THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES GIFT FEB 2 wo ELEMENTS OF DYNAMIC. ELEMENTS OF DYNAMIC MOTION AND REST IN SOLID AND FLUID BODIES W. K. CLIFFORD, F.R.S. LATE FELLOW AND ASSISTANT TUTOR OF TRINITY COLLEGE, CAMBRIDGE PROFESSOR OF APPLIED MATHEMATICS AND MECHANICS AT UNIVERSITY COLLEGE, LONDON. PART I. KINEMATIC. Uonfcon : MACMILLAN AND CO. 1878 [All Jliyhtg reserved.] Cambrfogc: PBIWTED BY C. J. CLAY, M.A. AT THE UNIVEESITY PRESS. Library I/,/ CONTENTS. BOOK I. TRANSLATIONS. CHAPTER I. STEPS. PAGE Introduction x 1 On Steps 3 Composition of Steps. Geometry . . . . . . 4 Composition of Steps. Algebra . . .... . . 7 Eesolution and Description of Steps 11 Eepresentation of Motion 14 Uniform Motion 15 Uniform Eectilinear Motion 16 Uniform Circular Motion 18 Harmonic Motion 20 On Projection 24 Properties of the Ellipse 27 Elliptic Harmonic Motion 31 Compound Harmonic Motion 33 Parabolic Motion , 38 CHAPTER II. VELOCITIES. The Direction of Motion. (Tangents.) 41 Exact Definition of Tangent 44 Velocity. Uniform 47 Velocity. Variable 51 Exact Definition of Velocity 56 Composition of Velocities 59 Fluxions 62 Derived Functions 64 Hodograph. Acceleration 67 The Inverse Method 68 vi CONTENTS. PAGE Curvature 73 Tangential and Normal Acceleration 77 Logarithmic Motion .... 78 On Series ; w ......*... 81 Exponential Series 83 The Logarithmic Spiral , ; . 85 Quasi-Harmonic Motion in a Hyperbola 89 CHAPTER III. CENTRAL ORBITS. The Theorem of Moments 92 Product of two Vectors 94 Moment of Velocity of a Moving Point .'.-. . . . 96 Belated Curves 100 Acceleration Inversely as Square of Distance .... 105 Elliptic Motion 107 Lambert's Theorem 108 General Theorems. The Squared Velocity 110 General Theorems. The Critical Orbit 113 Equation between u and 6 . . 116 BOOK II. ROTATIONS. CHAPTER I. Steps of a Rigid Body 118 CHAPTER II. VELOCITY-SYSTEMS. Spins 122 Composition of Spins ......... 123 Velocity-Systems. Twists 125 Composition of Twists . ' 126 Moments 132 Instantaneous motion of a Rigid Body 136 Curvature of Roulette . . . . . . . 140 Instantaneous Axis 141 Degree of Freedom 143 Involute and Evolute 144 CONTENTS. Vll CHAPTER III. SPECIAL PROBLEMS. PAGE Three-Bar Motion . . . ... . . . . 146 Circular Boulettes 151 Double Generation of Cycloidal Curves 152 Case of Eadii as 1 : 2 153 Envelop of Carried Roulette 155 BOOK III. STRAINS. CHAPTER I. STRAIN-STEPS. Strain in Straight Line 158 Homogeneous Strain in Plane 159 Representation of Pure Strain by Ellipse 160 Representation of the Displacement 161 Linear Function of a Vector 162 Properties of a Pure Function . . . . . . . 164 Shear . . . . 167 Composition of Strains 168 Representation of Strains by Vectors 170 General Strain of Solid. Properties of the Ellipsoid . . . 172 Representation of Pure Strain by Ellipsoid 176 Properties of Hyperboloid ........ 177 Displacement Quadric ......... 181 Linear Function of a Vector ........ 185 Varying Strain 188 CHAPTER II. STRAIN-VELOCITIES. Homogeneous Strain-Flux 191 Circulation 194 Strain-Flux not Homogeneous 197 Lines of Flow and Vortex-Lines 199 Circulation in Non-Homogeneous Strain-Flux .... 200 V1U CONTENTS. PAGE Irrotational Motion 203 Equipotential Surfaces 204 Motion partly Irrotational 205 Expansion 207 Case of No Expansion 210 Squirts 212 Whirls 214 Vortices 216 Velocity in Terms of Expansion and Spin 219 BOOK I. TEANSLATIONS. CHAPTER I. STEPS. INTRODUCTION. JUST as Geometry teaches us about the sizes and shapes and distances of bodies, and about the relations which hold good between them, so Dynamic teaches us about the changes which take place in those distances, sizes, and shapes (which changes are called motions), the relations which hold good between different motions, and the circumstances under which motions take place. Motions are generally very complicated. To fix the ideas, consider the case of a man sitting in one corner of a railway carriage, who gets up and moves to the opposite corner. He has gone from one place to another ; he has turned round ; he has continually changed in shape, and many of his muscles have changed in size during the process. To avoid this complication we deal with the simplest motions first, and gradually go on to consider the more complex ones. In the first place we postpone the con- sideration of changes in size and shape by treating only of those motions in which there are no such changes. A body which does not change its size or shape during the time considered is called a rigid body. The motion of rigid bodies is of two kinds; change of place, or translation, and change of direction or aspect, which is called rotation. In a motion of pure translation, every straight line in the body remains parallel to its original position ; for if it did not, it would turn round, c. 1 2 DYNAMIC. and there would be a motion of rotation mixed up with the motion of translation. By a straight line in the body we do not mean merely a straight line indicated by the shape or marked upon the surface of the body; thus if a box have a movement of translation, not only will its edges remain parallel to their original positions, but the same will be true of every straight line which we can conceive to be drawn joining any two points of the box. When a body has a motion of translation it is found that every point of it moves in the same way; so that to describe the motion of the whole body it is sufficient to describe that of one point. When a body is so small that there is no need to take account of the differences in position and motion of its different parts, the body is called a particle. Thus the only motion of a particle that we take account of is the motion of translation of any point in it. A motion of translation mixed up with a motion of rotation is like that of a corkscrew entering into a cork, and is called a twist. Bodies which change their size or shape are called elastic bodies. Changes in size or shape are called strains. The science which teaches how to describe motion accurately, and how to compound different motions to- gether, is called Kinematic (favrj^a, motion). We may conveniently reckon three branches of it, namely, (Points or particles (Translations). Kinematic of -JRigid Bodies (Rotations and Twists). (Elastic Bodies (Strains). It is found that the change of motion of any body depends partly on the position of distant bodies and partly on the strain of contiguous bodies. Considered as so depending, the rate of change of motion is called force; and the law just stated, expressing the circum- stances under which motions change, is called the law of force. The science which teaches how to calculate motions in accordance with the law of force is called Dynamic (Svva/jus, force). It is divided into two parts: Static, which treats of those circumstances under which rest or null motion is possible, and Kinetic, which treats of cir ON STEPS. 3 cumstances under which actual motion always takes place. Properly speaking, Static is a particular case of Kinetic which it has been convenient to consider separately. When change of motion depends upon the position of distant bodies, it is also, called attraction or repulsion; when it depends upon the strain of contiguous bodies, it is also called stress. Those elastic bodies whose shape may change without stress (i.e., without simultaneous change of motion in adjacent bodies) are called fluids; all others are called solids. There are no known bodies whose size can change without stress. The part of Dynamic which relates to fluid bodies is sometimes treated separately, under the name of Hydro- dynamic (Hydrostatic and Hydrokinetio). That part which relates to the changes of shape of solid bodies, considered in relation to the law of force, is called the theory of Elasticity. ox STEPS. When a body has a motion of translation, all the points of it move along equal and similar paths, For let a and b be two points of the body, and let a move along the path aa'a", and b along the path bb'b", so that when a is at a', b is at &', and when a is at a", b is at b", Then, by the definition of a trans- lation, the straight lines ab, a'b' and a"b" are equal and parallel. Consequently aa' is equal and parallel to bb', and aa" to bb". If therefore the path aa'a" be moved so that a comes to b, and the lines aa', aa", are kept parallel to their original positions, the points a, a" must come to b', b" respectively. But the point a' is any point on the path of a. Therefore every point on the path of a comes to coincide with the corresponding point on the path of b, or, which is the same thing, the path of a is equal and similar to the path of 6. That is, the paths of any two points are equal and similar. 12 4; DYNAMIC. Hence it is sufficient, in describing the translation of a rigid body, to describe the motion of any one point of the body. But the former is really simpler than the latter ; for the point starts from a definite place, which must be specified if its motion is fully described; but the fixing of this starting-point is unnecessary, as we have seen, when the motion of a point is only used to describe that of a rigid body. At present we shall attend only to the change of position which a body undergoes between the beginning and end of the time considered, without troubling our- selves about what has taken place in the interval. That is, we shall pay attention to the fact that a has got to a and b to &', without enquiring about the paths aa and bb', or about the time occupied in the transfer. A change of position effected by a motion of translation will be called a step. The step of the point a from a to a will be con- veniently denoted by the symbol aa; and we may re- present it graphically by the straight line aa, pro- vided we remember that the transfer takes place along any path whatever, and not necessarily along that straight line. This being so, the lines aa' and bb' will represent the same step of a rigid body if they are equal in length and in the same direction; that is, not merely parallel, but drawn in the same sense on two parallel straight lines. Thus a step of a rigid body is adequately represented by a line of given length and given direction drawn anywhere. We shall say that the step aa' is equivalent to the step bb' ; which may also be written shortly thus: aa bb'. Here the symbol =, which is commonly shorthand for equal, is used in the sense of equivalent It means more than that the length aa is equal to the length bb', namely, that the direction aa' is also the same as the direc- tion bb'. COMPOSITION OF STEPS. GEOMETRY. If, while a railway carriage moves along the line from the position 1 to the position 2, a man who was sitting on the seat a moves across to the seat b, the final position- COMPOSITION OF STEPS. of the man will be the same as the final position of b, namely, b'. The man is said to have made the step ab relative to the carriage ; and his actual step from a, to b' is said to be compounded of the step of the carriage, W, and of this step relative to the carriage. Thus the step ab' is compounded of the steps ab and bb r . In this case ab and bb' are called the components and ab' is called the resultant. Since aa' is equivalent to bb', we may equally speak of ab' as the resultant of aa' and ab. Thus we get two different rules for finding the resultant of two given steps : 1. Let the straight lines representing the steps be so placed that the end of the first is the beginning of the second ; then the step from the beginning of the first to the end of the second is the resultant (ab' resul- tant of ab and bb'}. 2. Let the straight lines repre- senting the steps be so placed that they have the same beginning, and let a parallelogram be constructed of which they are two sides ; then the resultant will be represented by that diagonal of the parallelogram which passes through the common beginning (ab' resultant of aa and ab). In the first rule we speak of the components as oc- curring in a certain order, first and second, viz,, the step relative to the carriage and the step of the carriage ; but in the second rule there is no such distinction. It appears from this that the two steps might be interchanged with- out affecting the result ; and it is indeed obvious that if the train had moved sideways by the step ab, and the man had moved along it by the step aa, he would in the end be at b' as in the case already considered. C DYNAMIC. It is sometimes necessary to compound together more than two steps. Thus, in the example just used, the train is moving relatively to the Earth, the Earth is moving round the Sun, and the Sun is moving on his own account through space, or rather, for this is all we can be sure about, he is moving relatively to certain stars. So that to get the actual motion of the man in the train relative to these stars, we must compound all these motions together. The rule for this is very easily found when the straight lines representing the steps to be compounded are so arranged that the end of each is the beginning of the next. Then the resultant is the step from the beginning of all to the end of all. Thus the steps ab, be, cd, de have the resultant ae ; for ab and be give ac, then ac and cd give ad, and finally ad and de give ae. But when the lines are all arranged so as to have a common beginning, the rule is rather more complex, and will be examined after we hare found a shorter way of writing about the composition of steps. What is true of two steps, that their resultant is inde- pendent of the order in which they are taken, is true of any number of steps. This we shall now prove. First, the resultant is unaltered by the interchange of two successive steps. For to inter- change the steps be, cd, that is, to take cd before be, we must draw be equal and parallel to cd, and then from c' a line equal and parallel to be. But this line will end precisely at d, because bcdc is a parallelogram. Nothing after the point d will be altered, and consequently the resul- tant ae will be the same as before. Next, any change whatever in the order can be pro- duced by a sufficient number of interchanges of successive steps. This statement clearly does not apply to steps only, but to any things whatever that can be arranged in order ; for example, letters or figures. The truth of the statement will be made clear by an example of the process to be used. Thus, let it be required to change the order 123456 into the order 314625. Bring 3 to the first place by successively interchanging it with 2 and 1. Then 1 will be in the second place as required. Bring 4 to the third place by interchanging it with 2, and then bring 6 to the fourth place by interchanging it with 5 and 2; lastly, interchange 5 and 2, and the required trans- formation is complete. As no one of these six interchanges has altered the resultant, it remains the same as at first. Thus the proposition is proved. COMPOSITION OF STEPS. ALGEBRA. When we have to deal with steps . < > which are all in the same straight line, as ab, be, cd, we may describe each of them as a step of so many inches to the right or to the left. To find the resultant we must add together the lengths of all the steps to the right, and also the lengths of all the steps to the left. The resultant is a step whose length is the difference between these two sums, and it is to the right if the former is greater, to the left if the latter is greater. Thus the resultant of the steps ab, be, cd is, as we know, ad ; and the length of ad is ab + cd cb. The resultant is a step to the right because the sum ab + cd is greater than cb. It is convenient to regard a step to the left as a ne- gative quantity, the addition of which is equivalent to the subtraction of its length from that of a step to the right. Thus + be is taken to be the same as cb. And thus we may write either ad = ab -f cd cb, or else ad = ab + cd + be. The symbol +, placed between two steps, is thus made to mean that their resultant is to be found, regard being had to their directions. The resultant ab + be is always ac, no matter how the points are situated ; but the length ac is a sum or a difference of the lengths ab and 6c, ac- cording as they are in the same direction or not. We shall extend this meaning of the symbol + to cases in which the component steps are not in the same 8 DYNAMIC. sti-aight line ; -that is to say, ab + cd shall always mean the resultant of the steps ab and cd, not the sum of their lengths unless this is expressly mentioned. Simi- larly ab cd will mean the resultant of ab and a step the reverse of cd, namely dc. After a little practice, the student will find that this extension of the meaning of the signs +, , and = does not cause any confusion, but on the contrary enables us to reason more clearly because more compactly. We shall now use this method to investigate the resultant of seve- ral steps the lines representing which are so placed as all to have the same beginning. In the case of two steps oa and ob, the rule is to complete the parallelo- gram oapb, and then the diagonal op is the resultant. But if we join the points ab by a straight line meeting op in c, both op and ab are bisected at the point c. Thus op is twice oc, which may be written op = 2oc. Observe that 2oc means a step in the direction of oc, of twice its length. We may now state our rule as follows : find c the middle point of ab, then the resultant of the steps oa and ob is twice oc ; or, more shortly, oa + ob = 2oc. We may extend this result. Let a ab be divided in c so that ac is to cb as m to I, where I and m are any two numbers. Then l.ac m.cb and (I + m) . ac = m . ab. Now oc = oa + ac ; that is to say, the step oc is the resultant of oa and ac. Therefore (I + m)oc = (1 + m) oa + (I + m) ac. But (I + m) ac = m . ab, and ab = ao + ob = ob oa. Substituting this value, we find (I + m) oc = (I + m) oa + m . ab = (l + m) oa + m (ob oa) That is, if ab be divided in the ratio m : I at the point c, tfon the resultant of I times oa and m times ob is l + m times oc. COMPOSITION OF STEPS. 9 We shall now write this proof in a shorter form. We have oa = oc + ca, ob oc + cb; therefore I . oa + m . ob = (I + m} ,oc + l,ca + m.cb = (7 + ra) . oc, because the point c was so chosen that l.ac = m. cb, or (which is the same thing), I . ca + m . cb = 0. The former investigation exhibits the process of finding oc in terms of oa and ob ; the latter is a shorter and more symmetrical proof of the result when it is known. We proceed now to the case of three steps, oa, ob, oc. Bisect ab in f, then 2o/= oa + ob, so that it remains to find the resultant of oc and twice of. This is a case of the last proposition, in which I = 1 and m = 2 ; we must therefore divide cf in the ratio of 2 : 1. Taking then a point g at two-thirds of the way from c to/, we find 3og for the resultant of oa, ob, oc ; or, more shortly, oa + ob + oc = 3og. This result is true wherever the point o is : whether in the plane abc or out of it. And the method of de- termining g is quite independent of the position of o. By making o coincide with g, so that og is zero, we find that ga + gb + gc = 0. This is independently clear, be- cause ga + gb = 2gf, and 2<7/+ gc = by construction. Hence also we see that g is f of the way from a to d, and from 6 to e, if d, e are the middle points of be, ca. Or the lines joining the angles of a triangle to the middle points of its sides meet in a point which divides each of them in the ratio of 2 to 1. To find the resultant of I times oa, m times ob, and n times oc, we must observe that whatever the point g is, oa = og + ga, ob = og + gb, oc = og + gc, and therefore .ob + n.oc = (l + m + n) . oj + 1 .ga + m.gb + n . gc. 10 DYNAMIC. If therefore we can find a point g such that l.ga+m.gb + n.gc=0, we shall have I . oa + m . ob + n . oc = (7 -4- m + n) . og. Now I . ga + m . gb = (I + ni) . gf^ if /is the point dividing ab in the ratio m : 1. Hence (I + m) . gf+ n.gc = 0, or g is the point dividing cf in the ratio I + m : n. We might equally well have found g by dividing be in the ratio n : m at d, or ca in the ratio I : n at e. That is, we have the equations l.fa = in.bf, n.ec = l.ae, and m.db=n.cd, between the lengths of the six segments into which the sides are divided. Multiplying these equations together, we find that the product Imn divides out, and that fa.ec. db bf. ae . cd. Hence if ad, be, cf meet in a point, then af.ce.bd=ea.dc.fb. This theorem is a useful criterion for the concurrence of three lines drawn through the vertices of a triangle. A similar set of theorems belongs to the composition of four steps. If/, g, h are the middle points of be, ca, ab, and /', g', h' of da, db, dc, then ff, gg', hti bisect one another at a point k, such that oa + ob + oc + od = For oa + ob = 2oh, and oc + od = 2oti ; also if Is be taken at the middle point of 7ih', oh + oh'= %ok; therefore oa + ob + oc + od = 2 (oh + oh') = 4o&. And the symmetry shews that this k is also the middle point ofgg'andofff. Moreover, if we take a ^ of the way from / to d, then k is of the way from a to a. For we know that ob + oc + od = Soot, and therefore oa + 3oa = 4o&, wherever o is : or taking o to coincide with k, ka + 3kx = 0, which shews that k divides aa in the ratio of 3 : 1. Observe that the points abed may either be in the same plane, or form a triangular pyramid, or tetrahedron. DESCRIPTION OF STEPS. 11 In general, if we have n steps oa 1) oa z , oa 3 ...oa n , it is always possible to find a point g such that n . off = o! + oa 2 + . . . + oa n = Soa, as this sum may be conveniently written. The position of the point g will depend upon the points a lf a 2 ...a n , but not in the least upon the point o. To prove this, suppose we take a point p, and draw the steps pa^ po.' 2 ...pa n . The resultant of these must be some step, which can be found by arranging them tandem as in our first process. Let pg be the ?i th part of this resultant, so that n . pg = 2pa. Now we know that og=op+pg, oa^op+pa^ ...oa Therefore n , og = n . op + n . pg = n . Op + Thus g being chosen so that n . pg = Spa for a particular position of p, we see that n . og = Soo. for any point o whatever. This point g is called the mean paint, or mid-centre, of the points 15 a z ...a h . Similarly, it may be shewn that there is a point g such that, if l lt l z ...l n are any numbers, (l t + l z + ... -f og = ^ . oa l + 1 2 . oa z + ... + l n . oa n , whatever point o is. And lastly, if we have n steps aj) v a 2 & 2 ,... B & n , anyhow situated, their resultant is n times the step from the mean point of a v a a ...a n to the mean point of b v b z ...b n . The proof of this is left as an exercise for the reader. RESOLUTION AND DESCRIPTION OF STEPS. We have already seen that a step in a known direction may be completely specified by describing its length. This may be done in two ways. First, approximately, by stating the number of inches or centimeters and parts of an inch or centimeter; if the parts are expressed in decimal fractions, the approximation may be carried to any required degree of accuracy by taking a sufficient number of places of decimals. But as the length to be described is generally incommensurable in regard to an inch or a centimeter, this method is very rarely anything 12 DYNAMIC. more than an approximation. Second, graphically, by drawing the length to scale. A certain line being marked out upon the diagram to represent a centimeter, another line is drawn bearing the same ratio to this one that the length to be described bears to a centimeter. Thus at the side of a map there is a scale of miles, by which the distance between two places may be estimated. The actual distance bears the same ratio to a mile that the distance on the map bears to the representative length on the scale. This is the theoretically correct way of representing all continuous quantities, except angles, which should also be drawn ; though it is sometimes convenient to describe an angle in terms of degrees, minutes and seconds ; or in circular measure, which is the ratio of its arc to the radius. When it is known that a step lies in a certain plane, it may always be resolved into two components which are in fixed directions at right angles to one another. Let oX, o Y be two fixed lines at right angles to one another. Let op be the step which it is required to resolve. Draw pm perpendicular to oX, then op om -f imp ; or the step op has been resolved into two, one of which is in the direction o X, and the other in the direction o Y. Let x be the number of units of length (e.g. centi- meters) in om, and y the number in mp. Let also i represent a step of one centimeter along oX t and j a step of one centimeter along o Y. Then om is # times i, or xi; and mp is y times j, or yj. Hence the step op = xi + yj ; and we may say that every step in the given plane may be described in the form xi + yj, where x and y are two numerical ratios, and i, j are fixed unit steps at right angles to one another. When the lengths x arid y are given either approxi- mately or graphically, the step (known to lie in a given plane) is completely described in the same way. It is to be understood that when m falls to the left of EESOLUTION OF STEPS. 13 o Y, x is a negative quantity ; and when p falls below oX, y is a negative quantity. When it is not known in what plane a step lies, we can still resolve it into three components along fixed directions at right angles to one another. Let o X, o Y, oZ be three lines at right angles to one another, op the step to be resolved. Draw pn perpendicular to the plane XoY, and nm perpendicular to oX. Then op om + mn + np, or the step op has been resolved into three, which are respectively in the directions oX, o Y, oZ. Let, as before, x, y be the number of centimeters in om, mn, and let z be the number in np. Let also i, j, k be three steps of one centimeter each in the directions oX, o Y, oZ. Then om = xi, mn = yj, np = zk, and op = xi + yj + zk. Thus we see that any step whatever can be described in the form cci+yj+ zk, where x, y, z are three numerical ratios, and i, j, k are fixed unit steps at right angles to one another. When the lengths x, y, z are given approximately or graphically, the step is completely described in the same way. It is to be understood that z is reckoned negative when p lies on the further side of the plane Xo Y. We shall find other quantities, besides steps, which can be resolved into components in three fixed directions, and completely described by assigning three lengths. All such quantities are called vectors, or carriers, from their analogy to a step of translation or carrying. They can always be described in the form xi + yj + zk, where i, j, k are fixed unit vectors at right angles to one another. Except these unit vectors, it is usual to represent a vector either by the beginning and end of the line representing it, as op, or by a single small Greek letter, as a, p. When the position of a point p is described by means of the step from a fixed point o to it, the point o is called the origin, and the components oc, y, K are called the co-ordinates of p. The lines oX, oY, oZ&re called axes cf 14 DYNAMIC. co-ordinates, and the planes which contain them in pairs the co-ordinate planes. The step or vector op is called the position-vector of the point p. REPRESENTATION OF MOTION. We go on to describe more completely the translation of a rigid body. Hitherto we have considered only the step from the beginning to the end of the motion ; we shall now take account of the path and of the time in which it is described. As before it will be sufficient to consider the motion of a single point of the body. To describe completely the motion of a point p from a to b it would be j necessary to assign the path and also the position of the point in the path at every instant of time. The path may be assigned by drawing it, or by stating its geometrical properties. The position of the point in the path may be assigned by giving the length ap measured along the path at every instant ; and this may be done in two ways. First, by the approximate or numerical method. We may construct a table, in the first column of which are marked seconds or fractions of a second, and in the second are written against them the number of centimeters in the length ap at that time. Tables on this principle are printed in the Nautical Almanac, giving the position at en -n- J ^^ r :ENTIMETER y \ :< i i i 4 5 6 r ' J SECONDS any time of the Sun and the planets; principally of the Moon. The method is imperfect, because it only gives the position at certain selected moments, and then only approximately. REPKESENTATION OF MOTION. 15 Secondly, by the graphical method. In this, the seconds are marked off on a horizontal line oX, and above every point of this there is set up a straight line repre- senting the distance traversed at that instant. Thus, at the instant t, about 3| seconds from the beginning of the motion, the distance traversed was tq, on the scale of centimeters marked on oY. Drawing qp horizontal to meet o Y, we find the distance about 7 centimeters. The tops of all these lines form a curve oqr, which is called the curve of positions of the moving point. The figure is equivalent to a table with an infinite number of entries, each of which is exact. The line oX is the first column, and the lengths tq, etc., answer to the second column. In certain ideal cases of motion, it is possible to get rid of one objection to the numerical method, and to make it partially describe the position of the point at every instant of time. This is when we can state a rule for calculating the number of centimeters passed over from the number of seconds elapsed; or, which is the same thing, when we can find an algebraical formula which expresses the distance traversed in terms of the time. Such motions do not occur accurately in nature; but there are natural motions which closely approximate to them, and which for practical purposes are adequately described in this way. We go on to consider some of these ideal motions. UNIFORM MOTION. When equal distances are gone over in equal times, the motion is said to be uniform. In uniform motion, the distances gone over in unequal times are proportional to the times (Archiraedes). For let t and Tbe unequal times in which the distances s and S are gone over. Take any two whole numbers m and n. Then if we take n intervals of time equal to t, there will be gone over in them n distances equal to s ; that is, a distance ns is gone over in the time nt. Similarly, mS will be gone over in the time mT. Now if nt is greater than mT, ns is greater than mS ; for in a greater time 16 DYNAMIC. a greater distance must be traversed. If nt is equal to mT, ns is equal to mS; and if nt is less than mT, ns is less than mS. Hence by Euclid's definition of proportion, 8 : s = T : t. Let v be the number of centimeters gone over (or described) in one second ; then s : v = t : 1, or s = vt, where s is the number of centimeters described in t seconds. Here all three numbers may be incommen- surable ; but the algebraic formula s = vt supplies us with a rule for calculating s when t is known ; viz., multiply t by v. The curve of positions in this case is a straight line. For, if we set up the length v above the point 1, and draw through o the straight line ovq ; then on drawing tq vertical through any point t, we shall have tq : v = ot : 1, * or tq correctly represents the distance described in the time ot. Uniform motion may of course take place along any path whatever. But there are two cases of special in- terest; when the path is a straight line and when it is a circle. UNIFORM RECTILINEAR MOTION. Let p be a point moving uniformly along the straight line alp, and let o be any fixed point. We shall com- pletely describe the position of the point p at any instant, if we specify the step which must be taken to go to p from o at that instant. Now op = oa+ap. Let ab be the distance traversed in one second, then ap, being the distance traversed in t seconds, is t . ab. Hence we have op = oa + 1 . ab, or, if we denote the step op by p, oa by a, ab by /3, then p = y. + 1@.- UNIFORM RECTILINEAR MOTION. 17 This is called the equation of uniform rectilinear motion. It is simply shorthand for this statement: the steps to be taken in order to get from o to the position of p after t seconds are, first, the step a (oa) which takes us to the position at the beginning of the motion, and then t times the step ft (ab}. Two uniform rectilinear motions compound into a uniform rectilinear motion. While p moves uniformly along the line ab, let q move uniformly, relative to p, along cd; and let cd be the dis- tance traversed in one second in the relative motion. Draw de equal and parallel to ab, then ce is the actual motion of q in one second. Draw qr parallel to ab, meeting ce pro- * dueed in r. Then, cq being traversed in the same time as ap, we must have cq : cd = ap : ab = t : 1. Now cq : cd = qr : de, so that qr = ap. Hence r- is the actual position of q at the end of the time t. It is in the straight line ce, and cr : ce = cq : cd = t : 1. Thus the actual motion of q is a uniform rectilinear motion. The same thing appears by considering the equations. Let p l be the step op, and p 2 the step pq ; then p = p t + p 2 is the step oq. Now we have = a where a : = oa, # x = ab, and therefore the equation to a uniform rectilinear motion. The curve of positions of any mo- tion whatever may be conceived to be constructed by help of a uniform recti- linear motion, in this way. Let the * original motion be that of a point p along the path a&; c. 2 18 DYNAMIC. Let a point p move along o Y at the same time, so that the distance op' is at every instant equal to the dis- tance ap measured along the path. While this motion takes place, let the straight line oY have a uniform hori- zontal translation of one centimeter in every second; then by this com- bination of motions the point p' will describe the curve of positions oq. Hence the curve of positions of any rectilinear motion is described by combining that motion with a uniform rectilinear motion of one centimeter per second in a direc- tion at right angles to it. UNIFORM CIRCULAR MOTION. In uniform circular motion every point p of the moving body goes round a circle so as to describe equal arcs in equal times, and therefore proportional arcs in different times. The radius of the circle is called the amplitude of the motion. The time of going once round is called the period. If the arcs measured on the circle are reckoned from a point a, and if the moving point started from e at the beginning of the time considered, the angle aoe is callecl the angle at epoch, or shortly the epoch. Strictly speak- ing, the epoch is the beginning of the time considered. The ratio of the arc ap to the whole circumference is called the phase at any instant. Let n be the circular measure of the arc described in one second, and a the radius of the circle; so that na is the length of the arc described in one second. Then not is the length of arc, ep, described in t seconds, and nt is its circular measure. UNIFORM CIRCULAR MOTION. 19 Let also e be the circular measure of aoe ; then circular measure of aop -nt + e. We shall now obtain an expression for the step op at any instant. Draw pm, ob, perpendicular to oa. Then op = om + mp. Now as far as lengths are concerned, om , mp ~ T = cos aop, and. - as sin aop. Or, since op = oa = ob in length, om = oa cos aop and mp = ob sin aop. In the equation om = oa cos aop, the quantities om and oa may be regarded as steps; for as they are in the same direction, one is equal to the other multiplied by the numerical ratio cos aop. The same may be said of the equation mp = ob sin aop. Now aop = nt + , and therefore op = oa. cos (nt + e) + ob . sin (nt + e), or if we write p for op, ai for oa, and aj for ob, so that , j are unit steps along oa, ob, then p = a {/cos (w + e) +j sin (n + e)}. This is the equation to uniform circular motion. The angle nt + e is called the argument of this expression for p. A circular motion which goes round like the hands of a clock, or clockwise, is said to be in the negative sense ; one that goes round the other way, or counter-clockwise, is said to be in the positive sense. Two uniform circular motions of the same period and the same sense compound into a uniform, circular motion of that period and sense. Suppose the circles so placed as to have the same centre. The motions of p and q relative to o may be combined by completing the parallelogram oprq ; then the motion of r is the resultant. We may consider the paral- lelogram oprq to be made of four jointed rods, of which op and oq turn round o. When these motions have the 22 20 DYNAMIC. same period and the same sense, the angle poq remains always constant ; therefore the shape of the parallelogram remains unchanged. Consequently or is of constant length, and makes always the same angle with op or og . Hence r goes round uniformly in a circle of radius or. Let op = p,, oq = p z , or = p. Then, if a, 6 are the amplitudes, i, j unit steps at right angles to one another, p l = ai cos (nt + e^) + af sin (nt + ej, p 2 = bi cos (nt + e 2 ) + bj sin (nt + e 2 ), which may be written ci cos (nt + e) + cj sin (nt + e), provided that a cos e^ + b cos e 2 = c cos e, a sin e 1 + 1 sin e 2 = c sin e. From these two equations we must find c and e. Dividing the second by the first, we find a sin e. + b sin e. tan e = - ! , -- " . a cos e 1 + b cos e 2 Squaring both sides of both equations, and adding them together, we find c 2 = 2 + 6 2 + 2a5 cos (e, - e 2 ). These formulae determine the amplitude and epoch of the resultant motion. It is left to the reader to verify them by comparison with the geometrical solution. Like the preceding theorem about uniform rectilinear motions, this theorem may be extended to any number of circular motions of the same period and sense ; by first compounding the first two, then the third with their resultant, and so on. Or the extended theorem may be proved directly, either by the geometrical or by the analytical method^ HARMONIC MOTION. While the point p moves uniformly round a circle, let & perpendicular pm be continually let fall upon a diameter SIMPLE HARMONIC MOTION. 21 aa. Then the point m will oscillate to and fro between a and a'. This motion of the point m is called simple har- monic motion. The amplitude, period, epoch, and phase of the simple harmonic motion are the same as those of the uniform circular motion of p. The epoch, how- ever, must be reckoned from one extremity of the diameter on which m moves ; i. e., either from a or from a. We may define these quantities solely in terms of the harmonic motion, thus : the amplitude is half the distance between the two extreme positions ; the period is the interval of time between two successive passages through the same position in the same direction ; the phase at any instant is the fraction of the period which has elapsed since the point was at its extreme position in the positive, direction ; the epoch is 2?r multiplied by the phase at the beginning of the time considered. The equation to the simple harmonic motion is om = oa cos aop = oa cos (nt + e) ; or p = a. cos (nt + e). 2-7T Here the amplitude is a, the period is - - (for since in time t the circular measure of the arc described is nt, it 2?r follows that in time the circular measure is 27r). the n i . -. ., , . nt -f e epoch is e, and the phase is , . ZTT Uniform circular motion is compounded of two simple harmonic motions of equal period, whose amplitudes are equal in length and perpendicular in direction, and whose phases differ by . Namely, the motion of p is com- pounded of the motions of I and m, which answer to this description. Any two diameters at right angles will serve for this resolution. 22 DYNAMIC. The curve of positions of a simple harmonic motion may be constructed by means of a right circular cylinder. (This surface is traced out by a straight line which revolves about a fixed parallel line ; the moving line is called a generator, the fixed line the axis, of the cylinder,) Cut the cylinder through obliquely by a plane cc', and through o the Centre of cc draw a plane perpen- dicular to the axis of the cylinder, which will cut the cylinder in a circle aba'b'. Let bb' be the in- tersection of the two planes. A plane through o perpendicular to bb' will contain cc and aa, and everything will be symmetrical in regard to this plane. The curve in which the plane cc' cuts the cylinder is called an ellipse. We shall shew that if a piece of paper be wrapped round the cylinder, marked along this curve, and afterwards unrolled and laid flat, the trace upon it will be the curve of positions of a simple harmonic motion*. Let q be a point on this curve; draw qp per- pendicular to the plane aba'b', meeting the circle in p; draAV pn, pi perpendicular to aa' and bb' respectively. Then the triangle qpl is similar to cao. Therefore pq : Ip = ac : oa, on or pq = ac. = ac cos aop. oa If then p moves uniformly round the circle aba'b' at the rate of one centimeter per second, we shall have pq ac cos (nt + e), where n . a = 1. * The reader should cut out in paper a wavy curve of the shape drawn in the figure, and then bend it into the form of a cylinder, when the plane elliptic section will become manifest. COMPOUND HARMONIC MOTION. 23 Hence pq will at every instant be the step from its mean position to a point which is moving in a simple harmonic motion of amplitude ac, period 2?r . oa. When therefore the figure is unrolled from the cylinder, the wavy curve (called the harmonic curve, or curve of sines because the 7T 6 f) ordinate pq is equal to ac . sin ^ ~ , that is, proportional to the sine of a multiple of the abscissa b'p) is the curve of positions of the simple harmonic motion aforesaid. The amplitude is the height of a wave, ac. The period is the length of a wave, b'b", every centimeter in that length representing a second of time. The curves of position of motions compounded of simple harmonic motions in one line may be constructed by actually compounding the curves of position of the se- veral motions that is, by adding together their ordi- nates to form the ordinate of the compound curve. Thus in the figures the height of the dark curve above the hori- zontal line is at every point half the algebraic sum (which is more convenient for drawing than the whole sum) of the heights of the other two ; as for example Zmq = mp 4- mr. A depth below the line is counted as a negative height. The first figure represents the com- position of two simple har- monic motions of the same period ; the second two such motions in which the period of one is half that of the other. The student should construct a series of these for different epochs of one of the motions, and then compare them with those figured in Thomson and Tait's Natural Philosophy, p. 43. In the case where the component motions have the same period, the resultant is a simple harmonic motion of that period. This follows at once from the corresponding theorem in regard to circular motions. Completing the parallelogram opqr, and drawing perpendiculars pi, qm, rn 24 DYNAMIC. upon aa, we see that ol=qs =mn, and consequently on=ol -t-om. Therefore the motion of n is compounded of the motions of I and m. But since r moves uniformly in a a circle, the motion of n is a simple harmonic motion. And we have seen that, when = acos then p = p l + p z = c cos (nt + e), provided that c 2 = a 2 + 6 2 + 2aJ cos (e x + ej, a sin e, + b sin e, and tan e = . a cos e t + o cos e 2 It follows at once that the theorem is true for any number of simple harmonic motions having the same period. The use of the jointed parallelogram opqr for com- pounding harmonic motions of different periods is exem- plified in Sir W. Thomson's Tidal Clock. The clock has two hands whose lengths are proportional to the solar and lunar tides respectively, while their periods of re- volution are equal to the periods of those tides. A jointed parallelogram is constructed, having the hands of the clock for two sides. If the clock is properly set, the height of that extremity of the parallelogram which is furthest from the centre will be continually proportional to the height of the compound tide. For this purpose a series of horizontal lines at equal distances is drawn across the face of the clock, and the height is read off by running the eye along these to a vertical scale of feet in the middle. ON PROJECTION. The foot of the perpendicular from a point on a straight line or plane is called the orthogonal projection of that point on the line or plane, or more shortly (when no mistake can occur) the projection of the point. Thus PROJECTION. 25 the point m is the projection of p on the straight line aa. We say also, by a natural extension, that the motion of m is the projection of the motion of p. Thus simple har- monic motion is the orthogonal projection of uniform circular motion on any straight line in the plane of the circle. When all the points of a figure are projected, the figure formed by their projections is called the projection of the original figure. Thus, for example (first figure of p. 22), the circle abab' is the projection of the ellipse cbc'b', for it is produced by drawing perpendiculars from every point of the ellipse to the plane. The point a is the projection of c, a of c', p of q, etc. ; b and b' are their own projections, being already in the plane of the circle. Instead of drawing lines perpendicular to a plane from all the points of a figure, we may also project it by draw- ing lines all parallel to one another, but in some other direction. This is called oblique projection. The ellipse cbc'b' is an oblique projection of the circle abab', for the lines ac, a'c, pq are all parallel to one another, although they are not at right angles to the plane of the ellipse. Orthogonal and oblique projections are both included under the name parallel projection, because in both cases the projection is made by drawing lines which are all parallel to one another. We may also project a figure on to a given plane by means of lines drawn through a fixed point; this is called central projection. It occurs whenever a shadow is cast by a luminous point. If we suppose the centre of projection c to move away to an infinite distance, the lines converging to it will all become parallel. Thus we see that parallel projection is only a particular case of central projection in which the centre of projection has gone away to an infinite distance. The shadow cast by a bright star is for all practical purposes a parallel projec- tion.. 26 DYNAMIC. The projection of a straight line is made by drawing a plane through it and through the centre of projection. Thus if we draw the plane cab and produce it to meet the plane of projection in ab', this line ab' will he the projection of ab. In parallel projection we must draw through the line a plane parallel to the projecting lines, like the plane aba'b' in the second figure. "We see in this way that the projection of a straight line is always a straight line, and that, since the line and its projection are in the same plane, they must either meet at a finite distance or be parallel (meet at an infinite distance). In parallel projection, parallel lines are projected into parallel lines, and the ratio of their lengths is unaltered. Through the parallel lines ab, cd we must draw the planes aba'b', cdc'd' both parallel to the pro- jecting lines, and therefore parallel to each other. These planes will consequently be cut by the plane of projection in the parallel lines ab', c'd'. Moreover the triangles pbb', ~ qdd', having their respective sides parallel, are similar ; therefore pb : qd= pb' : qd', and so also ab : cd = ab' : c'd'. The orthogonal projection of a finite straight line on a straight line or plane is equal in length to the length of the projected line multiplied by the cosine of its inclination to the straight line or plane. If pq is the projection of PQ, draw pq equal and parallel to PQ. Then Qq is parallel to Pp and therefore perpendicular to pq; therefore the plane Qqq is perpen- dicular to pq, and therefore q'q is per- pendicular to pq. Hence pq=pq'cos q'pq PQ x cosine of angle between PQ andpq. The orthogonal projection of an area on a plane is equal to the area multiplied by the cosine of its inclination to the PROJECTION OF AN AREA. 27 plane. This is clearly true for a rect- angle ABCD, one of whose sides is parallel to the line of intersection of the planes. For the side AB is un- altered, and the other, BC, is altered into Be, which is BG cos 6. Hence it is true for any area which can be made up of such rectangles. But any area A can be divided into such rectangles together with pieces over, by drawing lines across it at equal distances per- pendicular to the intersection of the two planes, and then lines parallel to the intersection through the points when they meet the boundary. All these pieces over, taken together, are less than twice the strip whose height PQ is the difference in height between the lowest and highest point of the area; for those on either side of it can be slid sideways into that strip so as not to fill it. And by increasing the number of strips, and diminishing their breadth, we can make this as small as we like. Let then A' be the sum of the rectangles, then A' can be made to differ from A as little as we like. Now the projection of A is A cos 6, and this can be made to differ from the projection of A as little as we like. Therefore there can be no finite difference between the projection of A and A cos 0, be- cause A' cos 6 can be made to differ as little as we like from both of them. PROPERTIES OF THE ELLIPSE. The ellipse may be defined in various ways, but for our purposes it is most convenient to define it as the parallel projection of a circle. This definition leads most easily to those properties of the curve which are chiefly useful in dynamic. Centre. The centre of a circle bisects every chord passing through it; such a chord is called a diameter. 28 DYNAMIC. The projection of the centre of the circle is a point hav- ing the same property in regard to the ellipse, which is therefore called the centre of the ellipse. For let aca be the projection of ACA' ; then ac : ca = AC : CA' \ but AC= CA', therefore ac = ca. It follows also that if any two chords bisect one another, their intersection is the centre. Conjugate Diameters. The tangents at the extremity of a diameter of a circle, A A', are perpendicular to that diameter ; if we draw another diameter BB' perpendicular to A A', and therefore parallel to these tangents, the tangents at the extremities of BB' will be perpendicular to BB, and therefore parallel to AA. It follows that in the ellipse, if we draw a diameter bb' parallel to the tangents at the ends of aa the projection of AA', this line bb' will be the projection of BB', for parallel lines project into parallel lines; therefore also the tangents at the extremities of bb' will be parallel to aa. Such diameters are called conjugate diameters; they are pro- jections of perpendicular diameters of the circle. Each of the diameters AA', BB bisects all chords parallel to the other ; thus A A' bisects P Q in the point R. Now PQ is projected into a chord pq parallel to bb', and the middle point R is projected into the middle point r. Hence also in the ellipse, each of two conjugate diameters bisects all chords parallel to the other. The assumption here made, that a tangent to the circle projects into a tangent to the ellipse, may be justified as follows. If we take a line PQ cutting the circle in two points, and move it away from the centre until these two points coalesce into one, as at A, the PROPERTIES' OF THE ELLIPSE. line becomes a tangent. Now when these two points coalesce, their projections must also coalesce ; therefore when the line becomes a tangent to the circle, its pro- jection also becomes a tangent to the ellipse. Relation between ordinate and abscissa. In the circle, if PM, PL be drawn parallel to CB, CA respectively, we know that CP*=CM* + MP*, and since CP = CA = CB, it follows that CM* , MP CB* CTIT ^ CCL Ttl'D* rr = CO CA* Hence it is equally true in the ellipse that For the ratio of parallel lines being unaltered by parallel projection, cm : ca = CM : CA, and mp : cb = MP : CB. The line mp is called an ordinate or standing-up line, and cm is called an abscissa or part cut off. If we write x for cm, y for mp, a for ca, b for cb, the equation becomes a? if __ I J 1 2 ^^ 12 a b The same relation may be expressed in another form which is sometimes more useful. Namely, observing that the rectangle am . ma = (ca + cm) (ca cm] = ca* cm*, we find that mp* : cb* = a'm.ma : ca 9 . This may also be proved directly by observing that it is true for the circle and that the Tatios involved are ratios of parallel lines. This relation shews that when two conjugate diameters are given in magnitude and position, the ellipse is com- pletely determined. For through every point m in aa' we can draw a line parallel to bb', and the points p, p where this line meets the ellipse are fixed by the equation mp* (or mp' 2 ) : cb* = a'm . ma : ca*. Axes. The longest and shortest diameters of an ellipse are conjugate and perpendicular to each other. We may shew, in general that if the distance of a curve from a #xed point o increases up to a point a and then decreases, 30 DYNAMIC. the tangent at a, if any, will be perpendicular to oa. We say if any, because the curve might have a sharp point at a, and then there would be properly speaking no tangent at a. Since the distance from o in- creases up to a and then decreases, we can find two points p, g, one on each side of a, such that the lengths op, oq are equal. Then the perpen- dicular from o on the line pq will fall midway between p and q. Now suppose p and q to move up towards a, keeping always the lengths op, oq equal; then the foot of the perpendicular on pq will always lie between p and q. When therefore the line pq moves on until p and q coalesce at a, the foot of the perpendicular will coalesce with them, or oa is perpendicular to the tangent at a. The length oa is called a maximum value of the dis- tance from o. It need not be ab- solutely the greatest value, but it must be greater than the values immediately close to it on either side. A similar demonstration ap- plies to a point where the distance, after decreasing, begins to increase; that IB, to a minimum value of the distance. Applying these results to the Ellipse, we see that the tangents at the extremities of the longest and shortest diameters (which of course must be points of greatest and least distance from the centre) are perpendicular to those diameters. Let bb' be the shortest diameter, and draw ad perpendicular to it, and therefore parallel to the tangents at b, b' ; then ad is conjugate to bb', and consequently the tangents at a, a' are parallel to bb r , and therefore perpendicular to ad. Therefore aa' and bb' are conjugate diameters per- AXES OF THE ELLIPSE. 31 pendicular to each other. Now describe on aa' as diameter a circle, aB a'B'. If this circle be tilted round the line ad, until B is vertically over 6, and then orthogonally projected on the plane of the ellipse, the projection will be an ellipse having aa' and bb' for conjugate diameters, which must therefore be the same as the given ellipse. Hence if Ppm be a line parallel to bb' meeting the circle, ellipse, and aa' in P, p, m respectively, we must have mp : mP= cb : cB. Hence the ellipse lies entirely within the circle, and therefore no other distance from the centre is so great as ca or ca ; that is, aa' is the greatest dia- meter. The diameters aa' and bb' are called the axes of the ellipse ; aa is the major or transverse axis, bb' the minor or conjugate axis. The circle on aa' as diameter is called the auxiliary circle. No other pair of conjugate dia- meters can be at right angles ; for they are projections cp, cq of per- pendicular diameters cP, c Q of the circle, and the angle pcq is always greater than the right angle PC Q. We see, then, that in every case of parallel projection, there are two sets of parallel lines, perpendicular to each other in the original figure, that remain perpendicular to each other in the projected figure. ELLIPTIC HARMONIC MOTION. A parallel projection of uniform circular motion is called elliptic harmonic motion. An elliptic harmonic motion may be resolved into two simple harmonic motions of the same period along any two conjugate diameters of the ellipse, these motions differing in phase . For we know that the uniform circular motion may be resolved into two such simple harmonic motions along any two perpendicular diameters. And the parallel projection of a simple harmonic motion is clearly another simple harmonic motion, with the same period and phase. 32- DYNAMIC. Conversely, any two simple harmonic motions on different lines, having the same period and differing in phase , com- pound into harmonic motion in an ellipse having those two lines for conjugate diameters. For let the ellipse be con- structed ; then we have shewn that a circle can be so placed as to have the ellipse for its orthogonal projection. Consequently the two given conjugate diameters are ortho- gonal projections of two perpendicular diameters of the circle, and the harmonic motions on them are projections of harmonic motions of the same period and phase on the diameters of the circle. But the resultant of these is uni- form motion in the circle ; therefore the resultant of their projections is the projection of uniform circular motion, namely, harmonic motion in the ellipse. The equation to elliptic harmonic motion is p = a. cos (nt + e] + ft sin (nt + e), where a, ft, are two semiconjugate diameters of the ellipse. For the equation to the motion of m is p = a cos (nt + e) if a = ca, and that to the motion of I, having ampli- tude ft, the same period, and phase differing by J (and therefore epoch differing by quarter circumference), must be ft cos (nt + e \TT) = ft sin (nt + e). - And the motion of p is compounded of these two. The resultant of any number of simple harmonic motions in any directions, having the same period, is elliptic har- monic motion. Let the equations to the different mo- tions be Pl = ctj cos (nt + ej, p 2 = or 2 cos (nt + e 2 ), . . ./> = a n cos (nt + ej. Expanding these cosines, we have, for example, PJ = otj cos e x . cos nt a 1 sin e a . sin nt ; and then, adding all together, P = Pt+ Pz+ + Pn= (otjCos^+agCos e a +. . . +a n cos ej cos nt (GJ sin ej-f- a 2 sin e 2 + ... + sin ej sin nt=a cos nt+ft sin nt, if a = 2a cos e, ft = ^a sin e. This is the equation to elliptic harmonic motion. COMPOUND HARMONIC MOTION. 33 It is worth while to notice the meaning of the steps iu this demonstration. The expansion (nt + j) = or 1 cos 6j . cos nt a t sin e : . sin nt is equivalent to a resolution of the simple harmonic motion into two in the same line, differing in phase . The epoch of one of these may be assumed arbitrarily, say 77 ; for x cos (nt + e a ) = ! cos (nt + 77 + P! 77) = a x cos fo 17) . cos (nt + 77) ^ sin fc 77) . sin (nt + 77). This is a particular case of the resolution of elliptic har- monic motion into two simple harmonic constituents, differing in phase \, the epoch of one being arbitrary (since any two conjugate diameters may be chosen). Then the summation Sacose. cosnt = acosnt means that the resultant of any number of simple harmonic motions of the same period and phase is a simple harmonie motion of that period and phase. Thus all the simple harmonic motions are reduced to two, which differ in phase | ; and the resultant of these, as we know, is elliptic harmonic motion. COMPOUND HARMONIC MOTION. If we combine together two simple harmonic motions in different directions with different periods, the resultant motion is periodic if the periods are commensurable, and its period is their least common multiple ; if they are in- commensurable the path of the moving point never returns into itself so as to form a closed curve. In either case the most convenient way of studying the resultant motion is to convert it into motion on a cylinder, by combining with it a simple harmonic motion perpendicular to its plane which forms a uniform circular motion with one of the components. Suppose, for example, that we wish to study the motion p = a cos (nt + e) + /3 cos mt (where a may be taken perpendicular to $) for different values of e. Then we should combine with it a motion p = 7 sin (nt + e), where 7 is perpendicular to both a and /3, and of the same length as a. The two terms a cos (nt + e) + 7 sin (nt + e) give a uniform circular motion in a plane perpendicular to /3. c. 3 DYNAMIC. Thus we have now to combine a uniform circular motion with a simple harmonic motion perpendicular to its plane. Or, we suppose a generating line to move uniformly round a cylinder, while a point moves up and down it with a simple harmonic motion. This is clearly the same thing as wrapping round the cylinder the curve of positions of the motion ficosmt. Hence the path of the motion on the cylinder may always be obtained by wrapping round the cylinder a harmonic curve. Now the original motion p a. cos (nt + e) + /3 cos mt is clearly the projection of this motion on the cylinder upon a plane perpendicular to 7; which plane we may suppose to be drawn through the axis of the cylinder. But by taking different planes through the axis for plane of projection we produce the same effect as by varying e. For this is the same as varying the diameter 2 a of the circle on which we project the uniform circular motion p = a cos (nt + e) + 7 (sin nt + e). And 'if the same circular motion be projected on two different diameters aaf and bb', the resulting simple harmonic motions will differ in epoch by the angle aob. We may illustrate this by the case m = 2w, when the motion is p = a cos (nt + e) + (3 cos 2n. The case e = is always one of special simplicity, being (like the simple harmonic motion) a case of oscillation on a finite portion of a curve. Let then om = /3 cos 2nt = ob cos 2nt, mp = % cos nt = oc cos nt. Then am = /3 (1 + cos 2nf) = 2/3 cos 2 nt = ab.cos'nt, but mp* = oc 2 . cos 2 nt. Therefore mp 2 : oc* = am : ab, or nip* varies as am. The curve in which this is the case is called a, parabola; its two branches extend indefinitely to the right, but only a finite portion of it is traversed by the harmonic motion. COMPOUND HARMONIC MOTION. 35 This finite curve, then, is the orthogonal projection of a curve on a cylinder; the axis of which may be (1) vertical, (2) horizontal. In case (1), the curve is made by wrapping round the cylinder a harmonic curve one wave of which will go twice round ; or, which is the same thing, by bending into a cylin- drical form the spindle-shaped figure here drawn. In case (2), we must wrap round a harmonic curve, two of whose waves will go once round; the result is some- thing like an ellipse whose plane is bent. The figure obtained by looking at the first along ab or the second along a direction making an angle of 45 with co in a vertical plane is here given. A series of intermediate forms is given in Thomson and Tait's Natural Philosophy, p. 48. The equation to the figure-of-8 motion is p = ot cos (nt + TT) + /3 cos 2nt, or p = a cos nt +(3 cos (2nt + ^TT). All the intermediate forms can be got by looking at the curve on the cylinder from a sufficient distance and turning it round the axis of the cylinder. For this purpose the curve should either be made in stiff wire or drawn on a glass tube. Whenever two simple harmonic motions in rectangular directions with commensurable periods are compounded together, there is a certain relation of the phases (viz. the equation is p = a.cosnt+pco$mt) for which the path resembles that on a parabola in the case just considered ;. namely, the path is a finite portion of a geometrical curve on which the moving point oscillates backwards and forwards. There is also a path which resembles the figure-of-8 in being symmetrical in regard to each of the perpendicular lines a, ft. From a knowledge of these two the intermediate forms may be easily inferred. The general shape of these two forms may be obtained by a very simple process, which will be understood from a particular example of it. The figures on the opposite 3-2 36 DYNAMIC. page represent the symmetrical curve and the curve of oscillation for the case m : n 3 : 4. Suppose the amplitudes of the two motions equal, so that the path is included in a certain square ; and draw circles equal to the inscribed circle of the square touching two of its sides at their middle points. Then if one point goes uniformly three times round ABCD while another goes four times round FQH, a horizontal line through the first point and a vertical line through the second will intersect on the curve which is to be drawn. The con- tacts of the curve with the sides of the square are projections on them of the four points ABCD, forming an inscribed square, and the three points FGH, forming an equilateral triangle. For the symmetrical curve these must be so disposed on their respective circles that their projections abed and fgh shall be all distinct and sym- metrically placed in regard to the sides of the square. For the curve of oscillation they must be so placed that the projections either coincide two and two or are at the corners of the square. When the contacts are de- termined, we must begin at any corner of the square say the left-hand bottom corner, Fig. 1, and join the nearest points c/by a piece of curve convex to the corner; then the next two, dh, in a similar manner ; then bg ; then ac', but as these points are on opposite sides of the square, the curve has a point of inflexion between them. The symmetry of the curve will now enable us to com- plete the figure ; or we may apply the same process, beginning at the adjacent corner gc'. The curve of oscillation has always to go through a corner of the square. If we fix upon the corner d, Fig. 2, this determines the position of A'B' C'D' and of FGH, shewing that the curve also goes through the corner I, but not through either of the other corners. This de- termines the position of ABCD, because their projections are thus obliged to coincide two and two. The motion takes place on a portion only of the geometrical curve*, whose continuations, indicated by the dotted lines, re- semble a parabola in shape. * On the equations and geometrical character of these curves, sec Braun, "Ueber Lissajous' Ctirven," Math. Annalen, vm. p. 5G7. 37 Fig. 2. Examples of harmonic motion, simple and compound, occur in the vibrations of elastic solids, the rise and fall of the tides, the motion of air-particles when transmitting sound, and of the ether in carrying radiations of light and heat. The important theorem of Fourier, that every motion which exactly repeats itself after a certain interval of time is a compound of harmonic motions, will be proved in the Appendix. 38 DYNAMIC. PAHABOLIC MOTION. It was observed by Galileo that if a body be let slide down a smooth inclined plane, the lengths passed over from the beginning of the motion are proportional to the squares of the times. That is, if the body goes a length a in the first second, then in the first t seconds it will go a length at z . It is easy to see that the lengths passed over in successive seconds are proportional to the successive odd numbers ; for in the nth second the distance travelled is . Thus if p = a. + fy3, then p = /9. In the uniform circular motion p = ai cos (nt + e) -f- aj sin (nt + e), the distance travelled in one second is na, and the direction of the motion is pt. Hence if oq be drawn parallel to pt, the velocity is represented by n.oq. Now oq is what op will become after a quarter period ; that is, after the angle nt + e has been increased by a right angle. Thus oq ai cos (nt + e + ^TT) + aj sin (nt + e + ^TT) ; and p = n.oq = nai cos (w + e -f | TT) + naj sin (n + e -f |TT). The rule to find p from p is therefore, in the case of uni- form circular motion : multiply by n, and increase the argument by \ TT, VELOCITY. VARIABLE. To make more precise the idea of a velocity which varies continuously with the time, let us consider the case of two parallel lines of rail, on one of which a train starts from rest and gradually increases in speed up to twenty miles an hour, while on the other a train runs uniformly in the same direction at 10 miles an hour. We will suppose the second train to be so long that a traveller in the first train has always some part of it immediately opposite to him. At starting, the uniform train will ap- pear to this traveller to be gaining on him at the rate of 10 miles an hour; but as his own train gets up speed, this rate of gaining will diminish. At the end, when the variable train is going 20 miles an hour, the uniform train will be losing 10 miles an hour. There must have been some moment between these two states of things at which the uniform train was seen to stop gaining and to begin 42 52 DYNAMIC. to lose. At that moment the variable train was going 10 miles an hour. In the same way if we suppose the uniform train to go at any other velocity less than 20 miles an hour, there will be an instant at which it will appear to a traveller in the variable train to stop gaining and to begin to lose. This will be the instant at which the variable train having hitherto been travelling at a less velocity, just acquires the velocity of the uniform train, and then, acquiring a still greater velocity, proceeds to gain upon it. When then we say that at a certain instant a train is going v miles an hour, we mean that a train moving uni- formly v miles an hour on a parallel line of rails would appear from the first train to stop. If the velocity of the variable train is continually increasing or continually de- creasing, the uniform train will appear to reverse 'its mo- tion ; but if the velocity after increasing up to that point began to decrease, or after decreasing began to increase, the uniform train would seem to stop momentarily and then go on in the same direction. By these considerations we have reduced the case of an instantaneous velocity of any magnitude to the case of stoppage or zero velocity, which can be readily observed and conceived. In the motion of a falling body, for example, we have s fa, where a is the distance fallen in the first second, and s the distance fallen in the first t seconds. Suppose another body to move uniformly downwards with velocity v ; in t seconds it will have passed over a distance s 1 = vt. Thus the distance between the two bodies is s r s = vt a?. Therefore 4a(s 1 - s) = 4avt - 4aV = i? - (v - 2a)l This quantity continually increases so long as v 2at diminishes by the increase of t, that is, until v = 2at ; then it begins to diminish again. Hence at the moment when v = "2at, or t = v : 2a, the uniformly moving body stops gaining on the falling body and begins to lose. Consequently the velocity of the falling body at the time t is 2at. Or if s = atf, then s = 2aL It appears therefore that a body falling freely in vacuo 4 DYNAMIC. tually made. If a body gets from a to 6, by any path, with any variation of speed, in t seconds, its mean velo- a ^ city during the interval is ab : t. The direction of it is the straight line ab. When the motion is rectilinear, the mean velocity is of course simply the distance traversed divided by the time of travelling. Now let k, I be two positions of a moving point, q, r the corresponding points of the curve of positions, qm, rn vertical. Then mn represents the time of travelling from k to I ; and conse- quently the mean velocity during this interval is represented by kl : mn. But this is the uniform velo- city whose curve of positions is the chord qr. Hence the chord joining two points on the curve of positions is itself the curve of positions of the mean velocity during the corre- sponding interval. If the chord cuts the line omn in t the mean velocity is mq : tm. The tangent at a point p is obtained by moving the chord qr till its ends coalesce at p ; and the tangent is curve of positions of the instantaneous velocity correspond- ing to the point p. Hence the instantaneous velocity at any instant (when there is one] may be obtained from the mean velocity of an interval by making both ends of the interval coincide with that instant. This appears to be nonsense, because there is no interval when the two ends coincide. But an example will shew what is the meaning of the rule. Let us take again the case of a falling body, s=af; it is required to find the instantaneous velocity at the time t. In the interval between ^ seconds and 2 seconds after the beginning of the time, the distance travelled is s i s 1 = at* at* ; therefore O . v u r p q r therefore v = pqr + pqr + pqr ; and it is clear that this theorem may be extended to any number of factors. Flux of a quotient of two quantities. Let p : q be the quotient ; then we have and the latter expression, when we cast out common factors and omit the suffixes, becomes pqpq ' . The ratio < : s is called the cur- vature of the curve at the point p. This ratio is the s-flux of < ; for we know that, since is a function of s which is a function of t, $, see p. 66. Thus we may define the curvature as the rate of turning round per unit of length of the curve. We may also define it independently of the idea of velocity, thus. The angle ty between the direction of the tangents at a and b is called the total curvature of the arc ab; the total curvature divided by the length of the arc is called the mean curvature of the arc ; and the cur- vature at any point is the value to which the mean cur- vature approaches as nearly as we like when the two ends of the arc are made to approach sufficiently near to that point. In a circle of radius a, the arc s = a$>; consequently s = a0, and < : s = 1 : a, or the curvature is the recipro- cal of the radius. (Observe that curvature is a quantity of the dimensions [X]" 1 .) It is in fact obvious that the arc of a small circle is more curved than that of a large one. When the point stops and reverses its motion, while the line goes on, we have a cusp in the curve ; at such a point 6- = 0, while < is finite, and the curva.ture is infinite. When the line stops and reverses its motion, while the point goes on, we have a point of inflexion ; at such a CIRCLE OF CURVATURE. 75 point

, but its direction is tc, parallel to the inner normal at p. Now the s-flux of p' (which we shall write p") is equal to the -flux of p divided, by s. But the -flux of p f , as we have seen, is in magnitude equal to <. Hence p", the second s-flux of p, is a line parallel to the inner normal at^>, of length equal to the curvature &ip. When a curve does not lie in one plane (in which case it is called a tortuous curve), a more complex machinery is required to describe it. We must then take a point, a straight line through the point, and a plane through the straight line ; and let them all move together so that the point moves along the line, the line turns round the point in the plane, and the plane turns round the line. The point is then a point on the curve, the line is the tangent at that point, and the plane is called the osculating plane at the point. The curvature is, as before, the rate of turning round of the tangent per unit of length ; and, in addition, the rate per unit of length at which the osculating plane, turns round the tangent line is called the tortuosity. In this case, however, we require somewhat closer at- tention to determine what we mean by the rate of turning round of the tangent line. Let ot be a line of unit length always parallel to the tangent; then the point t will always lie upon a sphere of unit radius ; but the curve not being now in one plane, t will not describe a great circle of the sphere (or circle whose plane goes through the centre). As p moves along the curve, t will describe some curve on the sphere, and the velocity of if will still be, in magnitude, the rate of turning round of ot, that is, of the tangent. COMPONENTS OF ACCELERATION. 77 But besides this, if tc be the tangent to the path of t, the plane otc will be the plane in which ot is turning, that is, it will be parallel to the osculating plane. Hence tc is parallel to the normal in the osculating plane at p ; this is called the principal normal. Since the curvature is a bending in the osculating plane, towards this normal, we may say that tc is the direction of the curvature. Now in this case, just as with a plane curve, p is the unit vector ot parallel to the tangent, and p" is a vector parallel to tc and in length equal to the curvature. Thus p" represents the curvature in magnitude' and direction. TANGENTIAL AND NORMAL ACCELERATION. We have remarked that the -flux of p is equal to the s-flux multiplied by the velocity, 6- or v. We may now find an expression for the second /-flux of p, or the accele- ration, by regarding it as the flux of this product, vp. Namely we have p = vp /. p = vp' + vp. But p'= sp" = vp" (as before remarked, p. 76), therefore p = vp' + v*p", that is to say, the acceleration p may be resolved into two parts, one of which vp is parallel to the tangent, and its magnitude is the rate of change in the magnitude of the velocity; the other v*p" is parallel to the (principal) normal, and its magnitude is the squared velocity multi- plied by the curvature. It appears also that when the path of the moving point is tortuous, the acceleration is wholly in the osculating plane. We may at once verify this proposition in the case of uniform motion in a circle, in which the hodograph is another circle (radius v) de- scribed uniformly. Since the two circles are described in the same time, the velocities in them must be proportional to their radii; hence the 78 DYNAMIC. acceleration of p, = velocity of u, : v = v : a, or accelera- tion = v* : a = v z x curvature. Thus the normal acceleration is the same as that of a point moving with the same velocity in the circle of curvature. The proposition may be further illustrated by means of the hodograph. Let ou represent the velocity of p, and uc be the velocity of u. This may be resolved into um in the di- rection of ou, which is the rate of change in its magnitude, or v ; and me perpendicular to ou, which is ou multiplied by its angular velocity, or v = v s x curvature. This theorem is of great use in determining the curva- ture of various curves. LOGARITHMIC MOTION. A point is said to have logarithmic motion on a straight line when its distance from a fixed point on the line is equally multiplied in equal times. When a quantity is equally multiplied in equal times, its flux is proportional to the quantity itself. Let mq, nr be two values of such a quantity, at the times represented by m, n ; and let mq = s, nr = s l . Then if we move mn to the right, keeping it always of the same length, the ratio of s to s t will remain constant; for the dif- ferent intervals represented by mn will be equal, and the quantity is equally multiplied in equal times. We shall have therefore s = ks v where k is this constant multiplier. Therefore s = ks L , and consequently s:s 1 = s:s 1 . Hence we may write s =ps, where p is a constant. Conversely, when the flux of a, quantity is proportional to the quantity itself, it is equally multiplied in equal times. For let s, s t be two values of the quantity, at times LOGARITHMIC MOTION. 79 separated by a given constant interval. Then we know that s : s t = s : s lt or ss 1 8^ = 0; that is (p. 65), the flux of the quotient s : s t is zero. Now a quantity whose flux is zero does not alter, but remains constant. There- fore s = ks 1 where k is constant; so that in any interval equal to the given one the quantity is multiplied by the same number k. A quantity whose flux is always p times the quantity itself is said to increase at the logarithmic rate p. If two quantities increase at the same logarithmic rate, their sum and difference increase at the same logarithmic rate. For if u =pu, v =pv, then u v =p (u v). If a quantity increases at a finite logarithmic rate, it is either never zero or always zero. For let such a quantity be zero at a and have a finite value bq at b. At the middle point c of ab it must have a value which is the geometric mean of zero and bq ; that is, zero. Simi- larly it must be zero at the middle point of 6c; and by pro- __, L __ ceeding in this way we may shew that it is zero at a point indefinitely near to any point on the left of b. If we make bd cb, the value at d is a third proportional to zero and bq ; that is, it is infinite. In the same way we may shew that the quantity is infinite at a point indefinitely near to any point on the right of 6. It appears therefore that the quantity sud- denly jumps from zero to bq and then to infinity ; so that at bq the rate of increase is infinite. Hence its ratio to bq is infinite, or the logarithmic rate is infinite. This case corresponds to the case in uniform motion when the velocity is infinite and the point is at a certain finite position at a given instant. At all previous instants it was at an infinite distance behind this position ; at all subsequent instants it is at an infinite distance in front of it. If two quantities increase at the same (finite) logarith- mic rate, they are either never equal or always equal. For their difference is either never zero or always zero. Let P be the result of making unity increase at the logarithmic rate p for one second ; then the result of 80 DYNAMIC. - making it increase at that log. rate for t seconds is P' when t is a whole number, for the quantity is multiplied by P in each second. It is also one value of P' when t is a commensurable fraction, say m : n. For let x be its value after t seconds, then the value after nt seconds is x n , for the quantity is multiplied by x every t seconds. But nt = m, and we know the result of growing for m seconds is P"\ Therefore a?" = P"', or as is an n th root of P m ; that is, it is a value of P'. If we spread out the growth in one second over p seconds, the number expressing any velocity must be divided by p ; hence if s was ps before, it must now = s. Hence the result of making unity increase at the log. rate p for one second is the same as the result of making it increase at the log. rate 1 for p seconds. Let e be the result of making unity increase at the log. rate 1 for one second ; then P is a value of e p whenever^? is commensurable. We now make this definition : the result of making unity grow at the log. rate p for t seconds is denoted by (P, and called the exponential of pt. The exponential coincides with one value of e to the power pt when_p is commensurable. Thus a? has two values, + ^a and \fa-, but e^ has only one value, the positive square root of the positive quantity e, whatever that is. If s = e pt , then pt is called the logarithm of s. The name logarithmic rate is given to p because it is the rate of increase of the logarithm of s. We have an example of a quantity which is equally multiplied in equal times in the quantity of light which gets through glass. If f of the incident light gets through the first inch, f of that f will get through the second inch, and so on. Thus the light will be multiplied by f for every inch it gets through; and, since it moves with uniform velocity, it is equally multiplied in equal times. The density of the air as we come down a hill is an example of a quantity which increases at a rate propor- tional to itself, for the increase of density per foot of descent is due to the weight of that foot-thick layer of air, which is itself proportional to the density. INFINITE SEEIES. 81 ON SEKIES. We know that when x is less than 1, the series is of such a nature that the sum of the first n terms can be made as near as we like to - by taking n large 1 ""* *C ' 1 X* enough. For the sum of the first n terms is - -- , and JL ~~~ C since x is less than 1, x* can be made as small as we like by taking n large enough. The value to which the sum of the first n terms of a series can be made to approach as near as we like by making n large enough is called the sum of the series. It should be observed that the word sum is here used in a new sense, and we must not assume without proof that what is true of the old sense is true of the new one: e.g. that the sum is independent of the order of the terms. When a series has a sum it is said to be convergent. When the sum of n terms can be made to exceed any proposed quantity in absolute value by taking n large enough, the series is called divergent. A series whose terms are all positive is convergent if there is a positive quantity which the sum of the first n terms never surpasses, however large n may be. For consider two quantities, one which the sum surpasses, and one which it does not. All quantities between these two must fall into two groups, those which the sum surpasses when n is taken large enough, and those which it does not. These groups must be separated from one another by a single quantity which is the least of those which the sum does not surpass; for there can be no quantities between the two groups. This single quantity has the property that the sum of the first n terms can be brought as near to it as we please, for it can be made to surpass every less quantity. The same thing holds when all the terms are negative, if there is a negative quantity which the sum of the first n terms never surpasses in absolute magnitude. c. <3 82 DYNAMIC. When the terms are all of the same sign, the sum of the series is independent of the order of the terms. For let P n be the sum of the first n terms and P the sum of the series, when the terms are arranged in one order ; and let Q n be the sum of the first n terms and Q the sum of the series, when the terms are arranged in another order. Then P n cannot exceed Q, nor can Q n exceed P; and P n , Q n can be brought as near as we like to P, Q by taking n large enough. Hence P cannot exceed Q, nor can Q ex- ceed P; that is, P = Q. When the terms are of different signs, we may separate the series into two, one consisting of the positive terms and the other of the negative terms. If one of these is divergent and not the other, it is clear that the combined series is divergent. If both are convergent, the combined series has a sum independent of the order of the terms. For let P m be the sum of m terms of the positive series, Q n the sum of n terms of the negative series, P, Q, the sums of the two series respectively ; and suppose that in the first m + n terms of the compound series there are m positive and n negative terms, so that the sum of those m + n terms is P m Q n . Then P P m , QQ n can be made as small as we like by taking m, n large enough ; therefore PQ (P m Q n ) can be made as small as we like by taking m + n large enough, or PQ is the sum of. the compound series. It is here assumed that by taking sufficient terms of the compound series we can get as many positive and as many negative terms as we like. If, for example, we could not get as many negative terms as we liked, there would be a finite number of negative terms mixed up with an infinite series of positive terms, and the sum would of course be independent of the order. If, however, the positive and negative series are both divergent, while the terms in each of them diminish with- out limit as we advance in the series, it is possible to make the sum of the compound series equal to any arbi- trary quantity C by taking the terms in a suitable order. Suppose C positive ; take enough positive terms to bring their sum above C, then enough negative terms to bring the sum below C, then enough positive terms to bring the sum again above C, and so on. We can always per- EXPONENTIAL SEEIES. 83 form each of these operations, because each of the series is divergent ; and the sum of n terms of the compound series so formed can be made to differ from C as little as we like by taking n -large enough, because the terms decrease without limit. Putting these results together, we may say that the sum of a series is independent of the order of the terms if, and only if, the series converges when we make all the terms positive. EXPONENTIAL SERIES. We shall now find a series for e x , which is the result of making unity grow at the log. rate 1 for x seconds. Suppose that e x = a + Ix + ex 2 + da? + ... that is, suppose it is possible to find a, b, c ... so that the series shall be convergent and have the sum e x . We will assume also (what will have to be proved) that the flux of the sum of the series is itself the sum of a series whose terms are the fluxes of the terms of the original series. Now the flux of e* is e x , because it grows at the loarithmic rate 1. Hence we have and this must be the same series as before. Hence b = a, 2c = b, Sd = c, etc. Now by putting x = we see that a = 1, because e is the result of making unity grow for no time. Writing then for shortness Tin instead of 1 . 2 . 3 ... n, we find SC vO ' CC / / \ S = l +x + -+-+- + . .. + + ...=f ft, say. This is called the exponential series. We shall now verify this result by an accurate investigation. The exponential series is convergent for all values of x. For take n larger than x ; then the series after the ?i th term may be written thus : 62 , | | , Un * '" 84 DYNAMIC. and each term after the first two of the quantity in the brackets is less than the corresponding term of x a? 1 + r-T+T TTT2+--- n + l (n + 1) which is convergent. And since it is convergent when the terms are all positive, the sum is independent of the order of the terms. The sum of the exponential series increases at log. rate 1 . Consider four quantities, x , x 1} # a , x, in ascending order of manitude. We find for the mean flux from x to x, M = i = ! + i * + ** , t l 11 ... Un (Observe that the order of the terms has been changed, and why this is lawful.) Each term of this series is less than the corresponding term of f(x), and greater than the corresponding term of /fa). Hence the series is convergent, and its sum M lies between /fa) and /(#). And since M is finite, -oO or /fa) -/(a;,), can be made as small as we like by making x 1 x 2 small enough. Hence also f(x) /fa) can be made as small as we like by making x x small enough. Consequently we can find an interval (from x to x) such that the mean flux M of every included interval (from x^ to # 2 ) differs from f(x) less than by a proposed quantity, however small. Therefore f(x) is the flux of f(x], or the sum of the exponential series increases at log. rate 1. It follows that f(x) = e* ; for both quantities increase at the log. rate 1, and they are equal when x = 0, there- fore always equal. It appears from the investigation above, that if f(x] denote the sum of a convergent series proceeding by powers of x, and /' (x) the sum of the derived series got by taking the flux of every term ; then /' (x~) will be the flux of/(#) whenever /'(#) f'(y) can be made COMPLEX NUMBERS. 85 as small as we like by taking x y small enough ; that is, when f'x varies continuously in the neighbourhood of the value x. By putting x = 1, we find the value of the quantity e\ it is 2718281828... THE LOGARITHMIC SPIRAL. "We may convert a step oa into a step ob by turning it through the angle aob and altering its length in the ratio oa : ob. But this opera- tion may be divided into two simpler parts. From b draw bm perpendicular to oa, then ob = om + nib. Now we may convert oa into om by simply increasing its length in the ratio oa : om. Let om : oa = x, so that om = x . oa. If oa is drawn perpendicular to oa, and equal to it in length, we can convert oa into mb by multiplying it by a nu- merical ratio y, such that mb = y . oa'. Now we can convert oa into oa by turning it counter-clockwise through a right angle. Let i denote this operation ; then oa = i . oa. Consequently mb = y . oa' = yi . oa. And finally ob = om + mb x . oa + yi . oa = (x + yi) oa. Thus the operation which converts oa into ob may be written in the form x + yi, where x and y are numerical ratios, and i is the operation of turning counter-clockwise through a right angle. This meaning is quite different from that which we formerly gave to the letter i. We shall never use the two meanings at the same time, in speaking of steps in one plane. If oa be taken of the unit length, every other step ob in the plane may be represented by means of its ratio to this unit ; for oa being = 1, ob = (x + yi) oa = x + yi. 86 DYNAMIC. The quantities x and y will then be the components of ob parallel to oX, oY. Since turning a step through two right angles is reversing it, z 2 = 1; thus i is a value of V( 1). The operation x + yi is called a complex number. The ratio ob ;.-: oa, which is + >J(a? } y*), i g called the modulus of the complex number x + yi. If a point moves in a plane so that p = qp, where q is a constant complex number, it will describe a curve which is called the logarithmic spiral. The velocity of the point p makes a constant angle with op and is proportional to it in magnitude. Let q = x+yi, then x .op is the component of the velocity in the direction op. If r denotes the length op, we shall have r = xr, and therefore r = ae xt , o -^ / where a is the value of r at the beginning of the time. Thus the magnitude of op in- creases at the log. rate x. The component of velocity perpendicular to op is yi.op; it is equal in magnitude to op multiplied by its angular velocity, or (if 6 is the angle Xop) it is op . 6. Hence 6 = y or the angular velocity is constant. Thus the motion of p is such that op increases at the log. rate x while it turns round with the angular velocity y. Since 6 = yt, while r = ae* 1 , it follows that r = ae, where k = x : y = cot opt. The position vector p of this point may be said to increase at the logarithmic rate q, because p = qp. Hence we may write p = ae gt , where a is the value of p when t = 0. The meaning of e*, when q is a complex number, is the result of making the unit step oa grow for t seconds at a rate which is got from the step at each instant by multiply- ing it by the complex number q. In other words, we must make a point p start from a and move always so that its velocity is q times its position-vector ; that is, its velocity must be got from the position-vector by turning it through a certain .angle and altering it in a certain ratio. We may now prove that, just as e x is equal to the sum SERIES OF COMPLEX NUMBERS. 87 of the series /(#), so e qt is equal to the sum of the series f(qt). To make our former proof available, we have only to premise some observations on complex numbers and on series formed of them. A complex number q alters the length of a step oa in a certain ratio (the modulus) and turns it round through a certain angle, so converting it into ob. Suppose that another complex number q l turns ob into oc, by alter- ing its length in some other ratio and turning it through some other angle. Then the product q^q is that complex number which turns oa into oc ; it therefore multiplies oa by the product of the two ratios, and turns it through the sum of the two angles. Hence q^q = qq^ ; or the product of two complex numbers is independent of the order of their multiplication; and the modulus of the product is the product of the moduli. The same thing is clearly true for any number of factors. Instead of operating on a step with a complex number, we may operate on any plane figure whatever. The effect will be to alter the length of every line in the figure in a certain ratio, and to turn the whole figure round a certain angle. Thus the new figure will be similar to the old one. Taking for this figure a triangle, made of two steps and their sum a + /3, we learn that q (a + /3) = qa. + q/3. The steps themselves may be represented by complex num- bers, namely their ratios to the unit step. Hence also (a + /3) q = aq + (3q. Thus complex numbers are multi- plied according to the same rules as ordinary numbers. A series of complex numbers may be divided into two series by separating each term x + yi into its horizontal (or real] part x and its vertical part yi. Neither of these parts can be greater than the modulus of the term ; and therefore both parts will converge independently of the order of the terms if a series composed of the moduli con- verges. To change the series f(qt) into the series of the moduli, we have merely to write mod. qt instead of qt ; viz. the series of the moduli is /(mod. qt); because the modulus of q n is the n ih power of the modulus of q. We have before noticed that when the step p grows at the complex log. rate x+yi, its length or modulus r grows 88 DYNAMIC. at the log. rate x. Hence p is either never zero or always zero. It may now be proved successively that the series f(gf) is convergent ; that if t 0> t lt t 2 , t are four quantities in ascending order of magnitude, the mean flux differs from gf(qt) by a complex number whose horizontal and vertical parts are severally less than the correspond- ing parts of qf(qt) qf(qt ], whose modulus may therefore be made less than any proposed quantity by making t t small enough ; and consequently that the flux of f(qt) is qf(qt). Hence it follows that /(qty^e*, because they both grow at the log. rate q, and are both equal to 1 when = 0. When the velocity of p is always at right angles to op, the logarithmic spiral be- comes a circle, and the quantity q is of the form yi. Suppose the motion to commence at a, where oa = 1, and the logarithmic rate to be i; that is, the velocity is to be always perpendicular to the radius vector and represented by it in magnitude. Then op = e if . Now the velocity of p being unity in a circle of unit radius, the angular velocity of op is unity, and therefore the circular measure of aop is t. But op = om + mp = cos t + i sin t. Therefore e u = cos t + i sin t, Euler's extremely important formula, from which we get at once the two others, cos t = i (e u + O , sin t = \ (e u - e~ u ] . Moreover, on substituting in these formulae the ex- ponential series for e u and e~ u , and remembering that i a = 1, we find series for cos t and sin t, namely, VERIFICATION OF EXPONENTIAL SERIES. 89 _ - f .0 C 6 The formula e* = cos 1 + i sin 1 may be graphically verified by con- struction of the several terms of the series 7-x The first term is oa ; then a> = i, be Ji . ab, cd = ^i . be, de = %i . cd, ef= $i . de, and so on. The rapid convergence of the series becomes manifest, and the point f is already very close to the end of an arc of length equal to the radius. QUASI-HARMONIC MOTION IN A HYPERBOLA. It is sometimes convenient to use the functions \ (e* e~ x ), called the hyperbolic sine of x, hyp. sin x, or .hs x, and \ (e x + e~ x ], called the hyperbolic cosine of x, hyp. cos x, or he x. They have the property he 2 a; hs 2 a; = 1 . Thus whenever we find two quantities such that the dif- ference of their squares is constant, it may be worth while to put them equal to equimultiples of the hyperbolic sine and cosine of some quantity: just as when the sum of their squares is constant, we may put them equal to equi- multiples of the ordinary sine and cosine of some angle. The flux of he x is x hs x and the flux of hs & is a; he x, as may be immediately verified. The motion p = a he (nt + e) + /3 hs (nt -f e) has some curious analogies to elliptic harmonic motion. Let ca = a, cb=/3, then cm ca. he (nt + e), mp = cb . hs (nt + e), so , 72 mi : ma . ma = cb : ca . The curve . Cm that - -=- ca cb = lj or having this property is called a hyperbola. once that We see at 90 DYNAMIC. p = nx hs (nt + e) + n@ he (nt + e) = n . cq, say; then cp + cq = (a + /3) e fl , and cp cq = (a /3) e- 9 where = nt + e. Thus pq is parallel to aJ, and en (where n is the middle point of pq) is parallel to ab'. Moreover pn . en = product of lengths of a + ft and a. ft ^cx.cy. Hence it appears that the further away p goes from cy, the nearer it approaches ex, and vice versa. The two lines ex, cy which the curve continually approaches but never actually attains to, are called asymptotes (aavinnwrai, not falling in with the curve). It is clear that the curve is symmetrically situated in the angle formed by the asymptotes, and therefore is symmetrical in regard to the lines bisecting the angles between them, which are called the axes. It consists of two equal and similar branches ; though the motion here considered takes place only on one branch. The acceleration p = n*p ; thus it is always proportional to the distance from the centre, as in elliptic harmonic motion, but directed away from the centre. The lines cp, cq, are conjugate semidiameters of the hyperbola, as are ca, cb. Each bisects chords parallel to the other, as the equation of motion shews. The locus of q is a hyper- bola having the same asymptotes, called the conjugate hyperbola. The hyperbola is central projection of a circle on a horizontal plane, the centre of projection being above the lowest, but lower than the highest, point of the circle. Let b, a be highest and lowest points of the circle, v the CENTKAL PROJECTION OF CIRCLE. 91 centre of projection, am the projection of db and pm of qn which is perpendicular to ab. We find an : am = nf : vf; also nb : gb = nf : vf, and gb : a'm=fb : af; multiply these three together, then an .nb : am . am = nf 2 .fb : vf 9 . of. But pm* : an.nb = pm* : qn* = af : nf*; therefore pm* : am.a'm = af.fb : vf, the property noticed above. Making a change in the figure, the same process shews that the ellipse is central projection of a circle which is wholly below the centre of projection. These three central projections of the circle, ellipse, parabola, and hyperbola, are called conic sections; being plane sections of the cone formed by joining all the points of a circle to a point v. CHAPTER III. CENTRAL ORBITS. THE THEOREM OF MOMENTS. THE moment of the finite straight line pt about the point o is twice the area of the triangle opt. Its mag- nitude is the product of the length pt and the perpendicular on it from o. Every plane area is to be regarded as a directed quantity. It is represented by a vector drawn perpen- dicular to its plane, containing as many linear centimeters as there are square centimeters in the area. The vector must be drawn towards that side of the plane from which the area appears to be gone round counter-clockwise. Thus om is the vector representing twice the area opt, p being the near end of pt and m on the upper side of the plane opt. The sum of the moments of two adjacent sides of a parallelogram about any point is equal to the moment of the diagonal through their point of intersection. That is, triangle oad = oac + oab ; each triangle be- ing regarded as a vector, in the general case when o is out of the plane abed. Taking first the spe- cial case of o in the plane, we observe that oad = ocd + cad + oac ; but ocd + cad = oab, because the height of oab is the sum of the heights of ocd and cad, while all three stand on the same base ab or cd. Therefore oad oab + oac. THEOREM OF MOMENTS. 93 Next, suppose o to be out of the plane. Then the vector representing oab will be a line am perpendicular to the plane oab, which may be re- solved into components an, nm, of which an is perpendicular to the plane of the parallelogram and nm parallel to that plane. Now an represents on the same scale the projection pab of oab on the plane abed, and nm its projection opq on the perpendicular plane. For the triangles oab, pab, opq, being on the same base ab or pq and having the heights respectively ao, ap, po, which are proportional to am, an, nm, must have their areas propor- tional to the lengths of these lines. Suppose, then, the vector representing each of the areas oab, oac, oad to be resolved into components per- pendicular and parallel to the plane ; the theorem will be proved if it is true separately for the components perpen- dicular and for those parallel to the plane. Now for the perpendicular components the theorem has been already proved, because they represent the triangles pab, pac, pad, which are projections on the plane abed of oab, oac, oad. For the components parallel to the plane, observe that mn represents opq', it is at right angles to ab and proportional to the product of ab by op the dis- tance of o from the plane. Hence the components parallel to the plane are lines ab', ac, ad respectively at right angles to ab, ac, ad, and pro- portional to their lengths multiplied by the distance of o from the plane. Thus the figure ab'c'd" is merely abed turned through a right angle and altered in scale; whence it is obvious that ad' = ab' + ac. Thus the proposition is proved in general. It is well worth noticing, however, that the proof given for the special case of.o in the plane applies word for word and symbol for symbol to the general case, if only we interpret 94 DYNAMIC. -i- and sum as relating to the composition of vectors. Thus oad = ocd + cad + oac, or one face of a tetrahedron is equal to the vector-sum of the other three faces. It is, of course, the sum, of their projections upon it; and the components of their representative vectors \vhich are parallel to its plane are respectively perpendicular and proportional to oa, ad, do, so that their vector sum is zero. Again, ocd { cad = oab, because the height of oab is the vector- sum of the heights of ocd and cad. For proving theorems about areas, the following con- sideration is of great use. We have seen that the pro- jection of an area on any plane is represented by the projection of its representative vector on a line at right angles to the plane. In fact, the angle oap between the two planes is equal to the angle man between the two- lines respectively perpendicular to them ; if we call this angle 0, the projection of the area A is A cos 0, and the corresponding projection of the line of length A is also A cos 6. Now it is easy to see that if the projections of two vectors on every line whatever are equal, then the two vectors are equal in magnitude and direction. Hence it follows that if the projections of two areas on any plane whatever are equal, then the areas are equal in magnitude and aspect. For example, the areas oad and oac + oab (figure on p. 92) are such that their projections on any plane are equal ; this projection is, in fact, the case of the theorem of moments in which o is in the plane abed. Hence the general theorem may be deduced in this way from that particular case. PRODUCT OF TWO VECTORS. On account of the importance of the theorem of mo- ments, we shall present it under yet another aspect. The area of the parallelogram abdc may be supposed to be generated by the motion of ab over the step ac, or by the motion of ac over the step ab. Hence it seems natural to speak of it as the product of the two steps ab, ac. We have been accustomed to identify a rectangle with the product of its two sides, when their lengths only are VECTOR AND SCALAR PRODUCT. 95 taken into account ; we shall now make just such an ex- tension of the meaning of a product as we formerly made of the meaning of a sum, and still regard the parallelogram contained by two steps as their product, when their di- rections are taken into account. The magnitude of this product is ab . ac sin bac ; like any other area, it is to be regarded as a directed quantity. Suppose, however, that one of the two steps, say ac, represents an area perpendicular to it ; then to multiply this by ab, we must naturally make that area take the step of translation ab. In so doing it will generate a volume, which may be regarded as the product of ac and ab. But the magnitude of this volume is ab multiplied by the area into the sine of the angle it makes with ab, that is, into the cosine of the angle that ac makes with ab. This kind of product therefore has the magnitude ab . ac cos bac ; being a volume, it can only be greater or less ; that is, it is a scalar quantity. We are thus led to two different kinds of product of two vectors ab, ac ; a vector product, which may be writ- ten V . ab . ac, and which is the area of the parallelogram of which they are two sides, being both regarded as steps; and a scalar product, which may be written S.ab. qc, and which is the volume traced out by an area represented by one, when made to take the step repre- sented by the other. Now the moment of ab a,bout o is V.oa.ab; that of ac is V.oa.ac; and that of ad is V. oa.ad, which is Voa . (ab + ac). Hence the theorem tells us that V. oa (ab + ac} = V.oa.ab + V. oa.ac; or if, for shortness, we write oa = , a& = /3, ac = y, the theorem is that Fa 08 + 7)= Fa/3 + Fa 1 p is equal to the moment of p lf that is the moment of the change in the velocity is equal to the change in the moment of velocity. Dividing each of these by the interval of time, we see that the moment of the mean flux of velocity is equal to the mean flux of the moment of velocity, during any interval. Conse- quently the moment of acceleration is equal to the flux of the moment of velocity. The same thing may be shewn in symbols, as follows, supposing the motion to take place in one plane. We MOMENT OF ACCELERATION. 97 may write p = re ia , where r is the length of op, and 6 the angle Xop. Then p=re io + r6.ie^, or the velo- city consists of two parts, r along op and r6 perpendicular to it. The moment of the velocity is the sum of the moments of these parts; but the part along op (radial component) has no moment, and the part perpendicular (transverse compo- nent) has moment r*0. Next, taking the flux of p, we find for the acceleration the value p^reM + ef. ie* + rd . ie* + r&* . tV + r'0 . ie* = (r- rfr) e* + (2f0 + r6) ie i9 . Or the acceleration consists of a radial component r r6' 2 , and a transverse component 2f$ + r0. The moment of the acceleration is r times the transverse component, namely 2rrd + t*Q. But this is precisely the flux of the moment of velocity r*0. Observe that the radial acceleration consists of two parts, r due to the change in magnitude of the radial velocity, and r&* due to the change in direction of the transverse velocity. We may also make this proposition depend upon the flux of a vector product. The moment of the velocity is Vpp, and the moment of the acceleration is Vpp ; we have therefore to prove that Vpp is the rate of change of Vpp. Now upon referring to the investigation of the flux of a product, p. 64, the reader will see that every step of it applies with equal justice to a product of two vectors, whether the product be vector or scalar. In fact, the only property used is that the product is distributive. Hence the rate of change of Fa/3 is Fa/3 + Fa/3. (Observe that the order of the factors must be carefully kept.) Apply- ing this rule to Vpp, we find that its rate of change is Vpp+ Vpp. Now the vector product of two parallel vec- tors is necessarily zero, because they cannot include any area ; thus Vpp = 0. Therefore 9, ( Vpp) = Vpp. This de- monstration does not require the motion to be in one plane. c. 7 98 DYNAMIC. The moment of velocity about any point is equal to twice the rate of description of areas about that point. When the motion is in a circle, twice the area aop being equal to r 6, and r constant, its flux is r*6, the moment of velocity. In any other path aq, having the same angular velocity, the area de- scribed in the same time is oaq, and the mean flux of area in the two cases is oap and oaq respectively di- vided by the time. The ratio of their difference to either of these is the ratio of apq to oap or oaq, which is approximately the ratio of pq to op or oq, and can be made as small as we like by taking p near enough to a. Thus the mean fluxes in the two cases approach one another without limit as they approach the true fluxes ; or the true fluxes are equal. Hence twice the rate of description of areas is always r*d, the moment of velocity. When the acceleration is always directed towards a fixed point o, the moment of velocity is constant, and equal areas are swept out by the radius vector in equal times. If the acceleration of p passes through o, its moment about o is zero; consequently the flux of the moment of velocity is zero, or that moment is constant. Because it is constant in direction, the path is a plane curve; for the plane con- taining op and the velocity has always to be perpendicular to a fixed line. Because it is constant in magnitude, the rate of description of areas is also constant, or, which is the same thing, equal areas are swept out in equal times. The following is Newton's proof of this proposition. Let the time be divided into equal parts, and in the first interval let the body describe the straight line AB with uniform velocity. In the second interval, if the ve- locity were unchanged, it would go to c. if Be = AB ; UNIFORM DESCRIPTION OF AREAS. 99 so that the equal areas ASB, BSc would be completed in equal times. But when the body arrives at B, let a velocity in the direction BS be communicated to it. The new velocity of the body will be found by drawing cC parallel to B8 to represent this addition, and joining EG. At the end of the second interval, then, the body will be at C, in the plane SAB. Join SC, then area tSCB = ScB (between same parallels SB and Cc) = SB A. In like manner, if at C, D, E, velocities along CS, DS, ES are communicated, so that the body describes in successive intervals of time the straight lines CD, DE, EF, etc., these will all lie in the same plane ; and the triangle SCD will be equal to SBC, and SDE to BCD, and SEF to SDE. Therefore equal areas are described in the same plane in equal intervals; and, componendo, the sums of any number of areas SADS, SAFS, are to each other as the times of describing them. Let now the number of these triangles be increased, and their breadth diminished indefinitely ; then their perimeter ADF will be ultimately a curved line; and the instantaneous change of velocity will become ulti- mately a continuous acceleration in virtue of which the body is continually deflected from the tangent to this curve ; and the areas SADS, SAFS, being always pro- portional to the times of describing them, will be so in this case. Q.E.D. The constant moment of velocity will be called h. It is twice the area described in one second. If^? be the length of the perpendicular from the fixed point on the tangent, we shall have h = vp=r*0. A path described with acceleration constantly directed to a fixed point is called a central orbit, and the fixed point the centre of acceleration. In a central orbit, then, the velocity is in- versely as the central perpendicular on the tangent, for v = h : p, and the angular velocity is inversely as the squared distance from the centre, for = h : r z . 72 100 DYNAMIC. RELATED CURVES. Inverse. Two points p and q so situated on the radius of a circle that cp . cq = ca?, are called inverse points in regard to the circle. If p moves about so as to trace out any curve, q will also move about, and trace out another curve ; either of these curves is called the inverse of the other in regard to the circle. 'The inverse of a circle is in general another circle; but it coincides with its inverse when it cuts the circle of inversion at right angles, and the inverse is a straight line when it passes through the centre of inversion. We know that cp.cq = c, which proves the second case ; the first is easily derived from it; and the third follows from the similarity of the triangles cmp, cqb, which gives cp.cq = cm . cb, which is constant and therefore = cd* = ca*. In the second case the circle clearly makes equal angles with cpq at p and q. In general, two inverse curves make equal angles with the radius vector at corre- sponding points. For we can always draw a circle to touch the first curve at p and to pass through q; such a circle is then its own inverse, and makes equal angles at p and q with cpq. More- over it touches the second curve at q, for as two points of inter- section coalesce at p, their two inverse points coalesce at q. Hence the two inverse curves make equal angles with cpq. PEDAL AND RECIPROCAL CURVES. 101 Pedal. The locus of the foot of the perpendicular frpm a fixed point on the tangent to a curve is called the pedal of the curve in regard to that point. Let two tangents to the curve intersect in p, ct, ct' be the perpendiculars on them. Because the angles ctp, ct'p are right angles, a circle on cp as diameter will pass through tt'. Now let the two tangents coalesce into one ; then p will become a point on the curve, and t't will become tangent to the pedal, and also to the circle on cp as diameter. There- fore the angle ctu = cpt, where tu is tangent to the pedal at t. Reciprocal The inverse of the pedal of a curve, in regard to the same point, is called the reciprocal .curve. Let s be the inverse point to t, and sn the tangent to the locus of s. We know that tu and sn make equal angles with cst ; therefore csn = ctu = cpt. Thus the triangles csn, cpt are similar, ens is a right-angle, and en : cs = ct : cp, or en . cp = cs . ct, so that n, p are inverse points. Hence p is a point on the reciprocal of the locus of s, or when one curve is reciprocal to a second, the second curve is reciprocal to the first. Hence the name, reciprocal. "We shall now shew that the reciprocal of a circle is always one of the conic sections. For this purpose it is necessary first to prove a certain property of these curves. 102 DYNAMIC. Two points s and h in the major axis of an ellipse, such that sb = lib = ca, and consequently that cs 2 = ch?= ca 2 cl*, are called the foci of the curve. Draw pm perpendicular to the axis from any point p of the curve, and take so that en : cm = cs : ca, en . ca cm . cs. Then we shall prove that sp = an. For c& 2 sp* = sm z +pm? = (cs crnf + ^ (ca 2 cm 2 ) ca = (cs cm) 2 + ca 2 cs 2 cm 2 + cn'= 2cs . cm + ca 2 + en 2 = 2ca . en + ca 2 + en* = an*. Similarly hp = na'. Therefore sp + hp = aa', or the sum of the focal distances of any point on the ellipse is equal to the major axis. If we take cd : ca = ca : cs cm : en, we shall have na : md ca : cd, and since pl = md, we have sp : pl= ca : cd = cs : ca. The ratio cs : ca is called the eccentricity of the ellipse, and sometimes denoted by the letter e, so that sp = e . pi. The line dl is called the directrix. Thus we see that the ellipse is the locus of a point whose distance from a fixed point (the focus) is in a constant ratio to its distance from a fixed line (the directrix). The distance from the focus is less than that from the directrix. FOCI OF CONICS. 103 A precisely similar demonstration applies to the hyperbola ; the points s and h being so taken that cs = hc = ce, and consequently cs 2 = ca 2 + c& 2 . Then c6 2 sp 2 = sm* + pm 2 = (cs cm) 2 -\ (cm 2 ca 2 ) Ca = (cs cm) 2 + en 2 cm 2 cs 2 + ca 2 = an as before. So lip = an, and hp sp = aa. On the other branch we should find sp hp = aa', or the difference of the focal distances of any point on the hyperbola is equal to the major axis. Taking cd : ca = ca : cs, and drawing pi perpendicular to dl, we find as before that : pl = cs sp ca. Thus in the hyperbola also the dis- tance from the focus is in a constant ratio to the distance from the direc- trix dl, but the ratio in this case is greater than unity. In the parabola we know that pm* varies as am ; take a point s on the axis so that pm 2 = 4as 2 = dm 9 , if da = as. Hence sp = pl, or the parabola is the locus of a point whose distance from the focus s is equal to its distance from the directrix. 104 DYNAMIC. We can now prove that the reciprocal of a, circle is a conic section, of which the centre of reciprocation is a focus. Let s be the centre of recipro- cation, st perpendicular to the tangent qt of the circle. Then the reciprocal curve of the circle is inverse to the locus of t ; and the size of the circle of inversion will evidently affect only the size, not the shape, of the curve. Let d be the inverse point to c, then if sp.st = sc. sd, p will be a point on the reciprocal curve. Now sc . sd = sp . st = sp (sn + cq) = sm . sc + sp . ca (since sp : sm = sc : sn) ; or sp . ca = sc (sdsm] = md . sc. Therefore sp : pl = sc : ca, or the locus of p is a conic section having s for focus, dl for corresponding directrix, and sc : ca for eccentricity. Hence if s is within the circle this conic is an ellipse, if on the circumference a parabola, if outside the circle a hyperbola. Since the reciprocal is the inverse of the pedal, and the inverse of a circle is a circle except when it passes through the centre of inversion, it follows that the pedal of a conic section in regard to a focus is a cirde in the case of the ellipse and hyperbola, and a straight line in the case of the parabola. We may prove this independently thus. The tangents to an ellipse or hyper- LAW OF CIRCULAR HODOGRAPH. 105 bola make equal angles with the focal distances r, r, , for since r r, is constant, r = + r l ; now r is the component of velocity of p along sp, and r a along Ap, and these being equal in magnitude, it follows that spy = Aps. Pro- duce hp to w?, making pw = ps, so that hw = ad. Then sw is perpendicular to j9y which bisects the angle spw. Hence sy is ^sw, and sc = ^sh, therefore cy = \hw = ca, or the locus of y is the circle on aa as diameter. This is called the auxiliary circle. ACCELERATION INVERSELY AS SQUARE OF DISTANCE. When the acceleration is directed to a fixed point, the hodograph is the reciprocal of the orbit turned through a right angle about the fixed point. Let py be tangent to the orbit, .9 the fixed point, su the velo- city at p, sy perpendicular to py. Then we know that su .sy = h, which is constant. Hence if we mark off sr on sy, so that sr = su, we shall have sr.sy = h, and therefore the locus of r is the reciprocal of the orbit. But the locus of u is the locus of r turned through a right angle. When the acceleration is inversely as the square of the distance from the fixed point, the hodograph is a circle (Hamilton). Let the acceleration f= /j> : r 2 , so that^/r 2 = p. We know that r*Q = h, therefore / : 6 ' = /j, : A, or the ac- celeration is proportional to the angular velocity. Now the acceleration is the velocity in the hodograph, whose direc- tion is that of the radius vector in the orbit ; so that the angular velocity, which is the rate at which the radius vector turns round, is also the rate at which the tangent to the hodograph turns round. Since then the velocity in the hodograph is in a constant ratio to the rate at which its tangent turns round, the curvature of the hodograph is constant and equal to h : fj,. Therefore the hodograph is a circle of radius /* : h. 106 DYNAMIC. Hence it follows directly that when the acceleration is inversely as the squared distance, the orbit is a conic section having the centre of acceleration for a focus. Now we have ys . sy = as . sa ca 2 cs 2 = c& 2 ; and moreover ys.v = h; whence sy : v = c6 2 : h. Hence the auxiliary circle is to the hodograph (in linear dimen- sions) as ctf : h; or h.ca : (j, = cb* : k; or h? : /j, = cb 2 : ca. If I'sl be drawn through s perpendicular to the major axis, II is called the latus rectum; and we have si 2 : cb z = as . sa : ca* = c& 2 : ca 2 or si = off : ca. Hence A 2 : p is the semi-latus rectum. The periodic time T in the ellipse is to be found from the consideration that h is the area described in two seconds, and the area of the ellipse (TT . ca . cb, orthogonal projection of area of circle, TT . ca 2 ) is described in T seconds, Hence T=27r.ca.cb : h; but c6 2 = h 2 . ca : p or cb = h. \/ca : \fjj,. Therefore T= Z ca = 4?r 2 . ca 8 . Consequently, in different orbits, if ytt is the same, the square of the periodic time varies as the cube of the major axs. Kepler stated three laws as the result of observa- tion of the planets. 1st, each planet describes about the Sun areas proportional to the times. 2nd, each planet moves in an ellipse with the Sun in one focus. 3rd, the KEPLER'S LAWS. 107 squares of the periodic times of different planets are to one another as the cubes of the major axes of their orbits. From these laws Newton deduced, 1st, that the acceleration of each planet is directed towards the Sun ; 2nd, that the acceleration of each planet is inversely as the square of its distance from the Sun ; and 3rd, that the acceleration of different planets is inversely as the square of their distances from the Sun, since /j, is constant. Kepler's laws and these deductions from them .are however only approximately true. . ELLIPTIC MOTION. Motion in an ellipse with acceleration always directed to one focus is called, par excellence, elliptic motion. The angle asp is called the true anomaly, and is denoted by 0. If qpm be drawn perpendicular to the axis, meeting the auxiliary circle in q, the point q is called the eccentric follower of p. Since the area asp is the orthogonal projection of asq, the latter is always proportional to it, and therefore to the time ; therefore q moves in the auxil- iary circle 1 with acceleration always tending to s. The angle acq is called the eccentric anomaly, and is denoted by u. The mean angular velocity is also called the mean mo- tion, and is denoted by n. The angle nt is called the mean anomaly. It is clear that nt : 2-Tr = area asp : TT . ca . cb = area asq : TT . ca 2 . Now asq = acq scq = ^u.ca z ^cs.qm = \u. ca 2 \e . ca . ca sin u = ^ca 2 (u e sin w). 1 On circular orbits with acceleration to a fixed point or points, see Sylvester, Astronomical Prolusions, Phil. Mag. 1866. 108 DYNAMIC. Therefore nt = u e sin u ; an equation connecting the mean and eccentric anomalies. The tangents at p and q meet the axis in a point t such that cm . ct = ca?. Let them meet the tangent at a in f, g respectively. Then the tangents fa, fp to the ellipse subtend equal angles 1 at the focus s, and the tangents ga, gq to the circle subtend equal angles at the centre c. Con- sequently angle asf= ^ 6, and acg = ^u. We find therefore tan ^0 = af : as, and tan \u = ag : ac, so that an : tanw = a. ac : as . ag = e : e. /I 4- e Therefore tan 1 6 = ^ / = -- . tan \ u ; y -L "~ an equation connecting the true and eccentric anomalies. We know that sp = an, if en = e. cm; so that, denoting sp by r, we have r = a(\ e cos u), which gives the distance in terms of the eccentric anomaly, and a the semi-major axis. LAMBERT'S THEOREM. The time of getting from a point p to a point q in an elliptic orbit may be expressed in terms of the chord pq, and the sum of the focal distances sp + sq ; a result which is called Lambert's Theorem. The following proof is due to Prof. J. C. Adams. Let r, r be the two focal distances, u, u the eccentric anomalies, Ic the length of the chord. Regarding the chord as the projection of the corresponding chord of the auxiliary circle, we see that its horizontal component is a (cos u cos u} and its vertical component is a V(l e 2 ) (sin u sin u) ; for the vertical component is reduced by the projection in the ratio a : b, which is 1 : V(l ~0 2 ) Hence If = a 2 (cos u cos u'f + a 2 (1 e 2 ) (sin u sin uj* = 4a 2 sin 2 \(u u'} sin 2 \(u + u') + 4a 2 (1 - e 2 ) sin 2 J (K - u'} cos 8 |(w + u} = 4>a* sin 2 \(u - u 1 ) (1 - e 2 cos 2 1 (u + u'}}. 1 Because cm.ct=ca z , it is easy to shew that ta:am=ts:an, and therefore that tf:fp = ts:sp, so that sf bisects the angle asp. LAMBERT'S THEOREM, 109 Now let u u' = 2a, and let /3 be such an angle .that e cos \ (u + u) = cos /3. Then & = 2a sin a sin ft, Moreover, r + / = 2a (1 - e (cos u + cos M')} = 2a{l-e cos (M - w') cos |(w + w')} = 2a (1 cos a cos /3). Therefore r + r ' + & = 2a (1 - cos 0) if = ft + a, and r + r' k = 2a (1 cos <) if ^ = ft a. Now nt = u u e (sin M sin w') = 2a 2e sin |(w w') cos f (w + w') = 2a 2 sin a cos ft 6 sin 6 (0 sin $). Thus nt is expressed in terms of 6 and <, which are them- selves expressed in terms of r + r, k, and a. Because nT= < 2 r jr, it follows that wV = ^; so that the time is given in terms of r + r, k, a, and /A, the acceleration at unit dis- tance. The angle' a is half the angle subtended at the centre by the corresponding chord of the auxiliary circle. If, keeping the focus and the near vertex fixed, we make the major axis of the ellipse very large, while the points p, q remain in the neighbourhood of the focus ; the ellipse will approximate to a parabolic form, and the angles u, u' will become very small; so therefore will a and /3, and consequently 6 and <. Hence we shall have approximately nt, =0-sin0-((-sm), =(0 3 _< s ), + k, = 2a (1 - cos ff), = a6\ 7c, = 2a (1 cos <), = a 2 , and w 2 a s = /i always. Therefore (r+r' + fc) f -(r + r- &)*, with an approximation which becomes closer the larger a is taken, and which becomes exact when a is infinite, or the ellipse becomes a parabola. This, therefore, is the form of Lambert's theorem for the parabola. An analogous 110 DYNAMIC. theorem for the hyperbola will be found in the paper referred to 1 , GENERAL THEOREMS. THE SQUARED VELOCITY. In general, if a point p be moving -with acceleration/ always tending from s, the resolved part of the acceleration along the tan- gent is /cos spt= /cos ty, say; there- fore v =fcos i/r. Now the resolved part of the velocity v along sp is r , so that r = v cos -x/r. It follows there- fore that fr = vi) = d t (^ v*). If the acceleration / depends only on the distance, so that f is a function of r, we may be able to find ffrdt or ffdr, and thence | v 2 to which it is equal. Suppose, for example, that f=p.r~ n , then (n-l)ffdr = /ir"** 1 + some constant c, or (n 1) v* + /ir~" +1 = c. Since vp h, this equation gives us a relation between r and p which determines the form of the orbit. In the elliptic motion we have J v* = p,r~ l + c, the acceleration being towards the focus ; and the constant c may be determined by means of the velocity at the extre- mity of the minor axis, where r = a and vb = h. Here | h 2 = \ v 2 b 2 = p.a~ l W + cb 2 , but we know that h 2 fjLa~ l b~, therefore c = iflf 1 and the formula becomes The analogous formula for the hyperbola is which may be found by considering the velocity at an infinite distance, when the point may be regarded as moving along the asymptote. Since a parabola may be regarded as an infinitely long ellipse or as an infinitely long hyperbola, we find the cor- responding formula for that case by making a infinite in 1 Messenger of Mathematics, 1877. VELOCITY DUE TO FALL FROM CIRCLE. Ill either of the two others, viz. J v* = p.r 1 ; in this case the velocity at an infinite distance is zero. We see then that when a point starts from the position p at a distance r from s, and moves with acceleration //-r" 2 always tending to s ; if the velocity at starting is / v /(2//r~ 1 ), the path will be a parabola ; if less than this, an ellipse with semi-major axis given by the formula paT 1 = 2/ir" 1 v 2 ; if greater, an hyperbola with semi-major axis given by the formula //.of 1 = v 2 2/tr" 1 . The major axis of the orbit depends only on the velocity, not at all on the direction, of starting. A special case of elliptic motion is that in which, the direction of starting being in the line sp, the ellipse re- duces itself to a straight line. The foci then coincide with the extremities of the major axis, the eccentricity e = 1, and the motion is the projection on aa of motion in the circle with ac- celeration tending to a. Writing x for ap, and u for the angle acq, we have x = a (1 cosu], nt = u sin u = 2 sin 1 = from which equations it may be verified with a little trouble that acx* = n*a\ It fol- lows that if from a point p in the ellipse a point be started with the velocity belonging to the elliptic motion in the direction sp, and have always an acceleration /tr~ 2 , it will ascend to a point r such that sr = aa', and then return to p with the same velocity; so that the velocity at any point of the ellipse is that due to a fall from the circle rk. If we join ph and pro- duce it to cut the ellipse at q, we have ph =pr, hy + sq 112 DYNAMIC. = aa' = sr, and therefore pq + sq =pr + sr. Hence if an ellipse be described with the foci s, p, to touch the circle at r, it will pass through q and touch the ellipse pq at that point (since both tangents must make equal angles with sq, hq). Thus all the orbits which can be described from p with given velocity touch an ellipse having foci s, p and major axis sr+pr. Or, in purely geometrical terms, given a focus, one point, and the length of the major axis of an ellipse, its envelop is the ellipse here specified. In the case of the further branch of a hyperbola de- scribed with acceleration yrora the focus, the velocity is that due to a fall out from the circle rk, from r to p. We have again ph = pr, and sq hq sr, therefore sq pq = sr rp, or q is on a hyperbola with foci s, p, touching the circle at r and the orbit at q. When the nearer branch is described with acceleration to the focus, the theorem be- comes rather more complex. If a point be started from p in the direction sp, with the velocity belonging to the hy- perbolic orbit, and acceleration from s, its velocity will approx- imate to a certain definite value more and more closely as it gets further and further away. If we now suppose a point to approach from an in- finite distance on the other side of s, with a velocity more and more nearly equal to the same value the greater the distance from s, but now with acceleration from s, this point will come up to the position r (where sr = aa'}, and there stop and go back. So that if now we reverse this process, start a point from VELOCITY IN PARABOLIC MOTION. 113 rest at r and make it fall through infinity to the point p, it will arrive at p with the velocity belonging to the hyper- bolic orbit. We have again ph = pr, sqhq = sr, therefore pq + sq=pr +sr, or q lies on an ellipse with foci s,p touching the circle at r and the further branch of the hyperbola at q. Returning to the case of the ellipse, we know that if it is lengthened out until one focus goes away to an infi- nite distance, it will become a parabola. If however we send away the focus h, the circle rk, having a fixed centre s and a radius increasing without limit, will itself go away to infinity ; and there will be no proper envelop of the different parabolic paths which pass through p. In a parabola described with acceleration towards the focus, therefore, the velocity at every point is that due to a fall from infinity ; or, as we may say, the velocity in the para- bola is the velocity from infinity. If, holding fast the focus h, we send s away to infinity, all the lines passing through it become parallel, and their ratios unity ; so that the acceleration becomes constant in magnitude and direction, and we fall back on the previously considered case of parabolic motion. Since ha is then = a'k, the circle rk becomes the directrix ; and we learn that in the parabolic motion the velocity at any point is that due to a fall from the directrix. The envelop of the orbits described by points starting from a given point with given velocity is a parabola having that point for focus and touching the common directrix at r. GENERAL THEOREMS. THE CRITICAL ORBIT. We have shewn that when the acceleration f= /*"", then \ (n 1) t? 2 -4- jj.r 1 '* is a certain constant, c. For con- venience, suppose that c = (n l)w a where u is a certain velocity ; then if we make r infinite, and suppose n greater than 1, r 1 '" will be zero, and we shall have v = u. Hence u is the velocity at an infinite distance ; and if the orbit has any infinite branch, u is the value to which the velocity of a particle going out on that branch would indefinitely approach. If however n is less than 1 or negative, r l ~" will be zero when r is zero, and in this case c. 8 114 DYNAMIC. u is the velocity of passing through the centre of accelera- tion. If we draw a circle with that point as centre, and radius a determined by the equation \ (n 1) u? = fta l ~ n , then in the case n > 1, the velocity at every point is that due to a fall from this circle, either directly or through infinity ; and in the case n < 1, the velocity is that due to a fall from the circle either directly or through the centre : it being understood that in passing through infinity or through the centre the sign of //, must be changed. Just as when n = 2 the parabola is a critical form of orbit, dividing from one another the ellipses and hyper- bolas, so in general, an orbit in which u = is called a critical orbit. When n>l, the velocity at every point of such an orbit is that due to a fall from an infinite distance (in this case /-t must be negative, or the acceleration towards the centre) ; and when n < 1, the velocity is that due to a fall from the centre, /-i being positive and the acceleration away from the centre. In both cases (w-l) v'^r 1 -*. Since vp = h, we find \ (n V)h z r n ~ l = pp* ; or the orbit is of such a nature that p varies as a power of r. Con- versely, in any curve in which p varies as a power of r, we can find the acceleration with which it may be described as a critical orbit. Now this is the case when r m = a m cos mO. For we know that the resolved parts of the velocity of P, along r and perpendicular to r respective- ly, are r and rO. Consequently -tan-|r=r0:r. But if r m =a m cosm0, we must have r" 1 " 1 r= a m d sin m.6, so that cot md = rd : r = tan ^. Therefore cos tnd = sin ty =p : r, or a m p = r m+1 . Comparing this with our previous expression, we find 2wi + 2 = n 1, or m = \(n 3), and a 2m =l(n- 1) A 2 : p, or a m = h A Changing the sign of m is equivalent to taking the inverse curve, since it replaces r by a 2 : r. We subjoin a list of curves CRITICAL ORBITS. 115 belonging to this class, observing that each is the pedal, the inverse, and the reciprocal of another curve of the series. m m Lemniscate. = 1, (n = 1) ; straight line (exceptional). = , n = 2 ; parabola with focus at s. m = 2, n 1 ; rectan- gular hyperbola with centre at s ; is its own. reciprocal. Cardioid. in \, n = 5 ; circle passing through s. m = |, n = 4t " cardioid," pedal of circle in regard to s ; inverse of para- bola. m = 2, n = 7 ; " lemniscate," inverse and pedal of rectangular hyperbola. (hyperbola with perpendi- cular asymptotes). The straight line, as we know, cannot be described with acceleration to any point out of it ; and in fact the case w = l, which the formula points to, is an exceptional one. From f= p : r, d t ( v z ) =fr = /*/* : r, we deduce- 1 \ w 2 = p log r or h* = 2yop 2 log r, which is not a curve of the kind here considered. Another exceptional case is the logarithmic spiral, in which r is proportional to p, and consequently n = 3, m = 0. A point started from a given position with the velocity from infinity and acceleration /*r~ 8 will describe a logarithmic spiral, in which the only thing that can vary is the angle at which it cuts all its radii vectores. In particular, if the point start at right angles to the radius vector, it will describe a circle. If we write z = x + iy, and = + it], supposing x + iy 1 If *=logy, then y=e", and y-x<.'=scy; lience x=y:y, or ^:j/ = d,log(/. 82 116 DYNAMIC. to be the position-vector of a point z in one plane, and + irj of a point in another plane, then any relation between z and will enable us to find one of these points from the other ; and if z move about describing any figure in its plane, will describe a corresponding figure in its plane. Now if = z n , and one of the points describe a horizontal or vertical line, the other will describe a cri- tical curve. For we may write z = re* 9 , f = se i(t> , then we shall have SB = r n e ine , whence s cos

= r n sin nO. Suppose then that moves in a verti- cal line, so that f, = s cos <, is kept constant, then r n cos nd is constant, or z describes a critical curve. If moves in a horizontal line, so that tj, = s sin , is kept constant, then r n sin nd is constant, which gives the same curve turned through an angle TT : 2n. EQUATION BETWEEN U AND 9. Let, as before, SP be denoted by r, ST by p, TP by q, and let be. the angle which PT makes with a fixed line. The components of velocity of T along 8T and TP, are p and p. But the components of velocity of T relative to P are q(j> and q in those direc- tions ; and the velocity of P is s along TP. Hence we have p = qj>, or q = d$p ; and s q = p$, or This value of the radius of curvature d$s is often useful. We may for example apply it to the hodograph of a central orbit. Let the letters r, p, 6, <, refer to the orbit, and let r lt p lt lt <$> l , mean the corresponding quantities in regard to the hodograph. Then we know that r t = v, and rp 1 =r 1 p = k, so that writing u for the reciprocal of r, or ur = 1, we have p r = hu. Moreover $ l = 6, and the radius of curvature of the hodograph f : ^ =f ; =f : hu*. REVOLVING ORBIT. 117 Making these substitutions, the formula becomes fcV (d e 'u + u) =f. This formula connects the law of acceleration with the shape of the orbit, independently of the time of descrip- tion. By means of it we may prove a useful proposition rela- ting to the effect of adding to the acceleration with which a given orbit is described a new acceleration, directed to the same centre, inversely as the cube of the dis- tance. We shall then have AV (d 6 *u + u] =f/j,u 3 , or if ^ = tf (n* - 1), then AV (dfu + tfu) =f. Now let = 0, then (> = n0 and d 6 u = u : 6 nu: = nd$u, so that 3/w = tfd^u. As hu z = 6, a change of into would change h into nh ; now the equation is n 2 hV (d^u + u) =f. Thus in the new state of things, when the value of u is the same as before, 6 is changed into nd. Therefore the same effect may be produced by letting the point move as before in its original orbit, while that orbit turns round the point s with n 1 times the angular velocity of the moving point. BOOK II. ROTATIONS. CHAPTER I. STEPS OF A RIGID BODY. THERE are two kinds of motion of a rigid body which are comparatively simple, and which it is convenient to study first by themselves. The first is the motion of a body sliding about on a plane (e.g. a book on a table), which may be completely described by specifying the motion of a moving plane on a fixed plane. The second is the motion of a body, one point of which is fixed ; which in practice is secured by a ball-and-socket joint, and which is most conveniently studied under the form of the slid- ing of a spherical surface on an equal spherical surface. When the centre of a sphere is very far away from the surface, the surface approximates to that of a plane. Thus the frozen surface of still water is approximately spherical, with its centre at the centre of the earth. In this way we may see that the first of our two motions is only a limiting case of the second, in which the fixed point is an infinite distance off. As in the case of translations we shall at first attend only to the change of position or step which the body makes between the beginning and end of the time considered, without troubling ourselves about what has taken place in the interval. In the case of a plane sliding on a plane, the motion is determined if we know the motion of two points a, 5, or the finite line ab. So in a sphere sliding on an equal sphere, the motion is determined if we know the DISPLACEMENT OF PLANE FIGURE. 119 motion of the arc of great circle ab. (A great circle on a sphere is one whose plane passes through the centre.) Every change of position in a plane sliding on a plane may be produced either by translation or by rotation about a fixed point. Let the straight line ab be moved to a'b'; it will be sufficient if we prove that this step can be effected in the way named, since the motion of all the rest of the plane is determined by that of ab. Join aa, bb' and bisect them at right angles by the lines co, do. First, let these meet in o. Then oa = oa', and ob = oV ; and of course ab = a'b'', so that the triangles oab, oa'b' have their sides respectively equal, and therefore the angle aob = a'ob'. Hence also angle aoa' = bob'. Therefore if the triangle oab be turned round the fixed point o, until oa comes to oa', ob by the same amount of turning will come to ob', and consequently the triangle oab will come to coincide with oa'b'. Next, suppose that the lines bisecting aa', bb' at right angles are parallel to one an- other. Then aa, bb' are parallel, and consequently either ab is parallel to a'b', and the required step is a translation, or else they make equal angles with aa', bb', and one can be brought to co- incide with the other by rotation round their point of intersection o. In the latter case the bisecting lines coincide, and the point o is not determined by their intersection. Tw.o figures which are equal and similar are called congruent. If they can be moved so as to coincide with each other, they are called directly congruent ; but if one is the image of the other in a plane mirror they are said to be inversely congruent,, or one is a perversion of the 120 DYNAMIC. other. Two plane figures which are inversely congruent can be moved into coincidence by taking one of them out of its plane and turning it over ; this does not make them directly congruent in regard to the plane. It is essential to the preceding demonstration that the two triangles oab, oa'b' should be directly congruent. Now V if they were inversely congruent, as in this figure, the lines bisecting aa and bb' at right angles would coincide, con- trary to the supposition. It is to be observed that the case of translation occurs when the lines co, do are parallel, I d r that is, when their point of in- *~ ' tersection o has been sent off to an infinite distance. Thus a step of translation may be regarded a as a step of rotation round an infinitely distant point. Every change of position in a sphere sliding on an equal sphere may be produced by rotation about a fixed point. The proof is exactly the same as before, except that straight lines are to be replaced by great circles of the sphere, and that the case of co, do being parallel does not occur ; for any two great circles intersect in two opposite points of the sphere, say o and o. Rotation about o is rotation about the axis oo', therefore also about o'. The theorem may be also stated thus: every displacement of a body having one point fixed may be produced by rotation about an axis through that point. The fixed point is of course the centre of the sphere. DISPLACEMENT OF EIGID BODY. 121 Every displacement of a rigid body may be produced by rotation about a fixed axis together with translation parallel to the axis (screw motion). Let a be any point of the body whose new position is a ; then we can produce the whole displacement by first giving the body a translation aa, and then turning it about a' as a fixed point. The latter step can be effected by rotation about an axis through a'. Now consider those points of the body which lie in a plane perpendicular to this axis. By the rotation they are merely turned round in that plane ; while by the translation the plane was moved parallel to itself. Hence the new position of this plane is parallel to its original position. Let then the body have first a translation per- pendicular to the plane, so as to bring the plane into its new position ; then the remaining displacement consists of a sliding of this plane on itself, which may be produced by rotation about a fixed point of it, or, which is the same thing, about an axis perpendicular to the plane. Thus the whole displacement is produced by rotation about that axis, together with translation parallel to it. If two plane polygons, which are perversions of one another, be rolled symmetrically along a straight line, one on each side, until the same two corresponding sides come into contact, the result will be merely a translation of each along the line through a distance equal to its perimeter. Hence successive finite rotations through angles equal to the exterior angles of a polygon, about successive vertices (taken the same way round) are equivalent to a translation of length equal to the perimeter. By supposing one polygon fixed, and the other to roll round it, we find that successive rotations about the vertices through twice the exterior angles will bring the plane back to its original position. The corresponding theorems for a spherical surface are easily stated. CHAPTER II. VELOCITY-SYSTEMS. SPINS. WHEN a body is rotating about a fixed axis with an- gular velocity &>, every point in the body is describing a circle in a plane perpendi- cular to the axis, whose radius is the per- pendicular distance of the point from the axis. Hence the velocity of the point is in magnitude &> times its distance from the axis, and its direction is perpendicular to the plane which contains the axis and the point. If ab be the axis, pm perpendicular to it, the velocity of p is a) times mp perpendicular to the plane pab. If, therefore, we represent the angular velocity (o by means of a length ab marked off on the axis, the velocity of p is ab multiplied by mp, which is the moment of ab about p, being twice the area pab. In the case of a plane figure, the rotation being about an axis perpendicular to the plane, or say about a point m in the plane (where it is cut by the axis), the velocity of any point p is a> . mp in magnitude, but perpendicular to mp; that is, it is i(o . mp, the angular velocity being reckoned positive when it goes round counter clockwise. When a body has a motion of translation, the velocity of every point in it is the same, and that is called the velocity of the rigid body. But in the case of rotation, the ROTORS. 123 velocity of different points of the body is different, and we can only speak of the system of velocities, or velocity- system, of its different points. Still, the velocity-system due to a definite angular velocity about a definite axis is spoken of as the rotation-velocity, or simply the velocity of a rigid body which has that motion. To specify it com- pletely we must assign its magnitude and the position of the axis ; it is thus represented by a certain length marked off anywhere on a certain straight line. For it clearly does not matter on what part of the axis the length ab is marked off; its moment in regard to p will always be the velocity of p. A rotation-velocity, so denoted, shall be called a spin. Such a quantity, which has not only magnitude and direction, but also position, is called a rotor (short for rotator) from this simplest case of it, the rotation-velocity of a rigid body. A rotor is a localised vector. While the length representing a vector may be moved about any- where parallel to itself, without altering the vector, the length representing a rotor can only be slid along its axis without the rotor being altered. Two velocity-systems are said to be compounded into a third, when the velocity of every point in the third system is the resultant of its velocities in the other two. COMPOSITION OF SPINS. The resultant of two spins I, m about the points a, b in a plane, is a spin (I + m) about a point c, such that I . ca + m . cb 0. For the velocities of p due to the two spins are il . ap and im . bp, and their resultant is consequently i (I + m) cp ; that is, it is the velocity due to a spin l + m about c. It should be observed that the result holds good what- ever be the signs of I, m\ but that, if their signs are different, the point c will be in the line ab produced. There is one very important exception, when the spins are equal but of opposite signs; the resultant is then a 124 DYNAMIC. l+m. translation-velocity. Let the spins be /, I, then il . ap il . bp = il (ap bjy) = il . ab. Thus the velocity of every point p is the same, namely it is of the magni- tude I . ab and is perpendicular to ab. Translating these results into language relating to axes perpendicular to the plane, we find that the resultant of two parallel spins Z, ra is a spin of magnitude equal to their sum, about an axis which divides any line joining them in the inverse ratio of their magni- tudes. But the resultant of two equal and opposite parallel spins is a trans- lation-velocity, perpendicular to the plane containing them, of magnitude equal to either multiplied by the distance between them. It follows that if we compound a spin I with a trans- lation-velocity v perpendicular to its axis, the effect is to shift the axis parallel to itself through a distance v : I in a direction perpendicular to the plane containing it and the velocity. A translation- velocity may be regarded as a spin about an infinitely distant axis perpendicular to it. Hence all theorems about the composition of translation-velocities with spins are special cases of theorems about the compo- sition of spins. The resultant of two spins about axes which 'meet is a spin about the diagonal of the parallelogram whose sides are their representative lines, of the magnitude repre- sented by that diagonal. In other words, spins whose axes meet are compounded like vectors. For if ab, ac represent the two spins, and ad is the diagonal of the parallelogram acdb, the velocities of any point p due to the two spins are the moments of ab and ac about p, and the resultant of TWIST-VELOCITIES. 125 them is the moment of ad about p, that is, it is the velo- city due to a spin ad. It follows from this that the resultant of any number of spins whose axes meet in a point is also a spin whose axis passes through that point. And that if i, j, k are spins of unit angular velocity about axes oX, o Y, oZ at right angles to one another, any spin about an axis through o may be represented by xi + yj + zk, where x, y, z are magnitudes of the component spins about the axes oX, oY,oZ. VELOCITY-SYSTEMS. TWISTS. If a rigid body have an angular velocity w about a certain axis, combined with a translation-velocity v along that axis, the whole state of motion is described as a twist- velocity (or more shortly, a twist) about a certain screw. We may in fact imagine the motion of the body to be produced by rigidly attaching it to a nut which is moving on a material screw. The ratio v : u> is called the pitch of the screw ; it is a linear magnitude (of dimension \L\ simply), and we may cut a screw of given pitch upon a cylinder of any radius. The pitch is the amount of trans- lation which goes with rotation through an angle whose arc is equal to the radius. For our present purpose it is convenient to regard the axis of the rotation as a cylinder of very small radius, on which a screw of pitch p is cut. The screw is entirely described when its axis is given, and the length of the pitch. The angular velocity to is called the magnitude of the twist. The velocity of a point at distance k from the axis is ko> perpendicular to the plane through the axis, due to the rotation, and v parallel to the axis, due to the translation. If the resultant-velocity makes an angle 6 with the axis, we shall have tan d = ko> : v = k : p. Thus for points very near to the axis, the velocity is nearly parallel to it ; for points very far off, nearly perpendicular to it; and for points whose distance is equal to the pitch of the screw, it is inclined at an angle of 45. A quantity like a twist- velocity, which has magnitude, direction, position, and pitch, is called a motor, from the 126 DYNAMIC. twist-velocity which is the simplest example of it, and which, as we shall see, is the most general velocity-system of a rigid body. COMPOSITION OF TWISTS. The resultant of any number of spins and translation- velocities is a twist. Take any point o, and let ab represent one of the spins. Then ab is equivalent to an equal spin about the parallel oc, together with a translation- velocity which is the moment of ab about o. In the same way every other spin of the system may be resolved into a spin about an axis through o and a translation- velocity. Then all the spins will have for resultant a spin about an axis through o, and all the translation-velocities will have for resultant a translation-velocity. Let os be the resultant spin, and ot the resultant translation- velocity ; then ot may be resolved into om along os and mt perpendicular to it. The effect of combining the spin os with mt is to shift its axis parallel to itself perpendicular to the plane sot through a distance mt : os. Thus we are left with a spin about an axis parallel to os and a translation along that axis ; that is to say, the resultant is a twist. It follows, of course, that the resultant of any number of twists is also a twist. We shall now determine the axis and pitch of the resul- tant of two twists 1 . It is convenient to suppose in the first place that the axes of the twists intersect at right angles. Let then oX, o Y be these axes, a, /3 the magni- tudes of the twists, a, 6 their pitches, OT, p, the magnitude and pitch of the resultant twist, k the distance of its axis 1 This theory, and most of the nomenclature of the suVect, are due to Dr Ball. CYLINDROID. 127 from the point o, the angle it makes with oX. Then a = o7cos0, /3 = OTsin0, the two spins a, /3 about oX, o Y compounding into a spin vr round oP. The translations due to these spins are ai, 6/3, or vracosO, ar&sintf, along oX, oY. The sum of their resolved parts along OP = -5j-acos0 . cos0 -1- OT&sin0.sin0 = 'ar (acos 2 + 6sin 2 0). The sum of their resolved parts perpendicular to OP = -sracos0.sin0 crfr sin . cos = ^-ar(a b} sin 20. The latter part shifts the axis OP parallel to itself in a direction perpendicular to the plane through a distance k, = i (a - 6) sin 20. The former part shews that the pitch of the resultant twist p, = a cos 2 + 6 sin 2 0. Now let a circle be drawn through o and two points A, B on oXand o Y equidistant from o. The centre c is the middle point of A B. Then since is the angle at the circumference AoP, 20 is the angle at the centre AcP, and sin 20 = Pm : cA. If a cylinder be drawn upon this circle, a plane through AB and a point vertically over C at a distance \ (a b) will cut the cylinder in an ellipse, and if Q be the point of the ellipse vertically over P we shall have PQ = k. For PQ : Pm = Cd : Cc, Pm = Cc sin 20, 128 DYNAMIC. and Cd=^(a b), whence PQ = $(a-b) sin 20 = k. Hence zQ, parallel to oP, is the axis of the resultant twist. The angle 6 depends upon the magnitude of the com- ponent twists, not at all upon their pitches. By varying this angle then, we shall obtain the screws of all twists which can be got by compounding twists upon the given screws. If varies uniformly, the line zQ, which is parallel to OP, turns round uniformly, being always per- pendicular to oZ; while the point z has a simple harmonic motion up and down oZ, whose period is equal 'to that of P in the circle. The surface traced out by the line zQ is called a cylindroid. It is clear that if we cut the cylin- droid by a circular cylinder having oZ for axis, the section will be the bent oval previously obtained by wrapping round the cylinder two waves of a harmonic curve (p. 35). The line oZ is called the directrix of the cylindroid. The pitch of each screw on the cylindroid depends only on its position and the pitches of the two component twists ; to represent therefore the distribution of pitch we may attribute to these twists any absolute magnitude that we like. We shall suppose their magnitudes to be inversely proportional to the square roots of their pitches. Let oa and ob be these magnitudes, and let the pitches be represented by numbers on such a scale that the pitch of oa is ob : oa, then the pitch of ob is oa : ob, since the pitches are as o6 2 : oa 2 . Then the translation accompany- ing the spin oa will be represented by i . ob, and that accompanying ob by i . oa' or i . oa according as the two pitches are of the same or different signs. In the first case construct an ellipse, in the second a hyperbola, with oa and ob for semi-axes ; then we shall shew that the translation accompanying a spin op, regarded as com- pounded of proper multiples of oa and ob, is i . oq, where oq is the semi-conjugate diameter. To prove this, we must observe that, pm and qn being drawn perpendicular to the major axis, om : oa = nq : ob, PITCH-CONIC. 129 and + on : oa= mp : ob. For the ellipse this follows by parallel projection from the circle, in which the property is obvious ; for the hyperbola we know that >n a where is written for nt + e of p. 89. v Thus the spin op being equivalent to om and mp, the translation due to om is to i . ob as om : oa, that is, it is i.nq; and the translation due to mp is mp : ob multiplied by i . oa,' and i . oa in the two cases respectively, that is, it is i . on. Hence the translation due to op is i . oq. If we draw qf perpendicular to op, of : op will be the height k of the screw which is parallel to op, and qf : op will be its pitch. Now in the harmonic or quasi-harmonic motion with acceleration towards the centre, n.po is the velocity at q, and fq is equal to the perpendicular from the centre on the tangent at q ; therefore the rectangle op .fq is constant, and consequently equal to oa . ob. Hence qf : op = oa.ob : op 3 , or the pitch of the screw parallel to op is inversely propor- tional to the square of op. This ellipse or hyperbola is called the pitch-conic. When the pitch-conic is a hyperbola, it follows that there are two screws of pitch zero, namely those which are C. 9 130 DYNAMIC. parallel to the asymptotes. Thus in two cases the result- ant twist is a pure spin. The distance from o of these is 1 and 1 respectively. Thus the scale on which the pitches have been reckoned is such that the unit of length is half the distance between the axes of pure spin. When the pitch of the screw on oX is zero, the pitch-conic re- duces to two lines parallel to oX; and there is no other screw whose pitch is zero, except when that of o Y is zero, and then all the pitches are zero, the cylindroid reducing to the lines through o in the plane Xo Y. In order to shift the figure of the pitch-conic through a distance k perpendicular to its plane, we must add Id . oa to the translation accompany- ing the spin oa, and ki. ob to that accompanying ob. Let at = k . ob, and bt' = k. oa' ; then the new translations are ot, ot', which are still along conjugate diameters, because by similar triangles we have mr : om = k.ob : oa and on : ns = k.oa : ob', whence om on oa ' oa' mr ns ~ob'ob > which is the condition. The resultant of two twists whose axes are anyhow situated is a twist about some screw which belongs to a cylindroid containing the axes of the given twists. This cylindroid we now proceed to find, supposing the two screws given. Find the line which meets both of their axes at right angles; this is the directrix of the cylindroid. Draw a plane through one of the axes perpendicular to the directrix, and a line in this plane parallel to the other axis meeting the directrix. Let ob be the first axis, oq per- pendicular to it, then i , oq will be the direction of the translation that goes with the spin about TO FIND AXES OF CYLINDROID. 131 ob; let oa be parallel to the second axis, and i. op the direction of the translation which together with a spin about oa is equivalent to a twist about that axis. If p be the pitch of the second screw, h the distance of its axis from o, tanao/> = j0 : h. Then the problem is to find an ellipse (or hyperbola) having oa, ob for conjugate dia- meters, and also op, oq. Or rather, having given that these are the directions of two pair of conjugate diameters, it is necessary to find the relative magnitudes of one pair. For this purpose we observe that if p, q are points on the conic, on : om = mp : nq, or the areas 'onq, omp are equal. Let po meet qn in q; then area omp : area onq = om? : or? since they are similar. But onq : onq' = nq : nq so that om* : ori* = nq : nq'. Given q, this determines p, so that the ratio op, oq is known. A conic described on these as semi-conjugate diameters is similar to the pitch-conic. Screws parallel to its axes compounded of the two given screws will be the oX, o Y of the cylindroid. The analytical solution is as follows 1 . Let p, q be the pitches, & t , & 2 the distances from o the centre of the cylindroid, I, m the inclinations to oX, of the two screws, h their distance and 6 the angle between, them. Then from the equations p = a cos 2 1 + b sin 2 I, q = a cos 2 m + b sin 2 m, Jc i = ^(a b) sin 2,1, & 2 = (a b) sin 2m, we have to find a, b, I, m, k lt & 2 in terms of p, q, h, and 0. Now h = k l k z = \ (a b) (sin 21 sin 2m] = (a b) cos (I + m) sin (I m), and k t + k a = \ (a b} (sin 21 + sin 2m) = (a b) sin (I + m) cos (I m). So also p q = ^(a b) (cos 21 cos 2m) = (b a) sin (I + m) sin (I m), 1 Ball, Theory of Screws, pp. 16, 17. 92 132 DYNAMIC. and p + q = a + b + (a - 1) cos (2Z + cos 2m) (a b)cos(l + m) cos, (I m) hcos0 (since l m = 0). Therefore 7r + (p - q) 2 = (a - 6) 2 sin 2 0, p q = h tan (Z + m) = /i tan (2 &) ; whereby ab, k 1 k a , I and m are expressed in terms of p, q, \, and 6. MOMENTS. "When a straight line moves as a rigid body, the com- ponent of velocity along the line of every point on it is the same. For consider two points, a, b ; the rate of change of the distance ab is the difference of the resolved parts of the velocities of a and b along ab. If therefore the length ab does not change, this difference is zero. This com- ponent of velocity of any point on the line may be called the lengthwise velocity of the line. The lengthwise velocity of a line due to a given twist is called the moment of the twist about the line. Let Im, = k, be the shortest dis- tance between the axis In of the twist and the straight line mr. It will be sufficient to determine the velocity of m along mr. Now m has the velocity kco perpendicular to the plane mln, and pa parallel to In, if 01 be the magnitude and p the pitch of the twist. Let be the angle between mr and In, then the resolved parts of these components along mr are ka> sin and + pw cos 0. Thus the mo- ment of the twist about the line is a (p cos k sin 0). The moment of a screw about a straight line is the moment of a unit twist on that screw about the line. Thus p cos k sin is the moment of a screw of pitch p about a line at distance k making an angle with its axis. COMPLEX OF A SCREW. 133 All the straight lines in regard to which a given screw has no moment, are said to form a complex of lines belong- ing to that screw. When a line belonging to the complex is moved by a twist about the screw, every point in it moves at right angles to the line. Ail the lines of the complex which pass through a given point lie in a given plane, namely, the plane through the point perpendicular to its direction of motion due to a twist about the screw. This plane passes through the perpendicular from the point on the axis, and makes with the axis an angle 6, such that tan 6=p : k. Conversely, all the lines of the complex which lie in a given plane pass through a certain point, at a distance p cot 6 from the axis along a straight line in the plane perpendicular to it. If any other line in the plane be- longed to the complex, every point in the plane would move perpendicularly to the plane, and the twist would reduce to a spin about some line in the plane. In the case when p = 0, or the twist reduces itself to a spin about its axis, the moment becomes k sin 8, and can only vanish if the line meet the axis (/c = 0), or is parallel to it (sin 9 = 0), which is the same as meeting it at an infinite distance. Hence the complex reduces itself to all the lines which meet the given axis. All the lines of the complex which meet a given straight line, not itself belonging to the complex, meet also another straight line. For, suppose the cylindroid constructed, which contains the given screw and the given straight line, considered as a screw of pitch 0. Then the pitch-conic must be a hyperbola, since there is one screw with pitch ; this is parallel to one asymptote, and there must be another parallel to the other asymptote. Hence every twist may be resolved into two spins, the axis of one of which is any arbitrary straight line, not belonging to its complex. Now, since the two spins are equivalent to the twist, the lengthwise velocity of any line due to the twist is the sum of its lengthwise velocities due to the two spins ; or the moment of the twist is the sum of the moments of 134 DYNAMIC. the two spins. If then a straight line belong to the com- plex and meet the axis of one spin, the moments of the twist and one spin are zero, consequently the moment of the other spin is zero, or its axis meets the line. Therefore a straight line of the complex which meets the axis of one spin, meets also the axis of the other. If however the axis of one spin belong to the complex, that of the other spin must meet it, since the moment of the twist about it is zero ; but in that case it must also coincide with it, since otherwise the pitches of all screws on the cylindroid would be zero. We have then the case noticed above, in which the pitch-conic reduces to two parallel lines. Erom the symmetry of the expression k sin 6 in re- gard to the two straight lines concerned, we perceive that the lengthwise velocity of a line A due to a unit spin about a, line B is equal to the lengthwise velocity of B due to a unit spin about A, Hence we may speak of this quantity as the moment of the two lines, or of either in regard to the other. We shall also define the moment of two spins as the product of their magnitudes into the moment of their axes. If one of the axes goes away parallel to itself to an infinite distance, and at the same time the angular velocity &> about it diminishes indefinitely, so that kw = v, the spin becomes a translation-velocity v perpendicular to that axis, making, therefore, an angle <, = ^TT 6, with the other axis ; and the moment becomes vco' cos <, if fa' is the magnitude of the finite spin. In the same way we may speak of the moment of a twist and a spin, meaning the magnitude of the spin multiplied by the moment of the twist about its axis. Suppose the twist resolved into two spins A, B; then its moment in regard to the spin C will be the sum of the moments of the component spins. Let us combine with C a spin D, making a second twist; then the sum of the moments of the twist A + B in regard to C and D will be equal to (A C} + (BO) + (AD] + ( CD), (where (A C) means the moment of A in regard to , about a point o situate on a line through a perpendicular to its direction of motion, at a distance such that a = ico . oa. To determine the motion of the instantaneous centre, we must find the acceleration of any point in the plane. The instantaneous centre shall be called c in the fixed plane, and c l in the moving plane; and at a certain instant of time it shall be supposed to be at a point o in the moving plane. Then at that instant c, c 1? o are the same point; but c means the velocity of the instantaneous centre in the fixed plane, c l its velocity in the moving plane, and 6 the velocity of o in the moving plane, which we know to be zero. ROLLING OF CENTRODES. 139 Now if p be any point in the moving plane, we know that at every instant p = ico. cp. To find the acceleration of p we must remember that the flux of cp isp-c. There- fore p ico . cp + ico (p c) = (id) w 2 ) cp ita . C. Now let p coincide with o, that is (for the instant) with c. Then 6 = ico . c, or the acceleration of o is at right angles to the velocity of c, and equal to the product of it by the angular velocity. If we suppose the moving plane to be fixed, and the fixed plane to slide upon it so that the relative motion is the same, then if p l is the point of the fixed plane which at a given instant coincides with p in the moving plane, the velocity and accleration of p : on one supposition are equal and opposite to the velocity and acceleration of p on the other supposition; also to becomes o>. Hence we shall have o l = + ico.c lt but o 1 = o. Therefore c t = c, or the velocity of the instantaneous centre in the moving plane is the same in magnitude and direction as its velocity in the fixed plane. Because these velocities are the same in direction, the two centrodes touch one an- other; and because they are the same in magnitude, the moving centrode rolls on the fixed one without sliding. For _ -C^. let s, s 1 be the arcs ac, be measured from points a, b which have been in contact ; then 5=^, and therefore (since they vanish together) s=s r The angular velocity G> is equal to 6- multiplied by the difference of the curvatures of the two centrodes. For suppose them to roll simultaneously on the tangent ct; then their angular velocities and ty will be respectively equal to their curvatures multiplied by s, and the relative angular velocity will be the difference of these. When the curva- tures are in opposite directions one of them must be con- 140 DYNAMIC. sidered negative. The same result may be obtained by calculating the flux of the acceleration of o. Thus if r, r t are radii of curvature of the fixed and rolling centrodes, we have , . and s = rr r r, CURVATURE OF ROULETTE. We may derive some important consequences from the expression just obtained for the the acceleration of a point in the moving plane, namely p = (id) to 2 ) cpico. c. This consists of three parts ; o> 2 . pc is the acceleration towards c due to rotation about it as a fixed point ; iw . cp is in the direction np perpendicular to cp, due to the change in the angular velocity ; and i(o.c is in the direction en, due to the change in position of c as the centrode rolls. Hence the normal acceleration of p, that is, the component along pc, is in magnitude af.pc a), c cos#. It vanishes for those points p for which w .pc = c cos#, or for which en = 6 : : c, that is, it is twice the difference between the curvatures of the centrodes. All the points of this circle, therefore, are at the given instant passing through points of inflexion on their paths. The path of any point p is called a roulette, as being traced by rolling motion. We can now determine the curvature of a roulette at any point. For since the normal acceleration is the squared velocity multiplied by the curvature, we have ,. ,, ,. co 2 . pc to . ccos0 curvature 01 path ot = : -, - = - rr pc pc r r 1 where r, r t are the radii of curvature of the fixed and rolling centrodes. The tangential acceleration of p is to . cp we sin#. If therefore we make ct=*wc : w, the locus of points whose tangential acceleration is zero is a circle on ct as diameter. The point at c belongs to both circles ; it is a cusp on its path, being a point where there is no normal acceleration, but also no velocity. It has, however, as we know, a tangential acceleration iw'c. The other intersection of the two circles has no acceleration at all. INSTANTANEOUS AXIS. In the case of a body moving with one point fixed, we may combine with its velocity-system a spin about any axis through the point, such that the velocity of a certain point a due to the spin is equal and opposite to its actual velocity in the motion of the body. The resultant velo- city-system is consistent with rigidity, and the point a is at rest; it is therefore a spin about the axis oa. Conse- quently the actual motion of the body is a spin about some axis in the plane of these two. DYNAMIC. Let oc, = w, be the instantaneous spin in magnitude and direction, op, = p, the position vector of any point p. Then we know that the velocity of p is the moment of oc about p, that is, twice the area of the triangle ocp. This quantity, which is in magnitude oc . op sin cop, and in direction perpendicular to oc and op, is what we have called the vector product of oc and op, and de- noted by Vwp. We have therefore p = Vwp. To find the acceleration of p, therefore, is to find the flux of the triangle ocp, due to the motion of p and c. Now suppose that c moves to c l in a certain interval ; then oc v p = ocp + coc 1 + c^pc, all the areas being of course regarded as vectors. -But if we draw pd equal and parallel to cCj, we shall have coc, + c^pc =pod, for the three tri- angles stand on the same base cc v or pd, and the height op is the sum of oc and cp. It follows that the flux of ocp, due to the motion of c, is equal to the moment about o of the velocity of c supposed to be transferred to p. That is, the flux of Vwp, due to the change of w, is Vwp. In a similar way it may be shewn that the flux due to change of p is Vwp. Hence 1 altogether, since p= Vwp, we have p = Vwp + Vwp = Vwp + V. w Vwp. The expression F. wVwp means the vector product of the two vectors, w and Vwp. Thus it appears that the acce- leration of p consists of two parts; V.wVwp along the perpendicular from p to the axis oc, due to the rotation w ; and Vwp, perpendicular to op and to the velocity of c, due to the change of w. Let a be the point of the moving body at which c is instantaneously situated, then d= VWW,OT the acceleration of a is equal to the moment about o of the velocity of c. 1 The flux of a vector product has been already found by a different method on p. 97. TO FIND INSTANTANEOUS CENTRE. 143 If we interchange the fixed and moving axodes, keeping the relative motion the same, we alter the signs of a and to ; therefore gof, ode be supposed rigid and jointed together at o, and let the other lines in the figure represent bars forming three jointed parallelograms. Then however the system is moved about inits plane (e.g. into the configuration of the second figure) the triangle abc mill be always of the same shape. Now 150 DYNAMIC. the system is one which in shape (independently of its position) has two degrees of freedom ; for if we fix one of the three triangles, the other two may be turned round independently. If therefore we impose a single condition, that the area abc shall be constant, the system will still have one degree of freedom. But this is equivalent to fixing the size of abc as well as its shape, so that we may fix the points a, 6, c; and still o will be able to move. In so moving it will describe a path which is due at the same time to three different three-bar motions. All that remains to be proved, therefore, is that the shape of abc is invariable. This can be made clear by very simple considerations. Xiet q be the operation (com- plex number) which converts hk into ho, so that ho = q. hk. Then the same operation will convert go into gf and od into oe, since the three triangles are similar. Consequently ac = ag +gf+fc = ho + gf+ oe = q .hk + q .go + q.od = q(hk + ah + kb) = q . ab, that is, ac is got from ab by the same operation which converts hk into ho ; therefore the triangle abc is similar to hko. Or in words, the components ag, gf,fc of ac are got from the components hk, ah, kb of ab by altering all their lengths in the same constant ratio and turning them all through the same constant angle. Therefore the whole step ac is got from ab by altering its length in a constant ratio and turning it through a constant angle. It is to be observed that the configuration in the first figure forms an apparent exception to the theorem. The area abc is then a maximum, and the path of o has shrunk up into a point, so that it is really not able to move. We may use Mr Roberts' theorem to transform motion due to the crossed rhomboid into that due to a figure called a kite by Prof. Sylvester. It also is a quadrilateral having its sides equal two and two, but the equal sides are ad- jacent. KITE. Now if the point o be taken iii the first figure so that gd is bisected at o, the triangles gof, ode will be equal in all respects, and Ik will equal ha. Now put the figure into such a configuration that dbc is equal to hko: then ahkb is a crossed rhomboid, and both the figures bdec, agfc are kites. For bc = ko = bd, and ac = ho= ag, while de = ec and gf=fc by construction. It follows that in the three-bar motion determined by a kite, the path of every point in the moving plane is the inverse of a conic ; since it may also be described by means of a crossed rhomboid. CIRCULAR ROULETTES. Considerable interest attaches to the case of plane mo- tion in which both centrodes are circles, or when one is a circle and the other a straight line ; the latter being a speciality of the former, obtained by making the radius of one circle infinite. The path traced by a point in the circumference of the rolling circle is called a cycloidal curve, that traced by any other point in the moving plane a trochoidal curve ; the names cycloid and trochoid sim- pliciter being applied to paths traced in the rolling of a circle on a straight line. 152 DYNAMIC. Two circles may touch each other so that each is out- side the other, or .so that one includes the other. In the former case, if one circle rolls on the other, the curves traced are called epicycloids and epitrochoids. In the latter case, if the inner circle roll on the outer, the curves are hypocycloids and hypotrochoids; but if the outer circle roll on the inner, the curves are epicycloids and peritro- choids. We do not want the name pericycloids, because, as will be seen, every pericy cloid is also an epicycloid ; but there are three distinct kinds of trochoidal curves. DOUBLE GENERATION OF CYCLOIDAL CURVES. Every cycloidal curve (except the cycloid par excellence) can be generated in two different ways. In the case of hypocy- cloids, let a and b be centres of two circles the sum of whose radii is equal to the radius of the fixed circle. Then if we complete the parallelogram oapb, p will be a point of intersection of these cir- cles, for ap = ob = od bd= radius of circle a, and similarly bp equal radius of circle 6. Hence the angles cap, pbd, cod are all equal, and therefore the arcs ap,pd, ad are the same portion of the circumferences of their several circles. But the radius of the large circle is the sum of the radii of the two smaller ones ; therefore its cir- cumference is the sum of their circumferences, and consequently the arc cd is the sum of the arcs cp, pd. Make cq = cp, so that qd = pd; then by the rolling of the circle a the point p would come to q, and by the rolling of the circle b the point p would come to q; hence the intersec- tion of the two circles is a fixed point on each of them, and the path of p may be described by 'the rolling of either. STRAIGHT HYPOCYCLOID. 153 In the case of epicycloids, the difference of radii of the rolling circles is equal to the radius of the fixed circle ; the arc cp is equal to cd + dp, and p would be brought to q by the rolling of either a or 6. CASE OF EADII AS 1 : 2. A very important case is that of internal rolling, the radius of one circle being half that of the other. Draw the straight line opq to meet both circles. Let cop = 6, then cap = 20 ; and if a be the radius of the smaller circle, 2a of the larger, arc cp = 2a0, and arc cq = 2ad ; therefore cp = cq, or p will come to q in the rolling. Hence every point in the circumference of the rolling circle describes a diameter of the fixed circle. The opposite point p' de- scribes the diameter perpendicular to the former. If we suppose the small circle to roll from c to the right, oa will turn counterclockwise into the position oa, while ac will turn clockwise into the position a'c. Hence the motion (supposing the rolling to take place uniformly) is a composition of two circular motions of the same period in opposite directions. Consequently the motion of any point can be resolved into simple harmonic motions all of the same period. It follows that the motion of every point 154 DYNAMIC. of the moving plane is harmonic motion in an ellipse, which in certain cases as we have seen reduces itself to simple harmonic motion on a diameter of the fixed circle. Hence if a line of constant length ab be moved with its extremities on two fixed lines oX, oY, the path of every point rigidly connected with ab will be an ellipse with centre o, unless the point is on the circumference of the circle circumscribing oab, in which case the path is a straight line through o. An apparatus for describing an ellipse by means of a pencil attached at a point p of a bar so moving, is called the elliptic compasses. The semi-diameters of the ellipse along o X and o Y are pa, pb respectively. When o X, o Y are at right angles, these are the semi-axes of the ellipse. If the small circle be fixed, and the larger roll round it, the motion is such that every diameter of the rolling circle passes through a fixed point on the small one. Now every line in the moving plane is parallel to some diameter of the large circle, and must therefore remain at a fixed distance from the point through which the diameter always passes ; consequently it always touches a circle whose centre is at that point. Hence every straight line in the moving plane, envelops a circle. Conversely, if a plane move so that two straight lines in it always touch two fixed circles, then every line in the plane will envelop a circle. For two lines parallel to them through the centres of the circles are fixed relatively to the moving plane ; thus a line of constant length in the fixed plane always has its extremities on two lines of the moving plane, and the motion is the one here considered. The curve traced by a point in the circumference of the large circle is a cardioid, which we have already met with as the inverse of a parabola in regard to the focus, or, which is the same thing, the pedal of a circle in regard to a point on the circumference. If the point q describe a cardioid, the line qt, tangent to the large circle, always touches a fixed circle whose centre is at p, and which ENVELOP OF ROULETTE. 155 therefore touches the fixed small circle at p. Hence q is the foot of the perpendicular on the tangent to this circle from the point p on its circumference. The cardioid may also be described by the external rolling of a circle on a fixed circle of equal size. ENVELOP OF CARRIED ROULETTE. When a circle rolls on a fixed circle, every diameter of the rolling circle envelops a cycloidal curve. Suppose a circle of half the size to roll together with the circle o, so as to have always the same point of contact ; then the relative motion of these two circles will be that which we have just considered, and a point p, fixed on the small circle, will be always on the diameter oq. The tangent to the cycloidal path described by p, in consequence of the rolling of the circle a on the fixed circle, is opq, since c is the instantaneous centre ; hence this line always touches the cycloidal curve. This theorem is a particular case of the following. Let a curve B roll on a curve A, carrying with it the roulette pq made by rolling C on B; then the envelop of this 156 DYNAMIC. roulette is a curve which may be described by rolling C on A. Suppose B and G to roll simultaneously on A, so as always to have the same point of contact; then the motion of (7 relative to B is that which describes the roulette qp. Now cp is perpendicular to the tangent both of this roulette and of that which p describes by the rolling of G on A. Hence the two roulettes always touch one another, as was to be proved. Observe that the point p is not necessarily on the curve 0. Returning to the case of the circles, we observe that the extremities q, r of the moving diameter describe similar and equal cycloidal curves, such that a cusp of one and a vertex of the other are on the same diameter of the fixed circle. Hence if a straight line of constant length move with its ends on two such cycloidal curves, starting from a position in which one end is at a cusp and the other at a corresponding vertex, it will envelop a cycloidal curve. The following are cases of this theorem : 1. The chord of a cardioid through the cusp is of constant length. (A point is a special case of a cycloidal curve.) 2. A line of constant length with its ends moving in two fixed lines at right angles envelops afour-cusped hypo- cycloid. 3. The portion of the tangent to a three-cusped hypo- cycloid intercepted by the curve is of constant length. The curvature of cycloidal curves may be calculated by means of the general theorem already given for the curva- ture of roulettes, or directly as follows. Let o be the centre of the fixed circle, take ce : dc=*do : co, draw a circle through e with centre o, and a circle on ce as diameter. Produce pc to meet this in q. If this circle roll on the circle through e, so that q is brought to h, we shall have eq = eh, and since eq : pd = ec : cd= oe : oc, pd is equal to the corresponding arc of the circle kc. Hence the two small circles may roll together on the two large ones, so that ce always passes through o, and pcq is a straight line. Then EVOLUTE OF CYCLOID. 157 pq is normal to the path of p and tangent to that of q, or the latter path is the evolute of the former. I It follows that the length of the arc kq is equal to pq, or s = de cos i/r. It is clear that ty is in a fixed ratio to the angle which pq makes with the normal at k, and consequently s = a cos m, if a = de and m is this fixed ratio. BOOK III. STRAINS. CHAPTER I. STRAIN-STEPS. STRAIN IN STRAIGHT LINE. WE have hitherto studied the motion of rigid bodies, which do not change in size or shape. We have now to take account of those strains, or changes in size and shape, which we have hitherto neglected. The simplest kind of strain is the change of length of an elastic string when it is stretched or allowed to con- tract. When every portion of the string has its length altered in the same ratio, the strain is called uniform or homo- ai 1 il geneous. Thus if apb is changed ' into a'p'V by a uniform strain, a'< ^ \l' ap : dp = ab : a'b'. The ratio ' dp : ap, or the quantity by which the original length must be multiplied to get the new length, is called the ratio of the strain. The ratio of the change of length to the original length, or dp ap : ap, is called the elonga- tion ; it is reckoned negative when the length is diminished. A negative elongation is also called a compression. Let e be the elongation, 8 the unstretched length ap, cr the stretched length dp', then cr s = es, or <7 = s (1 + e) . Thus 1 + e is the ratio of the strain. In general, a solid body undergoes a strain of simple elongation e, when all lines parallel to a certain direction are altered in the same ratio 1 : 1 + e, and no lines per- pendicular to them are altered in magnitude or direction. STRAIN OF PLANE FIGURE. 159 The strain is then entirely described if we describe the strain of one of the parallel lines. HOMOGENEOUS STRAIN IN PLANE. The kind of strain next in simplicity is that of a flat membrane or sheet. Suppose this to be in the shape of a square ; we may give it a uniform elongation e parallel to one side, and then another uniform elongation f parallel to the other side. It is now converted into a rectangle, whose sides are proportional to 1 + e, 1 +f. By each of these operations two equal and parallel lines, drawn on the mem- brane, will be left equal and parallel ; though, if not parallel to a side of the square, they will be altered in direction. We may prove, conversely, that every strain which leaves straight lines straight, and parallel lines parallel, is a strain of this kind combined with a change of position of the membrane in its plane. Such a strain is called uniform or homogeneous. Since a parallelogram remains a parallelogram, equal parallel lines remain equal. Then it is easy to shew, by the method of equi-multiples, that the ratio of any two parallel lines is unal- tered by the strain. Next, if we draw a circle on the un- strained membrane, this circle Avill be altered by the strain into an ellipse. For in the unstrained figure A'M.MA : CA*=MP S : CB\ and since these ratios of parallel lines are unaltered, it follows that in the strained figure also a'm . ma : ca? = mp z : cb*. Hence the strained figure is an ellipse, whose conjugate diameters are the strained positions of perpendicular diameters of the circle. It follows that there are two directions at right angles to one another, which remain perpendicular after the i6o: DYNAMIC, strain ; namely those which become the axes of the ellipse into which a circle is converted. If these lines remain parallel to their original directions, the strain is produced by two simple elongations along them respectively; in that case it is called a, pure strain. If they are not parallel to their original directions, the strain is compounded Qf a pure strain and a rotation. Two lines drawn anywhere in the strained membrane parallel to the axes of the ellipse into which a circle is converted, or in the unstrained membrane parallel to the unstrained position of those axes, are called principal axes of the strain. The elongations along them are called principal elongations; the ratios in which they are altered are called principal ratios. REPRESENTATION OF PURE STRAIN BY ELLIPSE. When the strain is pure, the new position of any step may be conveniently represented by means of a certain ellipse. Let the principal ratios be p, q, so that every line parallel to oX is altered in the ratio 1 : p, and every line parallel to o Y in the ratio 1 : q. Take two lengths oa, ob, along oX, o Y respectively, such that oa? : otf = q : p, and let m be the positive geometric mean of p, q, so that m* = pq. Then we shall have, so far as length is con- cerned, p . oa = m . 6b, and q . ob = m . oa. Hence, taking account of direction, oa becomes im . ob', and ob becomes im . oa, in consequence of the strain. Now construct an ellipse having oa, ob for pemi-axes ; then if p be any point on it and qq the diameter conjugate DISPLACEMENT-CONIC. 161 to op, the strain will turn op into im . oq'. For since it turns oa into im . ob', it will turn on into im . rq, because on : oa = rq : ob' (p. 129). And since it turns ob into im . oa, it will turn rip into im . or, because np : ob = or : oa. Therefore it will turn op, which is on + np, into im (rq + or), that is, into im . oq. Hence we see that the strained position of any vector is perpendicular to the conjugate diameter of a certain ellipse, having that vector as diameter, and is proportional to the conjugate diameter in length. For the ellipse used in this representation may be of any size, since all that is necessary for it is that its axes should be parallel to the principal axes of the strain, and inversely proportional to the square roots of the principal ratios. KEPRESENTATION OF THE DISPLACEMENT. The displacement of any point is the step from its old position to its new one. Thus if a vector op is turned by the strain into op, the displacement of p ispp'. When the two principal elongations e, f are of the same sign, the displacement may be represented by an ellipse, in the same way as we have represented the new position of any vector. The only difference is that we are now to draw an ellipse whose axes are inversely pro- portional to the square roots of the elongations, so that oa 2 : o5 2 =f : e, and to make ra 2 = ef, giving to ra the same sign as e or/ Then the displacement of a will be im . ob', and the displacement of 6 will be im . oa. Hence it follows (as before) that the displacement of p will be im . oq. In this case therefore the displacement of every point on the ellipse is perpendicular and proportional to that diameter which is parallel to the tangent at the point. But when e and /are of different signs, it is necessary to use a hyperbola to represent the displacement. Let ra 2 = ef, and oa 2 : o& 2 = / : e ; and let m be taken of the same sign as/ Then the displacement of a will be im . ob, and the displacement of b will be im . oa. If then a hyperbola be described with oa and ob as axes, and c. 11 162 DYNAMIC. op, oq be a pair of conjugate semi-diameters, the displace- ment of p will be im . oq and that of q will be im . op. The proof is the same as for the ellipse, depending on the property that np : ob = or : oa, and rq : ob = on : oa. The ellipse or hyperbola which is thus used to represent the displacement is called the displacement-conic of the strain. LINEAR FUNCTION OF A VECTOR. One vector is said to be a function of another, when its components are functions of the components of the other ; so that, for every value (including magnitude and direction) of one of them, there is a value or values of the other. Thus pi + qj + rk is a function of xi -\-yj-\-zk if 2), q, r are functions of x, y, z. We may express this rela- tion between them thus : pi + qj + rk = (f> (xi + yj + zJc). A function of a vector is said to be linear when that function of the sum of two vectors is the sum of the func- tions of the vectors. Thus the function * + <. At present we shall consider only linear functions of vectors which are all in one plane. It is clear that when a plane figure receives a homogeneous strain, the strained position of any vector is a linear function of the vector. For the triangle made of two vectors a, /3 and their sum a + /3 becomes after the strain a triangle made of the vectors y, and the displacement-function ty are connected by the equation, y. when the function < is linear; for since functions of equal lengths measured in the same direction are equal, and functions of multiples of such lengths are multiples of the functions of the lengths, it follows that functions of un- equal lengths are proportional to those lengths. Hence it follows that and therefore we know the function of every vector in the plane when we know the functions of i and j. Let then (f>i = ai + hj, is determined by its matrix. The matrix must be carefully distinguished from its deter- minant, which is the single quantity ab hti, calculated from the four constituents a, h, h' b of the matrix. PROPERTIES OF A PURE FUNCTION. The strain-function and the displacement-function of a pure strain are both called pure functions. We proceed to investigate what must be the relation between the quantities a, h, h', b in order that the function < may be pure. If is the strain-function of a pure strain, Spfa = Sa-fyp, where p, a- are any two vectors. Let op and or be semi- diameters parallel to p, a, of the ellipse which represents the strain; then if oq and os are the conjugate semi- diameters, <> (op) = im . oq and ^> (or) = im . os. Thus the cosine of the angle which op makes with (or) is the sine of the angle it makes with os. Therefore the scalar product of op and (or) is twice the triangle ops, and the scalar product of or and (op) is twice the tri- angle orq. But that these triangles are equal appears at once from intuition of the corresponding figure in the circle of which the ellipse is orthogonal projection ; where the angles POQ, EOS will be right angles. Therefore S . op . < (oq) = S . oq . (op) ; let then p=x.op, a = y . oq, and Sp (oq), = xy S . oq . (op) = Scrffrp. It follows immediately that the same property belongs to the displacement-function. For let p = p + typ, so that i|r is the displacement-function of the strain . Then we have Sp (a- + ^rcr) = Spfytr = Safyp = Sj = h'i-\- bj, = oq. Then a = om, h = mp, h' = on, b = nq. The. magnitude of '&'<;, the scalar product of oi and oq, is the pro- duct of the length of oi by the length of the component of oq along it ; that is, it is oi . on or h', since the length of oi is unity. (For reasons to be subsequently explained, the scalar product of two vectors is taken to be the negative product of either by the component of the other along it; this i& a convention, and does not affect the present argument.) Hence we have Sii=> h. If the function is pure, 8ii ; thus for the function (f> to be pure, it is necessary that h = h'. To shew conversely that when h h' the function is pure, we shall actually find the principal axes and elonga- tions of the strain of which it is the displacement-function. Let e,/be the principal elongations, 6 the angle between o X and the axis of elongation e. A step of unit length making the angle with oX is i cos +j sin 6, and a unit P 166 DYNAMIC. step at right angles to this is i sin 6 j cos0. One of these receives the elongation e and the other the elongation f, each in its own direction ; therefore $ (i cos 6 +j sin 9} = e (i cos 6 +j sin ff), (i sin 6 j cos 6) =f(i sin j cos 0). Multiply the first equation by cos 6, the second by sin 6, and add ; thus we get i = (e cos 2 +/sin 2 &) i + (j = (e f) sin 6 cos 6 . i 4- (e sin 2 +/cos 2 0) . J = Ai + 5J. It is now necessary to find quantities, e,f, 6, which satisfy the equations e cos a 6 +/sin 2 6 = a, esin 2 0+/cos 2 = J, (e -/) sin 6 cos 0, = $(e -f) sin 20 = h. Adding the first two, we have e +f=a + b; subtracting the second from the first, (e /) cos 20 = a b ; combining this with the third, (e-/y=4/i 2 + (a -*>)'. Consequently - 7 , a o tan 6 = 2/= a + b - V{4& 2 + (a - 6) 2 }. Compare with this the solution of an analogous pro- blem on p. 131, making in that, Q TT, and &,= &. SHEAK. 167 SHEAR. When the plane is as much lengthened along one principal axis of the strain as it is shortened along the other, so that (1 + e)(l +/) = 1, or e +f+ ef= 0, the strain is called a shear. In this case it is clear that the area of every figure in the plane remains unaltered. Let oa be changed into oA, and take ob= oA; then ob will be changed into oB, which is equal to oa. Hence the rhomb aba'b' will become the rhomb ABA'B', and ab, which becomes AB, will be unaltered in length. If we combine this pure strain with a rotation, so as to bring ab Atz A' a' B' to coincide with AB, then ab' may be brought to A'B' by a sliding motion along its line. Thus all lines parallel to ab will be slid along themselves through lengths pro- portional to their distances from ab. The amount of sliding per unit distance is called the amount of the shear. Since we have also ab = A'B, the shear might also be produced by the sliding of lines parallel to a'b ; but then 168 DYNAMIC. it would be combined with a different rotation. Thus there are two sets of parallel lines which are unaltered in length, and whose relative motion is a sliding along them- selves. The ratio oA : oa is called the ratio of the shear. If ob = ct. oa, the sliding of a'b relative to b'a is 2a& . cos aba and the distance between a'b and b'a is ab sin aba'. Hence the amount of the shear is 2 cot aba' = 2 cot 2$, if 9 = abo, so that cot = a. Now , a cos 2 d - sin 2 9 Q Q 1 2 cot 20 = . 7r - A = cot tan 6 = a . sin d cos 9 a Thus, if a be the ratio of a shear, its amount s is given by s a a" 1 . We have seen that e and / satisfy the equation e+f+ef=0, in the case of a shear. When e and f are very small fractions, ef is small compared with either of them, and we have approximately e +f= 0. The ratio e : f differs from unity, in fact, by the small fraction e. Thus the displacement-conic is approximately a rectangular hyper- bola. Now the ratio of the shear is 1 + e, and Hence the amount is 1 + 6- (!+/)='-/; this is accurate, whether the shear be large or small. But if the shear is very small, f is approximately equal to - e, and thus the amount is approximately = 2e. COMPOSITION OP STRAINS. When the displacement of every point, due to a certain strain, is the resultant of its displacements due to two or more other strains, the first strain is said to be the result- ant of these latter, which are called its components. If the displacement of the end of p in two strains respec- PRODUCT AND RESULTANT. 169 tively be $>p and typ, the displacement in their resultant s This must be carefully distinguished from the result of making a body undergo the two strains successively. Thus if p be changed into <^ t p by the first strain, and into -v/Tjp by the second, the effect of applying the second strain after the first will be to change p into ^ {<,(/?)} or v/r^p. To compare this with the preceding expression for the resultant, we must observe that 1 = I + and A/TJ = 1 + ty ; so that whereas in the one case the displace- ment is ((f> + i/r) p, in the other it is (< + ty + ^<) p. In one case only the addition, in the other the multiplication of functions is involved. For this reason we shall speak of the strain, whose effect is the same as that of two other strains successively applied, as the product of the two strains. A strain in which a b = 0, and h = h', is called a skew strain, and the displacement-function a skew function. It is the product of a rotation about the origin and a uniform dilatation ; for the displacement of every point p is perpendicular to op and proportional to it. Every plane strain is the resultant of a pure and a skew strain. For let a, h, h', b have the same meaning as before ; these numbers are the sums of o,i(A + A'), $(h + h'),b, andO, $(h-h'), }(h'-K), 0, of which the former belong to a pure, and the latter to a skew strain. But every plane strain is the product of a rotation, a uniform dilatation, and a shear. First rotate the plane till the principal axes of the strain are brought into posi- tion ; then give it uniform dilatation (or compression) till the area of any portion is equal to the strained area ; the remaining change can be produced by a pure shear. When two strains are both very small, their product and resultant are approximately the same strain. 170 DYNAMIC. REPRESENTATION OF STRAINS BY VECTORS. We have seen that if e, f be the principal elongations of a pure strain (a,h,h,b), then e +f=a + b. -Hence if a + b = o, we must have e +f= o. Hence the strain is made by an elongation in one direction, combined with an equal compression in the perpendicular direction. Such a strain is approximately a shear when it is very small ; we shall therefore call it a wry shear. Its characteristic is that its displacement-conic is a rectangular hyperbola. A wry shear accompanied by rotation shall be called a wry strain; that is (a, h, h', b) is a wry strain if a + b = 0. Every strain is the resultant of a uniform dilatation and a wry strain. For (a, h, h', b) = |(a + b, 0, 0, a + b) + (a - b, 2h, 2h', b - a). Every wry strain is the resultant of a skew strain and a wry shear. For |(a- b, 2h, 2h', &-a) =i(0, h- h', h'- h, 0) + I (a-b,h + h',h + h', b - a). The magnitude of a skew strain (0, h, h, 0) is h. Being the product of a rotation by a uniform dilatation, it is not specially related to any direction in the plane, and may therefore be represented by a vector of length h perpen- dicular to the plane. The -wry shear (a, h, h, a) has for its displacement- conic a rectangular hyperbola whose transverse axis makes with oX an angle 6 such that tan 26 = h : a (since in this case a b = 2a; the general value being tan = 2h : a b). Moreover if e, e are its principal elongations, we have in general (e f)* = (a b) 2 + 4A 2 , and therefore in this case e 2 = a 2 + A 2 . Hence if a wry shear be represented by a vector in its plane, of length equal to its positive prin- cipal elongation, making with oX an angle (26) equal to twice the angle (6) which that elongation makes with it ; the components (a, h) of this vector along oX and o Y will represent in the same way the wry shears (a, 0, 0, a) and SYMBOL OF A PLANE STEAIN. 171 (0, h, h, 0), having oX and oY respectively for axes and asymptotes, of which the given wry shear is the resultant. Let such a vector be called the base of the wry shear; then our proposition is that the base of the resultant of two wry shears is the resultant of their bases. This is obvious, because the base of (a, h, h, a) is ai + hj. This mode of representation is to a certain extent arbitrary, because it depends upon the position of oX. It will, however, be found useful in many ways. Combining this with our previous representation of a skew strain, we see that a wry strain in general may be represented by a vector not necessarily in its plane, the normal component of which represents the skew part of the strain, while the component in the plane represents the wry shear. When a figure receives a uniform dilatation, without rotation, we may regard it as merely multiplied by a numerical ratio or scalar quantity. Thus the whole opera- tion of any plane strain may be regarded as the sum of a scalar and a vector part. If we write, for example, 1 = (1, 0, 0, 1) ... (leaves the figure unaltered) J = (1, 0, 0, 1) ... (turns it over about oY) J= (0, 1, 1, 0) ... (interchanges oX and oY) K= (0, 1, 1, 0) ... (turns counter clock-wise through a right angle) then we shall have and it will be easy, by combining these operations, to verify that P = 1, J* = 1, K* = - 1, JK = I = - KJ, 172 DYNAMIC. GENERAL STRAIN OF SOLID. PROPERTIES OF THE ELLIPSOID. When a solid is so strained that the lengths of all parallel lines in it are altered in the same ratio, it is said to undergo uniform or homogeneous strain. It follows easily, as before, that all parallel planes remain parallel planes, and undergo the same homogeneous strain, besides being altered in their aspect. A sphere is changed into a surface which is called an ellipsoid, having the property that every plane section of it is an ellipse. We may easily obtain its principal pro- perties from those of the sphere, if we remember only that the ratios of parallel lines are unaltered by the strain. Thus we know that if a plane be drawn through the centre of a sphere, the tangent planes at all points where it cuts the sphere are perpendicular to it, and therefore parallel to the normal to it through the centre ; this normal meets the sphere in two points where the tangent planes are parallel to the first plane. A plane A drawn through the centre of the ellipsoid (a point such that all chords through it are bisected at it) is called a diametral plane. The tangent planes at all points where it cuts the surface are parallel to a certain line through the centre, called the diameter conjugate to the given plane ; this line cuts the surface in two points where the tangent planes are parallel to the given plane A. Any two conjugate diameters of the ellipse in which the ellipsoid is cut by the plane A, together with the diameter conjugate to that plane, form a system of three conjugate diameters; each of them is conjugate to the plane containing the other two. They correspond to three diameters of a sphere at right angles to one another. The CONJUGATE DIAMETERS OF ELLIPSOID. planes containing them two and two are called conjugate diametral planes. Let oa, ob, oc be three conjugate semi-diameters of the ellipsoid, p any point on the suri'ace ; draw pn parallel to oc to meet the plane oab in n, and then draw nm parallel to ob to meet oa in m. These points will be the strained positions of 0, A, B, C, P, M, N, when OA, OB, OC, are at right angles and P is a point on the sphere. Now OP 2 = OM * + MF* = OM * + MN* + NP* or, remembering that OA = OB = 00 = OP, we have OM* MN* NP* OA* + OB* + OC 2 _ ~ But the ratios of parallel lines being unaltered by the strain, OM : OA = om : oa, and MN : OB = mn : ob ; hence in the ellipsoid also we have om* mn* np* l_ i_ f oa? ob* oc? if x, y, z are written for om, mn, np, and a, b, c for oa, ob, oc. Let a plane be drawn through perpendicular to OP. It will cut the sphere in a great circle, whose area shall 174 DYNAMIC. be called A ; the area is of course the same for all sec- tions of the sphere by planes through 0. The angle between this plane and OBG will be the same as the angle between OP and OA, the straight lines perpendicu- lar to those planes respectively. Call this angle d. Then if we project the area A on the plane OBC, the area of the projection will be A cos 6. Now A is also ^the area of the circle in which the plane OBC cuts the sphere. Moreover OM OP cos 6 = OA cos 6. Thus we see that the projection of OP on O A bears the same ratio to A that the projection on the plane OBC of the section conjugate (at right angles) to OP bears to the section by OBC. The proposition thus proved for the sphere may be extended to the ellipsoid if we remember that the ratio of areas on the same or parallel planes is unaltered by the strain. The projections must now be parallel projections; that is, p is projected on oa by the line pin parallel to the plane ol>c ; and the conjugate area must be projected on obc by lines parallel to oa. The projected area will then bear the same ratio to the section by obc that om does to oa. We shall use this proposition in representing the strained position or the displacement of any vector, just as we used the corresponding property of the ellipse. At any point of a sphere, all the straighj; lines which touch the surface lie in one plane, called the tangent plane at that point. The same thing is therefore true for the ellipsoid. Now let a be a point on an ellipsoid, such that either oa is the greatest distance from the centre, and the dis- tance of all points immediately surrounding it is less than oa, or else some of these are equal to oa but none greater. There must clearly be such a point on the surface. If now we cut the surface by a series of planes through oa, the tangent lines to all these sections at a will be per- pendicular to oa ; for each of these sections is either an ellipse or a circle, and in the case of an ellipse oa must be its semi-major axis. Consequently the tangent plane at a is perpendicular to oa. Hence if ob and oc are the axes AXES OF ELLIPSOID. 175 of the section made by the plane through o perpendicular to oa, the three lines oa, ob, oc form a system of three conjugate diameters at right angles to one another. These are called axes of the ellipsoid. The planes con- taining them two and two are called principal planes of the surface, which is evidently symmetrical in regard to each of these planes. If oa is equal to either ob or oc say to ob then the section of the surface by the plane oab is a circle (being an ellipse with equal axes) and any two diameters at right angles in that plane are conjugate diameters. The surface may then be made by rotating an ellipse about its shorter axis oc. It is called an oblate spheroid, or oblatum, This is approximately the figure of the Earth. If ob and oc are equal, both being shorter than oa, the section obc is a circle, and any two rectangular diameters in that plane are conjugate. This surface may be made by rotating an ellipse about its longer axis oa ; it is called a prolate spheroid, or prolatum. It has two foci (those of the rotating ellipse) the sum of whose distances from any point of the surface is equal to the major axis. If, on the contrary, oa, ob, oc are all unequal, and in descending order of magnitude, we may derive the ellip- soid from a sphere having the same centre o and radius oa, by reducing all lines parallel to ob in the ratio OB : ob, and all lines parallel to oc in the ratio OC : oc. It will then be clear that every set of semiconjugate diameters op, oq, or on the same side of the plane oab lies entirely outside the solid angle formed by the rectangular lines oP, oQ, oR of which they ai*e the strained positions. Hence the axes are the only set of conjugate diameters at right angles to one another. It follows that in any homogeneous strain of a solid, there are three directions at right angles to one another, which remain perpendicular after the strain; namely those which become the directions of the axes of the ellipsoid into which the strain converts any sphere. Lines drawn through any point in these directions are 176 DYNAMIC. called principal axes of the strain, and the elongations along them are called principal elongations. If the axes remain parallel to their original directions, the strain is called pure ; if they are turned round, it is accompanied by rotation. REPKESENTATION OF PUKE STRAIN BY ELLIPSOID. We may now represent (in the case of a pure strain) the strained position of any vector by means of an ellip- soid, in a way entirely analogous to our previous repre- sentation of a plane strain by means of an ellipse. Let the principal elongations be e, f, g, and let p = 1 + e, , c are vectors repre- senting the conjugate areas, it follows that the displace- ment of any point) p on the displacement-quadric or its conjugate surface is a vector representing the area of section conjugate to op. For we have shewn that the components of that area, namely its projections on the principal planes, bear the same ratio to the principal areas Trbc, Trca, Trab, that the components of op, namely its pro- jections on the axes, bear to those axes. Now if om be om the projection of op on oa, the displacement of m is - om x the displacement of o, that is, it is x nbc. Con- oa sequently the displacement of m is a vector representing the projection on obc of the area conjugate to op. Now the displacement of^j is the resultant of the displacements of its projections on the axes ; and therefore it represents the area which is the resultant of the three projections here ^considered, namely, the area of section conjugate to op. The case of a point k lying on the asymptotic cone of the displacement-quadric requires some explanation. In that case the length of the line drawn in the direction ok to meet the surface is in- finite, and the displace- ment of its end is infinite also. The conjugate sec- . tion is made by a plane through ok touching the asymptotic cone, which cuts the conjugate surface in two parallel straight lines, In the case of sur- LINEAR FUNCTION. 185 faces of revolution it is clear that the distance between these lines is bb '; they lie on either side of ok in a plane through it perpendicular to the paper. Thus the dis- placement of p (the infinitely distant point on ok) is TT . ob . op, perpendicular to ok in the plane of the paper. Hence the displacement of k is tr . ob . ok in the same direction. And generally the displacement is TT . ok multi- plied by half the breadth of the conjugate section. In any other case if ot be the perpendicular on the tangent plane at p, the displacement of p is parallel to ot and equal to irabc : ot. For the perpendicular on a tangent plane, multiplied by the area of the parallel dia- metral section, is constant, and therefore equal to irabc. This follows at once for surfaces of revolution from the cor- responding property of the hyperbola ; and it is extended to any hyperboloids by the consideration that all volumes are altered in the same ratio by a homogeneous strain. We shall write H for rrabc or efg : 7r 2 , so that displace- ment of p =H : ot. LINEAR FUNCTION OF A VECTOR. Just as in the case of a plane strain, the strained position of a vector or the displacement of its end is said to be a linear function of the original vector when the strain is homogeneous. If the displacement of the end of p be denoted by <}>(p), the strained position of it is p + (f>(p) = (1 + <) p. When the strain is pure, is said to be a pure function. Let i, j, k be three unit-vectors at right angles to one another, and let i = ai -I- hj + g'k, j = h'i + bj k=gi+f'j 186 DYNAMIC. Then if p = xi + yj + zk, we shall have p = x$i -f yj + zk, so that the function of every vector can be ex- pressed in terms of these, and the strain is entirely spe- cified by means of the nine quantities a, b, c, /, g, h, f, g', h'. The equations just written down are sometimes conveniently abbreviated as follows : a h g' h' b f 9 f c or = a h g h' b f 9 f o and the form (a h g' ) is also called a matrix. Thus h' b f 9 f * every strain has a certain matrix belonging to it, which serves to define the strain by means of its displacement function. When the strain is pure, the scalar product of op and the displacement of p is H, if p is a point on the dis- placement-quadric. If pq is the displacement of p, the. scalar product is op . pq cos opq ; but we know that and ot, op cos opq = op cos put = ot, which proves the theorem. Hence if p is the step from the centre to a point on a quadric surface, Spp = H, whence H is TT times the product of the semi-axes of the surface. The scalar product of two vectors is the negative sum of the products of their components along the axes. For SCALAR PRODUCT. 187 op cospoq is the projection of op on oq, which is the sum of the projections of on, nm, mp on oq. Let x, y, z be the components of op, x t , y l , z t those of oq, and r, i\ the lengths of op and oq. Then if oq makes angles a, ft, 7 with oX, o Y, oZ, we must have cos a = cos = cos j z l : r r And op cos po<2 = on cos ^o X + mp cos #0 Y + ?i> cos = x cos a -f y cos /3 4- s cos 7 = xx^ + yy^ + zz 1 : r r Therefore Sop .oq op.oq cospoq = (xx^ + yy l + zz^. Let p, p. Since cr will be the volume of a cylinder standing on that section and having p for its axis. We have to shew that this volume is equal to that of a cylinder standing on the section p and having i, (f>j, j = h'i + oj +fk ; and therefore = h. Thus a pure strain depends only on the six quantities *> b, c, f, g, h, whereas a strain not known to be pure is specified by nine quantities. We may now prove that Sp(j>p = 8(xi + yj + zlc] (x< = - (ax* + I/ + C3 2 so that ax 9 + ly* + cz? + 2fyz + 2gzx + 2hxy = H. This is the relation which holds good between x, y, z for all points on the displacement-quadric. VARYING STRAIN. In a homogeneous strain, if we suppose one point of the body to be at rest, and draw any straight line through it, the displacements of points on this line will be all in the same direction and proportional to the distance from STKAIN IN TERMS OF DISPLACEMENT. 189 the fixed point. Hence the relative displacement of the two points at a unit distance along this line will be equal to the rate of change of the displacement per unit distance as we go along the line. Let z ; so that we may now write d a ar = is the displacement-function of this strain, d a cr = y. ;' where now 9 a , then at that instant the body is said to have the homogeneous strain- flux (f>. If we combine with this velocity-system a translation equal and opposite to the velocity of any point p, the resultant will be a new homogeneous strain-flux with the point p for centre. For if we keep all velocities constant for a second, we shall produce a homogeneous strain together with a translation restoring p to its place ; that is, a homogeneous strain in which p is not moved. It is clear that the resultant of two homogeneous strain- fluxes is again a homogeneous strain-flux ; but in this term we must include as special cases the motions con- sistent with rigidity. .A twist may be regarded as a homo- geneous strain-flux whose centre o is infinitely distant ; in the still more special case of a spin, the centre is indeter- minate, being any point whatever on the axis. The latter case is distinguished by the function $ being a skew function. For let the spin to=pi + qj+rk, then the velocity of any point whose position- vector is p will be Vwp. Consequently we have p = Vwp, and therefore fa = Vwi = -f rj qk, tf)j = Vwj = ri +pk, k = Vcok = + qi pj ; so that the matrix of < is ( 0, +r, -q) -r, 0, +p 3> -P> We may now separate any given homogeneous strain- c. 13 194 DYNAMIC. flux into the pure part of it and the spin. For it is evident that (a, h, #') = h',b,f a, b, + ( o, I (A'- A), !(/-/) Here the first of the matrices on the right hand belongs to a pure function, and the second to a spin, whose com- ponents are \ (//')> \ (ff ~9'}> z (h ~ ^')- The resolution cannot be effected in any other way ; for to change the spin into any other (not about a parallel axis) we must combine a spin with it. The resultant of the pure strain- flux and of this spin reversed will be no longer a pure strain-flux. CIRCULATION. Consider a plane curve joining two points p and q. Let a line be drawn through every point of the curve, per- pendicular to its plane, representing the component of velocity along the tangent to the curve at that point. All these lines will trace out a strip or riband standing on the curve. The area of this strip is called the circula- tion along the curve from p to q. When the resolved part of the velocity is in the direction from q to p, it is to be drawn below the plane, and that part of the area is to be reckoned negative. Hence the circulations from p to q and from q to p are equal in magnitude but of opposite sign. The circulation may also be described as follows. Divide the length of the curve into small pieces, of which SA, is one. Let a- be the velocity of some point included LINE-INTEGRAL OF A VECTOR. 195 in the piece SX, then Sv&\ will be the resolved part of this velocity along the curve, multipled by the length of 8X. The sum 2S, which is of course zero. Since the circulation round a closed curve is thus un- altered by the same velocity being given to all its points, wa may if we like reduce any one point to rest, without altering the circulation round any closed curve. The circulation round any two parallelograms of the same area is the same. We may change abdc into abfe by adding ace and subtracting bdf; and the circulation round these two tri- angles is the same. By repeating this process we may make one parallelo- gram into a translation of any other of ^ jr equal area. By equal area is of course implied that they are in the same or parallel planes. The circulation round any parallelogram is double of that round a triangle of half its area. Let o, the middle point of ad, be brought to rest. Then the circulation along ad is zero, and the velocities at correspond- ing points of ab and dc being equal and ^ opposite, the circulation along ab is equal to that along dc ; similarly that along bd is equal to that along ca. Thus (ab) + (bd) + (da) = (ad) + (dc) + (ca), or the circulations round the triangles abd, adc are equal, and therefore each half of the circulation round abdc. It follows that the circulation round any two triangles of the same area is the same. Hence the circulations round any two areas in the same or parallel planes are proportional to those areas. CIRCULATION IN TERMS OF SPIN. 197 For we may replace each of them by a polygon with short rectilineal sides, and these polygons may then be divided into small equal triangles. The areas will be nearly as the numbers of these triangles with an approximation which can be made as close as we like by making the tri- angles small enough. But the circulation round each polygon is the sum of the circulations round its compo- nent triangles ; therefore the two circulations are also as the numbers of the triangles approximately, and therefore as the two areas exactly. If a, /3 are the sides of a parallelogram, the circulation round it is Sftfa Sz/3. Let a = ab, (3 ac. Then the sum of the circulations along ab and dc is the difference of those along ab e i j"- and cd; which is the length of a multi- -/ / plied by the resolved part of j Sj(j>i = h Ji, which is twice the compo- nent of spin round oZ. Now any plane whatever may be taken for the plane of oXY; whence the proposition. STRAIN-FLUX NOT HOMOGENEOUS. In the case of a homogeneous strain-flux, if we take any point p of the body and draw a straight line pq through it, the velocities of points on this line, relative to p, will all be parallel and proportional to the distance from p along the line. Consequently the rate of change of the velocity, as we go along the line pq, is constant. When the strain-flux is not homogeneous, this rate of change of the velocity will no longer in general be con- stant. But we may imagine a homogeneous strain-flux 198 DYNAMIC. which is such that the rate of change of velocity due to it. in any direction, is the same as the rate of change at p when we are moving in that direction in the actual con- dition of the body. This homogeneous strain-flux will then be called the strain-flux at p. It will in general vary from one point of the body to another. In order that there may be a strain-flux at p at all, it is necessary that the velocity should change gradually as we pass through p in any direction. That is to say, there must be a rate of change up to p, and a rate of change on from p, and these must be equal. When this is the case, the entire strain-flux of the body may be said to be ele- mentally homogeneous, or homogeneous in its smallest parts. Any small portion of the body moves with an approxi- mately homogeneous strain-flux, and the approximation may be made as close as we like by taking the portion small enough. But if one portion of the body is sliding over another portion with finite velocity, this is not the case. In crossing the common surface of the two portions, we should find a sudden jump in the velocity. Such dis- continuities have to be separately considered. Let now a be any vector drawn through the point p ; and let d a v w u d t v d t w Consequently the spin G> is Y {(9jW d f v) i + (d z u d x w~)j + @ x v d t u}k}. It follows from this formula that if two velocity-systems are compounded together, the spin at any point in the resultant motion is the resultant of the two spins in the component motions. LINES AND TUBES OF FLOW. 199 LINES OF FLOW AND VORTEX-LINES. At every instant a moving body (to fix the ideas, con- sider a mass of water) has a certain velocity-system, i.e. every point in the body has a certain velocity tr. A curve such that its tangent at every point is in the direction of the velocity of that point is called a line of flow. It is clear that a line of flow can be drawn through any point of the body, so that at every instant there is a system of lines of flow. If the body has a motion of translation, the lines of flow are straight lines in the direction of the translation. If it rotates about an axis, the lines of flow are circles round the axis. If fluid diverges in all direc- tions from a point, the lines of flow are straight lines through that point. It is important to distinguish a line of flow from the actual path of a particle of the body. A line of flow relates to the state of motion at a given instant, and in general the system of lines of flow changes as the motion goes on. Thus while the path of a particle touches at every instant the instantaneous line of flow which passes through the particle, it does not in general coincide with any line of flow. The particular case in which the system of lines of flow does not alter, and in which, therefore, each of them is actually the path of a stream of particles, is called steady motion. In that case, the lines of flow are called stream-lines. Thus, if a rigid body move about a fixed point, we know that its velocity-system at every instant is that of a spin about some axis through the fixed point, and conse- quently the lines of flow are circles about that axis. But in general the axis changes as the motion goes on, and the path of a particle of the body is not any of these circles. If we take a small closed curve, and draw lines of flow through all points on it, the tubular surface traced out by these lines is called a tube of flow. In the case of steady motion all tubes of flow are permanent, and the portion of the body which is inside such a tube does not come out of it. 200 DYNAMIC. In general, a body has also at every instant a certain spin-system; i.e. at every point of the body there is a certain spin &>. In fact, if the strain-flux is elementally homogeneous, there is at every point a homogeneous strain- flux which is the resultant of a pure strain-flux and a spin &>. A curve such that its tangent at every point is in the direction of the spin at that point is called a vortex-line. If we draw vortex-lines through all the points of a small closed curve, we shall form a tubular surface which may be called a tube of spin; the part of the body inside the tube is called a vortex-filament. In the cases of fluid mo- tion which occur most often in practice, there is a finite number of vortex-filaments in different parts of the fluid, but the remaining parts have no spin. CIRCULATION IN NON-HOMOGENEOUS STRAIN-FLUX. If we consider any small area 8x, which may be taken to be approximately plane, the strain-flux in its neighbour- hood is approximately homogeneous ; and if &> be the spin at a point inside of the area, the circulation round the area will be approximately equal to its magnitude multiplied by twice the component of spin perpendicular to it; that is, it will be approximately 2So)&x, where Sa is regarded as a vector representing the area, and therefore perpen- dicular to it. This approximation is closer, the smaller the area is taken. Now let abed be any closed contour, whether plane or not, and let us suppose it to be covered by a cap, as aec, so that the contour is the boundary of a certain area on the surface of this cap. If this area be divided into a great number of very small pieces, asjf, each of these may be taken to be approxi- mately plane. And the circulation round abed will be the sum of the circulations round all the small pieces. Thus it will be approximately equal to SURFACE INTEGRAL. 201 where Sz is one of the small pieces, and &> the spin at some point within it. The approximation may be made as close as we like by taking the pieces small enough, and therefore the circulation is exactly equal to the integral 2fSa)Sy. If SA, be a small piece of the contour abed, we know that the circulation is also equal to $S(?d\ ; and consequently we have 2 fSmdot. = In general, if &> is any vector having a definite value at every point of space, the integral fSwdct, taken over any area, plane or curved, is called the surface-integral of the vector over that area. We may therefore state our proposition thus : the line-integral of the velocity round any contour is equal to twice the surface-integral of the spin over any cap covering the contour. Let us now draw another cap, age, covering the con- tour. Then the surface-integral of the spin over age must be equal to that over aec, because each is half the line-integral of velocity round abed. But in one case the vectors representing small pieces of area will all be drawn inwards, and in the other outwards. If then we suppose them all to be drawn outwards, the surface-integral over the entire closed surface aecg will be zero. It is in fact obvious that if we divide the area of any closed sur- face into small pieces, and suppose each of these to be gone round in a counter-clockwise direction, as viewed from outside, the sum of all their circulations will be zero, since each boundary line is traversed twice, in opposite directions. We learn, therefore, that the surface-integral of the spin over any closed surface is zero. The closed surface may be that of a body having no holes through it, as in the figure, or it may be that of a body with any number of holes through it ; for example, the surface of an anchor- ring, or of a solid figure-of-eight. Let us now apply this proposition to a portion of a tube of spin, cut off at a and 6 by surfaces of any form. 202 DYNAMIC. This closed surface consists of the two ends at a and b, and of the tubular portion between them. At every point of the tubular portion the axis of spin is tangent to the vortex line through that point, which lies entirely in the surface ; conse- quently it has no component normal to the surface. Therefore the tubular portion of the surface contributes nothing to the surface-integral. It follows that the sum of the surface- integrals over the two ends is zero. Now the surface- integral over either end is half the circulation round its boundary ; but since the lines representing pieces of area are to be drawn outwards in both cases, these boundaries must be gone round in opposite directions. Since then, when they are traversed in opposite directions the circu- lations are equal and opposite in sign, it follows that when they are traversed in the same direction the circulations are the same. Or, the circulation is the same round any two sections of a tube of spin. When the tube is small, the spin at any part of it is inversely proportional to the area of normal section. For then the surface-integral over the section is approximately equal to the spin at any point of it multiplied by the area of the section ; and we have seen that this surface-integral is constant. Hence a vortex-filament rotates faster in pro- portion as it gets thinner. This shews us also that a vortex-filament cannot come to an end within the fluid, but must either return into itself, each vortex-line forming a closed curve, as in the case of a smoke-ring, or else end at the surface of the fluid, where the velocity no longer changes continuously ; and consequently our previous reasoning does not apply. Such a vortex-filament may be formed by drawing the bowl of a teaspoon, half immersed, across the surface of a cup of tea; the filament goes round the edge of the sub- merged half of the bowl, and the two ends of it may be seen rotating as eddies on the surface. VELOCITY-POTENTIAL. 20$ IRROTATIONAL MOTION. If it is possible to cover a contour by a cap such that there is no spin at any point of it, the circulation round the contour will be zero, since it is equal n to twice the surface-integral of the spin, taken over the cap. Let p and q be two points on such a contour paqb, then the circulation from p to q is the same along paq as along pbq. For (paq) + (qbp) = Q, or (paq) = (plq}. Therefore Of two paths going from p to q, if it is possible to move one into coincidence with the other without crossing any vortex-line, the circulation along them is the same. Where there is no spin, the motion is called ir ro- tational. If there is no spin anywhere, so that the motion is irrotational throughout all space, the circulation from one point to another is independent of the path along which it is reckoned. Let a point o be taken arbitrarily, then for every point p in the body there is a certain definite quan- tity, namely, the circulation along any path from o to p. This is called the velocity-potential at p. If p be moved about so as to keep its velocity-potential constant, it will trace out a surface which is called an equipotential surface. It is clear that we may draw an equipotential surface through every point of space, and in this way we shall have a system of equipotential surfaces. There is no cir- culation along any line drawn on an equipotential surface ; because the circulation from one point to another is equal to the difference of their velocity-potentials. (Circulation from p to q = circ. from o to q circ. from o to p.) Suppose, for example, that a body has a motion of translation. Then a plane perpendicular to the direction of motion will be an equipotential surface ; for there is no component of velocity along any line in such a plane, and therefore the circulation along that line is zero. If we 204 DYNAMIC. choose any point in this plane for the point o, the velocity- potential for all points in the plane will be zero ; and for all other points will be proportional to the distance from this plane, being positive on the side towards which the body is moving, and negative on the other side. EQUIPOTENTIAL SURFACES, In general, the equipotential surfaces are perpendicular to the lines of flow. We have already seen that if we suppose the velocity of every point of the body to be marked down at that point, so as to constitute a per- manent diagram of the state of motion of the body at a given instant, then the rate of change of the circulation from o to p, when p moves in the diagram with unit velocity, is the component along the tangent to the path of p of the instantaneous velocity at the point p. Hence if we now use P to denote the velocity-potential at p, viz. the circulation from o to p, we shall have d,P = v cos 0, where v is the magnitude of the instantaneous velocity at p, and 6 the angle it makes with the direction of s. Now those directions which lie in the equipotential surface through p are such that there is no change of potential when p moves along them, or d t P = 0. Hence either v = or cos 6 = ; that is, if there is any velocity, it is perpen- dicular to the equipotential surface. If the motion of p is along a line of flow, cos 0= 1$ and d t P = v ; that is to say, the velocity at any point is the rate of change of potential per unit of length along a line of flow. Hence if we take two equipotential surfaces very near to one another, the velocity at various points on one of these surfaces will be inversely proportional to the distance between them, with an approximation which is closer the nearer the surfaces are taken to one another. For the difference of velocity-potential between a point on one and a point on the other is constant ; and the rate of change of P per unit of length is inversely proportional to the distance required to produce a given change in P. MANY-VALUED POTENTIAL. 205 Hence if we draw surfaces corresponding to the values 0, 1, 2,... of the velocity-potential, this system of surfaces will constitute a sort of diagram of the state of motion of the body. The velocity is everywhere at right angles to the equipotential surfaces, and where these are close together the velocity is large, where they are far apart it is small. MOTION PARTLY IRROTATIONAL. Suppose that in a mass of fluid there is a single vortex- ring of any form (i.e. a vortex-filament returning into itself), but that there is no rotation in any other part of the fluid. Consider a closed curve which is once linked with the ring, such as abc. The circulation round such a curve is equal to the circulation round a section of the vortex-filament, which we know to be the same for all sections ; for the curve can be moved until it coincides with the section without crossing any vortex-line. Let the circulation round abc be called G. We will now consider the circulation from a point o to a point p. Let the circulation along a path which goes from o to p entirely outside the vortex-ring be called (op). A path like oxp, which goes through the ring, can be altered, without crossing any vortex-line, into the form orsr'p, in which it is made up of a path orr'p outside the ring, and a path rsr linked with the ring. Hence the circulation along oxp is made up of the circulations along these two paths, or it is (op] + C. A path such as oyp, which is twice linked with the ring, may be altered into a path going outside the ring together with two such closed paths as rsr', and conse- quently the circulation along 206 DYNAMIC. it is (op) + 2 C. And generally, the circulation along a path which is m times linked with the ring, the same way round as rsr, will be (op) +mC. If it is linked with the ring by going the other way round it, the circulation will be (op) m C. Thus the circulation from o to p has not, as in the case of irrotational motion, a single definite value inde- pendent of the path pursued, but an infinite number of values included in the formula (op) + mC, where m is an integer number positive or negative. We may still speak of the velocity -potential at p, but it is now a many-valued function of the position p. We may compare it with the angle which has a given tangent ; if be one value of the angle, there is an infinite number of other values, included in the formula + mir, where m is an integer positive or negative. If there are any number of vortex-rings, and the circu- lations round paths linked once with them are respectively OjG^..., then the circulation along a path from o to p linked ra, times with the first, m. 2 times with the second, etc., is (op) + m^C^ + w 2 (7 2 +... In such cases we may still find equipotential surfaces. The equipotential surfaces through a point p contains all those points q which can be got at from p by a path along which the circulation is zero. Then the system of values of the velocity-potential at p is the same as the system of values of the velocity-potential at q. Every equipotential surface meets every section of each vortex-filament, and breaks off there. Tims if there is only one vortex-filament, the equi- potential surfaces partially consist of caps, covering the contour of the ring, as indicated in this figure. They must of course break off at the sur- face of the filament, because there is no velocity-potential inside the re- gion where the motion is rotational. We say they par- DIAPHRAGMS. 207 tially consist of such caps, because some of them may be in separate portions. When there are two or more vortex- rings, they may be joined by equi- potential surfaces, as in this figure ; or an equipotential surface may consist of caps covering different rings, together with detached closed portions. The proof of this pro- position is very simple. Consider any section of a vortex- filament, and a point p on the boundary of the section. Choosing a single value P of the velocity-potential at p, let this vary continuously as p moves round the boundary of the section. Then it will gradually increase from P to P+ C, where C is the circulation round the filament. Consequently every possible value of the ve- locity-potential will be represented on the boundary, and therefore every equipotential surface will meet it. We may restore a one-valued velocity-potential by drawing caps to cover all the vortex-rings, and then defining the potential at p to be the circulation from o to p along a path which does not cross any of these caps. The caps are then called diaphragms. On the two sides of a diaphragm covering a vortex-ring the circulation round whose section is C, the velocity-potential will differ by C. Thus in crossing the diaphragm we should find a sudden jump in the velocity-potential. When a vortex- ring has two ends in the surface of the fluid, we must join these ends by any line in the surface, and then draw a cap covering the contour thus formed. EXPANSION. In general, the volume occupied by a finite portion of the moving body will increase or diminish in consequence of its motion. We proceed now to find, in the case of a homogeneous strain-flux, the rate of increase of a unit of volume of the body. 208 DYNAMIC. For the unit of volume we take the cube, three of whose sides are i, j, k. Let ol, om, on be unit lengths measured on the axes oX, oY, oZ. Then we shall have velocity of I = (j)i = ai + hj + g'k ) m = j = h'i+ bj+ fk, n=k=gi+fj+ck. And in consequence of these velocities the cube olmn will change itself into a parallel- epiped whose sides will be the new positions of ol, om, on. Now considering first the motion of I, we see that the volume of the cube will not be altered by any component of velocity- parallel to the plane omn ; because parallel- epipeds standing on the same base and between parallel planes are of equal volume. Hence the only part of the motion of I which can produce change of volume is the component of its velocity parallel to ol, which is a. And it is clear that the rate of change of volume due to this motion is precisely the velocity of I along ol, since the area omn is unity. Similarly the rate of change of volume due to the motion of m is b, and that due to the motion of n is c. As these three changes take place simultane- ously, the whole rate of change of the volume is a + b + c. Volume is poured into the cube, as it were, through three faces of it, and so the whole change of volume is the sum of the changes due to the three faces. The quantity a + b + c is called the expansion. If we consider any other volume, containing V units, the rate of increase of that volume will be V(a + b + c). When the strain-flux is not homogeneous, we must divide the volume of the fluid into very small parts, one of which shall be called 8V. Throughout this part the strain-flux is approximately homogeneous, and the ex- pansion is d x ii + d v v + d t w, if the velocity a- = ui + vj + ivk. Hence the rate of increase of 8 V is approximately (d x u + d v v + 9 z w) 8 V, and consequently the rate of increase of the whole volume considered is 2 (d x u + d u v + d t w) 8 V, EXPANSION. 200 with an approximation which is closer the smaller the parts B V are taken. Hence this rate of increase is exactly equal to the integral / (d x u + d v v + d,w) d V, since this also is represented by the same sum with an approximation which is closer the smaller the parts B V are taken. We shall write E for the quantity d x u + d v v + d,w, so that the rate of increase of a volume V is $EdV, Of course, if the expansion E is a negative quantity at any point, the volume of the moving body is diminishing at that point. From the value just found for the expansion it follows that if two velocity-systems are compounded together, the expansion at any point in the resultant system is the sum of the expansions at that point in the component systems. The rate of increase of a finite volume of the fluid may also be calculated in another way. Consider a portion By. of the surface of that finite volume, so small that it may be regarded as approximately flat. Then if the fluid is flowing out of the volume at that part, there will be a rate of increase of volume equal to the magnitude of By. multi- plied by the component of the velocity er perpendicular to it ; that is, equal to S&Bz. And this will be negative if the fluid is flowing into the volume. The whole rate of change of the volume will be the sum of the rates of change due to all the small parts of the surface, and is therefore equal to the integral JScrdtx. Hence we have or, the surface-integral of the velocity is equal to the volume- integral of the expansion. The sealer quantity E is derived from the vector , if the path go round in the direction rqpr. If therefore we consider a piece of the contour so short as to be approximately straight, the motion in its immediate neighbourhood will be like that round the axis of a^ whirl for which X = 2z/. As in that case, we may draw a small tubular surface enclosing the contour, and substitute for the actual motion inside of it that of a small vortex-filament; so that any small length of this filament rotates like a cylinder about its axis. In this way we may make the velocity vary continuously, yet so that the motion is everywhere conceivable. If we suppose the contour to be covered by a cap, and that the area of this cap is divided into a number of small areas, then the solid angle subtended by the contour at any point is the sum of the solid angles subtended at that point by all the small areas. Consequently the velocity- system just described, which may be called a vortex, is the resultant of a number of smaller vortices, whose vortex- lines are small closed curves which may be regarded as approximately plane. We shall now, therefore, examine more closely the case of such a small plane closed curve. Take a point a within the area, and draw ax perpen- dicular to its plane. Let the angle xap = 6, and the mag- nitude of the area = vl. If we draw a sphere with centre p and radius pa, the area marked off on it by a cone with vertex p standing on A will be A cos 6 nearly. For if the area is small, the portion of the sphere cut by the cone may be regarded as approximately plane, and the generating lines of the cone are approximately parallel, so that the spherical area is very nearly an orthogonal projection of A. Hence the solid angle subtended at p is 218 DYNAMIC. nearly equal to A cos 6 : r 2 , if r = ap. Consequently the potential at p is vA cos : r 2 approximately. Now we can produce the same potential in another way. Let us put at b a source of strength /i, and at a a sink of the same strength ; then the potential at p due to this combination is fj. IJL ap bp IccosO i = p ff- = u, ^ , nearly. op ap ap.bp r Hence if we make vA=p. be, and then let the area A and the length bo diminish continually, increasing v and /t so as to keep vA, = /JL . be, = Jc, a finite quantity, both the vortex and the combination of a positive and negative squirt will continually approximate to the motion in which the ve- locity-potential is h cos 6 : r*. Now the source-and-sink combination gives no expansion except at a and b ; con- sequently the limiting motion gives no expansion except at a. But we have seen that every vortex may be made up of component- vortices, whose vortex-rings are as small as we like. Hence these two conclusions : 1. There is no expansion anywhere due to a vortex. 2. A vortex is equivalent to a system of squirts con- structed in this way. Let two caps be drawn covering the vortex-ring, so as to be everywhere at a very small distance from each other. Let one of them be continuously covered with sources and the other with sinks, so that the source and sink on any normal are equal in strength, and so also that if pSA be the total strength of the sources on the small piece of area BA, and t the thickness of the shell at that part, the product jj,t is constant all over the shell and equal to v the constant of the vortex. Then, keeping all these conditions satisfied, this system of squirts will the more nearly approximate to the vortex the more nearly we make the two caps approach one another. For if we di- vide the cap into small areas, we have already seen that this is true of all the vortices whose vortex-rings are the boundaries of those areas. Such a system of squirts is called a double shell. In- side the shell itself the velocity is not that due to the POTENTIAL OF EXPANSION. 219 vortex, but is very large and in the contrary direction, namely, from the source to the sink. In crossing the shell the velocity potential is changed by VELOCITY IN TERMS OF EXPANSION AND SPIN. We are now able to resolve the velocity-system of an infinite mass of fluid, having no velocity-potential at in- finity, into squirts and vortices. Let E a be the expansion at a point a. Suppose the entire volume divided into small portions, of which SV a is the one including the point a. Place at the point a a source whose strength is EJbV a : 4?r. Then the rate of increase of SV a , due to this source, is S a BV a . And the velocity-potential at a point p, due to this source, is EJ> V a : 4<7rr ap , where r ap means the distance between a and p. If a similar source be placed at some point inside each of the small pieces & V into which the volume is divided, the velocity-potential due to all of them will be V V And if we indefinitely diminish the size and increase the number of the 8 V a , this quantity will ap- proximate to the integral J -."- ". The meaning of the integral, however, requires ex- amination. It supposes that every point where there is expansion is a source, so that in a region where the expansion is constant, the sources will be uniformly dis- tributed. The strength of the source at each point must be zero, since the aggregate strength of all the sources in a portion of the volume is the rate of increase of that portion of the volume, which is finite. We have therefore to form the conception of a continuous distribution of source over a volume, so that the aggregate strength is a finite quantity, and yet there is a source at every included point. If, for example, sources are uniformly distributed in the interior of a sphere, the effect will be a homogeneous strain-flux, consisting of a uniform expansion of the sphere ; 220 DYNAMIC. so that the velocity relative to the centre is proportional to the distance from it. When the distribution is variable, the rate of distribution at any point is what would be the aggregate strength of a unit of volume in which the dis- tribution was uniformly what it actually is at that point. If S is the rate of source-distribution at any point, E the expansion at that point, then E=4<7rS. For the rate of increase of a sphere of unit volume is 4nr x the aggregate strength of the sources within it. We have given, then, a velocity-system in which the expansion at any point a is E a , and the velocity-potential is zero at an infinite distance. We construct the system whose velocity-potential at a point p is / - - ; and we shew that this system also has expansion E a at every point a (for only the sources in the immediate neighbourhood of a point can produce expansion at the point), and its velocity-potential is obviously zero at an infinite distance. If therefore in the given system there is no spin, the given and the constructed systems are identical ; for if we subtract one from the other, we get a system in which there is no spin, no expansion (for the expansions in the two systems are everywhere the same), and no velocity- potential at infinity. And this, we have already proved, means no motion at all. Next, let there be spin in the given system, and let Q p be the solid angle which a certain vortex-line subtends at p. Let a very small curve of area 8A be. drawn em- bracing the vortex-line, and let w be the spin at that part ; then 2a)$A will be the circulation round this curve. Let 2w&A = &k; then a motion whose velocity-potential at n Sk p is - - will have no expansion anywhere and no spin except at the vortex-ring. If we suppose the spin to be confined to isolated vortex-filaments, we may draw a sur- face across each filament, divide this surface into small areas 8 A, and draw a vortex-ring through some point in 1 / each one of these small areas. The sum 2-, p " will 47T POTENTIAL OF SPIN. 221 approximate to the integral /-/ if we indefinitely in- crease the number and diminish the size of the areas SA. But this integral expresses a continuous distribution of vortex-lines throughout the filament. If we suppose straight vortex-lines to be uniformly distributed parallel to the axis in a circular cylinder, the motion relative to the axis will be rotation of the cylinder as a rigid body about it. If then in the middle of the vortex-filament we draw a very small vortex-filament of circular section, so that a short piece of it may be regarded as a circular section, the motion of this small filament will be com- pounded of two motions. First, an irrotational motion, ffi dk whose velocity-potential is I - , calculated from all the rest of the vortex-filament. Secondly, a rotation as of a rigid cylinder about the axis with the spin &>. Let us now suppose the integral fl p dk : 4-rr to be extended over all vortex-filarnents ; whereby we may also admit the possibility that these vortex-filaments continu- ously fill the whole of space. Then a motion constructed so as to have this for its velocity-potential, and in places where there is spin to be determined as just described, will have everywhere no expansion and a spin equal to that of the given motion. If therefore we have given a motion in which E a is the expansion at any point a, and H p the solid angle subtended at p by a vortex-line, while the velocity-potential is zero at infinity, aud if we construct a motion having the ve- locity-potential I -~ " + I I " , the two motions will be identical. For if we subtract one from the other, we shall get a motion in which there is no spin, no expansion, and no potential at infinity ; that is, no motion at all. Thus we have shewn that if the expansion and the spin are known at every point, the whole motion may be determined. And the result is, that every continuous motion of an infinite body can be built up of squirts and vortices. CTamlmtigc : TBINTED BY C. J. CLAY, M.A. AT THE UNIVEKSITY PRESS. MACMILLAN & CO.'S PUBLICATIONS. BY I. TODHUNTER, M.A., F.R.S. Algebra for Colleges and Schools. New Edition. Crown 8vo. 7s. 6d. Key, 10s. 6d. The Theory of Equations. Third Edition. 7s. Qd. Plane Trigonometry. Sixth Edition. Crown 8vo. 5s. Key, 10s. 6d. Spherical Trigonometry. Third Edition. 4s. 6d. Conic Sections. Fifth Edition. Grown 8vo. 7s. 6d. Differential Calculus. Seventh Edition. 10s. 6d. Integral Calculus. Fourth Edition. Crown 8vo. 10s. Gd. Examples of Analytical Geometry of Three Dimen- sions. Third Edition. Crown 8vo. 4s. 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