANICS Electromagnetism, it will suffice here to observe briefly that the total electric current is always and everywhere solenoidal,* and that an electrostatic field, for example, is irrotational. We have already admired the efficiency of the Hamiltonian V in two cases, namely in getting the curl and the div of a vector. Now, to see another performance of the same operator, i.e. a third property of V, let us consider & purely irrotational field R, that is such that the condition curl R = o is satisfied throughout the whole space. Then, by Theor. V., if s be any closed path, or circuit, i R d$ = o. Now, taking on s any pair of points i, 2 (Fig. 24), we have -^(.x) = ^i2 + *l = MaJ "~ MM j /,,=[ J(s FIG. 24. Thus, the line-integral of R has for the two paths a, <, and hence also for all possible continuous paths leading from i to 2, one and the same value. Or, in other terms, the integral is a single-valued function of the position of the terminal points i and 2 in the field. Hence, if any fixed point O be chosen once and for ever as the starting point, and if p be any other point of the irrotational field R, PR ,& = <, (17) Jo where is a single-valued function of the position of p alone. This function is called the (scalar) potential! of the vector R. / *And this is one of the most characteristic features of Maxwell's theory. t This is the common use in general mechanics and hydromechanics, whereas f in electrostatics and magnetostatics, not the above but - 0, i.e. \ Rds, is called the potential. This, of course, is a matter of convention. VECTOR ALGEBRA AND ANALYSIS 43 If the region of irrotationality of the field R does not occupy ithe whole space but only a certain portion r of space, and if this portion of space be cyclic or multiply-connected, the vector R has still a potential in this region T, but then < is no longer a single- valued but generally a many-valued function of position. For ! further particulars regarding this subject see Chap. VI. But, at any rate, whether < be a single- or a many-valued function of position, we have, by (17), the differential property which holds good at such places where the field is irrotational. Here, ds means the tensor of ^s, so that, if R s be the (scalar) component of R along does not require any further explanation. Generally, V< is called the slope or gradient of the scalar function <. Thus, in hydrokinematics the velocity of fluid motion is the slope of the 'velocity-potential,' if, of course, such a potential exists, i.e. if there are no vortices at the place considered (see Chap. V.). Any surface < = const, is called an equipotential surface. Now, taking d$ tangential to such a surface, we have and consequently, by the above formula, R ds = o ; 44 VECTORIAL MECHANICS and since this is true for any tangential direction, we see that the vector. R is normal to the equip otential surfaces, At the same time it is seen that the (positive) direction of R is the direction of the most rapid increase of <. Hence, denoting this direction of most rapid increase by the unit-vector n and using the common symbol 3/3 of partial derivation taken in the direction n, we can write also Thus, the Hamiltonian, in its application to a scalar function, might have been denned from the beginning as the slope of this function or, symbolically, as the operator V = n|- (ao) Then, developing this into V = i9/9^+j 3/9j + kd/9z and applying it either scalarly or vectorially to a vector R, we could arrive in this manner to the divergence and curl of R. But this did not seem to me as natural a way as that chosen above ; it would be essentially more artificial, and then it would not enable us to see the truth of the Theorems V. and VI. in such an immediate manner as the method adopted. But without entering any further into similar comparisons of didactic character, let us return to the Hamiltonian itself. Having already recognised it in its whole generality, i.e. in its threefold character as slope, divergence and curl, V = slope of , VR = div R, VVR = curl R, what we need still are but a few remarks regarding the application of this marvellous operator. First of all, the operator V is distributive, since, apart from its vectorial peculiarities, it is a simple differentiator. This property holds good not only for a sum of scalars, as , ^, but also for a sum of vectors, as R, S, i.e. not only V(< + ^) = V<#n-V& (21) but also V(R + S) = div(R + S) = VR + VS = divR + divS, (22) and similarly VV (R + S) = curl (R + S) = curl R + curl S. (23 ) Then, the application of V to a product of a pair of scalar functions VECTOR ALGEBRA AND ANALYSIS 45 does not present any difficulty ; since : etc., we have simply (24) i.e. slope (<^) = < slope ^ + ^ slope <. Similarly, the application of V to a scalar product of a pair of ^vectors does not bring in anything new; V(RS) is simply the slope iof the scalar magnitude RS or of jRScos 6, if the angle included by 'R, S is 6; thus we may write V (RS) =, XSV (cos 0) 4- S cos BVR + R cos 0VS. ;But, as far as I know, it is never met with in practical applications. Of greater importance is the application of V to the vector- product of a pair of vectors, VRS. Since this is a vector we r may operate on it by V either scalarly or vectorially, thus giving rise to div VRS = VVRS and curl VRS = VVVRS. Let us consider the first of these expressions. If V were a real vector, say A, then we should have, by Theor. III., AVRS = RVSA = SVAR = - RVAS ; consequently, if the vector S were constant, in space, we should have, writing again V instead of A, VVRS = SVVR = S curl R; in the same way, if R were constant and S variable, we should VVRS = - RVVS = - R curl S. Hence, both R and S being variable, we have VVRS = div VRS = S curl R - R curl S. (25) This formula is of particular importance in Electromagnetism ; it iserves, for instance, to show almost immediately that, R = E and S = M being the electric and the magnetic force, respectively, their vector product VEM, multiplied by the (scalar) velocity of light in vacuo, gives the flux of electromagnetic energy, per unit time and unit area, i.e. the so-called Poynting-vector, which already has been mentioned. In the same way, the application of the formula (4) of Vector Algebra to the second of the above expressions gives V V VRS = R ( VS) - S ( VR) + (S V) R - (R V) S, *.*. curl VRS = R. div S-S. div R + (SV) R - (RV) S, (26) 46 VECTORIAL MECHANICS where (SV) is an operator composed of S and V in the same way as the scalar product of a pair of real vectors, that is in Cartesians, for instance, (8V).^|^ S | + 5,|. (37) The same meaning is to be attributed to (RV) in (26). Having thus obtained the required formulae (25), (26) by a method which may seem dubious to the reader, it is desirable to verify the validity of these formulae, and consequently the legitimacy of the short and almost brutal method adopted. Now, this is done in a moment by Cartesian development. Thus div VRS = n o + etc. - & ( - -2 ) - etc. \ty 3* / = S curl R - R curl S. Q.E.D. In the same way the reader may verify the formula (26). In the above we had the opportunity of encountering the operator (SV), as developed in (27). As it occurs rather often in practice, it may deserve here a few remarks. This operator is composed, scalarly, of S, which means any real vector, and of the Hamiltonian V, which we may call, after the example of Oliver Heaviside, a ' 'fictitious vector' All directional peculiarities of S and V, if considered separately, disappeared after they have been melted together into the scalar composition. That is the reason why the operator (SV) has a purely scalar character. It does not change the nature of the magnitude operated on ; that is to say, when applied to a scalar it gives a scalar, and if applied to a vector R it gives a vector; thus or simply =SV< = scalar, the parentheses in this case being superfluous ; again, = i (SV) R^ + j (S V) R, + k (S V) R z - vector. Here the parentheses are necessary, since S(VR) has a different meaning from (SV)R, namely S(VR) = S.divR. VECTOR ALGEBRA AND ANALYSIS 47 Observe that V< is the slope of , and consequently (SV)< or ; SV is the compo?ient of the slope of < taken along 8 and multiplied by the tensor S. Particularly, if s be a unit-vector, sVc/> is simply the component of the slope of < along s or so that sV is the symbol of what is commonly called axial differ- entiation, i.e. differentiation along the direction of s. And if s = n be the direction of the most rapid increase of <, i.e. the direction of the resultant slope, then On and this follows also immediately from the previous formula (20), after scalar multiplication of both sides by n, and remembering that n-= i. Finally, let us consider some iterations or repetitions of the Hamiltonian V, such at least as are met with most often in the physicist's practice. In the first place, if < be any scalar functign, then, as we already know, V< is a vector. Now, it may again be operated on by V, either vectorially or scalarly. In the first case, we have VVV = curlV< = o, (29) p^o j "p^2 / identically, since ~ ~ ^ ^- = o, etc. Thus, a vector which has a (scalar) potential is irrotational. Observe particularly the form Y W< = o, and compare it with the identity V AA< = o which occurs for any real vector A ; thus, also in this regard, V behaves like a real vector. Again, in the second case, we have in this case the omission of the parentheses cannot give rise to any misunderstanding, and consequently we may write also VV$, or more shortly V' 2 < = div V<. (30) As to the Cartesian expansion, remember that, by (15), div v9 = ^-(^- . dx\dx Again, taking the scalar square of V, i.e. 48 VECTORIAL MECHANICS and remembering that ij = o, etc., i-=i, etc., we get that is, when applied to , precisely the same thing as above for divV<. Thus, the last operator, V 2 which is widely known as the Laplacian, and which has a purely scalar character, may be denned simply as the scalar iteration of the Hamiltonian V. It is of very great importance in nearly all branches of mathematical physics, especially in gravitational problems, in Electromagnetism, in non-rigid Dynamics, in Conduction of Heat and in Diffusion. Since V 2 is an operator of scalar character, it may also be applied to a vector R, without any further explanation. Thus (32) this is a vector as well as R itself. In the same way V-c/> was a scalar, as was itself. In (32), the operand being any vector R, the operator V has been taken both times scalarly. Now, what remains still to do is to apply V to R : i first vectorially and then scalarly, giving rise to divcurlR, 2 both times vectorially, giving rise to curl curl R. Remark that the third eventuality, curl div R, is pure nonsense, since, divR being a scalar, its curl is meaningless. Now, in the case i we have seen that div curl R = o, identically. Thus, what remains to be considered is the case 2. Now, curl curl R, i.e. the curl of the curl of R, which is usually denoted by cur! 2 R, is the same thing as VVVVR. Hence, treating V as a vector and applying the formula (4), or rather its special case (5), we get cur! 2 R = V(VR)-V 2 R, Le. curl 2 R = VdivR-V 2 R, (33) or, in words : curl of curl = slope of divergence minus Laplacian. This formula, which is also of considerable practical importance, may again be verified by Cartesian expansion. As regards the application of the above elementary notions of Vector Method to Mechanics, it may be remarked here that the VECTOR ALGEBRA AND ANALYSIS 49 Hamiltonian V will be needed in Chapters II., III., IV. only in its simplest aspect, i.e. as the slope of a scalar function. The operators curl and div or VV and V in application to vector functions will not occur before Chap. V., which treats of deformable bodies. Meanwhile, therefore, let us here put together and mark with Roman numbers the few vector formulae which will be completely sufficient for the Mechanics of a particle or of a system of material particles and, in particular, for Rigid Dynamics Chap. II., III., IV., respectively. To avoid digressions, I shall cite them shortly, if necessary, by reference to the Roman numeral. AB = BA. (i.) A(B + + ...) = AB + AC +.... (ii.) VAB=-VBA. (in.) .. (iv.) (v.) VAA = o. (vi.) AVAB = o, BVAB = o. (VH.) AVBC = BVCA = CVAB. ( vm.) VAVBC = (AC)B - (AB)C. (ix.) V* = n|*. (x.) d Other formulae belonging to pure vector calculus, i.e. independent of any mechanics, will be given, on the basis of the above preparatory sketch of this modern mathematical language, always under Roman numerals, (XL) etc., in the following chapters, as the need arises in the subject under consideration. V.M. CHAPTER II. GENERAL PRINCIPLES. D'Alembert's Principle. CONSIDER any system of material particles i, 2, etc., of which the masses are m lt m 2 , etc., and which are acted on by what are technically termed the 'impressed' forces F x , P 2 , etc., respectively. The position of the particle i at the instant of time /, relative to some point of reference (9, chosen at random but once and for ever, will be determined completely by the vector r 15 of which the length Oi is the tensor and O -* i the direction (Fig. 25). Similarly will the vectors r.,, r 3 (generally T t ) determine the instan- taneous positions of the other particles, viz. 2, 3 (generally i). Omitting, for the sake of brevity, the index, we shall denote simply by r the vector corresponding to any one of these particles, and by m its mass. We could say, shortly, that, for our purposes, a material particle is characterised by one vector and one scalar (position and mass). GENERAL PRINCIPLES 51 The instantaneous velocity of such a particle, as regards both direction and absolute value, will be given by the vector di v =^ = r ' and, similarly, the acceleration by the vector It is scarcely necessary to say that the directions of the vectors r, r, if will, generally, be different from one another. If the system be free, we should have simply for every particle separately, by Newton, mf = F (a) or mi - F = o. But if the system is constrained, these vectors, say mi - F = S, will generally not vanish. The name of ' lost forces ' was long ago given to the vectors - S.* Let Sr be an infinitesimal virtual displacement of one of the particles, i.e. a displacement permitted by the connections of the system. If, to give an example, the particle i is constrained to remain always at a constant distance from the point O, we have r \ = r i 2 = const., so that r x 61^ = 0; that is to say, the virtual displacement ST-^ is perpendicular to the vector T I or tangential to the sphere of radius r l = const., a state- ment which after all is only a slightly changed enunciation of the original supposition. If, to take another example, two other particles 2, 3 are connected by a rigid bar of length /, we have (r 2 - r 3 ) 2 = r 2 = const., whence (r 2 - r 3 ) (Sr 2 - or 3 ) = o, an equation which tells simply that the difference of the displace- ments of the particles 2 and 3, or their relative displacement, must be normal to the bar. Observe the scalar character of the connections in both examples given above: ^ = const., I = const. Every one of such scalar con- 'ditions takes away one degree of freedom of the system. ' We * For the history of this subject, see for instance : E. Mach, Die Mechanik in ikrer Entwickelung, Leipzig, 4th edition, 1901 ; Encyklop. d. mathem. Wissenschaften, Vol. IV. Heft I, Leipzig, 1901. 52 VECTORIAL MECHANICS emphasise this point, because the conditional equations, expressing the connections, can also be of a different kind, namely vectorial. Let, for instance, one of the particles be constrained to remain on a given straight line ; this condition can be expressed by two scalar equations, for instance by the equations of two planes intersecting along that straight line ; these two equations will deprive the particle of two degrees of freedom. But we can do this in a simpler way, viz. by writing down a single vector equation r = jca + b, where a is a constant vector taken in the direction of the straight line in question, b any other constant vector (which runs from O to any arbitrary point chosen once and for ever on the given straight line), and x a freely variable scalar. Then we shall get which is the most immediate expression of the condition requiring the particle to remain on the given straight line. This one vector equation deprives the particle of as many degrees of freedom (i.e. two) as did the two scalar equations mentioned before. But these are, obviously, pure questions of form. Further on, when considering the equations of Lagrange (in their ' first ' form), it will be convenient to express all the connections of the system in the scalar form, i.e. by scalar functions of vectors. But meanwhile the choice of the form is quite indifferent. As to the definition of virtual displacements, the ' following further circumstance must be emphasised here : The conditional equations, i.e. the equations expressing the connections, may contain the time / explicitly. If this be the case, we call virtual only those displacements which satisfy these equations, after we have put in them t = const. From this additional explanation of the definition it follows, for instance, that the real displacements di occurring during the motion of the system, in the time interval dt> are virtual displacements when and only when the connections do not involve the time explicitly. To give an example, if one of the particles is constrained to remain always on the surface of a sphere (of radius R\ whose centre C is moved about in space in a given way, the corresponding conditional equation will contain / explicitly. 'In fact, denoting by 1 the vector drawn from the fixed point O to the centre C, at the GENERAL PRINCIPLES 53 time being (Fig. 26), we have as the expression of the supposed constraint : / r _ m _ j?i FIG. 26. Now, 1 being an explicit function of the time, this condition contains explicitly the variable /. In this case, then, a virtual dis- placement of the particle (m) will be a displacement Sr, which leaves it on the sphere supposed to be stopped, i.e. in which 1 is treated as a constant ; thus, the condition for Sr will be (r-l)Sr = o, or Sr _L to the vector r-1, in the instantaneous position of the sphere-'s centre. In this case the real displacement dx = v dt will generally not be contained among the virtual displacements, for it may have a component (not only normal but also) parallel to the vector r-1. Having thus explained the precise meaning of the virtual dis- placements Sr x , Sr 2 , etc., generally Sr, we can now write down the principle of d'Alembert in the vector form. This principle, expressed briefly in the old-fashioned manner, is that the virtual work of all the lost forces is equal to zero* The virtual work of the * lost force ' S = mi - F is the product of the absolute value (or tensor) of the virtual displacement and of the " component of S along this displacement, or, by the definition of the scalar product of two vectors, is equal to S Sr. The ' virtual work ' of all the 'lost forces' Sj, S 2 , etc., is the algebraic sum of similar * For its history see Encyklop. d. math. Wiss., loc. cit. p. 78. 54 VECTORIAL MECHANICS products for all the particles of the system, i.e. S 1 6r 1 + S 2 8r 2 + ... or, more shortly, 2SSr. Thus, the vector expression of d'Alembert's Principle is 2(ir-F)'5r = o, (i) where the summation extends to all the particles. Henceforth, deducing from (i) particular corollaries, and also transformations not less general than (i), we shall try to travel by the purely vectorial road, i.e. without recurring to any artificial splitting of forces, or of accelerations or of the virtual displacements themselves, into components, rectangular or other. Lagrange's Equations.* These equations are wholly equivalent to d'Alembert's Principle, i.e. they express the same thing in a different form. When the system is free, the displacements Sr are perfectly arbitrary and independent of one another, so that it follows immedi- ately from (i) that for every particle mi - F = o, as in (a) above. The equations (a) are already Lagrange's equations for a free system. But let the system be constrained. Let its connections be expressed in a finite form (holonomic system) by K mutually independent equations ^ ^ = Q> x = 0> etc ., (2> where , ^, \, etc., are scalar functions of the positions of all, or of some, particles of the system, which functions may also contain explicitly the time /. We have no need to trouble ourselves whether the positions of the particles i, 2, etc., on which , ^, etc., depend in a given manner, are expressed by rectangular or polar, or any other co- ordinates of the particles, or by the above r's. It is sufficient to know that <, for instance, depends in some given way on the position of the particle i, and say also on the position of 2, 3, 4, etc., i.e. that changes its value when one or more of these particles change their position in space. The reader may imagine, if he wishes, that < is from the beginning given as some scalar function of the vectors r 1? r 2 , ... ; or, if it is not given in this form, he may imagine it to have been reduced to this form. But for the general * Of the * first ' form. GENERAL PRINCIPLES 55 consideration of our subject all such questions are completely indifferent. It is sufficient to know that , and similarly j/', etc., vary in a manner which depends on the positions of the particles i, 2, and so on. Now, if depended only on the position of one particle, we should denote the gradient or slope of this function by V<, and if r be the vector corresponding to this particle, then, from the condition = const. = o, would follow V.Sr = o as the condition for the virtual displacement Sr. But, as (f> may depend on the positions of several particles, i, 2, etc. (generally 2), of which the corresponding vectors are r lf r 2 , etc. (generally r^), we cannot of course speak simply about * the gradient ' of this function <, but we must specify also the particle, relatively to whose displacement in space the gradient has to be taken. For this purpose we shall add to the symbol V the index (1) , or (2 ), or generally (0 , so that V ( i)<, V (2) <, generally V (i) <#>, will be the gradients of <$> corresponding to change of position of the particle i only, or 2 only, or / only, while all the remaining particles are considered to be fixed. These gradients could be called partial gradients, in the same way as we speak of partial differential quotients in ordinary, scalar analysis. Each of the partial gradients V (t -)< is, of course, a vector, namely a vector normal to that surface which is expressed by < = const, on the supposition that all the particles, implied in , are fixed with the exception of the z th particle. Using this notation, we get from < = o the following condition for the virtual displacements : V(i)< . &! + V (2 )< . <5r 2 + . . . = o or, written shortly, 2V (i) . S^ = o, where the summation extends to all the particles of the system. Similarly for the remaining conditions \^ = o, x = > an d so on. Hence, by considering all the K connections (2) imposed on the system, we get as many equations for the virtual displacements : (K equations) - **(W bT i = k (3) 56 VECTORIAL MECHANICS To satisfy d'Alembert's principle (i) and also the conditional equations (3), the well-known method of Lagrangian indeterminate multipliers can be used. Multiplying, then, the equations (3) respectively by the scalar coefficients A, //, etc., then adding them to the equation (i) and equating to zero the coefficient of each 8i t separately, we get the required Lagrangian equations of motion niii = F { + AV (0 < + pV (i) \/> + . . . . (4) The number of these equations is equal to the number of particles, say n. We have then n vector differential equations (4) and K scalar equations (2) for the n vectors r } , r. 2 , ...r,, and the K scalar multi- pliers A, /*, .... (In Cartesian language we should say : we have 3/2 scalar differential equations and K equations expressing the con- nections, that is 3/2 + K equations for the 3^ coordinates and the K multipliers A, /A, ... .) r The equations (4) are of the 2 nd order with respect to the time /; if then for the initial instant /=/ the positions and the velocities, i.e. the vectors / r \ / r ^ (T \ (T } \ X 1'0' \ X 2/0>'"; \*1/0> \ X 2/0'"' be given, the whole motion of the system, or its time-history, will be determined, on the supposition, of course, that the forces F^ are given functions of the positions and velocities of the particles con- stituting the system. These forces, as well as the connections, may contain the time / explicitly. The reader will observe that in deducing (3) from (2) the time has not been varied, according to the observation made above in regard to the complete definition of virtual displacements. As an example of the application of (4), take the case of a system consisting of a single particle constrained to remain on the surface < = . ( 2 ^) In this case (4) gives mi = F + AV<. (4) As the vector V< is normal* to the surface in question, the additional force . 9 N = ^h = AV< = A n, (5) expressing the ' reaction ' of the surface, is normal to the surface. To find its intensity 3^, , 'dn we have only to determine the Lagrangian coefficient A from the given condition = o. * By (x.), Chapter I. GENERAL PRINCIPLES 57 Let, for instance, the surface < = o be a sphere at rest, of radius R and centre O ; then < = Ar 2 - J? 2 = o so that V< = n ^ = nr = IL/? = r. (6) Hence, the Lagrangian equation of motion will in this case become ;;;r = F + Ar. To find X, write simply, according to (v.), r-' = r 2 = const. = J? 2 , and differentiate with regard to t; then rr = o, and differentiating again : r'r + r 2 = o. Now substitute r from (4^) ; then o = Fr + Ar* + mi 2 = #zz> hence A^?=-Fn--^- (7) where v = r is the instantaneous velocity of the particle. From (7) we get, by (5) and (6), the final expression of the force of reaction o x (8) Remembering that Fn is the normal (or radial) component of the impressed force F,"the reader himself will translate the formula (8) into physical language and recognise it as a well-known proposition of elementary dynamics. To get from (4^) the differential equation of motion of a simple pendulum, of length R, we have only to write F = flZ-a, (9) where g is the 'terrestrial acceleration' and a denotes a vertical unit vector, pointing downwards. It will be observed that multiplying scalarly the Lagrangian equations (4) by the corresponding Sr/s and adding them, we get, by (3), the Principle of d'Alembert, which was our point of departure. Thus, the equations of Lagrange, with given equations of the con- nections of the system, are wholly equivalent to d'Alembert's Principle. 58 VECTORIAL MECHANICS Hamilton's Principle. This also is equivalent to d'Alembert's Principle. The vectorial road, again, leads very easily from the second to the first, and vice versa. Denoting the virtual work of all impressed forces F by 6' W* i.e. writing, for the sake of brevity. S'W=2FSr, (10) the Principle of d'Alembert becomes 2;rdr = S'ZF. (i') Now, for each particle separately, at at r Sr + v Sv ; where v = r is the velocity of the particle in question. Multiply both sides by its mass m and sum up for all the particles of the system ; then, by (i'), ^tmfi + StV=~2*ar*t, or, denoting the vis-viva or kinetic energy of the whole system by T, i.e. putting V^mv-=T: at hence, by integration from' any instant t = a to any other arbitrary instant /-*: ^ (8T+8 ' JV} ^p mv S^ () where [ ] denotes, as usual, [ ] (=b -[ ] t=a . Now, if the displacements Sr, virtual but otherwise arbitrary during the interval of time a -> b, vanish for the terminal instants a, b themselves, the right side of equation (u) is =o, so that o, (12) and this equation (together with the condition of vanishing 3r's for the instants a, b) is an expression of what is called Hamilton's Principle. * We write 5' W (and not d IV) to emphasise that this infinitesimal work is, generally, not a complete variation of a function of position and time. GENERAL PRINCIPLES 59 Observing that the instants a and b can be chosen in an arbitrary manner, the reader will easily pass back from Hamilton's to d'Alembert's Principle. The deduction, from (12), of the so-called 'second' form of Lagrange's equations of motion is a purely scalar question, and can therefore be omitted here. Moreover, it is accomplished in a few lines by introducing the con figu rational coordinates ^, ^ 2 > '/ (j = number of degrees of freedom) and the corresponding velocities 7*i IP ^i> ?2 anc ^ integrating by parts the terms - 8# it so that , /'~^'ri\ tfi - -=-( -^-JS^come in instead; in this way, writing B' W^TZQify^ so that <2 is the (generalised) force corresponding to q it the reader will get immediately from (12): which is the well-known second form of Lagrange's equations of motion. Still one remark. In Hamilton's Principle (12) not a single heavy (clarendon) letter occurs, which circumstance is the most evident sign of its scalarity ; all space-directional properties have disappeared, and this is what constitutes the advantage of Hamilton's Principle, inasmuch as it permits us to introduce directly any configurational variables or independent parameters equal in number to the degrees of freedom of the system, without regard to the particular type of its connections. CHAPTER III. SPECIAL PRINCIPLES. LET us now consider three principles or propositions, less general than d'Alembert's Principle, which follow from this principle under particular conditions. 1. The Principle of Vis-viva. If the equations (2), or the 'connections' of the system, do not contain the time t explicitly, then among the virtual displacements Sr, are also contained the actual displacements d^ of the particles, occurring during their motion in time, viz. in the time-element dt. For a system, then, which satisfies this condition we can put in d'Alembert's Principle (i) : 6r i = r t -^/ = v i ^/, where dt has the same value for all particles *'= i, 2, etc. Hence, writing r = v, and omitting the common factor dt, we get 2Fv or, denoting again the kinetic energy of the system by T, (13) This equation, true for any instant /, can be read as follows : The increase of kinetic energy of the system, per unit time, is equal to the activity of the impressed forces or to the work done by them on the system, per unit time. Integrating both sides, we have T b -T a =W ab , (14) where W ab is the work done on the system in the interval from t=a to t=b. SPECIAL PRINCIPLES 61 This is the principle of Vis-viva. In particular, if the forces F f have a (scalar) potential independent of /, i.e. if P J = V/x^7 (M\ * t v (t) ^ \ ji and 9^7/9/=o, then = o which is moved about in space in any given way, so that < contains t explicitly. Then (a) reduces to J but, returning to the equation of motion mi = F + AV<, we see that N = AV<, as in (5), is the force of 'reaction' of the surface. Hence, the above equation may be written or in words : the increase of kinetic energy is equal to the work done by the total force, impressed phis reactive. Observe that N is always normal to the surface < = o, be it fixed or moved. Now, if the surface is fixed, then v is tangent to it, and Nv vanishes, so that the * reaction ' does no work ; but if the surface be moved, then v is generally not tangential, and N does work, the activity of N being If, for instance, the moved surface be a sphere, then (as in a previous example, to which corresponds the Fig. 26) and _(r-l)-= where v c is the velocity of the centre C of the sphere. Now, nv c or the normal component of the velocity of C is generally different from zero, and so also will be the supplementary activity, 2. The Principle of Centre of Gravity.* If the connections be (none or) such that it is possible to displace all the particles of the system simultaneously by one and the same = centre of mass. SPECIAL PRINCIPLES 63 length 8e in one and the same direction a, or, in mathematical language, if it is allowed to put Sr 1 = 8r 2 =. ..=Sr H = a&, where a is some unit-vector and Se an infinitesimal scalar, it follows from d'Alembert's Principle (i), by omitting the common factor 8e ' that a2( W r-F) = o. (a) Here Fa = aF is the component of the force F along a ; denote it by F a and write Z^ r = j/ S , M=?m, (17) also S a for the component of the vector S along a; then, by (), The point defined by (17), i>e. the end of the vector S, is called the centre of gravity or, more correctly, the centre of mass of the whole system; S a is its coordinate measured along a, the initial point of S, as of all the r's being O. Bearing this in mind, the reader will easily recognise ( i Sa) as the expression of the principle of motion of the centre of mass, for the direction a. It is scarcely .necessary to say that the position of the centre of mass is independent of the choice of the point of reference (9, or of any auxiliary framework (or system of coordinates), and that it depends exclusively on the distribution of mass in the system.* If this distribution be continuous in space, the summation 2 be- comes an integration : \dm=\p dr, MS = I r/o dr, p being the density of mass and dr an element of volume. If some other direction b has the same property as a, we have not only (a), but also b2(wr-P) = o (6) or But then this property belongs also to any direction k parallel to the plane a, b ; in fact, any such k can be put into the form where x, y are scalars ; hence we have only to multiply the equation (a) by x and (/>) by y, and add, in order to obtain * The vectorial proof of this statement is left to the reader as an exercise. 64 VECTORIAL MECHANICS If, finally, a third direction c, not coplanar with a, b, also possesses the property of a, b, we have or J/S e = 2/7. Then, and only then, it follows, from the three equations (a), (b} and (c), that the whole vector 2(/r-F) vanishes, so that J/S = J/^ = 3F, (18) which is called the principle of motion of the centre of mass, without any reservation, i.e. for all directions in space. In particular, if all the impressed forces are in mutual equilibrium, i.e. if 2F = o, or in other terms if the vectors F, arranged in a chain, form a dosed polygon, S = o; ;. S = A/ + B, (19) A and B being constant vectors. Under these conditions the centre of mass has a uniform motion along a straight line. This is the principle of conservation of the motion of the centre of mass. 3. The Principle of Areas. If the connections of the system permit every particle of it to be simultaneously turned through the same angle 80 about the same axis a, i.e. if (taking the point of reference O on this axis) or^Sfl.VaTi, *'=i, 2, ...71 is a virtual displacement, then introducing it in d'Alembert's Prin- ciple (i), and omitting the common factor 86, which is a simple scalar, we get v/ .. 2(#/rVar-FVar) = o. Now, by (vin.), FVar = aVrF, and similarly r'Var = aVrr, and as the axis of rotation a is, by supposition, the same for all the particles of the system, it can be put before 2, so that a2(*Vrf-VrF) = o. The vector sum 2VrF is the resultant moment, about O, of the impressed forces, as regards both direction and intensity; denote it by L. Then, remembering that Vff = o, identically, we have SPECIAL PRINCIPLES 65 so that the last equation takes the form -L} = o. (a) Similarly, if the whole system can be turned also about another axis b passing through O, and also about a third axis c passing through O and not coplanar with a, b, we have b{as above} =o, (b) c{as above } = o, (c) and from the three equations (a), (b} and (c) it follows that the bracketed vector expression must vanish, or that 2 m Vrv = L. (20) dt The vector product jVrv expresses,* by its tensor, the area swept out by the vector r in unit time in the plane r, v, its direction being normal to this plane. Whence the name principle of areas given to what is expressed by (20). It is the principle of areas, without reservation, i.e. for any con- ceivable axis of rotation. The equation (a) alone, for example, is the principle of areas belonging to the axis a, viz. for areas swept out in the plane normal to a. The vector sum SwVrv is what is called the moment of momentum of the whole system, about O. Thus, (20) may be enunciated : The rate of time variation of the moment of momentum is equal, in absolute value and direction, to the resultant moment of the impressed forces, both moments being taken about the same point O, which, of course, can be chosen in a perfectly arbitrary manner. In particular, if L = o, we have the conservation of areas, i.e. or ^wVrv = C, (21) the vector C being constant both in its tensor and direction. The plane normal to this vector C is called the invariable plane of the * As shown in an example, given in Chapter I. ; see page 27. V.M. E 66 VECTORIAL MECHANICS system. If the system consists of a single particle, then its orbit lies in this plane; in this case (21) reduces to Vrv = const. (Kepler's II. law, for instance). If not the whole moment L, but only some component of it, say La or L a , vanishes, the principle of conservation of areas applies only to the axis a, i.e. to the areas swept out in the plane normal to a. In such a case we have only one scalar invariant, viz. a2/ Vrv = const. But if L = o, as above, we have (21), i.e. one vector invariant, which is equivalent to three scalar invariants. The principle of conservation of areas, in its full extent, holds, for instance, for a system of free particles acted on by 'central forces/ i.e. by forces F^ coinciding in direction with the corresponding r/s, because then for each particle separately VrF = o, and hence L = o. Kepler's second law, already alluded to, is the same thing as conservation of areas for a single particle, viz. a planet in its motion round the sun. Its equivalence to ' centrality ' of motion has been already shown in Chapter I. To illustrate the simultaneous validity of the principle of con- servation of areas and of vis-viva, let us consider a single particle, not subjected to any constraint and acted on by a force F, not only central, i.e. directed towards or from the fixed centre O, but also such that its intensity is a function of the distance r alo?ie (r being the tensor of r), say F=F(r). The unit-vector drawn from O to the particle, at the time being, may be written T/r, hence Now, put (p(r)dr= U= U(r so that F(r) = -^- ; then U will be the potential of the force F, i.e. In fact, since U depends only on r and since V U means simply the slope of U, we have, by Chapter L, . r dU F(r) MU=- r- = r -- - = F. Q.E.D. r dr r *The additive constant of integration being unessential. SPECIAL PRINCIPLES 67 Thus, our force F has a potential, and since it is besides a central force, both the principles mentioned above hold, i.e. \mv*-U(r) = c, (a) Vrv = a, (/?) where c is a scalar constant, namely the total energy, and a a vector consta?it\ its tensor, a, is twice the area swept out by the radius vector, per unit time ; and since also the direction of a is constant, the vectors r, v are always in the same plane containing O, i.e. the particle describes a plane orbit round O. If the revolution round O, in the plane of Fig. 27, be clockwise, then a is normal to the paper, pointing away from the reader. In other words, a, r, v is a right-handed system. FIG. 27. To obtain the equation of the orbit, we have to eliminate the velocity v from (a), (/3). Multiply both sides of (/?) vectorially by v; then, by the fundamental formula (ix.), Vva = VvVrv = z> 2 r - (vr) v ; multiply this scalarly by r, getting thus or, since by (vm.) rVva = aVrv = a 1 , z;V 2 -(vr) 2 = a 2 . If 1 be a unit-vector tangent to the orbit, drawn in the direction of motion, then v = z>l, and consequently z> 2 {r 2 -(rl) 2 }=a 2 . Now, if e be the angle included by r and 1, rl = r cos e, r 2 - (rl) 2 = r 2 sin 2 c ; hence wsine = fl or vD = a = const., (p ) where D is the length of the perpendicular from O upon the tangent. Thus, the velocity is inversely proportional to this perpendicular, 68 VECTORIAL MECHANICS which is the known theorem of Newton, holding for any central motion. Observe that (/3') has been, deduced uniquely from (/3), without recurring to (a), i.e. to the existence of a potential. Combining (/3') with (a), the required elimination of v is effected immediately, leading to ma TT . / -. ~U(r)=e t (7) which already is the equation of the orbit, for any law of depend- ence of the central force on the distance r, i.e. for any form of the function F(r). In particular, if this be the Newtonian law of gravitation, say j. = Mm r 2 (where M is a constant), then f Mm U= \Fdr = , - and the equation of the orbit, (7), becomes rm where everything but D, r is constant. The proof that this is a conic, with one of its foci in O, may be left to the reader. CHAPTER IV. RIGID DYNAMICS. General Remarks. FOR studying the motion of a rigid system, let the vector i=O^-m determine, as in the preceding chapters, the instantaneous position of a material particle m of the system in question, relatively to the point O, which can be regarded as a point belonging to some ' fixed ' system of reference* (see Fig. 28). The instantaneous velocity of FIG. 28. the particle m, also relative to O, will be, as before, * The reason being that in order to determine a vector r in absolute value and direction, a single point of reference is not sufficient ; to do this, some extended, non-symmetrical system is necessary, to which may belong as one of its points. This is the reason why in Fig. 28, and also in Fig. 25, round the point has been drawn a (dotted) contour, representing such a reference system. In the course of our reasoning we shall often refer to the ' point O,' but always in the sense here explained. 70 VECTORIAL MECHANICS We shall also choose once and for ever some point O fixed in the rigid system itself, as the initial point of a system of reference in- variably attached to the rigid system, and we shall denote the vector O'^>m by r'. The vector r' will thus determine the position of the particle m as seen by an observer attached to the rigid system, while r will do the same office for an observer attached to O. In other words, the vector r' will distinguish the individual material particle m of the rigid system from all other particles of this system. Thus, the rigidity of the system to be considered in this chapter will be completely expressed by saying that r' does not vary with the time, or by writing fa' f ' = *=- For a rigid system which is perfectly free to move about in space (having 6 degrees of freedom) all the three principles treated in Chap. III. hold: i the principle of vis-viva, because the connec- tions do not contain the time / explicitly ; 2 the principle of centre of mass, because the whole system can be moved (displaced) in any direction; 3 the principle of areas, because the rigid system can be turned as a whole about any axis. Differential Equations of purely Rotational Motion. As the treatment of translatory motion of the whole system does not offer any difficulties, we shall consider here only the case of purely rotational motion of the rigid system about a ' 'fixed' point, i.e. about a point which does not move relatively to the point O. The rigid system will then have only three degrees of freedom. The principles of vis-viva and of areas hold, but the principle of centre of mass will lose its applicability, since one of the points of the rigid system is fixed. Let O be this fixed point of the system. We can in this case, if we wish, let O coincide with O', but then we must not forget that the dotted contour surrounding the point O (Fig. 28, as explained above) will not keep company with the rigid system in its rotational motion. In other words, notwithstanding that the points O, O are made to coincide with one another, the vectors r, r', though overlapping in space, will behave differently in time. We shall still have p = Q while f = v = o, in general ; but r 2 or r 2 , and hence also r, will be constant in time for every RIGID DYNAMICS 71 point of the rigid system, r being simply its scalar distance from O, or, what is the same, from O'. Just as with r, r', we shall also denote any vector, say w, in a twofold way, viz. by w, if it is treated in regard to the fixed system of reference (O), and by w', if it is taken in regard to the moving system ((7), i.e. in regard to the rotating rigid system. It will be convenient to refer to these two ways of treating a vector by saying shortly: 'relatively to <9,' in the first, and 'rela- tively to 6>V in the second case. Let now p be the instantaneous angular velocity of the rigid system, in absolute value and direction (i.e. so that p is the absolute value of the angular velocity, and the direction of the vector p is in the positive sense of the axis of rotation*). Then, considering any vector w, and its alter ego w', and observing that w - &' is simply the resultant velocity with which the end-point of the vector w moves on account of the rotation of the rigid system alone, we have the important relation w-w' = Vpw (22) or dvrjdt dw'/dt + Vpw. f The formula for the velocity v of a point of the rigid system, v = r = Vpr, (23) is only a particular case of the more general formula (22); for we have only to write in (22) r instead of w and remember that r' = o, to get immediately (23). After these, slightly lengthy but necessary, preliminaries we pass to the dynamics of a rigid system. We can apply to it at once the principle of areas, as expressed by the formula (20), Chap. III. Denoting the moment of momentum of the system about O by q, i.e. writing q = 2 OT Vrv, (24) and calling, as before, the moment of the impressed forces L, we have instead of (20) ^ 5T L - {25 > *Viz. in that direction in which a person, looking along the axis, would find the rotation about it to be right-handed or clockwise. t This is the vector form of what in Routh's Advanced Rigid Dynamics is called a ' Fundamental Theorem,' the proof of which covers there one page and a half. This, by the way only, to show the convenience of vector language, as compared with scalar or Cartesian. 72 VECTORIAL MECHANICS This is the differential equation for q relative to O, i.e. relative to the ' fixed ' system of reference. Applying the general formula (22), we get from it immediately the differential equation for q', relative to the rotating rigid system itself: ^' = Vqp + L. (26) The moment of the impressed forces L is given, and between the moment of momentum q and the angular velocity p there exists at every instant a certain relation based only on the properties of the rigid system, which also can be considered as given ; this relation will be developed in the next section. Taking into account this relation between q and p, peculiar to the given system, we have ultimately in (26) one vector differential equation of the first order for one vector, namely for q if p be elimi- nated, and vice versa. If then p, say =P , be given for any par- ticular instant, /=/ , then the whole motion of the system will also be determined. Kinetic Energy. Principal Axes. Euler's Equations of Motion. By(23) ' ^ = vVpr = pVrv; hence the kinetic energy T= %2mv 2 of the rotating rigid system will be r=| or,by( 24 ), r i.e. equal to half the scalar product of the angular velocity and the moment of momentum. Now for the mutual dependence of p and q, promised at the end of the preceding section. This is also a very simple matter. By (23), Vrv = VrVpr = r 2 p-(pr)r, by (ix.) ; hence, by (24), q = p2 ^2 _ 2w(pr)r . (a8) Here both the members on the right side contain p to the first power. Thus, the moment of momentum q is a linear vector function of the angular velocity p, and this, i.e. (28), in fact, is already the required relation. Denoting by K the operator implied in (28), which turns p to q, we can write shortly RIGID DYNAMICS 73 K is called a linear vector operator. It changes not only the tensor, but in general also the direction of the vector which is operated on. The linear vector operator has found its chief application in the researches of electromagnetic phenomena in crystals ; q is related to p just as the electric displacement is related to the electric force, or as the magnetic induction to the magnetic force.* In the case before us the reader has to look on the operator A' as being, for the time, only a short symbol for certain operations which are fully determined by the preceding formula (28). Further on we shall see that, in fact, all the properties of the operator K can be deduced from this formula. Introducing (280) in (27), the kinetic energy of the rigid system can be expressed by i.e. as a (quadratic) function of the angular velocity alone. In the same way, q = ATp can be substituted in the equation of motion (26). From (28) we see at a glance that all properties of the operator K depend only on the mass of the rigid system and on its distribution about O ; whence it follows that K does not vary with time, so that it can be written before the differentiator d/df. We have, then, instead of (26) : Ar^ = L-VpAp, (26a) and this form of the equation of motion is the vectorial incar- nation of the three celebrated (scalar) equations of Euler, as will be seen explicitly further on, by an appropriate decomposition of (26a). Meanwhile let us look for the properties of the operator A'. In the first place, then, what are the principal axes of the operator AT, or in mechanical terminology the principal axes of the rigid system, i.e. what are the directions x, for which the moment of momentum q coincides with the axis of instantaneous rotation, * For the theory and application of the linear vector operator see, for instance, Heaviside's Electromagnetic Theory, Vol. L, Chap. III. Linear vector functions were first introduced by Hamilton in his calculus of quaternions. It may be remarked here that the above operator A", which converts p into q, is a symmetrical (or self-conjugate) operator. The more general, non-symmetrical linear vector operators, which have nothing to do with rigid dynamics, will appear in the next chapter as very useful for the treatment of problems connected with non-rigid bodies. 74 VECTORIAL MECHANICS or with the direction of p? This fundamental question can be answered easily. We may consider x as a unit vector, so that P=/x, Q = ^x, and the required principal axes will have the property q = p = /x, (A) where n is a scalar. Introducing (A) in (28) and dividing both sides by /, we get jEmr* - ~m (rx)r = x or 2w(rx)r = x(2w/- 2 -). (B) If +x satisfies (B), -x will also, with the same value of . Hence a principal axis can be denoted shortly by writing simply x instead of x. Suppose now that x x and X 9 are two different principal axes, to which correspond the particular values 15 n. 2 of the scalar n. Then, by (B), ?m (rxj) r = x x (2mr 2 - n^ ), ( p^ ?m (rx 2 ) r - x 2 (IW 2 - 2 ), whence, multiplying, scalarly, (BJ) by X 2 and (B 2 ) by x x and sub- tracting, ( 2 - 1 )x 1 x 2 = o. Hence, if !=#.,, then X 1 x = o, i.e. Two principal axes with different values of n are perpendicular to one another. Hence, if X 3 be a third principal axis to which belongs a value n = n s different from both n^ and ;* 2 , this axis X 3 will again be perpendicular to the first two axes, so that x x , X , X 3 will constitute a normal system of unit vectors. We shall choose the order of the indices i, 2, 3 so that X 15 x. 2 , x 3 is a right-handed system, i.e. x 3 = Vx 1 x 2 , as for i, j, k. Generally, n lt n^ n^ will be different from one another. But if it happens that, say, n 2 = n lt then multiplying (BJ) by any scalar a and (B 2 ) by any other scalar b and adding, RIGID DYNAMICS 75 so that the vector ax : + te. 2 also satisfies (B). Hence : i If X 15 X 2 are two principal axes having equal values of , then any vector (a^ + x 2 ) taken at random /'// the plane Xj , x. 2 is also a principal axis, with the same value of n. And from this proposition follows immediately : 2 If three, non-coplanar, principal axes have equal values of n then any direction in space, drawn from <7, is also a principal axis with same value of n, i.e. the directions of q and p coincide for any direction of p. But if to three principal axes belong three different values of n, it is obvious that no other principal axes can exist, as a fourth principal axis would, by the above properties, be identical with either X 15 x 2 or X 3 . Hence, in the most general case, a rigid system has three principal axes (x l5 X 2 , x s ) mutually perpendicular, to which corre- spond different values ti lt 2 , n z of the scalar n.* By this reasoning we see that the vector equation (B) admits in i \e most general case only three different roots n corresponding to principal axes. Indeed, it is not difficult to deduce from (B) a cubic scalar equation for n. But we can find all roots from the equation (B) itself. Let us denote the components of r along x x , X 2 , X 3 by r^ r 2 , r Bt or write VT r rv r rv - 1X3 rj , rx 2 r 2 , rx 3 r% , more generally, let w lt w 2 , w s be the components of any vector w along x 15 x 2 , X 3 . (It may be observed that this is not an artificial decomposition of vectors into scalars ; for X 15 X 2 , X 3 represent essentially peculiar directions, and hence properties belonging in- trinsically to the given rigid system, and are no more artificial than are, say, the optical axes of a crystal.) Now, substituting in (B), successively, x=-x : , and so on, we get 2=1,2,3 *If n l = n^^n 3t we have the above case i, or axial symmetry as regards rotational inertia; in this case, the third axis only, i.e. X 3 , corresponding to n 3 , is usually called the principal axis of the rigid system. If n 1 = n^ = n 3 , we have the above case 2, viz. complete isotropy as regards rotational inertia ; in this case, all directions having the same property, none of them is called (in the accepted terminology) a principal axis. In this case the operator K degenerates into an ordinary scalar factor. + r 3 , etc., or, finally, 76 VECTORIAL MECHANICS then, multiplying respectively by X 15 x 2 , X 3 and remembering that Y 2_ T 2_v2_ T T 2_: > -2_ 1 _^24.^2 X l ~~ X 2 ~~ X 3 ~~ I ' r ~~ ' 1 + r \ 2 ~r ^3 ) 2 + r 3 2 ), etc., v n s = 2.mp B 2 , (D) Pi 5 /2' /s being the shortest distances of a particle w of the system from the three principal axes. At the same time it follows from (c), by remembering that x u X 2 , X 3 are mutually perpendicular, 2#zr 2 r 3 = ^mr^r^ = ^mr^r^ o. This is a well-known property of the principal axes, and determines their directions if the distribution of mass be given. The scalar quantities ?z 15 ^ 2 , n 3 (D) are called the principal moments of inertia of the rigid system about the point O'. As they are, at the same time, the principal values of the linear vector operator K> we may conveniently denote them by K^ K^ K^, Remember that these are ordinary scalars. The operator K could be called the inertial operator of the given rigid system (the point O' being tacitly assumed). If the structure of K is known, all that regards the dynamics of rotation of the particular system is given. Returning to (A), we can write ?i = K \P\ > 2z = K ?. Pv ?s = ^"3/3 or, vectorially, ^^p.^^+^^ + ^r^, ( 29 ) where (30) Also ft 2 = (VrXj) 2 , and so on. The above equation may also be inverted, thus P = A-iq, the operator A'" 1 being the so called inverse of K, i.e. also a linear vector operator having the same principal axes as K and the principal values K -^ K -^ K -^ as is seen immediately from the developed right side of (29). Remembering that x^i, etc., X 2 x 3 = o, etc., we can write, by (29), the kinetic energy expressed by (2701) in the form T= RIGID DYNAMICS 77 Substituting (29) in (260), splitting this vectorial equation into three scalar ones, along the principal axes, and remembering that Vx 2 x 3 = x 1 , etc., we get which is the usual form of Ruler's equations of motion. The com- ponents pvPvP z of the angular velocity being already taken along the (moving) principal axes of the rigid system itself, no dashes are necessary. The equation (260), i.e. with q = ^Tp, is the complete vectorial equivalent of the three scalar equations of Euler. Multiplying this vector equation, scalarly, by p' = p and remem- bering that pVqp = o, identically, we have but by (270), dT dT 7 rrt hence ^7 = Lp: and Lp being the activity of the impressed forces, this is the expression of the principle of vis-viva. In fact, as we saw at the beginning, this principle is true for any rigid system. * Since ~5LA'p' = p'A'-?-, and, in general, for any pair of vectors A, B, at at provided that the linear vector operator AT be a symmetrical operator, as in our In the next chapter it will be seen that this property does not belong to non- symmetrical operators. 78 VECTORIAL MECHANICS Motion under no Forces (Motion k la Poinsot). If there are no impressed forces or, at least, if their resultant moment L vanishes (for the given fixed point (9'), it follows immediately from (25) that --, (3.) i.e. the moment of momentum q. has a constant value and constant direction ''in space] i.e. relatively to the system of reference O (the system of 'fixed' stars or any other moving relatively to it with uniform translational velocity). The plane normal to this invariable direction is called the invariable plane of the rigid system. Euler's equations of motion, or rather their vectorial equivalent (260), reduce in. this case to *?-$-v (33) where q = A"p, as before. If the rigid system rotates at a given instant about one of its principal axes, i.e. if q = p, then Vqp = o and, by (33), i.e. the system will continue to rotate for ever about this same principal axis with constant angular velocity. Besides the principal axes no other axis of rotation has this important property. In fact, Vqp vanishes only if q || p (the trivial case of p = q = o* not being worthy of notice). On account of this important property, the principal axes are also called the free axes or the permanent axes of rotation of the rigid system. The eq. (33), multiplied scalarly by p = p', or also (31) for L = o, gives T i.e. T= lpA"p = k (Krf* + K z p + A' 3 / 3 2 ) = const. (34) Thus, the kinetic energy is now an invariant of the rigid system, which could have been foreseen. Again, (33) multiplied scalarly by q = q' gives qVq'/ A A tne same meaning as the common V in ordinary space. But 2T t> = 2 T= pq, for all values of p and of the corresponding q ; .'. Q=VT;, ( 3 6) whence it is seen at once that the vector q is normal to the plane which touches the ellipsoid T in the end-point of the vector p (see Fig. 30). From the same formula (36) it follows also that ? = ^ (37) where D is the length of the perpendicular from the centre of the ellipsoid to the above tangent plane.* This is the usual geometrical representation of the relation of any two vectors, one of which is a linear (symmetrical) function of the other, like p and q = K"$. Thus, the ellipsoid T, which is invariably attached to the rigid system, always touches the invariable plane; but the point of tangency is the extremity of the axis of instantaneous rotation, and has therefore no velocity relatively to this plane, i.e. does not glide on it. Hence, in the case of no impressed moment (L = o), the motion of the rigid system is such that the ellipsoid T= const., with its , centre fixed, rollsjn the invariable plane. The discovery of ^ this beautiful property is due to Poinsot. Whence also the motion* of^a rigid system about a fixed point under no impressed forces is usually called Poinsofs motion or motion a la Poinsot. The most general motion of this kind can be represented analyti- cally, as depending on the time and on the initial conditions, by elliptic functions. To obtain dpjdt as a function of p itself and of the given values of the invariants T and q, which is the starting point for the analytical solution of the problem alluded to, namely for the expression of t * In fact, by purely geometrical reasons, L but, by (36), :^f?, where V = n^, as in (x.); on on V.M. 82 VECTORIAL MECHANICS as a function of p and then, by inversion, of p as an (elliptic) function of.the time /, go back to the equation of motion (33), A' = Vqp, say = w J r*w, where A'~ 1 or i/A'is the inverse of A", as explained above. Multiply, scalarly, both sides of the last equation by p' = p, and remember that p' dv'ldt = J d(p'*)ldt = i d(jP)!dt =p f . Then p 4 = p^-i w = (AT-ip) w = (A'- 1 ?) Vqp, since AT" 1 , exactly as A' itself, is a symmetrical operator.* But (AT- 1 p)Vqp = pV(A'- 1 p)q, by (VIIL), and q = Ap; hence: pV(A-ip)(A'p)' or, if the time t be counted from the instant for which /=/ , Ci> -ft dfi t which is the required expression. The denominator, if appropriately developed (see 'Appendix'),'^ proves to be the square root of a cubic function of p- alone, so that f is expressed by an elliptic integral ; whence, by inversion, the scalar angular velocity / reduces to elliptic functions of the time /. It may be observed here that the integration according to (a) (or rather to its developed form, given in the 'Appendix') has been effected in finite terms in two cases, namely when AT 2 = A 3 , l A' 1 >A 2 , i.e. for an uniaxial rigid system, and when 2Tf by (39), and remembering that as = cos (a, s), it is not difficult to recognise in the product - ^as the usual RIGID DYNAMICS 85 expression for potential energy, i.e. the weight of the system x height of its centre of gravity. Thus, (42) expresses the constancy of the sum of kinetic and potential energy, or the constancy of the total energy of the whole system (consisting of the rotating rigid system and the attracting earth). By (41) and (42) we have, then, two scalar invariants-. q a being the vertical component of the moment of momentum and E the total energy. Both of these functions of the state of the system conserve constant values during its motion. Other invariants, or integrals, for a rotating heavy rigid system, in the general case, i.e. for any principal moments K^, K^, K z , no one has yet succeeded in finding. Observe that in the absence of impressed forces the whole vector q has been constant, while, in the present case, only its vertical component is constant. Instead of three we have now only one scalar invariant, that is, together with the energy, two scalar invariants, whereas before we had four. Two, due to the force of gravity, are gone, or at least have hidden themselves so deeply that mathematicians cannot find them. In fact, the general problem of a rotating heavy rigid system is still awaiting its solution, i.e. a reduction to quadratures. A third invariant necessary, in addition to (I T ) and (I 2 ), for the solution of the problem, has been discovered only in a few special cases, viz. under special suppositions as to the moments of inertia and to the position of the centre of gravity of the system. The most remarkable is the special case associated with the names of Lagrange and Poisson, viz. in which the centre of gravity C lies on one of the principal axes (passing through O), say s x t , while the moments of inertia corresponding to the other two axes .are equal, i.e. K^ = K^ The ellipsoid of inertia becomes in this case an ellipsoid of rotation, X 1 being its axis of symmetry. A third integral has also been found, in some other special cases, by Hess, Mrs. Kowalewski and Tschapligin.* Here we shall limit ourselves to the consideration of the case of Lagrange and Poisson, which may be defined, shortly, thus : s = x 15 K. = K. 2 . (43) *See Routh's Advanced Rigid Dynamics, or P. Appell's Traiti de nitcanique rationale, Vol. II. Paris, 1904. 86 VECTORIAL MECHANICS Multiply (40'), scalarly, by s = s' and remember that sVsa = o, identically; then , , ,, ,. ,- sWA#=sVqp; but d^ldt=o; hence ~dt ( 4/8/ ) = ~dt ^ = SVQP ' ^ In this equation, which is generally true, we could write qs instead of q's' after the symbol djdt, since the scalar product of two vectors is certainly independent, also as regards its rate of variation, of any system of reference. Now, in the special case considered, the product sVqp on the right side of (44) vanishes. To see this, remember that, by (43), the ellipsoid of inertia is an ellipsoid of rotation, so that the vectors p, q are in the plane passing through the axis of symmetry x 3 ; hence sVqp = x x Vqp = o, since Xj, p, q are coplanar. Thus, by (44), (qx 1 ) = ' l = o or q^ = A'J/J = const., whence also p^ = const. The third invariant, in the case of Lagrange and Poisson, is thus P*i=A (Is) i.e. the angular velocity of the system about its axis of symmetry. In other words, the component of the angular velocity taken along the axis of symmetry retains its initial value. Combining (I 3 ) with the two other, general, invariants (Ij), (I 2 ), the complete solution may be reached at once, that is to say, the angle # = , ^ ' see the same work or Routh's Rigid Dynamics. RIGID DYNAMICS 87 turn only about a fixed horizontal axis, having thus one degree of freedom only, may be obtained immediately from the invariant T- ^as = E = const. This simple system, the state of which is determined completely by two scalars, namely by the angle 0=<&, a and by the angular velocity Q, has but 2 - i = one essential invariant, and this is the energy E, or any function of E alone, which is exactly the same thing. Denoting the pendulum's moment of inertia about the fixed horizontal axis by B and remembering that as = cos#, we have E = \B& - c . cos = const., whence, by differentiation, JBti=-c.smO, (45) a well-known result. The coefficient c is, by (39), equal to MgS or to the so-called directive moment of the pendulum. Kinematical Relations. To determine the instantaneous position of the rotating rigid system relative to a * fixed ' system of reference, the well-known angles of Euler 0, ^, < are usually employed. The component angular velocities A> A>A are t ^ ieii expressed by linear functions of the fluxes 0, ^, < with coefficients depending on simple trigo- nometrical functions of the angles 0, ^, < themselves (cf. one of the works cited above). These relations are known under the name of Hauler's kinematical equations. Euler's angles are very convenient parameters, especially as their number is three^ i.e. equal to the number of degrees of freedom of a rotating rigid system. Nevertheless their choice is based on distinguishing one of the three axes from the other two ; in other words, the angles 0, ^, are not ma y De replaced by other scalars, but their number will not thereby be changed. Instead of nine scalar data three vectors may be substituted ; 3 x 3 = 9. The formula (XH.) is a short expression for the three scalar equations (xn.a). Omitting the vector A, to be operated on, we may conveniently write for the operator : In order to get from this the vector B = wA, we have only to write A l as a factor in the whole first column, A 2 in the second, A 3 in the third column; then, the first horizontal line (its terms * The above definition of a linear vector function may also be deduced from another, in some regards more convenient and less artificial definition, viz. as stated by Gibbs- Wilson (loc. /. p. 262) : ' A continuous vector function of a vector is said to be a linear vector function when the function of the sum of any two vectors is the sum of the functions of those vectors. That is, the function/ is linear if /(ri + r 2 )=/(r 1 )+/(r 2 ).' 94 VECTORIAL MECHANICS being joined by the signs + ) gives B^ the second B^, the third B^. Now, it is not necessary to write the ' A ' nine times ; it is sufficient to imagine it done. With this understanding the scheme or symbol (xn.^) will be very useful. Generally, <*> 23 =f o> 32 , etc. But if it happens that OJ 23 = W 32> W 31 = W 13> W 12 = W 2l then o> is called a symmetrical linear operator, and the vector B = wA is called a symmetrical linear function of A. The symmetrical operator will henceforth be denoted by a capital omega (12). Thus, the operator 12 will be completely characterised by 9 - 3 = six mutually independent scalars, say by the six co- efficients J2 n , 12 22 , 12 33 , 12 23 , I2 31 , 12 12 . It may be written The diagonal, joining u and 33 is an axis of symmetry of the whole square table ; remembering this, it is unnecessary to fill out the vacant places. By permutation of the indices in the general operator 32 instead of o> 23 , etc., another linear operator w' is obtained, which is said to be conjugate to w. This relation is, of course, reciprocal, i.e. w is also conjugate to o>'. Now, by the definition of the symmetrical operator, fl 32 = I2 23 , etc. ; thus, 2 is a self-conjugate operator. Let and $ denote any two linear vector 'operators ; suppose that then we may write also B = o>A, w being the linear operator, the coefficients of which are equal to the sums of the corresponding coefficients of , ^ *-( w n = ^n + ^ii W i2 = ^12 + ^12' etc - This may be expressed by writing (0 = (f) + ^ and by calling w the sum of the operators , ^. It follows at once that MECHANICS OF DEFORMABLE BODIES 95 The same holds for subtraction. Whence follow also the rules for the multiplication or the division of a linear vector operator by a scalar number. If n be an arbitrary scalar, then not is simply a linear operator which has the coefficients w> n , w 12 , etc.; hence we may write also A may always be split up into a symmetrical linear function of A and a vector product of A by another vector c which is characteristic for the operator 12 ) 96 VECTORIAL MECHANICS but the sum of all the terms on the right side is the vector product of the vector C = i( w 32 - W 23) + J( W 13 - W 3l) by the vector A, i.e. or, omitting again the vector operated on, t = Vc, which completely proves the theorem. Thus, we have, for any linear vector operator : w = ft-f JVc, where \ ft = i() has first of all to be multiplied scalarly by Oj, or 2 , or 3 , and that the scalars thus obtained are to be multiplied, respectively, by i, j, k, and added. As to the symmetrical operator 12, we know already that it has, generally, three mutually perpendicular principal axes to which correspond the three principal values, say 12 1} 12 2 , 12 3 (ordina/y scalars). For this operator, it is most convenient to take the ortho- gonal system of reference i, j, k along the principal axes themselves ; then fi n = 12 i> fl i2 = etc -> or in the tabular form : = fl A = 1 (XV.) as we wrote previously, when we were considering the dynamics of a rigid system. (The ' K" 1 has, of course, been a special case of 12.) We then saw also that, for any pair of vectors A, C, C12A = A12C, (xvi.) i.e. the scalar product of a vector by a symmetrical linear function of another vector is equal to the scalar product of the second by the same function of the first. Observe that this property does not belong to a non-symmetrical operator w; in fact, as may easily be shown, we have in this case the more general relation CwA = Aw'C, (xvi'.) w' being the conjugate of w. The relation (xvi.) follows from (xvi'.) as its special case ; since 12 is .^//-conjugate, i.e. 12' = 12. * In Heaviside's denotation, whereas Gibbs uses the dots as symbols of scalar products (and little crosses as symbols of vector products). V.M. G r v 98 VECTORIAL MECHANICS Strains. Let the vector a define (relatively to the system of reference O) the position of an individual point or 'particle' of a body or medium before the deformation, and the vector r the position of the same particle after the deformation, or, if we prefer it : the respective positions of the same material particle at the initial instant / and at any other instant /. The individual particle, the position of which before the deforma- tion is determined by a, we shall call the particle a, or the point a. Fig. 33 represents the position of such a point, i before, and i' after the deformation. Similarly, 2, 3 are the positions of two otRer points before, and 2', 3' after the deformation. FIG. 33- We shall assume that the vector r is a continuous function of the vector a, admitting definite and continuous derivatives with respect to any component of a. In speaking of the continuity of a vector, we mean, of course, continuity as regards both tensor and direction. Discontinuities will be considered later on. Let i and 2 (Fig. 33) represent the positions of two points at an infinitesimal distance from one another, a = a a and a = a 2 , before the deformation of the body. The infinitesimal vector 1 = a 2 - a x , drawn from i towards 2, will characterise some linear element or j one-dimensional assemblage of individual points of the body. MECHANICS OF DEFORMABLE BODIES 99 The positions i', 2' of these points after the deformation will be denoted by r = r t and r = r 2 respectively, so that l / = r 2- r i is what 1 becomes after the deformation, namely the vector i'->-2' (Fig- 33)- The infinitesimal vectors 1 and 1' represent the length and the direction of one and the same linear element before and after the deformation, respectively. What, then, becomes of the element 1 after the deformation ? In other words : What is the relation between 1' and 1 ? Let us denote by V the Hamiltonian in * the space a,' i.e. in that space in which the position of a point is given by a. Then the change of any magnitude < (scalar or vector) due to passing from the point a to the point a + 1 will be expressed by infinitesimals of higher order than/ being neglected.* The operator IV has (by Chapter I.) a scalar character, expressing simply the component of the gradient or slope in the direction of 1, multiplied by the length / of this line-element. This can be applied immediately to the vector r, i.e. to < = r. Thus we get the required expression for r 2 - r^ or 1', namely l' = (lV)r. (48) If r be given as a function of a, i.e. if we know precisely the deformation to be dealt with, then the operation indicated on the right side of (48) may be performed, for any a and for any 1, i.e. for any line-element having its origin in any point of the body. Thus, (48) gives the length and the direction of any linear element after the deformation in terms of its length and direction before the deformation. The direction of the vector (IV) r is, of course, generally different from that of 1. The element 1 will, in other words, change not only in length / but also in direction. In addition, it will be also displaced as a whole, this purely translational displacement being namely i -> i' or r-a (Fig. 33). * If it should happen that (1V)0 = O, then, of course, infinitesimals of the second, and eventually of higher orders, have to be considered. ioo VECTORIAL MECHANICS The general term 'deformation' implies all these changes. The proper deformation of a linear element will be only its elongation (contraction = negative elongation) ; the deformation of an element of surface or of volume will imply change both of dimensions and shape of the elementary area or volume. But all the changes of two- or three-dimensional elements may be easily deduced from the changes of linear or one-dimensional elements. Therefore the formula (48), or its scalar equivalents, has a most fundamental importance in the whole theory of deformations or strains. Instead of the vector r, determining the position of the point a after deformation, the displacement of this point, i.e. the vector D = r-a, (49) may easily (and conveniently) be substituted in the fundamental formula and in those which will follow from it. In fact, IV being a distributive operator, we have 1' = (1V)(D + a) = (1V)D + (IV)a ; now, the second term on the right side is the change of a itself due to passing from the point a x to the point a 2 , i.e. simply (!V)a = a 2 -a 1 = l; hence But in what follows the preceding formula (48) will be used, as being somewhat simpler. In the results, when once obtained, it will always be very easy to pass from r to the displacement D by means of the simple relation (49). By (48) the vector 1' is already seen to be a linear function of 1 : hence it may be written r = wi. But we have still to find the properties which in this case belong to the linear vector operator w, i.e. to split o> into its symmetrical and non-symmetrical parts, to determine the precise form of each of these parts as dependent on the given deformation and, finally, to give their physical interpretation. By (xni.) we may write at any rate : l' = Ql + iVcl; (50) thus, the problem is reduced to developing the expressions for the symmetrical operator 12 and for the vector c. Both must be deduced, of course, from the particular linear relation (48) : (48 bis] MECHANICS OF DEFORMABLE BODIES 101 In order to avail ourselves of the reasoning and the symbols of the last section, let us employ, for the moment, as a system of reference, the normal unit vectors i, j, k, i.e. let us write and similarly Vj, etc., being the 'components' of V, i.e. the derivators in the space a or if a = i< Then ' iv=/^ hence, by (48), 1' =

    + w'), by (XIIL), will be determined by the coefficients and the vector c = i(w 32 - o> 23 ) + ... will be, by (510): c = 102 VECTORIAL MECHANICS but the sum of the terms on the right side is the curl of the vector r taken in the space a; thus c = curl (a) r, where (a) is an abbreviation for ' relatively to the space a,' that is c \ 2 Relatively to the space r we should have curl r = o, i.e. curler = o identically, since all r's are radial vectors, drawn from the common origin O, so that r is irrotational ; using the coordinates r 19 r 2 , r s we should have simply cV 3 /3r 2 = o, etc., or curl (r) r = o again. All this is to justify the index (o) , in the above expression of c. It must be remembered that our V has also been an abbreviation for V (o) , and the curl is nothing else than VV. Hence, both are to be taken in regard to the space a. But now, after these explana- tions, we shall omit the index (a j, and assume it tacitly without writing it, both for curl and V. The curl so understood, applied to a itself, gives, of course, curl a = o identically ; hence, if D = r - a is the displacement, as in (49), we shall have c = curl r = curl D. (51) Having determined both the symmetrical operator ft and the vector c, we can now apply fully the theorem (xin.). Thus, we have ultimately as the expression of the relation between the deformed line-element 1' and the original line-element 1 : l' = o>l = Gl + Vcl, ' (50) where the vector c is given by c = curfD (51) and the symmetrical linear operator ft by its six coefficients ft tl = V t r t ; Q M = (V t r K + V K r L ) = ft Kl , (52) for i, K = i, 2, 3. Each differential operator is to be taken with respect to the space a. The vectors D and I are connected by the simple relation D = r - a. (49 bis] The interpretation of this result for the case of an infinitesimal c is seen at a glance. , ; . . / ; ; ; MECHANICS OF DEFORMABLE BODIES 103 The term *Vcl expresses then a rotation of the line-element 1 by an angle \c about the axis c ; it is this rotation which is given, in amount and direction, according to (51), by the vector \ curl D, D being the displacement of a point. The angle and the direction of the axis of rotation will, of course, be generally different for different a's, i.e. for different parts of the strained body. If curlD = o, the strain is called irrotationai. This does not mean that no line- element 1 experiences a change of direction, but only that no volume- element, as a whole, does experience a rotation, as will be better understood from the following explanation. The symmetrical part of the operator to in eq. (50), i.e. the operator 2, produces by itself a change, not only in the length, but generally also in the direction of a line-element 1. Hence, if also curl D = o, then some particular line-elements only do not experience a change of direction. 12, being symmetrical, has in general three principal axes, mutually perpendicular, and three corresponding principal values ft 1? 12 2 , 12 3 . Only those line-elements which before the deformation coincided with one of these axes will conserve their direction notwithstanding the deformation. If the values flj, & 2 , 3 are all different from one another, then only three such special directions will exist, but at any rate not less than three, mutually perpendicular, directions. Let us imagine a bundle of (oc 2 ) line- elements diverging from a given point a in all directions, but having, before the deformation, one and the same length /, so that the surface /= const, will be the boundary of a volume-element of the body, initially spherical. Now, by the deformation, this sphere will become an ellipsoid, of which the principal axes will coincide with the principal axes of the operator 12. Hence, if curl D = o, the line- elements which coincide with the axes of this ellipsoid initially will remain coincident with them after the deformation. Thus, we see that in this case the elementary sphere will be transformed into an ellipsoid, that generally the diameters of the sphere will experience a change of direction, with the exception however of three mutually perpendicular diameters, which will maintain their direction and, consequently, that 'the sphere as a whole will not be turned.' This is the reason why a strain satisfying the condition c = curl D = o is called irrotational. 104 VECTORIAL MECHANICS Returning now to the more general case, i.e. for A 3 = V 3 > 3 , (55) where D l> etc., are the components of displacement taken along the principal axes of 12 or of the strain characterised by 12. Using ordinary symbols, (55) would be : *, etc. ' If the strain be heterogeneous, it must be remembered that the lines along which a lt # 2 > a s are measured are not straight, but generally curved lines which may be considered as the limits of chains consisting of the corresponding principal axes;* we have then a threefold curvilinear, orthogonal system, and the indices i, 2, 3 are symbols of components taken tangentially to such lines, passing through the given point a. Similarly, if c = curlD : fo, the rotational lines, i.e. the lines showing everywhere the direction of the axis of rotation, will generally be curved lines. These lines may traverse the triple system of elongational lines, generally, in all imaginable directions. * The homogeneous strains ' tangent ' to the given heterogeneous strain being different for different points a. MECHANICS OF DEFORMABLE BODIES 107 Having learned the meanings of the different terms and having become familiar with the operator ft and its companion Vc, we shall henceforth make free use of the short formula (50). Let us imagine three non-coplanar line-elements 1, m, n diverging from one and the same point a (Fig. 35) ; these define an elementary L... r ' m m FIG. 35. parallelepiped, of which the volume dr before the deformation is g iven b y ^T = lVmn. (56) After the deformation of the body, let 1, m, n become 1', m', n', respectively, and the volume of the parallelepiped dr ' ; then dr = I'Vm'n'. It is required to compare dr' with the original volume dr. Let again the tensor of c be infinitesimal ; then the part -|Vc of the operator o> in (50), representing a pure rotation of every line-element (with the point a as origin) by one and the same angle \c about one and the same axis, will, of course, exercise no influence on the volume of the parallelepiped. In other words, 1 the change of volume will be the same as if c were equal to zero. Thus, it will suffice to take into account the symmetrical part of the operator only, and to write 1' = 121, m' = 12m, n' = 12n ; dr ftlVftmftn -- Now, this ratio of scalarly-vectorial products is an invariant characterising 12 itself, i.e. it does not change at all if instead of 1, m, n any other set of three (non-coplanar) elementary vectors, diverging from the same point a, be taken. This property belongs also to the general linear operator w, i.e. IVmn is an invariant of w. This theorem is proved in the following way : *Cf. Heaviside, loc. cit. io8 VECTORIAL MECHANICS The most general vector may be expressed by a > A 7 being freely variable scalars. Now, if we take al instead of 1, both the numerator and the denominator will be simply multiplied by a, so that the quotient x will not be changed; again, adding /3m, we get in the numerator the complementary term ft . wmVcomcon, which vanishes identically, by (vn.), and similarly also in the denominator (3 . mVmn = o ; hence x will still be unaltered. And it will remain invariable if the term yn be added. Hence, if 1 be changed in the most general way, the value of x will not change ; and, similarly, if instead of m, n any other vectors be taken, x will still keep its value. Q.E.D. Applying the theorem to ft (as a special kind of w), we see by (57) that the ratio dr'jdr is independent of the dimensions and of .the shape of the volume-element dr., and depends only on the properties of ft, i.e. on the properties of the given strain, at the place where dr is taken. Hence, the ratio n dr' - dr dr' -3r- = #-*' which is called the cubic dilatation, or simply the dilatation, has a perfectly definite meaning, requiring no further explanatory speci- fications. We can speak simply of ' the dilatation at a point? Now, the value of the ratio (57) being independent of the choice of the line-elements 1, m, n, these may be, most conveniently, taken along the principal axes of 12 at the given point ; hence, denoting the directions of these axes by i, j, k, and taking 1 = /i, m = mj, n = k, so that mWfiji, etc., we have dr fr~ iVjk the principal values fi t , Q 2 , ft 3 being ordinary scalars. Thus, by (520), the cubic dilatation will be : = (i +V 1 Z?,)(i + V 2 > 2 )(i +V 3 ^ 3 ) - i, (58) or, by (55), -Ag)-!, ( 5 8a> where \, A 2 , A 3 are the principal linear elongations. Notice that we have seen already that an elementary sphere of radius /, and consequently of volume ^r/ 3 , is transformed intOj MECHANICS OF DEFORMABLE BODIES 109 an ellipsoid having ftj/, 12 2 /, 12 3 / as semiaxes, and consequently the volume so that, in fact, the ratio of volumes is fi^o^y, as above. Infinitesimal Strain. A strain is said to be infinitesimal if the products and squares of the derivatives 'd/'da^ 3/3a 2 > ^/^% f anv components of the displacement D be negligible in comparison with their first powers (after the final rejection of any arbitrary displacement of the whole body in space). Under such conditions, the cubic dilatation becomes, by (58^), a#, 9 A neglecting A 1 A 2 = -^ -^* and a jortion A^Ag, OCL-^ OClf) 6> = A 1 + A 2 + A 3 , ( 59 a) or, by (58) : 6 = VjZJj + V 2 Z> 2 + V 3 Z> 3 = VD, i.e. (see Chapter I.) : = divD; (59) thus, the cubic dilatation is equal to the divergence of the dis- placement. This result may also be obtained more directly, without (58), by simply recurring to the definition of divergence by means of the surface-integral, i.e. by the original definition of div as given in Chapter I. For, let T be the volume of any portion of the body, before its ! deformation, bounded by the surface 3 = ?g/3=/3, and similarly V 3 Z> 2 = /7=y ; hence so that Similar meanings will be found for 12 31 , 12 12 with respect to k, i ; i, J- Thus, the expressions in (A) are, for an infinitesimal strain, the shears parallel to the planes j, k; k, i; i, j respectively. If i, j, k be taken along the principal axes of 12 in the given point a, the shears fi 23 , etc., vanish, of course, i.e. the right angles between these axes remain right angles after the deformation. The Equation of Continuity. Let p be the density of mass of the body, i.e. let p dv be the mass of its volume-element dr, before the deformation. Let us denote by 8 the change, due to an infinitesimal strain* of any magnitude belonging to an individual element of the body. Then, the so-called equation of continuity, which is the expression of the invariability of the mass of every individual element, will be 8 (p dr) = o or dr . fy> + p 8 (dr) = o, or, substituting the dilatation = 8(dr)/dr, 8p + p6 = o. Hence, by (59), the equation of continuity may be written 8/3 + p div D = o. (60) Introducing, as in the beginning, the vector r, and remembering that D is to be infinitesimal and that 8a = o (because a itself individualises a given element), we may write D = 8r, and consequently, instead T) = o. (6oa) If Sr is to be any virtual displacement, then it must in the first :e satisfy this condition. * Henceforth only infinitesimal strains will be considered. H2 VECTORIAL MECHANICS Classification of Strains. If the displacement D satisfies throughout the strained body, or throughout the whole ' region a ' considered, the condition curl D = o, then, as we have seen already, no element dr experiences a rotation. Consequently, such a strain has been called irrotational, or longi- tudinal. The meaning of the last name will be explained in the next section. On the other hand, if D satisfies, throughout the whole region a, the condition div D = o, no element dr experiences a change of volume. Such a strain, which will be met with in the study of incompressible elastic solids or liquids, is called a purely transversal strain. It might also be called a solenoidal or circuital strain, these epithets being usually attached to any vector w, or better, vector-field w, satisfying the condition div w = o ; but we prefer the name ' transversal strain,' the meaning of which will be cleared up further on. It is not difficult to prove that every vector-field w may be decomposed, and this, moreover, in one single way, into a purely irrotational and a purely solenoidal part. Thus, the most general strain may be represented as a superposition of a longitudinal and a transversal strain. In the superposition of any two (or more) infinitesimal strains given, say, by D I} D 2 , i.e. in the strain given by D^Dj + Do, we have a simple superposition both of the dilatations and the rotations of any element. For, both curl and div are distributive operators, so that = div (D x + D 2 ) = div Dj + div D 2 = ^ + 2 , and similarly c = curl (D x -f D 2 ) = curl D x + curl D 2 = c x + C 2 . If the first strain is purely longitudinal, and the second purely transversal, then = lt c = c 2 , i.e. the resultant strain owes its dilatational properties wholly to the first, and its rotational properties entirely to the second strain. Thus, it is convenient, and possible, to treat separately the two kinds of strain, longitudinal and transversal. Such a splitting is not artificial at all. If the time-development of the phenomena is MECHANICS OF DEFORMABLE BODIES 113 considered, then the two kinds of strains are, generally, actually separated, i.e. the respective disturbances are propagated with different velocities, so that if also emitted by the source simultaneously, the first kind will soon lag behind the second, or vice versa. Longitudinal Strain. The condition curl D = o being fulfilled everywhere, the displacement has a scalar potential, say ^, i.e.\ D = V^. (61) This potential is not necessarily a single-valued function of the position of a point in the region a, say, for example, of the co- ordinates flj, a. 2 , a 3 . It will be such a function for an acyclic or simply-connected region, as for instance for a medium occupying all the (Euclidean) space, or for a sphere, for a cylinder and for similar bodies. But generally it may be a many-valued function, viz. for a cyclic body, as for an anchor ring. The displacement is in the present case, by (61), normal to the surfaces ^ = const. ; for, by (x.), we may write Two equipotential surfaces, to which correspond the values ty and $ + d$ of the potential, cut out from the body an infinitely thin sheet or lamina; thus, the whole body may be split into a series of such sheets. If e is the (infinitesimal) thickness of one of such sheets, in a \ given place, before the deformation, and ' after; then, putting e' - e = 8e, we shall have as the formula for the relative thickening of a sheet. The cubic dilatation will be, by (59), or 6 = V 2 ^; (62) in ordinary rectangular coordinates, for example, we should have H4 VECTORIAL MECHANICS But, in general considerations, it is best to retain the short formula (62) without specifying the choice of any system of coordinates. V 2 is, at any rate, quite independent of any system of reference. (See Chap. I.) Writing again, as in (50), for an arbitrary line-element, the wJwle operator w will, in the present case of irrotational strain, be a symmetrical linear operator : w = ft, for c = curlD = o. By (520), the coefficients of 12 will now be, with any directions of the axes of rectangular coordinates a lt 2 , # 3 , i/ y 3_ y+ 23 2 \daa B 'dafia Vaki daa B 'dafia so that the shears will be given by -, etc. If all the shears vanish for any coordinate planes, the strain con- sists in a pure change of volume ; for then we have, for any choice of the rectangular coordinate axes, In other words : any direction whatever plays in this case the part of a principal axis ; hence so that or I' \l=N, independently of the direction of the line-element. Thus, in this particular case, the operator fi degenerates into an ordinary scalar multiplier. The reader may verify this result by returning to the equations 3 2 ^/? 2 3 3 = o, etc., and deducing from them the conclusions which easily suggest themselves. MECHANICS OF DEFORMABLE BODIES 115 If the displacement is the same (as regards both absolute value and direction) for all points situated on any plane 0^ = const. and varies only on passing from one such plane to another parallel plane, or in other words if D depends only on a lt then At the same time it follows from the condition curlD = o, that i.e. DI = const., Z> 3 = const. Thus, the transverse components of displacement, i.e. the components normal to the axis of a lt are the same for the whole body, giving by themselves only a displace- ment of the whole body in space, as if it were a rigid body. But as we are here concerned rather with relative displacements of some parts of the body in regard to others, we may, without impairing the generality, write simply Thus, what remains will be D l only, i.e. a purely longitudinal displacement. Hence the name given above to strains satisfying the condition curlD = o. Transversal Strain. If, in the whole body, so that the volume of any of its portions remains invariable, we may put D = curlA, (63) with the supplementary condition for the auxiliary vector A : div A = o. (64) Generally, if a (solenoidal) vector X is put into the form curlY, then Y is called the vector-potential of X. Thus, A will be the vector-potential of the displacement D. For a longitudinal strain, we had D = V^, whereas now we have, somewhat analogously, writing curl = VV, n6 VECTORIAL MECHANICS The scalar potential of displacement does not exist for a transversal strain; it is replaced here, mutatis mutandis, by the vector-potential. By (63) we have for the vector c = curlD, i.e. for twice the rotation of an element dr, c = curl 2 A. Now, for any vector, say w, we had, in Chapter L, the identical formula cur! 2 w = V div w - V 2 w. (xvn.) In our case, div A being =o, we have, more simply, c=-V*A, (65) and this is only a vectorial generalisation of the well-known equation of Laplace-Poisson. Using, for example, rectangular components i, 2, 3, we may split (65) into Now, as is well known from the elementary theory of potential functions, with similar expressions for A^ A z (if A lt etc., and their first derivatives are finite, continuous and vanish 'at infinity' in the well-known way). Hence, recombining again, i f c , . = - dr, 47r)r (66) where dr is any volume-element of the body (in which the rotation c does not vanish) and r the distance from dr to that point P^ for which A has to be calculated (see Fig. 38), the integration, FIG. 38. of course, being extended to all rotating particles, or, what turns out to be the same thing, to the whole strained body. To this formula and to the analogous one for the scalar potential, *=-- MECHANICS OF DEFORMABLE BODIES 117 which follows from the differential equation (62) precisely in the same way as (66) from (65), we shall return later, namely in the kinematical part of the subject, where not the displacement itself but its velocity or time-rate of change will be treated. We shall then give also the interpretation of these formulae. At any rate, it must be observed that both the scalar and the vector-potentials are auxiliary notions, while the displacement, the rotation or dilatation have immediate physical meanings. Supposing again (as for the longitudinal strain) that D, in a transversal strain, depends only on a lt we have from divD = o: or D l = const., so that, by the above remarks, we may put, in this case : A=; thus, the remaining displacement is entirely normal to the axis of aj, or Di = o. This is the reason why a strain satisfying the condition divD = o has been called a transversal strain. Surfaces of Discontinuity. Till now the displacement D has been always assumed to be a continuous function of a(aj, a 2 , 3 ) ; in other words, it has been supposed that on passing from one particle to others the vector D changes in a continuous manner. But now let us admit that at some surfaces (i.e. on traversing some surfaces from one side to the other) the displacement itself or its derivatives, or the derivatives of some of its components, may jump or experience a discontinuity. FIG. 39. Let o- (Fig. 39) represent such a surface of discontinuity. Let the unit vector n be the normal of one of its elements d ., O)O1O)O) ^^^4^^^ FIG. 40. figure being n, S. The axes of these tubes, not necessarily straight, will coincide with the rotational lines, i.e. with the lines indicating everywhere the direction of the vector c = ^-curlD. In regard to this equivalence of surfaces of gliding and of sheets of rotation, the following remark may not be superfluous. When we say that a given strain is purely irrotational (or ' longi- tudinal') in the whole body or medium in question, we mean that not only curl D = o in continuous regions but also that there are no infinitesimally thin sheets of rotation or surfaces of gliding, i.e. that not only curlD = o, where continuity reigns, but also [T] = o and 120 VECTORIAL MECHANICS consequently [D] = o at any surfaces of discontinuity. The con- dition that the body should not be torn requires the continuity of the normal component, and the supposed complete irrotationality of the strain requires the continuity of the tangential part of the displacement ; thus, the whole displacement must be continuous, [D] = o, for irrotational strains. Nevertheless, certain surfaces of discontinuity are possible for such strains ; in fact, the displacement D itself must then be continuous, but different magnitudes deduced from it, say, by the process of differentiation in regard to space, may be discontinuous. Such discontinuities have played, since the times of Riemann, a very considerable part in many branches of mathematical physics. In order to become acquainted in the easiest way possible with some of their fundamental properties, let us begin with the simpler case of a scalar. The passage to discontinuities of a vector will then follow almost immediately. Let be any scalar function of position in the space a ; suppose that, through the surface cr, the function c/> remains continuous, i.e. M = , but that its first derivatives in regard to space, say 3/'da 1 , etc., and consequently also its slope V<, is discontinuous, or that Then the discontinuity is called a discontinuity of the first order. Similarly, if \ -^ \=h o. (The time, however, will come in only in the next section of this chapter.) Generally, if the function < itself and all its derivatives up to the (n- i) st order, inclusively; are con- tinuous, and the derivatives of the # th (with or without the deriva- tives of higher) order become discontinuous, then the discontinuity is called of the th order, The mere fact that the discontinuities are supposed to be dis- tributed over some surface cr, or /= o, leads of itself to certain con- ditions, which Hadamard calls the identical conditions.* Limiting ourselves to the consideration of discontinuities of the first order only, we shall now proceed to develop the corresponding identical conditions. * J. Hadamard, Lemons sur la propagation des ondes et les Equations de Fhydro- dynamique, Paris, 1903, p. 81. Compare also G. Zemplen, Encyklop. d. math. Wissenschaften, Vol. IV. 2 Heft. 3, 1906, and P. Appell, TraiM de mecanique^ Vol. III. Chap. XXXIII. iii. 1903. MECHANICS OF DEFORMABLE BODIES 121 Let, then, [] = o, (68) but [V<] * o, (69) at the surface a-. This surface divides the space into two regions ; we shall suppose that in each of these not only the gradient V< exists and is continuous, but also that on approaching the surface a- from one or from the other side, V< and consequently also 3", whence, by subtraction, and remembering that [<] = o on the whole surface, and consequently d[$\ = d<$ d<$' = o, This equation is true for every 1 lying on o- ; hence : The vector [V<] or the jump of the vector V is normal to the surface of discontinuity (provided always, as has been supposed, that < itself remains continuous). Denoting, then, by A a scalar quantity, the values of which will generally be different for different points of cr, we may write [Vfl = An (70) or, remembering that n = Vf:'df/'dn, A. being again a scalar, viz. A = A : 'df/'dn, and /= o the equation of the surface. The equation (70) or (700) is the vector equivalent of Hadamard's Jthree scalar identical conditions for a discontinuity of the first order. The value of the scalar A, i.e. the absolute value of the jump, remains arbitrary ; but if it be given, all about the discontinuity is known, since the direction of the jump is determined, being, by the above theorem, everywhere normal to the given surface f=o. 122 VECTORIAL MECHANICS Of course, it has been tacitly assumed here that this surface has at the points considered a determinate .normal. The above theorem may now be applied to each of the com- ponents of any vector, for instance, of the displacement D. If all the components of D itself are continuous, i.e. if and hence also [D] = o, ( 7 ifl) then we have, by (70), writing simply < = Z> 15 or D. 2 , or Z> 3 , and denoting by m lt m. 2 , m z three (mutually independent) scalars, every one of which takes in turn the place of the above A, (71') where we may write also n = V/:'df/?)?t. The three vector equations are equivalent to Hadamard's nine scalar identical conditions for a discontinuity of the first order of a vector. Now, the scalars m lt m. 2 , m 3 may be considered as the com- ponents of a vector, say m : m = im l + jm 9 + kw 3 , this vector being characteristic for the given discontinuity. In fact, if we know the form of the surface f= o and the vector m for every point of this surface, then we know everything about the discon- tinuity. The reader will easily show that m is independent of the choice of the system of reference (and it is this that proves its true vectorial character). Consequently we have no need of such a system, and thus instead of the three equations (71') we may write a single equation. In fact, let x be any vector whatever, say a unit vector, of a quite arbitrary direction ; then we may write instead of (71') : [V(Dx)] = n(mx). (71) We here reach the extreme of condensation, viz. nine scalar identical conditions of Hadamard packed closely into a single formula. From this the formulae for the jump of the (double) rotation or of c = curlD and for the jump of the cubic dilatation # = divD may immediately be deduced. MECHANICS OF DEFORMABLE BODIES 123 In fact, divD = V 1 Z> 1 + V 2 Z> 2 + V 3 Z> 8 = VD and curl D = i(V 2 Z> 8 - V 8 Z> 2 ) * hence [0] = [divD] = nm, (72) [c] = [curlD] = Vnm. (73) These formulae show an instructive analogy to one another, especially if they are written in a slightly different way, [VD] = nm, [VVD] = Vnm. From (72), (73) the truth of the following propositions is seen immediately : i. If the vector m be tangent to the surface of discontinuity, then mn = o and [0] = [divD] = o, i.e. the cubic dilatation remains continuous. 2*. For any direction of m, (73) multiplied scalarly by n gives : [nc] = [n curl D] o ; the normal component of the rotation remains continuous, so that only its tangential part may jump. 3. Similarly, multiplying by m, [me] = [m curl D] = o, i.e. the component of the rotation taken along m also remains continuous. Thus, the whole jump of the rotation, considered as a vector, is normal to the plane m, n. If m be given, both [6] and [c], or [divD] and [curlD], are given ; conversely, if [0] and [c] are known, also m, the vector characteristic of the discontinuity of ist order, is completely known, since [0] determines its normal and [c] its tangential part. Kinematics of a Deformable Body. Until now the strain has been considered without regard to the -interval of time in which it has been produced. The displacement D, which in the last sections has been supposed infinitesimal, was simply the difference of the vector r determining the position of a given particle 'after deformation' and the vector a determining the position of the same particle ' before deformation,' independently 124 VECTORIAL MECHANICS of the time-interval that has elapsed. Let us now bring in the time /. The infinitesimal displacement, till now denoted by D, will henceforth be written d"D and considered as taking place in the infinitesimal time-interval dt. Since a characterises a given indi- vidual particle of the body, so that d& = o (a being simply the vector defining the 'initial' position of the particle, say for /=/ ), we may write di instead of d'D, remembering that we had generally D = r - a. The infinitesimal vector dT> = di will express the element of the path described by the individual particle a during the interval of time / -> / + dt. Thus, denoting by v the instantaneous velocity of motion of the particle, we shall have _ be any (scalar or vector) quantity belonging to a given particle of the body. Then, following the individual history of the particle, viz. watching it in its motion, we shall denote the time-rate of change of ^ by , is called the vector-potential of velocity. The reader may compare this with what has been said about the corresponding potentials of displacement. The interpretation, promised on that occasion, of the formulae i fcurlv . i fw , B = dr = dr 47rJ r 2Trjr i fdivv _ and < = -- dr 4 7rJ r will be given later on, in the chapter on Hydrodynamics or rather in its purely kinematical part. There we shall say also a few words about ' lines of flow ' and ' vortex lines.' For the present these general remarks about the kinematics of a non-rigid body (whether it be a fluid or, more particularly, a liquid or an elastic solid) will suffice. But before passing finally to dynamical considerations it will be useful once more to consider surfaces of discontinuity, this time from the kinematical point of view. Kinematics of Surfaces of Discontinuity. Let again f o be the equation of a surface of discontinuity o-. If D be the displacement, we have, for a discontinuity of the first [D] = o, while the vectors VZ> 15 VZ> 2 , VZ> 3 and dDjdt=v are, generally, discontinuous : We have already seen that the first three of these discontinuities are completely defined by a single vector m. In order to determine the fourth discontinuity, i.e. that of the velocity v, another vector must be given, say [v] = m'. Thus, the discontinuity of the first order will be defined entirely by two vectors, m and m', which, of course, may be different for different points of the surface o-. These two vectors, however, are not mutually independent. That is to say, if the discontinuity, distributed over some surface at the instant /, is to exist also at the instant t+dt on some surface, i.e. if it is required that the surface of position and time, such that, on traversing the surface a- or f=f(a^ a%, 3 , = o > [*]-. If this condition is to be satisfied not only at the instant /, but also at t+dt, we have i.e. where 1 is any line-element satisfying the condition j-f dt=o. (*) Here we have written for the partial derivative with respect to the time the symbol , since a or ,, 2 , a~ have to be kept constant at on differentiating with respect to /, i.e. we have to follow the same particle of the body ; and it is for such derivatives that the symbol has already been used above. Now, take in the eq. (a) instead of , say, the first component of displacement D^ and remember that, by the 'identical conditions,' [V> 1 ] = m 1 n' } (71') thus or, writing again n = V/":3//3, * See Hadamard or Zemplen, loc. cit. 130 VECTORIAL MECHANICS hence, by (), r n n/zn M Similarly, we shall have w-s hence This is the relation between the vectors m' and m, alluded to. Remember that ^- is the same thing as the tensor of Vf or, in Cartesians, A discontinuity is called stationary, if the surface a-, though moving about in space and changing its form, is composed always of the same material particles, i.e. if f is a function of a lt 2 , a 3 not con- taining the time /, which we may write shortly /=/(*) = o. If, on the other hand,/ As tne components with equal indices being, thus, normal and those with different indices tangential to the surface acted on (Fig. 43). For any given direction of n, the normal component of the pressure will be A* = nP> and the tangential part of the pressure, in intensity and direction, p n -n(np M ). Finally, the components of p w along the conventional i, j, k will, according to the above, be denoted by lPr = Al > JPn = As > kP = As The stress is determined completely, if the vector p n be known for every point of the body or medium and for every orientation of da- or, what is the same thing, for every direction of n. Now, assuming that the resultant force of the stress, on an element of volume, is a magnitude of the same order as (or at least not lower than) this volume itself, it is easily proved that A i = Ai n \ + Ai a + Ai s > } Aa =A21 +Aa8 +A23 ( 8 3) A 3 = As *1 + As 7Z 2 + As ^3 J J where j, 2 , 3 are the components of the unit vector n along i, j, k, or the direction cosines of the normal n relatively to the same system of axes. Thus, in the most general case, the stress will be determined by nine scalars, say : Ai A2 As Ai A2 As Al /32 A3' (Also, it will next be shown that, for dynamical reasons, say, by d'Alembert's Principle, this number is at once reduced to six.) According to (83) it may be said that the pressure p n is a linear vector function of the normal n of the surface-element acted on, which fact may be expressed shortly, denoting by / a linear vector- operator, the stress-operator, and writing P=/n (84) in the same way as, in the general theory of such operators, B = o>A. Thus, the theory of stress is reduced at once to the theory of the linear vector-operator; hence it would be superfluous to dwell 134 VECTORIAL MECHANICS more on this subject. It will be sufficient to add a few words as to the terminology. Thus, principal axes of a stress are called those directions n, for which the pressure becomes purely normal ; hence, the principal stress-axes and the corresponding principal pressures (or tensions) are synonyms of the .principal axes and the principal values of the linear vector-operator /. If, in particular, this operator degenerates into a simple scalar, then the pressure is purely normal and equal for all directions of n, or becomes what is called a ' hydrostatic' pressure, i.e. an isotropic pressure. Taking / 32 instead of/ 23 , etc., another stress is obtained which is called conjugate with respect to that given by the operator p let it be denoted by q, i.e. ?=/ ( l > * = I 2 > 3)- Then (83) may be written thus : Ai = nq l5 / n2 = nq 2 , A 3 = nch (85') or, in a single vector equation, p n =/n = i.qan + j.qon + k.cbn, (85) so that the operator /, denning the stress completely, assumes again the form of a dyadic, as mentioned before, namely Now, let dr be a volume-element of the deformable body, dv a surface-element of its boundary, with normal n directed outwards, p the density of mass, and, finally, F the impressed (or external) force, per unit volume. Among the virtual displacements of the body, considered as a system free to move about in space (i.e. relatively to our original system of reference <9), are included : i the displacement of the whole body in any direction whatever, as if it were a rigid body ; 2 the rotation of the body as a whole about any axis. Thus, applying d'Alembert's Principle (i), Chap. II., first to the displacement i and then to 2, we have, respectively, Lr dr = LF dr - lp,X-> (a) (p Vrr dr = f/oVrF dr - f Vrp n <*r, (b) the integrals extending to the whole volume and over the whole MECHANICS OF DEFORMABLE BODIES 135 bounding surface of the body, and r being a short symbol for /22> As (normal pressures), As> Ai> A2 (tangential pressures). This property, as we have seen, belongs to every dynamical stress, i.e. to every stress obeying d'Alembert's (or any other equivalent) Principle. * For this purpose it suffices to recur to the formula div (sA.) = s div A + AJJs, which is satisfied identically for any scalar s and any vector A, and which gives fsdivA.dT= div (sA) . dr - (&.^s.dT= fsAnda- JAVs.dr. t This name is based on the circumstance that the differences Aj-As etc -> would be the components of the moment of a couple of forces (per unit volume) tending to turn an element dr ; and since / 32 =^ 23 , etc., this moment vanishes. MECHANICS OF DEFORMABLE BODIES 137 It may, by the way, be remarked here that Maxwell's 'electro- magnetic stress' (for isotropic bodies is, but) for crystalline bodies is not irrotational and, consequently, does not fulfil the above mechanical requirements. Henceforth, by the last proposition, we may write in all our formulae, p instead of q. Thus, the general differential equation of motion (86) becomes p-^ = /o^| = />F-idivp 1 -jdivp 2 -kdivp 3 . (87) This is the vectorial condensation of the three commonly used scalar equations, of which it will suffice here to write the first only : } (87-) r \t r 2"> r z being the same as the ordinary *#, y, z.' For a non-viscous fluid the tangential pressures vanish and those normal become equal for all directions of n, that is to say : As = Ai = A2 = and Ai-Aa^As* say=/. The linear operator, which we have in the general case denoted also by /, becomes now a simple scalar, i.e. a scalar function of position and of time. In this simplest case eq. (870) reduces to "and the vector equation (87) becomes '$-'%-*-** < 88 > To this fundamental equation of Hydrodynamics we shall return in the next chapter. For viscous liquids or (more generally) fluids, and for elastic solids the stress loses this idyllic simplicity ; the pressure (or tension) is no longer isotropic and, in general, not normal, but becomes normal for three particular directions only the principal axes, varying from point to point. In this case we must return to the general equation of motion, (87), taking into account the six different components of stress Ai> A2> As As> An A2 which depend generally in a rather complicated way on the elemen- tary deformations of the body. 138 VECTORIAL MECHANICS We shall not enter here into the details of this subject, but shall limit ourselves to the brief remark that, for infinitesimal deforma- tions, the normal and the tangential components of stress have a potential, i.e. may be represented as the partial derivatives of a scalar function f with respect to the six quantities which determine the strain. That is to say, denoting again by D the displacement and by # n , etc., jc 23 , etc., the principal linear elongations and the shears, we have , _3/. , .. a/__y_, 1 * u ~5.' ** S"S; *? [ (8 9 ) t, K=I, 2, 3, J where / is a quadratic homogeneous function of the six ' strain- characteristics ' x u , x lK . This function, taken with the negative sign, constitutes the so- called energy of elastic deformation, per unit volume. Remembering that the virtual work of the pressures or tensions will then be given, per unit volume, by Sf, the differential equations of motion of an elastic body may be easily deduced from Hamilton's Principle. Finally, it may be observed here that for non-viscous fluids the energy of deformation depends only on the cubic dilatation, i.e. the above function / reduces to whence it follows immediately that A3=Ai=A2 : = and Al=A2=As> as stated above. CHAPTER VI. HYDRODYNAMICS. Fundamental Notions and Equations. IN the case of a non-viscous fluid, of which the dynamics shall be the subject of the present chapter, the stress, as has been mentioned above, reduces to a pressure / purely normal and isotropic, not only in the state of equilibrium, but also in any state of motion. Under these conditions the equation (88) holds, where v is the velocity, p the density of the fluid, F the impressed force, per unit volume. We shall limit ourselves to the consideration of the so- called Eulerian form* of the hydrodynamical equations, and shall, consequently, choose as independent variables the time / and any three coordinates which determine the position of a point in space relatively to some system of reference which does not participate in the considered motion of the fluid, and which is mechanically equivalent to our previous 'system O.' Thus, understanding by V the gradient in this space and using the relation (740), we have, as the fundamental differential equation of Hydrodynamics, = | + (V V)V = F-IV / . ' (90) In addition, we have, by (78) and (78^), the equation of continuity \ ^ + pdivv = ^ + div(pv) = o. (91) If then the relation between the pressure and the density g(ft/) = , (92) * As distinguished from the so-called Lagrangian form, which will be omitted here, and in which a lt a 2 , a 3 , t are the independent variables. 140 VECTORIAL MECHANICS characteristic of the fluid considered, be given, we have three equations : one vectorial and two scalar, for one vector v and for the two scalars /, p. Thus, the initial data V , /> and the corre- sponding / , together with eventual data for the bounding surfaces, will completely determine the course of phenomena in time ; the forces F being always considered as given for every point and for all times. It may be observed here that, in the more general case, the relation between density and pressure may contain also the tem- perature; in this case, of course, the above equations do not constitute by themselves a complete system. But we shall suppose that the relation (92) contains p, p only (with a single exception to be considered later, in a section on vortex motion). If the fluid be, in particular, an incompressible liquid, we have simply, as a special case of (92), p const, and, consequently, divv = o. But in general we shall admit compressibility. The free surface forming the boundary (or a part of the boundary) of the fluid in motion, if it exists, is easily shown to be composed always of the same material particles; hence, if its equation be /=o, we have I-- or, by what has been explained previously, In all the points of a rigid, fixed wall, bounding the fluid, com- pletely or partially, the velocity v is tangent to the wall, so that vn = o, n being the (say, unit) vector normal to the wall. If a rigid body, being immersed in the fluid, moves about in it, then, at any point of its surface, vn is equal to the normal component of the velocity of the rigid body at the given point. The differential equation to a line of flow, i.e. a line having at every point the direction of v, will be, in vector form, d = Av ds, where ds is any element of the line, regarded as a vector, and A a scalar. HYDRODYNAMICS 141 The differential equation of a stream line or of the path of an individual particle is simply <* *- T> where r is to be considered as an unknown function of the time. The stream lines coincide with the lines of flow when and only when the motion of the fluid is steady, or locally invariable, i.e. if 9v/3/=o. But in the general case they must be carefully distinguished. The vortex velocity (' molecular rotation ') will be denoted, as before, by w ; thus w = J curl v. Whence the equation of a vortex line (having everywhere the direction of the vector w), ds = Aw ds, the meaning of the symbols being as above. A tubular portion of the fluid bounded by a surface generated by vortex lines is called a vortex tube (Fig. 45), and in particular, if it be of infinitesimal FIG. 45. FIG. 46. section, a vortex filament. The product of the cross section into the vortex velocity, taken scalarly, is called the moment or the strength of a vortex filament. Any vortex tube may be decomposed (mentally) into filaments, and thus its moment will be the sum, or the integral, of the elementary moments of all filaments. Since VVVv = o or div curl v = o, .identically, we have divw = o, and consequently, for any closed surface cr, r wn da- = o, n being the surface-normal. Applying this to a portion of a vortex tube contained between two cross sections 144 VECTORIAL MECHANICS and also that the density p is given as a function of the pressure alone. Under these assumptions . where Q is a scalar, namely and, in particular, for an incompressible fluid, Q = - $ -pfp- The differential equation of fluid motion will then become ^ = V<2, (93) or, by (a), W). (93) Thus, under the stated conditions, the acceleration dv J dt = d^i I dfi has a scalar potential Q, so that the curl of dvjdt is permanently equal to zero. Conservation of Energy. Let us consider an individual portion T of the fluid, mentally separated from the whole mass of fluid and bounded by the surface o-. If/= o be the equation to this surface, we have since o- itself, equally with the whole portion r, consists always of the same fluid particles. The kinetic energy of this portion of fluid is and since (p d-r) = o, the change of Z, per unit time, will be dt dL f dv i.e., by (90), r- JvV/.^r. Now, the first integral on the right side is the work done per unit time on the portion of fluid under consideration by the impressed forces F ; let us denote it by W. Again, as to the second integral, HYDRODYNAMICS 145 \\ t have vV/ = div (v/) -/ div v, whence, denoting by n the normal of o- directed towards T, - I vV/ . df = l/vn do- + \p div v . dr = J/vn^r + J/-^( + AV^, (96) (f>, A, \ff being three, in the general case mutually independent, scalar functions of position and time. In hydrodynamics this form was used by Clebsch (1856-58), in order to effect the corresponding transformation of the differential equations of motion.* Clebsch, of course, made use of scalar language. Writing curl = VV and remembering that W . Vi/' or curl Vi/' vanishes, identically, we get immediately from (96) the vortex vector w = lcurlv, namely w = iVVA.V (97) which is the vector equivalent of the three scalar formulae of , /3A "d^ 3A 3^\ Clebsch, fii(o - ) etc - 2 \dy ^z ^z dy/ Since the vectors VA, V\ are normal to the surfaces A = const., j^ = const., respectively, the vector w is tangent to both surfaces, so that by (97) we see almost immediately that the vortex lines are the intersections of the surfaces \-fonst.j ^ = const. If, in particular, these surfaces coincide with one another, i.e. if ^ is a function of A only, we have, by (97), w=o. Hence, if ^ is a function of A only, the motion is irrotational, and (as is easily seen) vice versa. Now, since in the general case the fluid motion may be rotational, we shall consider A, \j/ as mutually independent functions (of position * For the literature of this subject see A. B. Basset's Treatise on Hydrodynamics^ Cambridge, 1888, Vol. I. p. 28. HYDRODYNAMICS 147 and time). The third function is also, in the general case, independent of the first two. It may be observed in passing that, by (96) and (97), vw = V.VVA.V& (98) or vw = |- volume of the parallelepiped constructed on the vectors Vc/>, VA, V^ as edges ; whence it is easily seen that the lines of flow constitute an orthogonal network with the vortex lines when and only when the surfaces < = const., A = const, and ^ = const. have common lines of intersection, or, in other words, when the vortex lines lie wholly on the surfaces const. In order to introduce Clebsch's functions in the equation of motion (93) or (93^), let us take the vector product of (96) and (97). We have - 2 Vwv = VvVVA. V^ = (vV^)VA-(vVA)V^, by (ix.) ; using this and (96), the equation (93^) becomes (vVA)Vi/'-(vVi/')VA Xfvx Ultimately, then, the differential equation of fluid motion is o, (93*) where the scalar H is given by "-W+^+V-Q- (99) Multiplying scalarly by w and remembering that, by (97), wV^ = o, wVA = o, we get, from (93^), or H= const, along any vortex line. And since such a line is an intersection of a surface A = const, with a surface $ = const, we have, in the first place, - But then it may be proved also that H depends neither on A nor on \l/ t and is, consequently, a function of / alone. 148 VECTORIAL MECHANICS To see this, take the curl of (93^) ; then hence, multiplying by VA or by Vt/', scalarly, and using (97), ^d\ ^d^ wV-^=o, wV-^ = o. (a) From this point we may repeat verbatim the reasoning of Basset (/of. fit.) : A vortex line, as we know already, lies on a surface A = A = const. ; but by (a) this line is contained at the same time in the surface , dX X + Tt= A '> now, this is possible only if -j- = o ; similarly we shall have ;rr = Thus, (93^) reduces to W7=o, ( 93 ,) so that the function H has everywhere one and the same value, which can vary o?ily with the time ; in other terms, we have what is called a first integral of the differential equation of motion, H=Q + l t + W-Q = ff(t). (100) At the same time we see, from the equations dX dt Tt = > ^ = ' that any surface \ = const, or ^ = const, is composed always of the same particles of fluid; hence, the same thing is true for any vortex line, since it is the intersection of a pair of such surfaces. Thus is proved one of Helmholtz's celebrated theorems, according to which any vortex line is composed always of the same particles. (To this subject we shall return.) It must be remembered that the above deductions are based on the assumption that the density is a function of the pressure only and that the impressed forces have a (scalar) potential. Fluid in Equilibrium. To obtain the necessary and sufficient condition for the equilibrium of a fluid, it is enough to assume simply that the initial velocity HYDRODYNAMICS 149 V vanishes throughout the fluid and that for all times and for all particles the meaning of Q was Neglecting an irrelevant additive constant, we may write *= -n, and as II depends only on p, the surfaces of constant potential of impressed forces coincide with those of constant pressure, which are also surfaces of constant density. In the absence of all impressed forces, other than pressures exerted on the bounding surface, we have, by (1010), vn = or 11 = const., and consequently p = const., i.e. a pressure uniform throughout the fluid and equal to that exerted on its surface (Pascal's Law). If p be not a function of p only, we must return to the more general equation (101), which gives pF = V/, whence, by curling, P curlF i.e. p curl P - VFV/J = o. 150 VECTORIAL MECHANICS Multiplying scalarly by F, we have FcurlF = b. ( I02 ) Thus, in the more general case of equilibrium, in which / is not a function of /> only, the impressed forces may have no potential ; but this being the case, the equilibrium of the fluid is possible only if the ' vortex ' of the impressed force-field, i.e. curl F, is everywhere normal to the force F itself. The reader will observe that (102) is but the vectorial form of the well-known condition, necessary and sufficient in order that should have an integrating factor, which in our case is the density. Irrotational Motion. Returning again from rest to motion, let us consider in the first place irrotational motion, i.e. motion free from vortices. We shall still limit ourselves to the case in which the impressed forces have a (scalar) potential and in which the density is a function of the pressure alone. Then (93), or (930), holds ; hence, assuming w = Jcurlv = o, ( I0 3) so that v can be derived from a scalar potential <, called the velocity potential, i.e. v = V<, (104) we shall have, by (930), But the operators and V are commutative ; hence where z> 2 = (V) 2 . Thus, the scalar expression in the parenthesis has a constant value, i.e. constant in space but not necessarily in time ; but since an arbitrary (additive) function of the time alone is implied HYDRODYNAMICS 151 already in the velocity potential,* we may simply, without impairing the generality, omit that 'constant,' and thus write <2 = o (105) or + J(V<)2 + n + = o. (1050) The reader will observe that this first integral of the equation of motion, in the case of irrotationality, would also follow immediately from the more general integral of Clebsch, (99) ; for, in the absence of vortices, A is a function of $ alone, so that putting we have A 3/ = 37 ; thus, to get (105) from (99), we have only to incorporate ' into the velocity potential <. The equation of continuity becomes, by introducing the velocity- potential, according to (104), 4- div ( P V) = + P V* = o (106) . at For an incompressible fluid (liquid) this becomes V^ = o, (107) which is the well-known equation of Laplace. If the (irrotational) motion be steady, (105) reduces to Jz/ 2 + n + <> = const, (108) where ' const.' is constant in space and in time. For an incompressible fluid or, shortly, for a liquid, under the action of gravity alone, n=AV>, $>=gh, where h is the height above some level chosen conventionally, as a reference level, and g the terrestrial acceleration. Hence, by (108), for any two levels h^ h 9t where the resultant velocities are v x , v. 2 and the pressures p l9 / 2 , respectively, ^+A + V = ^ + A + V (lo8a) Pg 2 PS 2 S * For, putting + F(t) instead of 0, we have still as in (104). 152 VECTORIAL MECHANICS This property is known commonly under the name of Bernoulli's Theorem. We shall see, later on, that the expression \v*-Q has a constant value for any (not necessarily irrotational) steady motion, but constant only along any given line of flow and generally not throughout the whole mass of fluid. * Bernoulli's Theorem ' properly refers to this somewhat more general property. Putting /j=/ 2 and # 2 = o, which satisfies the conditions for a liquid issuing through a small aperture from a large vessel (Fig. 48), Fie;. 48. we have, by (io8#), where v^ is the velocity of issue and h. 2 h-^ the depth of the aperture below the free level (Torricelli's Law). For compressible fluids, the equation as has been remarked, gives , (I0 9 ) , 4 7T J r 477- J r dt r being the distance of a volume-element d-r from the point for which the value of is to be found. This formula may be interpreted by saying that every element dr in which the fluid density is changing contributes to the velocity-potential the ele- mentary amount dr d log p i 4?r dt r and consequently to the velocity v the elementary vector quantity i dlogpdr r* 4?r dt \r r z dt 4?r r which is radial and inversely proportional to the square of distance. *Here r/r is a radial unit vector, directed from dr to the place at which the value holds. HYDRODYNAMICS 153 Multiplying by p we see from the above that an element dr behaves as a source, or sink, of productivity proportional to -T-^ T > acting isotropically, i.e. emitting fluid isotropically during its dilata- tion, and absorbing fluid during its compression. It is scarcely necessary to point out that such a splitting of the expression (109) into elementary summands is wholly artificial and that the corresponding 'inverse square law' is not at all characteristic of fluid motion. Any irrotational vector-field might be as well represented in exactly the same way. It has already been remarked that the velocity-potential may be either a single-valued or a many-valued function of position in space. If the fluid, in irrotational motion, occupies a simply connected or acyclic region T, then < is certainly a single-valued function. By the definition of the velocity-potential, <#> = I v = o, and let P be any other point of the region r. Any pair of paths OAP, OBP in T, leading from O to P may be composed to form a closed curve or a circuit OAPBO, which also does not pass outside the boundary of the region T (Fig. 49). Hence O -LOAPBO -IOAP "t~ -*fgo \ but Ip BO = -Soup', consequently -*OAP = J-OBPl so that independently of the path chosen we arrive at the point P with the same value of , provided only that none of the paths pass outside the boundary of the region. In other words, the 154 VECTORIAL MECHANICS velocity-potential is, in the case of an acyclic region, a single- valued function. Q.E.D. On the other hand, if the irrotationally moving fluid occupies a cyclic region, as for instance the core of an anchor-ring (doubly- connected region), the above surface o- may be drawn only through FIG. 49. some but not through every possible closed path s contained in the region, so that in this case we shall have generally o, w i.e. a circulation generally differing from zero, and hence also, generally, IOAP^ IOBP' Thus, the velocity-potential < will for such a region be generally a many-valued or ' polycyclic ' function. An (n+ i)-ply connected region contains n essentially different, or mutually independent, cycles, i.e. closed paths which cannot by con- tinuous deformation without leaving the region be reduced either to a point or into each other. Call these cycles s lt s 2 ,...s n . Through none of these is it possible to draw a surface o- of the above description. Hence, the integral I vds or the circulation, Jw notwithstanding the irrotationality of motion, may be different from zero for each of these cycles. Let the circulation round s lt s 2 , ... s n be denoted by 7 15 / 2 , ... /, respectively, i.e. 1 2 and let us again start from the initial point O with the value < = o ; by following a certain path OAF we shall arrive at the point P with some determinate value of the potential, < = A , namely with the value where m lt m^ ... m n are integers, positive or negative, indicating the number of, positive or negative, cycles s lt s 2 ,...s n (respectively), into which the entire circuit OBPAO may be decomposed. If this circuit can be contracted to a point, by continuous defor- mation, without passing out of the region in consideration, then m jt m z , ... m n all vanish, and < = < B = <. 4 . But if OBPAO cannot be contracted in this way but is reducible, for instance, to a single cycle jj taken in the positive sense and to the cycle s 2 repeated three times in the negative sense, then m l =ij ^2= - 3) m s = m i = ... =m n = o, so that < = )Vr = - Lvn^o-- fov* = |< the unit vector n denoting the normal to the shell is in general many-valued ; and certain supplementary terms are required, viz. integrals of vn extending over both sides of each barrier or 1 diaphragm' necessary, and sufficient, to convert the multiply- connected into a simply-connected region. The reader will find a fuller account of this subject in Vol. I. of Basset's Hydrodynamics, and in Vol. I. of Maxwell's Treatise on Electricity and Magnetism. For the purposes of this book the above remarks on irrotational motion will suffice. PROPERTIES OF VORTEX MOTION. Kinematical Properties. Any motion of a fluid may be represented as a superposition of purely irrotational motion characterised by curl v = o ; .*. v = V$, but div v = o, and of solenoidal vortex-motion, for which curlv^o, but divv = o, and consequently v = curlB. For the first kind of motion a scalar potential (<) exists every- where, for the second kind such a potential exists only outside the vortices, whereas the vector potential (B) exists everywhere. Hence, in the general case, to be considered in this section, we may put v = V< 4- curl B (m) with the supplementary condition for the auxiliary vector B, which it is always possible to satisfy, divB = o. ( II2 ) It follows for the vortex velocity, by (in), w-^curlv = icurl 2 B = JVdivB-JV 2 B, i.e., by (112), V 2 B=-2W, (113) the solution of which is, as has already been remarked, the integration extends through all the elements d-r of the vortices, and r denotes the distance of such an element dr from the point HYDRODYNAMICS 157 in which the vector potential B holds. It may be easily shown that (114) satisfies not only the differential equation (113) but also the supplementary condition (112). The vector potential B is thus completely determined by -the vortices alone. As to the scalar potential <, we have, by (in), since divcurlB = o, identically. Hence, as before, by the equation of continuity, and < = - ^^ dr. (109 bis] 4 7T J r dt This potential and the corresponding velocity have already been interpreted, in the section on irrotational motion. Thus it will suffice to say a few words only in regard to the second part of the velocity, viz. that associated with the existence of vortices, v = curl B. If the fluid is incompressible, the whole velocity is reduced to this. Now, by (114), i r . /w\ _ V = I curl }dr. 27rJ \rj where, on curling, w is to be considered a constant vector, while r is variable, being the distance of dr from that point P, for which v has to be calculated. Thus, denoting by r tt a unit vector pointing from dr towards P, it follows that curl (-} = VV (- w) = ^ -^ Vr u w - 4 Vwr u ; \r/ \r J r- r- hence , p , (s) Taking as the volume element dr a portion of a vortex filament of cross section d = plfw = cja-w = f/fjij and consequently d / But vr/p = zv\//p = W(r .l/c = pl/, identically. Hence, multiplying both sides of (#) by the constant mass alone, the last term would vanish ; hence, when a given particle of the fluid does not rotate at the instant / , then for this instant d"\fr dt thus, the particle will begin to rotate, namely round an axis which initially (i.e. for /=/ ) w iH coincide in direction with the vortex of the force F, i.e. with the vector curlF. Also the moment p of fluid vortices already existing will generally vary with time. From the last equation we see also that if the motion is to be irrotational, i.e. if we require that w = o permanently, we must have curlF = o. But let us suppose that the forces F have a potential, and that p is not a function of p alone. It may, for instance, depend also on the temperature which may be different for different particles and at various instants of time. In this case the first term on the right side of (125) vanishes, but the second remains, so that If n, n' are unit vectors normal to the surfaces p = const., p = const. respectively, drawn in the directions of increasing density and of increasing pressure, then (126) may also be written dt \ p The vector on the right side, and consequently also the vector sum on the left side, is tangent to the line of intersection of the surfaces p = const, and p = const. Considering again an element of a vortex filament l = ew, we may transform the left side of (126) into _ or c dt c dt c dt' c being the mass of the element of length / and of cross section o or sin < o. Thus, a rotating particle may in time lose its vortex motion. On the other hand, if at a given instant / , for some particle, or for the whole fluid, ze/ = o, then, by (126), Vnn' ; for /-/ , (128) dt that is to say : In a fluid devoid, at the instant / , or till the instant / , of every vortex motion, vortex lines would be created, coinciding initially with the intersections of the surfaces of constant density and of constant pressure* If such surfaces, intersecting mutually during a certain time T, should then once and for ever cease to intersect one another, the vortices created during that time would afterwards keep their moments constant. The period T might be brief, yet the moments of the vortices which have been formed in this short time might be considerable when the gradients of pressure and of density are correspondingly large. The Wave of Acceleration; Hugoniot's Theorem. The surfaces of discontinuity, for which the identical and the kinematical conditions have already been established (pp. 122, 131), must also fulfil certain conditions following from the dynamical properties of a non-rigid body, and, in particular, of a fluid. We shall limit ourselves here to the consideration of a surface of discontinuity of the second order (in r, i.e. of the first order in v = difdt} or of the so-called wave of acceleration, which was generally investigated by Hugoniot^ Let p be a function of/ only; then p J dpp * Cf. the author's paper, ' Ueber Entstehung von Wirbelbewegungen in einer reibungslosen Fltissigkeit,' Bulletin, Acad. Sc. Cracow, 1896. t Comptes rendtis, Vol. 101, p. 1119; Paris, 1885. 168 VECTORIAL MECHANICS hence the equation of motion : Let, again, be the equation of the surface of discontinuity cr, and n its normal. Let the impressed forces F, the velocity v, the pressure and density, as well as dpjdp be continuous, whereas the acceleration (and hence also Vlog/o) is discontinuous at the surface cr. Then, writing (129) for the two sides of cr and subtracting, we shall have [Wl 4 [_dt\ d f Since [v] = o, we have, in exactly the same way as in eq. (72) for D, [divv] = mn = m^/ f : ^ (l>) where m is the vector which characterises the discontinuity. On the other hand, by (70), since [log/o] = o, [V log p] = An, (c) A being a scalar which we shall eliminate immediately. Again, the kinematical condition of compatibility will be, as in (82), where t) is the velocity of propagation of the wave, the value of which is to be found. But, by the equation of continuity, (91), d\ogp/dt = -divv, and consequently so that, by (b) and (d), A=imn; hence, by (c), [Vlogp] = ^(mn)n. (130) This is the vector equivalent of the three scalar formulae for the jump of the components of slope of density, obtained by Hadamard by a considerably longer process.* *See Hadamard, loc. cit. t or also AppelPs Mdcanique, Vol. III. p. 312. HYDRODYNAMICS 169 On the other hand we have, as in (82), |~^n = - tan ; hence, introducing this value and also (130) into the dynamical equation (a), (mn)n. (131) There are only two possible ways of satisfying this equation : i) the vector m is normal to the surface in terms of orthogonal curvilinear coordinates u, v, w. That is to say, u = const, v = const, w = const, being a threefold orthogonal system of surfaces in space, show that where the vectors U, V, W are the slopes of u, v, w respectively, i.e. U = Vw, V = Vp, W = Vze/. Do not employ the expression of V in terms of 'dj'dx, etc. Use only (x.), Chap. I., as the definition of slope. If a, b, c, be the units of U, V, W respectively, and ds^ ds.^ ds 3 the elements of the lines of intersection of the surfaces v = const., w = const., etc., measured in the direction of a, to, c, then V< = a3 &> C 2 ; 2 A, T.B, iC are the principal axes of the ellipsoid =o. If x,y, z are the ordinary, rectilinear coordinates measured along the principal axes of the whole PROBLEMS AND EXERCISES 177 system of confocal quadric surfaces, u>v>w are the roots of the equation r i V 2 -2 oo>u> - so that f(u\ f(iv) are positive, while f(v) is negative. Remembering all of this, use 35, 37 and 36, respectively. Here it will be enough to write the Laplacian, leaving the curl and div wholly to the reader. Introducing the elliptic integrals du ~ C dv C dw the Laplacian will be where N' 1 =(v - w)(u - w)(u - v). 39. Write div, curl, V 2 in polar coordinates u = 6, z> = e, w r. In this case, 9, e being the latitude and longitude respectively, ds^r .dt, ds z = dr, so that U= i/r, V= i/rsmQ, W= i, and so on. 40. Write down div, curl, V 2 in cylindrical coordinates u = z, Here, p being the distance from, and 2 being measured along, the axis, /=i, V=\, W=ijp. Consider the peculiarly important special case of axial symmetry, i.e. 3/9=o. 41. Verify, by Cartesian expansion, the formula (26) of Chap. I. Show, without splitting, that the divergence of its right-hand side vanishes identically. 42. Verify, by Cartesian expansion, the formula (33) of Chap. I. Write down curl 3 , curl 4 , etc., for a solenoidal vector. 43. Show that d'Alembert's Principle, (i), Chap. II., is invariant with respect to the ' Newtonian transformation ' r/ = T f - &t, where a is an arbitrary constant vector. Assume, of course, that the constraints of the system imply only the positions, and velocities, of its constituent parts relative to one another. 44. Formula (i), Chap. II., being the expression of d'Alembert's Principle relative, say, to the ' fixed-star ' system of reference, write this principle using the earth as a framework of reference. Compare it with (i) and interpret the supplementary terms. V.M. M 178 VECTORIAL MECHANICS 45. Integrate the equation of motion of a particle (say, an electron contained in an atom) drawn back to its position of equilibrium, O, by an 'elastic' force, i.e. r = - 2 r, where a 2 is a positive scalar. Investigate the properties of the orbit for any given initial state, r = r , r = v , for /=o. Write down the principle of areas and that of vis-viva and interpret them. Elliptic harmonic motion. See 25. 46. Work out, vectorially, the elementary theory of the Zeeman- effect, that is to say, integrate the equation of motion r = - 2 r - 2 VvM, where M is a vector constant in space and time (homogeneous magnetic field), < 2 a positive scalar (the specific charge of an electron, e/m, divided by the velocity of light in vacuo), and v = r. Consider the projections of the orbit i on a plane normal to M, and 2 on a plane parallel to M. Compare the periods with that corresponding to the above example. 47. Show directly that the position of the centre of gravity (mass-centre), i.e. of the end-point of the vector S = 2#zr/2#z, (17), Chap. III., is independent of the choice of any auxiliary framework. 48. Find the mass-centre of three equal masses placed at the corners of a triangle. 49. Find the mass-centre of four equal masses placed at the corners of a tetrahedron. 50. If M be the momentum of a system (2wv), q its moment ! of momentum about O and q' that about O ', show that q' = q + VaM, where a is the vector drawn from O to O. Interpret this simple property dynamically. 51. Prove that the resultant of two angular velocities, p 15 p 2 , and hence also of infinitesimal rotations, of a rigid body, round axes which meet one another, is given by the vector Pi + Pg. First show, by elementary considerations, that the velocity v of any point P of the rigid body is expressed by v = Vpr, where r is the vector to P from any point O on the axis of rotation. Then, using the additive property of velocities, write v = v x + v 2 = Vp t r + Vp 2 r, and so on. 52. Show that by any parallel shifting of the axis of rotation only a translation-velocity, of the body as a whole, is added. PROBLEMS AND EXERCISES 179 Instead of r take r' = r + a, and so on. Notice that if a||p, the supplementary translational velocity vanishes ; in other words, the shifting of p in its own line is a matter of indifference. Thus, p, the angular velocity or * spin] if taken with all of its kinematical implications, is a vector localised in a (straight) line, or a rotor. But if we are concerned with rotational motions only, this localisation becomes, of course, unessential, and spins can be manipulated with as free or ' unlocalised ' vectors. There is, at any rate, no danger of misunderstanding arising from the above peculiarities. Another example of a vector localised in a line is a force applied to a rigid body. 53. Prove that the most general instantaneous motion of a rigid body is a screw motion or twist about a ce'rtain screw, i.e. an angular velocity p about a certain axis, combined with a translation velocity V along that axis.* First write v = v ' + Vpr', where v ' and p will be generally oblique ; then decompose v ' into components parallel and normal to p, and so on. 54. Show directly that div v = o, thus verifying the incompressibility (Chap. V.) of a rigid body. Write v=v +Vpr. Use for div the Hamiltonian applied scalarly. 55. The meaning of v being as above, prove directly that curl v=2p. 56. Show that, for the most general motion of a rigid body, Also, prove that (vV)v = Vpv, so that = :=r--fVpv. [Notation as in (74), Chap. V.] Interpret the last result. Cf. (22), Chap. IV. 57. In a rigid body (or system), let the mass be distributed symmetrically round an axis a. Show that the inertial operator of the body, about a point on a, is of the form K= s^ - s 2 a . a, where s^ s 2 are positive scalars. *The ratio V Q \p is called the pitch of the screw. A quantity which, like a twist-velocity, has ' magnitude, direction, position [in a line], and pitch ' has been called by Clifford a motor. i8o VECTORIAL MECHANICS The above K is written as a dyadic, the dot being a separator. Thus, if q be the moment of momentum and p the angular velocity, q=^ 1 p J 2 a(ap). Determine the values of s lt s 2 in a few simple cases, suqh as a pair of point-masses and a massive (homogeneous) circle. 58. Let tu = i.0 1 +j .0 2 + k. 3 be any linear vector operator, as in (xiv.), Chap. V. Show directly that its principal values n are roots of the cubic equation Develop this, by vector algebra, into the form ;z 3 - c^ri 1 + c^i - ** = o, and interpret this result. Show that WH + W22 + W33, and so on, are invariants of the operator, i.e. independent of the orientation of the framework i, j, k. Discuss as far as you can the general case and then the case of a symmetrical operator. 59. Using (XIIL), prove (xvi/), Chap. V., i.e. that for any pair of vectors A, B, where w is any linear vector operator and o>' its conjugate. Show first that if w = 12 + Vc, then w' = 0-Vc, and so on. 60. Investigate the properties of an operator w, whose inverse is identical with its conjugate, A / 7} i y*" \ ^ sA i n i /~* \ etc., by cyclic permutation of i, 2, 3. 6*! = (A^^ + A 2 C 2 + A S C Z )J5 1 185 VA(BC). = VAVBC = (AC)B-(AB)C. Line integral'. Surface-integral : Curl or Rotation : Divergence : div R = VR. Theorem of Divergence^ called also Gauss' Theorem : I divR.^r= iRndb-, n being the outward normal. Slope or gradient-. R = V<, being a scalar function. Stokes 1 Theorem : n curl R da- = I R d$. = I J () etc., by cyclic permutation of i, 2, 3. supposing fixed Cartesian axes. -[ S= I (7?!^! + ^? 2 2 + R&^&TI n^ n 2 , n z being the direction-cosines of n. Ks^e, 3^0 3 -^r-+-^- + - 3^ Sy = f (Rfr + R^ + R^} d, and consequently V0.V^ = (V0) 2 . i88 VECTORIAL MECHANICS Hamilton's Principle, (12), Chap. II. , has nothing vectorial about it. Thus, it will be sufficient to write only the right-hand side of equation (n), from which the principle followed, along with its Cartesian equivalent : v l Sx + v^ 6> Then, continuing our survey, we have the following equivalences : Centre of mass, (17), Chap. III.: where M= The resultant moment impressed forces : of The principle of areas, (20), Chap. III. : dt Fundamental kinematical re- lations for a rigid system, (22), Chap. IV. : w = w + Vpw. Moment of momentum, (24), Chap. IV. : q = 2w Vrv. Equations of motion of a rigid system, (25), (26), ibid. : ~ S lt etc., being simply the Cartesian co- ordinates of the centre of mass. d_. ~di dw-, etc. etc. etc., w^, w z , w s being the components of w along fixed axes, and w^, etc., the com- ponents of the same along axes moving with the rigid system.* etc. etc. * Remember that A> A> Pz-> the components of the angular velocity p, have been taken always along the moving axes, namely along the principal axes of the rigid system. (Chap. IV.) APPENDIX CARTESIAN EQUIVALENTS Kinetic energy, (27), (270): T= 189 Eulers eq. of motion, (260): K^, etc., being the principal values of the (symmetrical) linear operator K. 3 , etc., where the accents could be omitted, since /!, etc., are already taken along the (moving) principal axes. Time as a function of angular velocity, formula (a), Chap. IV., page 82 : /-f *, ' w JP9 2V where To obtain the usual, Cartesian, development of this denominator, apply the determinantal form of the product AVBC (see First Part of this ' Appendix ') ; then P\ Pi A where A== ^ 2 -A- 3 + ^ 3 -^ + ^ 1 -^ 2 ^i AT K. Now, / 1? / 2 , / 3 may readily be expressed by /, and by the invariants 2T and q, namely by solving the three equations, Thus, writing we get immediately A .-fifit iii --\ -**-2 3 * K* K whence , etc., A ^ 3/2 _ X 3 ), i go VECTORIAL MECHANICS which is the usual form given in text-books on rigid dynamics, the constants under the radical being given by etc.; -q*}, etc. Introducing the last expression of N into the above integral (a), we get what is alluded to in Chap. IV. p. 82. But let us continue the columnar system of our Vectorially- Cartesian dictionary. Fundamental formula of strain-theory, (48), Chap. V. : Developed form of the above, (50), (51) : Cubic dilatation, (59) : Rotation (infinitesimal) : c = J curl D. Equation of continuity, (60) : Sp + p div D = o. where x t y, z are the coordinates in the unstrained body, identical \vith our a lt a 2 , a s') A> A' etc -> are tne projections of 1, 1' on the axes of x, y, z. /3Z). where x, y, z and the components of 1, 1', D are taken along the principal axes of fl at the point x, y, z. " = ^ - ~^ ox oy oz APPENDIX CARTESIAN EQUIVALENTS IQI Surface of discontinuity ; ddentical conditions, (71) : [V(Dx)] = n(mx); put x = i, j, k respectively. Idem, (72), (73): [0] = [divD] = Individual and local time- variation, (74) : Variation of a line-element, 75): Variation of a volume- lement, (77): (^r) = divv.^r. f- at Eg. of continuity, (780) : these are the nine scalar conditions of Hadamard. [0] = n l m 1 +/+/3J ' 1 ' etc + - + -W 192 VECTORIAL MECHANICS Kinematical conditions of compatibility, (82) : General eq. of motion of a non-rigid body, (87) : dv - k div p 3 the dots being separators. Hydrodynamical differential equation, (90), Chap. VI. : [Equation of continuity, see above.] Circulation, ibidem : 7= I v d$. Jw Hydrodynamical eq. for conservative forces and p =f(p\ (93*) : where w = J curl v. CkbscKs Transformation, (96), (97): v = etc - 2S > \ 3s/ ^7 3/ r(fdx = T W Q = INDEX ( The numbers refer to the pages. ) Acceleration, rectorial, 25-26 Activity, 1 1 Acyclic space, region, 153 Addition of vectors, 4 d'Alembert's principle, 53-54 Antisymmetrical linear operator, 96 Appell, 85, 1 20, 1 68 Area, of a plane figure, 174 Associative property of addition of vec- tors, 6 Atled, 28 Axial symmetry, 75 Continuity, of a vector function, 22 Continuous field, 28 Creation of vortices, 165-167 Cubic dilatation, 108 Curl, defined, 34 in Cartesians, 35 in curvilinear coordinates, 176 Current, through surface, 31 Curvilinear coordinates, 175-177 Cycles, of a cyclic region, 154 Cyclic space, region, 43 Cylindrical coordinates, 177 Basset, 146, 148, 156 Bernoulli's theorem, 152, 160-161 Biot-Savart's law (hydrokinematic ana- logy of), 158 Cartesian method, 2 Central motion, 27, 68 Christoffel, 118, 129 Circuit, 29 Circuital strain, 1 1 2 Circulation, 31, 142, 154 Clebsch's transformation, 146-148 Clifford, 173, 174, 179 Commutative property of scalar multi- plication, 12 of addition of vectors, 6 Compatibility, kinematical conditions of, 129 Components, of a vector, 21 of acceleration, tangential and nor- mal, 26 Confocal quadric surfaces, 176-177 Conic orbits, 68 Conjugate linear operator, 94 stress, 134 Conservation of areas, 65 of energy in a fluid, 144-146 of the motion of the centre of mass, 64 of vis- viva, 61 Desargues' theorem, 171 Diaphragms, in a cyclic region, 156 Difference of vectors, 8 Differentiation, of vectors, 22 partial, 27 Dilatation, 1 08 Directive moment, of a pendulum,. 87 Discontinuity, surfaces of, 117-123,. 128-132, 167-169 Displacement^ in a strained body, 100 virtual, 51 Distributivity of scalar multiplication,, 12 of vector multiplication, 17 Divergence (div), denned, 38 in Cartesians, 39 in curvilinear coordinates, 175 Dyadic, 97 Elastic solids, 137 Electric energy, 12 Electromagnetic force, 18 stress, 137, 181-182 Electromotive force, 30 Ellipsoid of inertia, 80 Elliptic coordinates, 176-177 Elliptic harmonic motion, 173 Elongation, 106 INDEX 195 Energy, as an invariant, 61 electric, 12 kinetic, of a rigid system, 72 of a fluid, 144-146 of elastic deformation, 138 magnetic, 12 Equation of continuity, ill, 126-127 Equation of Laplace, 151 Equipotential surfaces, 43 Euler's equation of motion, of a rigid body, 77 hydrodynamical equations, 139 kinematical equations, 87 Field, solenoidal, 41 irrotational, 41 vector-, defined, 28 Flux of electromagnetic energy, 18, 45 Free axes. 78 Free surfaces, 140 Gauss' theorem, 38 Gibbs, 3, 10, 93, 96 Gradient, 43 Gravitation, 68 Green's theorem, 155, 186-187 Hadamard, 120, 129. 168 Hamilton, 73 Hamilton's principle, 58 Hamiltonian, V, 28, 36, 39, 43-44, 46, 9.1 Heaviside, 3, 10, 35, 46, 73, 91, 172 Helmholtz, 162 Helmholtz's theorems, 148, 164-165 Henrici and Turner, 171 Hess, 85 Heterogeneous field, 28 Hodograph, 173 Holonomic system, 54 Homogeneous field, 28 strain, 107 Hugoniot, 129, 167 Hugoniot's theorem, 169 Hydrodynamical equations, 139 Identical conditions, 120-123 Incompressible fluid (liquid), 151 Indestructibility, of vortices, 164 Individual change, 124 Inertial operator, 76 Infinitesimal strain, 109-111 Integral, line-, 29 surface-, 30 Invariable plane, 65, 78 Inverse operator, 76 Inverse square law, 152 Irrotational field, 41 motion, of a fluid, 150-156 strain, 103, 112 stress, 136 Joly, 171 Kepler's second law, 27, 66 Kinematical conditions of compatibility, 129 Kinematical relations, for a rigid system, oo oo Kinetic energy, of a rigid system, 72 of vortex motion, 160 Kowalewski, Sophy, 85 Lagrange and Poisson, 85-86 Lagrange's equations of motion, 56, 59 Laplace's equation, 151 formula, 169 Laplacian, y 2 , 48 in curvilinear coordinates, 176 Law of Biot-Savart, 158 of inverse squares, 1 52 of Kepler, second, 27, 66 of Pascal, 149 of Torricelli, 152 Line-integral, 29 Linear vector function, 72, 92-97 operator, 73, 93-97 Lines of flow, 140 Local change, 124 Localised vectors, 179 Longitudinal discontinuity, 169 strain, 112,113-115 Lorentz, 91 Mach, 51 Magnetic energy, 12 Magnetomotive force, 30 Maxwell, 18, 42, 91, 137, 156, 181 Moment, directive, of a pendulum, 87 of a curve about a point, 174 of a vortex filament, 141 of impressed forces, 64 of inertia, 76 of momentum, 65 of rotational sheet, 119 Momental ellipsoid, 80 Motion, central, 27 Motor, 179 Multiply connected space, 43 Nabla, 28 Newton's theorem, 68 196 INDEX Newtonian transformation, 177 Orthogonal 175-177 curvilinear coordinates, Parabolic motion, 173 Parallelepiped, volume of, 16 Part, of a vector, 21 Partial gradients, 55 Pascal's law, 149 Pendulum, mathematical (simple), 57 physical (compound), 86-87 Permanent axes, of rotation, 78 Phoronomic conditions, 129 Pitch, of a screw, 179 Poinsot's motion, 81 Polar coordinates, 177 Polycyclic potential, 154 Positive sense, of a circuit, 33 Potential (scalar), 42 of stress components, 138 Poynting vector, 18, 45 Pressure, 132 Principal axes, of a rigid system, 73 of a stress, 134 of inertia, 76 Principal elongations, 106 pressures or tensions, 134 Principle, d'Alembert's, 53 Hamilton's, 58 of areas, 65, 71 of motion of centre of mass, 64 of vis- viva, 60 Product of vectors, scalar, 10 vectorial, 13 Propagation, defined, 130 Quadric surfaces, 176-177 curl, 91 Reaction, 57 Right-handed system of vectors, 14 Rigidity, defined, 70, 181 Rotation = curl, 34 Rotation, infinitesimal, 103 Rotor, 179 Routh, 71, 85 Scalar, I Scalar character, of an operator, 46 Scalar product, 10 square, II Screw motion, 179 Self-conjugate operator, 94 Sense, positive and negative, of a cir- cuit, 33 Shears, in Sinks, 153 Slope, 43 in curvilinear coordinates, 175 Solenoidal field, 41 strain, 112 Sources, 153 Sourceless field, 41 Square, scalar, II Stationary discontinuities, 130 Steady motion, 141, 160-161 Stokes' theorem, 34 Strains, 98-117 Stream lines, 141 Strength, of rotational sheet, 119 Stress, stress-operator, 133 Subtraction of vectors, 8 Sum of vectors, 4 Surface, equipotential, 43 Surface-integral, 30 Surfaces of discontinuity, 117-123, 128- 132, 167-169 Symmetrical linear vector operator, 73, 94 Tension, 132 Tensor, 2 Theorem of Bernoulli, 152, 160-161 of Desargues, 1 7 1 of Gauss (divergence), 38 of Green, 155, 186-187 of Helmholtz, 148, 163-165 of Hugoniot, 169 of Newton, 68 of Stokes, 34 of Thomson, 161-162 Torricelli's law, 152 Transversal discontinuity, 169 strain, 112, 115-117 Triple product, scalarly-vectorial, 16 vectorially-vectorial, 20-21 Tschapligin, 85 Twist, 179 Unit vector, 3 Veblen and Young, 171 Vector, defined, i, 4 Vectors, equality of, 3 Vector field, 28 Vector function, linear, 72 Vector potential, 128, 157 Vector product, 13 Vector sum, 4 INDEX 197 Velocity (vectorial), 23 Vortex motion, 156-167 Velocity of propagation, defined, 131 Vortex velocity, 126 of sound in gases, 169 Velocity-potentials, 127-128, 153, 156- Wave o f acceleration, 167 Volume, general formula, 174 of a parallelepiped, 16 Zeeman's effect, 178 Vortex lines, tubes, 141 Zemplen, 120, 129 GLASGOW : PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLRHOSE AND CO. 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