Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031240199 Comall University Library arV17339 Uniplanar kinematics of solids and fluid 3 1924 031 240 199 olin.anx Claattiroit '§tm Btxm UNIPLANAR KINEMATICS OF SOLIDS AND FLUIDS g: m. minchin Honiton HENKY FEOWDE OXPOED TTNIVEESITT PKESS "WABEHOUSE 7 PATERNOSTER ROW €hxznlam |pr«ss Sot«s UNIPLANAR KINEMATICS OF SOLIDS AND FLlilDS WITH APPLICATIONS TO THE DISTRIBUTION AND FLOW OF ELECTRICITY GEORGE M. MINCHIN, M.A. PROFESSOR OF APPLIED MATHEMATICS IN THE ROYAL INDIAN ENGINEERING COLLEGE, COOPER's HILL AT THE CLARENDON PRESS 1882 [A/l rights reserved] PREF IlgF^CE. The present work aims at supplying a deficiency in the course of Mathematical Physics usually pursued by the higher class students in our Colleges and Universities.' In the majority of cases, I think, little attention is paid to the special study of Kinematics, and, as a consequence, the student enters into the study of Kinetics with somewhat misty notions of acceleration and other leadjng con- ceptions which belong to the province of Kinematics. In view of the great progress which has been made in this country in Experimental Physics, under the leadership of Faraday, Thomson, and Clerk Maxwell, it appears to me very desirable to include in the ordinary course of higher mathematics as much as possible of the application of mathematics to physical problems — at the expense, no doubt, of its applications to manifest unrealities. Hence a large portion of the work deals with the theory of fluid motion and electric distribution and flow. Anything like a complete treatise on general (or three-dimen- sional) Kinematics would be a much more difficult task, and would be addressed to a very limited class of students. When, however, the leading results for motion in one plane are properly mastered, the student will have little difficulty in understanding what they become when a third co-ordinate is introduced. Thus much as to the general aim of the work. The subjects treated of call for a few special remarks. In Chapter I considerable attention is devoted to the Instan- taneous Centre (of velocities), and in Chapter II much prominence is given to the Instantaneous Acceleration Centre, with the accom- panying Centrodes, some results with regard to which will probably be new to most readers. Applications of these results are made to several problems, the solutions of which are thus simpUfied. Pro- fessor Wolstenholme kindly placed his Book of Mathematical Problems at my disposal, and I have freely availed myself of his permission to select illustrative questions. Chapter III deals largely with Roulettes, and aims at bringing under one general principle the discussion of many isolated results. VI PREFACE. Some of the theorems in this chapter are, I believe, new ; and for several of them I am indebted to the kindness of Mr. M^Cay, Fellow of Trinity College, Dublin. My obligations are very great to such an elegant and sliilful geometer for the assistance which he has rendered. In Chapter IV I have advocated a reversion to the use of the term force of inertia which was employed and clearly explained by Newton, and which is employed by many French physicists, in- cluding Delaunay. The term was shown by Newton to convey a definite physical idea, to which, it seems to me, greater pro- minence ought to attach, now that the notion of actions propagated through a medium has been brought into such importance by the experiments of Faraday and the great work of Clerk Maxwell. The expression ' effective force,' used by D'Alembert, seems but a poor substitute for Newton's "vis inertiss. Chapter V follows closely the order and method of the chapter on three-dimensional strain in my Statics. Chapter VI contains the subjects which, undoubtedly, are the most difficult of treatment in the book ; and I could not venture, with any degree of confidence, to publish it without the criticism of some able physicist. Happily, Professor O. J. Lodge and his brother, Mr. Alfred Lodge, Fereday Fellow of St. John's College, Oxford, rendered their invaluable aid, and many alterations and improvements in the original MS. of the chapter are the result of their criticism. I hope that the student will find, in particular, the section on Conjugate Functions an assistance in his reading of Clerk Maxwell. Lamb's admirably simple Treatise on the Motion of Fluids has been of much assistance to me in this chapter. I must thank my friend Professor G. Carey Foster for the free use of his papers on Electrical Flow, and his revision of the pages dealing with this subject. Some Notes treating more generally of kinematical questions are added, in the belief that they will be of assistance to the more advanced student. I may particularly mention Note D, on Vectors and their Derivatives. Captain Eagles has rendered me efficient service in revising the proofs, and I hereby acknowledge my obligations to him. G. M. MINCHIN. Cooper's Hill, November, 1882. TABLE OF CONTENTS. PAGE CHAPTER I. Displacement and Velocity ... i CHAPTER II. Acceleration of Velocity . . . ■ .52 CHAPTER III. General Theorem of Epicycloidal Motion 79 CHAPTER IV. Mass-Kinematics of Solid Bodies . . 106 CHAPTER V. Analysis of Small Strains . . . .121 CHAPTER VI. Kinematics of Fluids 137 Section I. General Properties 137 Section II. Multiply-Connected Spaces 193 Section III. Motion due to Sources and Vortices. Electrical Flow .... ... 200 Section IV. Conjugate Functions 226 NOTES:— A. ' Centrifugal Force ' 255 B. Strain Invariants 256 C. Conjugate Functions ... .... 257 D. Vectors and their Derivatives 259 E. Flow of Electricity in one plane 263 F. Current Power . . 263 INDEX .... 265 ERRATA. Page 5, line 5 for PO read Po, inside the brackets. For read P from line 20, p. 17, to line 6, p. i8. In Art. 112, and accompanying figure, read \l/,ip — a, ^— 2a, . . . instead of ^, it + a, \j/ + 2a,... i since the sum of the separate stream functions must be constant along the resultant stream lines. The figure, as it stands, suits the discussion of equipotential lines (p. 180), the additive property- being attended to. CHAPTER I. Displacement and Velocity. 1. ITniplanar Motion. By uniplanar vioiion, or one-plane motion, is understood in the following pages motion which takes , place in one plane or parallel to one plane. Thus, a cylinder rolling down an inclined plane, its axis always remaining hori- zontal, has uniplanar motion, because the motions of all the particles of the cylinder take place in parallel planes. 2. Linear Velocity. When a point moves in any manner, its velocity has a component in every direction. The velocity in any direction may be defined as the rate of describing space, per unit of time, in that direction. Linear velocities follow the same laws of composition and resolution as Forces in Statics; and with these (such as the parallelogram and polygon of velocities) the student is assumed to be already familiar. If a point, P, moves in any manner in the plane of the paper, its velocity may be com- pletely defined (both in magnitude and in direc- tion) by its components along two fixed axes Ox and Oy, just as the po- sition of P at any instant may be completely de- fined by its two co- ordinates in these directions. The resultant velocity of P takes place along the tangent at P to the path AB, along which P is moving. The magnitude of this resultant velocity is-^. where f = the % Displacement and Velocity. length of the arc of the path measured from some fixed point, A, on it up to P; for if, in the time A/, /'moves from /" to a As very close point Q, the velocity is the limiting value of — when the length PQ (or As) is indefinitely diminished. The velocity PR Ax in the direction Ox is the limit of -— j or of -— ; and the velocity At At QR c ^y in the direction Oy is the limit of — or of — • The velocities parallel to the axes of x and j/ are therefore 3. Diagram of Space described. If a point moves with constant velocity, u, whether in a right line or in any curve whatever, the length of path described in time t is u.t. If the velocity of the point is perpetually changing, and if v is its velocity at any instant, we may assume that the velocity remains constant and equal to v during an indefinitely small time, A/, Now the length described (whatever be the path) in this time is v.At; so that the whole length described in any interval of time is / vdt I' taken throughout this interval. This may be graphically represented as follows. Draw two rectangular axes. Ox and Oy (fig. i), and take the first as axis of times and the second as axis of velocities ; i. e. measure off OM to represent the time /, and MP, perpendicular to it, to represent v, the magnitude of the velocity at the time /. Let ilfiV represent A/; then v.Ai is represented by the area of the little rectangle MPQM By laying off the different times along Ox, and drawing at the extremity of each abscissa a perpen- dicular (as above) to represent the corresponding velocity, we trace out a curve, AB, of velocities ; and it is clear from the above that the area included between this curve, the axis Ox^ and two extreme ordinates, Aa and Bb, will represent, on the scale adopted, the length of path (whether straight or curved) Angular Velocity, 3 travelled over by the moving point between the time represented by Oa and the time represented by Ob, 4. AugiUar Velocity. When P moves in any manner, the line joining /" to a point revolves about 0, or, in other words, changes its direction in fixed space. The direction of OP may be measured by the angle, POx, which OP makes with any fixed line Ox. This angle is to be expressed in absolute or circular measure. Denote, then, its circular measure by Q. If after the interval of time A/ the point P comes to Q, and if the circular measure of the angle QOx is 5 + Afl, the circular measure of the small angle QOP through which OP has turned is Afl ; tS and the limit of the ratio — , viz. A/' dQ It' is called the dngular velocity of P about O. Angular velocity means rate of describing angle (in circular measure) per unit of time, just as linear velocity means rate of describing space per unit of time. If P moves in a circle whose centre is 0, the linear velocity of P is at every instant equal to its angular velocity about the centre multiplied by the length of the radius, — or V = ra>, where v = linear velocity, co = angular velocity about the centre, r = radius of the circle. Even when P moves in a circle, this equation does not hold between the linear velocity and the angular velocity about a point which is not the centre. For if the direction of the resultant velocity is PQ (or rather the tangent to the path of P at P), and if PR is drawn perpendicular to OP and OQ, we have Pli^0Px^6,. where ^0 is the circular measure of the small angle QOP. B 2 4 Displacement and Velocity. Now PR = PQ. sin OQP = V sin OQP x A/, so that if OP = r, r tS = v sin OQPx^, dd V . ^ .-. -=-sm.^, where <^ is the angle between the radius vector OP, and the tangent at P. Hence it is only when the direction of velocity is at right angles to the radius vector OP drawn through a fixed point O, that — linear velocity = angular velocity x distance. The velocity of the moving point P perpendicular to the , ^ PR . rAd . radius vector, OP, is the limiting value of — -> or of — -; i.e. , , . . dd ^' ^' this component of velocity is r— ■ The velocity component along the radius vector is the limit , RQ . dr of — -J I.e. ^« At dt 5. Composition of An- gular Velocities. An an- *^/> gular velocity . PA and to' .PB is a magnitude of the same kind repre- sented by (ft) + (/) . PO, where divides AB so that — = ~ (see OB CO ^ Statics, Art. 20). If instead of being measured along PA and PB the directed magnitudes are measured perpendicularly to them, u / Composition of Angular Velocities. 5 or making any constant angle with them, the resultant will be perpendicular to FO or inclined at the supposed angle to PO. Hence the resultant of the two velocities (represented by Pa and Pb, perpendiculars to PA and PB, and equal to m.PA and m' .PB) is a velocity (represented by iga) equal to (w + w') ./"O and perpendicular to PO. Hence the two angular velocities are equivalent to an angular velocity co + m' about O. These angular velocities compound, therefore, exactly like two parallel forces proportional to to and to' acting at A and B. If the angular velocities are in opposite senses, the point must be taken on AB produced. If to' = — to, the point O is at infinity and the resulting angular velocity is zero. The motion of P in this case is a velocity equal to .AB in a direction perpendicular to AB. For, the re- sultant of a velocity to . PA along PA and a velocity o) . PB along BP produced is a velocity parallel to AB and equal to m . AB, as is easily seen. Turning through a right angle the lines representing these motions, we get the resultant of the rotatory motions of P, with equal and opposite angular velo- cities, round A and ^ to be a motion perpendicular to AB with velocity m . AB. Hence all points which have these equal and opposite angular velocities round A and B have exactly the same velocity, (a . AB, in magnitude and direction ; so that if a lamina had these angular velocities, the resulting motion would be a simple motion of translation of the lamina in a direction perpendicular to AB. Angular velocities and motions of translation in kinematics are therefore analogous to forces and couples, respectively, in Statics. Again (in exact analogy with Statics, p. 93), an angular velocity of a lamina i:~\ \^~^ round any point B is equivalent to an A M "^ .^ equal angular velocity, in the same sense, round any point A, together with Fig. 4. a motion of translation. For, let two equal and opposite angular velocities round A (fig. 4) be introduced, each equal to that about B. This will not alter the motion. Then we may take the angular velocity 6 Displacement and Velocity. about B with the equal and opposite angular velocity about A as equivalent to a motion of translation with velocity o) . AB ; and in addition we have the angular velocity about A of same sign as that about B. Therefore, etc. Examples. 1. A point P moves in a circle with constant linear velocity; find its angular velocity at any instant about any given point. Let C (fig. 5) be the centre of the circle, ^ the point about which the angular velocity ^-- ^~iv-P at any instant is required, CP = a, OC — — 5 y^ yy\ '"■ / ■ ,( y^/ \ » = P's linear velocity; and let 0/'= n Then / y' / 0^ \ ' the component linear velocity of P perpen- I -/ '"' I dicular to OP is z). cos OPC; therefore the \ "v / angular velocity about O is \ / V. cos OPC Fig- 5- But cos OPC = ^^ ll — ; therefore the 2ar angular velocity about is — fi + f^j __^ ^-l 2 a L \ nV r^S The angular velocity about is therefore variable with the distance OP; tut if is on the circumference, the angular velocity becomes — , i. e. 2a constantly half the angular velocity about the centre — which is obvious without calculation by the proposition that the angle at the centre is double the angle at the circumference. 2. A point moves along a right line with constant velocity, prove that its angular velocity about any fixed point varies inversely as the square of the distance from this point. 3. Prove that in general the angular velocity of a point, P, moving with velocity v in any cunre about any fixed point, O, is ^, where r = OP, and/ = perpendicular from O on tangent at P to the curve. 4. A point moves in an elUpse so that its angular velocity about one focus varies inversely as the square of the distance from this focus; find its angular velocity, in any position, about the other focus. Ans. If the angular velocity about the first focus is -,, the angular velocity about the second is — at the same instant, the focal distances being r and t'. Hence if the ellipse (like a: planet's orbit) be one the Harmonic Motion. square of whose eccentricity can be neglected, the angular velocity about the second focus would be approximately constant. 5. Find the curve such that if a point describes it with constant velocity, the angular velocity of the point about a. given fixed point shall vary in- versely as the distance. Ans. An equiangular spiral having the fixed point for pole. • 6. A point moves in -a. parabola; compare, in any position, its angular velocities about the vertex and focus. Ans. If r is the distance of the point from the focus, a and w' the angular velocities about the focus and vertex, and m one-fourth of the latus rectum, — = ■ 7. A point describes an ellipse in such a manner that its angular velocity about the centre varies inversely as the square of the distance ; show that the sum of the reciprocals of its angular velocities about the foci is constant. [Use the expression in example 3.] 8. In the same case show that the sum of the reciprocals of the angCilar velocities about the extremities, whether of the axis major or of the axis minor, is constant. 9. A point moves in an ellipse so that its velocity varies inversely as the perpendicular from one focus on the tangent ; show that the velocity can be resolved into two (oblique) components, each of constant magnitude — one perpendicular to the focal radius vector and the other perpendicular to the major axis. (Prof. Adams.) [The velocity is directly proportional to the perpendicular from the other focus on the tangent ; therefore, etc.] 10. If at every instant the velocity of a point along its path is propor- tional lo the time, find the length described in any interval. [Use a velocity diagram. Art. 3. It will be a right line,] 6. Harmonic Motion. If a point, R, (fig. 6) moves con- tinuously round in a circle with constant velocity, the foot, P, of the perpendicular from the point on any diameter of the circle moves backwards and forwards along the diameter with a mo- tion which is called a simple harmonic motion. Fig. 6. Let O be the centre of the circle, and let the time be 8 Displacement and Velocity. reckoned from the instant at which R occupied any position, Rf,\ then if a= iRfiA, a = radius of circle, to = angular velocity of R about O, and OP^x, we have X = a cos ( and AT = (z cos(— - + a). (i) In the same way, if OB is the diameter perpendicular to OA, and if ^e is the perpendicular on OB; OQ being denoted by>', we have ,„ . j/ = asm(-^+t?). (2) The time T is obviously also the time taken by P to complete its excursion, i. e. the time taken to move from A to A' and back again to A ; and T is called the period of the motion. The velocity of P at any instant is, of course, 317a . ,2itt , --^sm(— + a); and this, which vanishes at A and A'y attains a maximum at O. Similarly for the motion of Q. If we imagine another point, R', to start from /"/, so that / R^OA = a', when R starts from R^ , and to go round the circle in the same time as R, its projection, P^ , on OA will have the harmonic motion , ,2jr/ ,, X =a cos( ha )> and the extreme limits of the excursion of P' will be A and A', just as before ; and P" will occupy all the positions occupied by i'— but, of course, at different times. What essentially dis- tinguishes the displacement (from 6) of P' from that of P is the angle — - + a', as compared with the angle \-a; and the displacement of the projection of a point which started from Harmonic Motion. 9 A at the origin from which time is reckoned would be a cos The angles — + a and 1- a' which at each instant thus essentially distinguish the two previous displacements from each other and from this last are therefore called the phases of those motions. The expression, a sin ( h a) > for the displacement along the perpendicular axis, can be written , 2 n-/ w. y= — acos( ha + -)> whose phase is 1 (-a. Thus we see that if the position T 2 of a moving point, S, be compounded, according to tiie parallelo- gram law, from two displacements along two rectangular lines, the displacements differing in phase by -, the motion of the 2 point will take place in 3 circle ; or two simultaneous rectangular harmonic motions of the same period and amplitude (presently defined) and differing in phase by — are equivalent to circular motion. Harmonic motions play an important part in Physical Optics, the motions being performed by particles of the luminiferous ether. The excursions of the particles from their positions of rest (in the ether when it is unstrained, i.e. not transmitting light) are, of course, extremely small; and the motions are called vibrations. We shall speak of them as such in what follows. The maximum excursion of the harmonic vibration ,21st , .a? = a cos \—=r-\-a) is a, and this magnitude is called the amplitude of the vibration. The angle a is sometimes called the epoch angle, or simply the epoch. The three essential characteristics of any vibration are, there- fore, its amplitude, its phase, and \\s>period, T. Another magnitude lo Displacement and Velocity. which plays an important part in ' the vibration is the mean square of the velocity in the time of a complete vibration. This is, of course, CT dx / ^%ydt Jo ^ dt ' 2'tV'C? which is half the square of the maximum velocity in the vibration. 7. Superposed Equiperiodic Eectilinear Vibrations. Suppose several disturbing causes to impress simultaneously on a particle at a corresponding number of harmonic motions of the same period, all superposed in the direction OA. Then if (/ is the position of the particle at any time /, and if 0(f = f, we have 2 17 i 2 tri f=<7, cos(— +ai)+ajCOs(— + a,) + «3C0S(— +03) + .., (i) where flj, a^, a,,... are the amplitudes, and Oj, a^, a^,... the epochs of the several independent vibrations. This equation for ^ is of the form i=Acos{~-Jr^), (2) for it gives ^ = (a^cosa^-\-a^cosa^+ ...) cos — - , . , . , \ . 21:/ — (aismai + fl2sma2+...) sm — t and comparing this with (2), we have A cos \/f »= fli cos Oj + ffj cos 0^+ ... = 2a cos a, for shortness ; ^ sim|f = flj sin Oj + a^ sin a^+ ... = 2a sin a. Hence A = ■/(2a cos af + {2.a sin 0)=*, (a) , 2a sin o tani/f== 5 IS) 2a cos a \ ' where obviously A is the amplitude of the resultant vibration, and t// its epoch, or — + i/f its phase at any instant. Graphic Circular Superposition. 11 These values of the resultant amplitude and phase, when compared with the values of the resultant of any number of coplanar forces and the tangent of its angle of direction, show that amplitudes and epochs of any number of equiperiodic vibrations, superposed in the same right line, answer exactly in their compo- sition to magnitudes and directions of coplanar concurrent forces. Just as in the composition of forces it is only the angle between two forces that is important, and not the separate angles which their directions make with an arbitrary line, so here it is only difference oi epoch, or of phase, that is important ; and any one of the vibrations may be arbitrarily selected as that of zero epoch. We have already seen how the phase of a rectilinear vibration 2 TT/ x = a cos ( — + a) can be exhibited as an angle subtended at the centre of a circle having the whole excursion 2 a as diameter. To compound amplitudes and phases of such vibrations graphically, we draw lines equal to the different amplitudes each in the direction given by the corresponding phase or corre- sponding epoch ; and thus the graphic methods of Leibnitz and the polygon of forces can be applied (see Statics, pp. 15, 17). Hence the resulting agitation will be nil in certain cases; and, in particular, when two vibrations of equal amplitude and period, and differing in phase by ir, are superposed. 8. Graphic Circular Superposition. Taking the case of two equiperiodic vibrations in the same right line, the resultant position of the moving point at every instant may be graphically represented by the following method. Let the circle (fig. 6) in Art. 6 completely represent in amplitude, phase, etc. the vibration (a, T, a), and draw another circle having the same centre and completely representing the second vibration («', T, a'). Let 1^ be the position at time / 'of the point which is supposed to travel over this second circle, R being, as before, the position of the point travelling over the first. Then the diagonal. Op, of the rigid parallelogram whose two adjacent sides are OR and or! will be the resultant amplitude of the two vibrations, and its extremity p will determine by its motion in I a Displacement and Velocity. a circle (in the manner explained in Art. 6) a simple harmonic motion which is at every instant the resultant of the two super- posed motions. This is perfectly obvious from the graphic construction given at the end of last Article. 9. Harmonic Motions of Different Periods. If two harmonic motions of different periods, T, T', are superposed in the same right line, the resultant motion will be expressed by the equation f = a50s(^^+a)+a'cos(^+o'). (i) instead of equation (i). Art. 7. For simplicity put n and n' for — and -p' respectively. Then, in general, the resultant vibration may be put into the form ^ = Aco%{mt+'>^) (2) in an infinite number of ways ; but A and -^ will be no longer constant magnitudes, but functions of /. In particular, let us choose m equal to ^ (« + »'). Then putting (/) = 4 («—«')/, we have ^= a cos(»z/+a + (^) + a'cos(w?/+a'— (/)), which gives A^ = c?—zaa' cos(a— a'+ a ^) + a'^, a sin a + {x) is any periodic function of period X, and let it not become infinite within its period. Assume d(j:) = ^o + ^, COS2ir-r + ^sCOS2W-— + ...+A^C0S,2TS-r- + ... A A A X . 2X . nx , . +S,sm2-n^+B^sm2ii'-r-+ ...+B„sm2'ir-r' + ... (2) A A A where -4„, ^j,... Si,-., are all constant coefficients which are to be determined. Multiply both sides of this equation by 2cos2ir— , and then integrate both sides from j; = o to jc = X. The coeflScient of A^ alone will remain, as is at once obvious ; and we have ^ rK pix ^m = lj {x)cOS21I-^dx- (3) Similarly, if we multiply both sides of (2) by 2sin2ir -j-> and integrate, we have ^m = ^J (l>{x)sia2iT—dx; (4) I C' while "^""X/ ^^^^'^^' ^5) 14 Displacement and Velocity, The coefficients, A^, A^,... are therefore constant quantities depending simply on the form of the function <^; and we have finally I C^ -KT"/ nz . nz. I \ {z) = - (}){x)dx + 2^^{A^C0SZTi-^+B„sm237j-),... {a) (the variable being, for clearness, denoted by z), the suffixes signifying that n is to receive all integer values from i to oo, and the coefficients having the values determined in (3), (4), (5). nz . fiz , The typical term ^„cos2 7rY+^„sm2ir-^may, ot course, I nz . be put into the usual simple harmonic form ^ cos(2w y+a); and thus the Theorem enunciated is proved. Observe that the periods of the successive component simple harmonic vibrations — wave-lengths as we shall call them, for a reason to be presently explained — are A., ^A., ^A [For a complete discussion of the proof of Fourier's Theorem the student is referred to Thomson and Tail's Nat. Phil. vol. i. PP- 65; etc. ; Price's Infin. Cat., vol. ii. ; or Donkin's Acoustics, pp. 51, etc.] Fourier's Theorem has a very wide application in Physics — and notably in the cases of periodic motion which present them- selves in the theory of Sound, A body — a tuning fork, for example — ^when thrown into vibration throws the air into some kind of periodic motion; and Fourier's Theorem shows that this motion can be exhibited as the superposition of a (theoreti- cally infinite) number of simple harmonic vibrations whose wave-lengths are represented by A, J A, |A, ... and whose amplitudes are represented by the values of Va^+B^ just given. It is pointed out by Donkin {Acoustics, p. 60) that Fourier's Theorem and the Ear of an animal perform exactly the same function, the first analytically and the second physically ^-each resolves a vibration into a number of simple harmonic vibrations; but whereas Fourier's Theorem takes account of an infinite number of these simple harmonic vibrations, with diminishing wave-lengths and gradually dying amplitudes, the Ear is capable of appreciating comparatively few of them. We may put Fourier's Theorem to the purely analytical use: Fourier's Theorem. 15 of expressing the ordinate of a given curve within given limits as the sum of a number of simple harmonic values. Thus, take the arc, OA, of a parabola cut oif by a line OA perpendicular to the diameter and at a distance A from the vertex. Taking O as origin and OA as axis of x, the equation of the curve, if 0A = 20, is _y = A(^ j). 7tX fix Assuming y = /*„ + S(^„ cos 2ts^- + B„ sin 2ir — ) > we h n<= ,2x x^, ^ 2, have .4n = — / ( — — ^)ax = -h. h C^" .2X a:\ nx j aH h r^<=,2x x'. . nx J 2 , ih -^rao I n-nx Hence y=-n » .Z,, -jcos -^ 3 ti^ ■"^^ »" c As another example (see Donkin's Acoustics, p. 54) take a broken line, consisting of a right line 0.4 and a right line ^C. Take the right line OC as axis oix; let OC = A ; from A let fall .4iV perpendicular to OC, and let 01^= a, AN= b. It is required to express the ordinate of the broken line O^C in a Fourier series. 7 From x = o to x = a, we have y = -x, and from x = a to .a; = \, we have y = t— ^ (A— x). Assume j' = ^o+ 2^x UnC0S2w-j^+^„sm2w — ) (.jb) - xcos2'!i^-dx+r--T / {\—x)cos2it—-dx aJo A A \-aJa A Then 2 A ajo 3A' , 2«irfl . (cos-^^ 0- " 4»V^fl(A-a) ^ A ^A' . 2«ira Similarly ^„ = ^„.^,^^j,_^^ ^'n -^ , 1 6 Displacement and Velocity. and A^ = \h. Substituting these values in (<^), and forming a single harmonic term with A^ and B^, we have ^A." •%:» I . n-na . 2nv(x—\d) It is obvious that in all cases the first term, A^, in Fourier s series is the mean value oiy within the given limits. Examples. I. Exhibit the ordinate of the right line j/ = m{x--) between the limits X = and jc = A. in a Fourier series. m\ ri . 2irx I . 4irx 1 . 6irx Ans. m\ ri , 2Trx I . iirx 1 . bitx , ■. y = i— sin + — sin — — + — sm — ; — + ... )• e / r > \ -•^^jy R ^ X X 2. Exhibit the ordinate of the curve r =23 smh - cos — between jc = o ' c c and ;i; = — Tt in a Fourier series. 2 . , i6a ■ vs " -D — 32o« . IT .^TM. .i4„=— -— — 2^-^r- smh" - ; B„ = — ^-^r-sinh-. " 7r(i + i6»') 4 " m(i + i6«2) 2 11. Wave Dis- turbance. Suppose a disturbance of every jf particle on the right Fig_ y. line Oz (fig. 7) to take place perpendicularly to Oz; letO be chosen as origin, and let 0/'=z = the distance from of a particle at P. Let £ be at any time the displace- ment of P from its position of rest, and suppose the value of f, as dependent upon both s and /, to be given by the equation f = asin— (»^— 2), (a) where A is a constant linear magnitude and v a constant (ex- pressing necessarily a velocity). We might with equal propriety have chosen the form a cos— (o/— s) for f ; but the above is chosen because we wish to reckon the time, /, from the instant at which our origin particle, 0, was in its undisturbed position; a change to the Wave Disturbance. 17 cosine form would amount simply to a change m the origin of time. Now consider what the values of ^ are at the same time and at different distances. They are exactly the same at the distance z as at the distances z + \, 2+2A, 2+3X, ; and they are the same in magnitude but opposite in sign at the distances z and z + JX. Now let us draw a curve representing all the displacements of particles at the same time. The displacement of is asin— 7j — } and is represented by Oo (fig. 7); that of /"by Pp\ and we have just seen that the particles at distances A., 2 A., 3 A, .. from P are in exactly the same state of disturbance as P. Let PR = A j then, in other words, all the disturbances of particles from R onwards are exact reproductions of the dis- turbances between R and P. Let ^^ = ^ A ; then at Q we have a particle whose disturbance, Qq, is the same in magnitude as that of P but different in sign. The history of the disturbances at different points at the same time is therefore represented by some such curve as that figured. Next let us confine our attention to one particle,?)^?, and represent its disturbances at different times. It is quite obvious that if z is constant and / variable, we shall 2 TT get from the expression a sin— (»/— z) exactly the same series of values as we got from it when / was taken constant and z variable ; in other words — the curve which represents the history' of the displacements of all particles at the same time represents also the history of the displacement of any one particle at different times. And the values of the displacement ofW will reproduce them- selves after an interval of time equal to — • v The distance A, which intervenes between two particles which are always in the same phase is called the length of the wave. We may now put the matter thus : — f is at the time t the same at l^and R, and after this instant its value changes at both points, being, however] always of equal amounts at both ; f will t8 Displacement and Velocity. , A again be the same at R as it was at time / after the interval — ; we may suppose this second disturbance f at ^ to have travelled up to R from' iS(; and since it has gone over^^, or \, in time — > ei is the velocity with which the disturbance travels. When this disturbance has got up to iP, '!?.is again in exactly the same state of disturbance as at time /; i.e., i\has just gone throjigh all its cycle of disturbances. Hence — every particle takes exactly the same time to go through all its phases of dis- turbance as that taken by any given disturbance to travel over a wave length. When any one disturbance (value of f ) has advanced a wave length, the wave is said to have travelled over a wave length. A point (such as A) at which a disturbed particle is at its maximum distance from its position of rest is called a crest of the wave. Two particles, such as Q and P, which are always in opposite phases are separated by -i or half a wave length ; for if Og = 2', the difference of phase of Q and i" is — {z'—z), and if this = it, or /— z = ^A, the value of f for Q= — that for P. In the expression a sin— {vt—z), z is called the retardation of the vibration. A retardation of a whole wave length (or difference of phase equal to pir) is tantamount to no retardation at all. 12. Composition of two Bectangular Vibrations. Sup- pose a particle at (fig. i) to receive a displacement £ along Ox and a displacement »j perpendicular to Ox, the periods of both being the same, the axes Ox and Oy being in the plane of the paper, while an axis Os may be imagined as drawn upwards from, and perpendicular to, the plane of the paper ; and suppose that / f = asin(3Tr- +o)> ?j = 3sin(27r- +/3)« Composition of Rectangular Vibrations. 19 Eliminating /, we get ■^-^cos6+^ = sin''8, a^ ah Ir where 6 = a— /3 = difference of phases of the two* constituent vibrations. Thus the resultant vibration is one in an ellipse having the undisturbed position of for centre. • To determine the direction 0/ vibration in this ellipse, find the value of ~ when f = «. Now -J = ^ cos (277- +/3); and at at yivy.'-/' if f=a, 27r- +a = -; T 2 and -^ will be positive if 8 >o and < -j i. e., the motion is from the positive axis of f to that of jj if 8 > o, < - • The re- 2 ^g tardation of the i) component behind the f component is — » so that if this retardation is between o and a quarter of a wave length the motion will be from the positive axis of £ to that of 1J. Cor. I. Two rectangular vibrations of same wave length and period, one being a quarter of a wave length in advance of the other compound an elliptic vibration, the axes of the ellipse being in the directions of the constituent vibrations. For then d = -> and the equation of the ellipse is ^ + 7^ = 1. Cor. 2. If the amplitudes, a, b, of the constituent rectangular vibrations are equal, the resultant vibration is, in general, elliptic; but if the phases differ by -» i.e., if the retardation be \\, the resultant vibration is circular. If the periods of the two rectangular components are not the same, the equations will be f = asin(27r- +a). »j = i5siii.(2w-p + ^), or, for shortness, let us write $=awa.ni, ?j = 3 sin (»«/+ e). (a) c 2 30 Displacement and Velocity. The Cartesian equation of the curve described by the dis- turbed particle is obtained by eliminating t from these equations. •Now it is clear that if n and m have a common multiple, this curve will be closed, the particle performing complete revolu- tions in the curve in a time equal to the least common multiple of the two constituent vibrations; but if the periods are in- commensurable, the particle will never return to its original position — it describes a curve which never closes. Figures of the curves which will be described for different 71% values of e and — will be found in Thomson and Tait (p. gi) n ^ and in several works on Physics. Clifford proposes to study all cases of motion expressed by equations (a) by converting the motion (which, of course, is uniplanar) into motion on a cylinder. This is done by in- troducing a third component harmonic motion along a line perpendicular to the plane of the two components £, t;, and choosing this third motion, f, so that it produces, when taken in conjunction with one of the components — r\, suppose — uniform circular motion. In this case we should have f =3cos {jnt-\-f), and the total resulting motion is motion in a circle perpen- dicular to the axis of f combined with motion along this axis ; which comes to the same thing as imagining a cylinder described on the circle, and a generating line to be carried round it with uniform angular velocity m, while a particle per- forms the harmonic motion f along this generating line. The curve of positions on the cylinder is obviously obtained by wrapping round it the wave curve which is the curve of positions of the motion f=asin«/; and the given motion, expressed by equations (a), is obtained by projecting the cylindrical motion on a plane perpendicular to f. Keeping the same cylinder, with the same curve of positions wrapped round it, if we project the cylindrical motion on different planes passing through the axis of the cylinder, we obtain the same effect as is produced by varying e in the equations (a). '. [See Clifford's Kinematic, p. 33.] Resolution of a Rectilinear Vibration, 21 13. Besolution of a Bectilinear Vibration. If in the preceding Article the two rectangular vibrations have the same phase (i.e., if there is no retardation), or a = /3, the resultant vibration is rectilinear ; for then 7} h which is the equation of a right line ; and the amplitude of the resultant is ^c^JrU^. If the difference of phase is w, instead of o (i. e., if the retardation is X), the resultant vibration is again rectilinear j for then -= = • C a Conversely, a rectilinear vibration of amplitude a along any line OA (fig. 8) can be resolved into two rectilinear vibrations of amplitudes a cos a and a sin a, along any two rectangular lines, Ox and Oj/,the phases of the components and that of the resultant being all the same, and o being the angle between Ox and OA. Reckoning from Ox as an initial line, the angle a is called the azimuth of the vibration in OA. Again, any rectilinear vibration is equivalent to two circular vibrations, the rotations in these being in opposite senses. 2 Ttt Suppose a vibration f = a sin along the axis of x, and a vibration -q = o — in other words, no vibration — along the axis of y. We may write these Fig. 8. f=i . svt fl sm — + : a sm - 2-nt ■^1 + ^2. suppose; »j = |asm{— +-)+4«sm( — +— ) =??i+»jj. Take the vibration (fi r/i) separately ; it is obviously circular, smce 2 7r/ f, = iasin-— . ^iH^J,^ ■ni ■h 2irt = i«cos— , 22, Displacement and Velocity. And by finding the value of -^ when fj = Ja, we see that the sense of rotation in the circle is from the positive part of the axis of 77 to that of the axis of f. * In the same way, (fj I2) '^ ^^ equal circular vibration, the sense of rotation being the reverse of the preceding. Conse- ,quently the (ether) particle which describes the vibration along the axis of x (i.e., along any line) may be supposed to be agitated by — or to be compounding — two equal and opposite circular vibrations. This mode of resolving a rectilinear vibration is important in the theory of the rotatory polarisation of qttariz. Thus, suppose that the quartz retards the circular vibration (f, t/j) more than it retards the circular vibration (f j ijj) of opposite rotatory sense ; and let the amount of this retardation be - A, where A is a wave n ' length and n any number. Then the vibrations (fj tjJ and (f , ?/,) on emergence from the quartz may be written . . ,2Ttt 217. . . 27r/ , ,2Tlt 2Tr. , 217/ 77, = iacos(— -— ); 7,^,= -Jacos— • The resultant disturbance of the ether particle at the place of exit of the light from the quartz has then for components ^ 17 . ,217/ 17. f = a cos — sm ( — ) , n ^ T n' . IT . ,2lr/ 17, 7; = asm -sm ( ); n ^ T n' and since ^ = tan - , the vibration at emergence is rectilinear, of the same amplitude, a, as at incidence, but along a line 17 making an angle - with its original direction. The same method of resolution of a rectilinear vibration into two oppositely rotating circular vibrations will serve to explain the phenomenon observed first by Faraday — ^viz., that the plane Transverse and Longitudinal Vibrations. 23 — - — . U AAA P is a point of zero disturbance these lines will answer to an equilibrating system of forces. Obviously, there- fore, in order that there may be zero- disturbance at any point, the ampli- tudes of the interfering vibrations must be such that the sum of any two exceeds the third. Denote the angles a^ Oa, , a, Oa, , Kig. g. fli Och by Oi , Oj, Uj. Now ^ (.'■i-r,) = "i ; x (rs-''i) = 2T-«2 ; -^ ('a-n) = «>. any one of which equations follows from the other two. Hence the points of intersection of the hyperbola] r,-r,^-^K (I) 2ir 26 Displacement and Velocity. with the hyperbola ^3-1 = -— 27r \ (a) are points of zero disturbance. But now observe that the direction angle of (say) Oflj may be just as well —-— ±2mr, where » is any integer, as — - — . SO that we can have and therefore all the points on the fixed hyperbola (2) which lie on the series of hyperbolas obtained by varying « in (3) are points of rest on the surface of the water. But in the same way as ra may be altered to r, + n\, any other distance, as yvj, may be similarly altered; and thus we shall obtain a series of hyperbolas r^-ri = -^\ + mK (4) round Ai and A^ as foci. Hence finally we obtain a net-work of points of rest by taking all the points of intersection of the series of hyperbolas (3) obtained by varying n with the series of hyperbolas (4) obtained by varying m. Of course these points are continuously at rest, and not merely so at a particular time. [We have in the solution assumed that the amplitude of each wave re- mains constant — a supposition which is allowable if we consider only points not very far removed from the origins of disturbance.] 3. If the motion of a point consists of a harmonic vibration of period T, and X denotes its distance at any time from its mean position, show that iPx 4^ [Conversely an equation of motion of this form indicates harmonic vibra- tion of period 71] 4. Prove that the resultant of any number of simple harmonic vibrations, in different directions, with different phases, but of the same period, is elliptic harmonic motion. [Arts. 13 and 12.] 5. Prove that the motion expressed by the components (a), Art. 12, when m = 2n and e = — , is oscillatory motion in a parabolic arc. 16. Eelative Motion. When any two points, O and P, move in any manner, we must distinguish three different motions which take place, viz., the absolute motion of P (i. e., its motion in fixed space), the absolute motion of 0, and their relative motion. The relative motion of two points, whether both moving or not is the motion which each appears to the other to possess. Relative Motion. 27 If a point, P, moves round a circle with a velocity either constant or variable, the centre, 0, of the circle, although fixed, appears to P to move round P in a circle of the same radius, and with a velocity either constant or varying exactly as the • T ' " a' Fig. 10. velocity of P varies. [Two observers may be supposed to be stationed at P and 0.] Let there be two points, whether both moving or not ; then the angular velocity of the first with respect to the second is exactly the same as the angular velocity of the second with regard to the first, and it is measured at any instant by the rate per unit of time at which the line joining them revolves, — i.e., by the rate at which this line describes angle (in circular measure) with a fixed Une in space. Let O and P be the points at any instant, and after a very short time let them come to O' and P", respectively, the distance between them being altered or unaltered. Let LM be a fixed line in the plane of motion. An observer at O might measure the direction of one at P by the angle 6 which OP makes with LM; and an observer at P might measure the direction of one at by the same angle. If OP" is drawn equal and parallel to O'P', the point P will appear to the ob- server at O, at the end of the motions considered, to occupy the position P", and the change of P's direction is measured by the angle P^'OP, which is M. If A/ is the small time occupied by the motions 00' and PP', 38 Displacement and Velocity. the limiting value of the ratio ;T-7>or — j is the angular velocity of P about in the positions and P. Exactly in the same way, if PCf' is drawn equal and parallel to P'O', the point will, at the end of the time A/, appear to P to occupy the position O", i. e., it will have revolved about P through the angle OPC/', which is also A9 ; so that the angular velocity of O about P is the same thing exactly as the angular velocity of P about O. To find their relative linear velocities, we observe that P appears to to have described the little straight path PP" in the time A/, so that the limiting value of the ratio —r— repre- sents the relative linear velocity of P with respect to 0. In the same way the limiting value of the ratio — — repre- sents the relative linear velocity of with respect to P; and it is obvious that PP" is equal and parallel to OO", and that they are in opposite senses. Two points may be moving in such a way that at some in- stant each appears stationary to the other. This will happen, if at the end of the small interval A/, the line joining them remains parallel to its direction at the beginning of the interval ; or, in other words, when for a short time the line joining them does not re- volve. When this is the case /^«fOOT- ponents of their absolute velocities per- pendicular to the line joining them are equal and in the same sense. Fig- 1 1- For if O', y are the positions oc- cupied by O and P at the end of a very short time At, and if c/p' is parallel to OP, the com- ponent of 00' perpendicular to OP is equal to that of PP' in QQ' pr/ the same direction; and — and — are in the limit, the absolute velocities of and P. Thus, a figure will easily show that one planet must at a Relative Motion. 39 certain time appear stationary to another, and their positions at this time can be roughly indicated without calculation. The relative path, or relative orbit, as it is called, of one moving point with respect to another is a curve obtained by drawing /ram a fixed origin lines parallel and equal to the simul- taneous distances of the two points. Thus, let the moving point O (fig. 12) occupy positions O^, Oj, O3, ... along any path in fixed space at the same times that the moving point P occupies positions P.^, P^, P,, ... along any 6ther path in fixed space, the time intervals between these simultaneous positions being very small. Then, starting from any fixed point A, draw AP^, AP^, AP^', ... equal and parallel to Oj^j, O^P^, Og^g, ... and we shall obtain a number of points Pi, P^, -?/, ... sufficiently close and numerous, if the time intervals are small enough, to enable us to draw a new curve, perhaps widely diflFerent in shape and equation from either of the absolute paths, and this is the curve which P appears to O to describe. Fig.. 1 2. Obyiously, also the rate at which P^ moves along this relative path is exactly the relative velocity of P with respect to O. [Con- sider fig. 10, and the definition of relative velocity before given.] In particular, consider O to be the centre of a circular wheel, which rolls along a horizontal plane without slipping, and P to be any point on its rim. The absolute path of is a horizontal right line, the absolute path of 7' is a cycloid, but the relative path (of O with respect to P, or of P with respect to o) is a circle. 30 Displacement and Velocity. 17. Problem. Given the absolute velocity of a morning point, O, and the absolute velocity of another moving point, P, to find the relative velocity of P with respect to 0. This problem has been already solved, but for clearness we reproduce the solution. Let OCf (fig. 1 3) represent the magnitude and direction of the absolute velocity of 0. \OCf may either be drawn equal to the velocity of O, or equal to this velocity x A/. If the latter, Cf is actually the position occupied by at the end of A/ ; if the former, O will, of course, never be at (f unless O moves in the right line 0(/, and continues to do so.] Let PJP' represent in magnitude and direction the absolute velocity of P. Then drawing PQ equal and parallel to OCf, but in the opposite sense, the relative velocity of P with respect to O will consist of the two components PF^ and PQ, which give as a resultant PP'', the diagonal of the parallelogram determined by PP' and PQ. In other words, reverse the velo- j^" city of on P, and compound it with the absolute velocity of P ; and the , . , , resultant of these is the relative veto- city of P with respect to 0. ^\ \ 18. Problem. Given the abso- \ A ■ lute velocity of a moving point, 0, \ and the relative velocity of another p. moving point, P, with respect to 0, to find the absolute velocity of P. In other words, given OCf and PP" in the above figure, to find PP'. At P draw Pi^ equal and parallel to OC/ and in the same sense; then compound P(^ and PP" and we obtain PP' ; so that we give P the absolute velocity of (not reversed), and compound it with the relative velocity of P with respect to O, and the resultant is the absolute velocity of P. Such construction is evident from common sense. 19. Problem. Given the absolute motion of one moving point 0, and the relative path, together with the law of its description, of another moving point, P, with respect to 0; find the absolute path of P. Relative Velocity of Two Points. 31 From each point, O^, Oj,... (fig. 12) of the path of draw a line, Oji'j, O^P^,... equal and parallel to the corresponding radius vector, AP^ , AP^,.. of the relative path, and the locus of the extremity of this radius vector is obviously the absolute path of P. As a particular case, let the relative path be a right line passing through the moving point and described with constant velocity, /3. The point P always appears to O to be coming straight towards it, so that if a is the angle made with the axis of X by the line of relative motion (which is that joining the initial positions of the points) and \i c = initial distance between the points, the co-ordinates of P at any time will be X + (f — /3/) cos a, J/ + (c—^i) sin a, if X and y are the co-ordinates of O. 20. Cartesian equation of Relative Path of two Points. Let the co-ordinates of the point with reference to rectangular axes fixed in space be (a, 0), those of P with reference to the same axes {x, y), and those of P with reference to axes drawn through parallel to the fixed axes (.a/,_/). Then x = a-\-x',\ , y = ^+y, ) ^ ' and in (fig. 12) the co-ordinates of I^ with reference to axes drawn- through A parallel to the fixed-space axes will be obviously (x', jf), so that these are the co-ordinates of the point in the relative orbit. The relation between x' and j/— i.e., the equation of the relative path — ^is to be found by eliminating ;*;, y, a, /3 between the above equations and those which will be given as defining the absolute motions of and /"—such as, for example, x=/AO, y-/M « = '^i(/). /3 = «^2W. where / is the ime, which must be also eliminated. 21. Components of Relative Velocity of two Points. The components, parallel to the fixed axes, of the relative dx' <*■"' velocity are —r- and ---J wnere dt dxf dx da dt dt ~ dt 3^ Displacement and Velocity. It ~ H ~ Tt' 22. Path of a moving point relatively to a moving plane. Quite distinct from the path of the moving point P relatively to a moving point Q is the path of P relative to a moving plane space containing Q and P. This latter would be got by drawing any two axes in the moving plane — i.e., fixed in the moving plane — and laying down at each instant the co- ordinates which P has at that instant with reference to these axes. The curve thus obtained would be the path oi.P relatively to the moving plane ; and, of course, the axes may be drawn at any point of the plane. Suppose that A (fig. 14) is the point chosen in the moving plane through which to draw axes, A^, At\, fixed in the plane, and let Ox and Oy be axes fixed in space. Then the mo- tion of the moving plane will be defined by the values (a, ^) of the p. co-ordinates of A and the angle, (^, which A f makes with Ox. Obviously if (f, ij) are the co-ordinates of the moving point P with reference \.o A^ and A-t\, and {x, y) the co-ordinates of P with reference to Ox and Oy, we have f=(j;-a)cos^-|-(j;-/3)sin^, ) »? = — {x—a) sin (^ -1- {y—fi) cos <^, J and in addition we shall be given some such equations as where / is the time. Hence the relation between f and tj may be found. 23. Velocity of a moving point relatively to a moving plane. In the last case — and -j- are the components of the velocity of P relatively to the moving plane, or,' what is the same thing, relatively to the moving system of axes A^ and At{. Examples. 33 They are known by diflferentiating the values of ^ and ij just given when the other quantities are assigned as functions of /. Obviously for the purpose of finding the resultant relative ve- locity we may take the axes A £ and Ar\ parallel to Ox and Oy, i. e., we may put (|) = o after differentiating the general expressions (a). Examples. I. Two points, P and Q (fig. 15), starting at the same instant from given positions, move each in a right line v?ith constant velocity; find their relative angular velocity at any instant, and their relative path. Let their lines of motion meet in ; suppose them to have been originally (i. e., when t = 6) at / and q ; let Op == a, Oq = b, u = velocity of P, V = velocity oi Q,a = angle POQ, and e = the angle OPQ. Then -=■ is their relative angular velocity. If / is the time at which they are at P and Q OP = a + ui, OQ = i + vt, and (fl + uf) sine = (i + vt) sin (9 + a), a + ut . . . .•. ■■ — = cos o + sm a cot 9. b-i-vt Differentiating with respect to t, bu—av sin a dd {b + vtf ^ ~ si^ di ' But if >- = PQ, we have ?- sin e = (i + ■vt) sin o, d6 av—bu . :■ -n = — 5— sm o. dt r" At any instant the direction of the relative velocity of Q with respect to P is got by reversing P% velocity on itself and on Q, and compounding this reversed velocity with ^'s absolute velocity. If at Q we draw a parallelogram with adjacent sides equal to v along OQ and » parallel to PO, its diagonal passmg through Q is the direction of relative velocity, i.e., the tangent to the relative path at Q. But this diagonal, in all positions of /"and Q, makes a constant angle vrith OQ, viz., ,0— «cosa tan""' : ■ 2 be the angular velocity of the . q body about G at that instant. \ ^'C'TV We must now find the direction ^V~~>- Displacement and Velocity. proportional on the same scale to v; then, completing the- parallelogram FqQp, the diagonal PQ. represents in magnitude and direction the absolute velocity of P (§ i8). Hence, drawing CI perpendicular to GT and PI perpendicular to PQ, their point of intersection, /, is the instantaneous centre (§ 26). To find the distance GI, we have GI sin GPI _ sin pPQ _ ^ _ » . GP ~ sin GIP ~ sin gPQ ~ Qq therefore __ v . GP' GI ^ and since the length GI\% independent of r, i.e., of the point P, we get the same point /, no matter what point P we choose. This proves the existence of an instantaneous centre, and, at the same time, ifinds its position. 29. Analytical Proof. Let Ox and Oy (fig. 19) be two rect- angular axes fixed in space in the plane of motion ; let a particle of the body at G have co-ordinates a arid /3, at any instant, with refe- rence to Ox and Oy; let to be. the angular velocity of the body at this instant — i.e., the rate per unit of time at which the line join- ing any two particles is describing an angle with a fixed-space line. Ox ; let /■ be a particle at a distance r from G ; draw Gx' and Gy parallel to Ox and Oy ; and let PG make an angle Q with Gx', or Ox. If X and y are the co-ordinates of P with reference to Oii and Oy, x^a-^r cos 6, (i) J/ = /3 -I- r sin 0. (2) dx Denote time-rates by dots — i.e., -j- by x, &c. Then Fig. 19- x = d— cor sin 0, y = $ + (orcos6, (3) (4) Space-Points and Body-Points. 41 since fl is to; and observe that we have diiferentiated on the supposition that r is of invariable length. If x' and y are used for r cos Q and r sin Q, the co-ordinates of P with reference to Gx' and Gy', we have X = d—a>y, (g) j/ = /3 + a)y. (6) Hence .:«■ and y will both be zero if we choose that is we shall have chosen the particle which for the instant has no velocity — i.e., the particle which is at the instantaneous centre. It is very easily seen that this analytical determination of the instantaneous centre from the known motion of a point G and the angular velocity of the body agrees with the geometrical method of last Article. The co-ordinates, with reference to axes fixed in space, of the instantaneous centre are, therefore, x^a-^, (8) y-^+i- (9) 30. Notation for Space-Points and for Body-Points. Consider any one definite position of a moving body, and let the position of a point be determined according to any assigned law — such as, for example, that the point shall always be taken on the normal to the path of the body's centre of mass, at a distance from this point proportional to the time. At the position of this pflint we shall find a particle of the body (or of a rigid membrane imagined as prolonging the body). Denote the par tide by Pj, and the space-point occupied by it by P,. Now beyond the bare fact that they coincide — i.e., that their co-ordinates are the same — P^ and /", have, in general, nothing in common. Their velocities, and as we shall subsequently see, other properties also, are wholly different both in magnitude and in 43 Displacement and Velocity. direction — notwithstanding that their positions coincide. In fact, their motions must be conceived to be as distinct as those of two moving points, describing wholly different curves, which happen to reach at the same instant a point at which their two paths cross each other. Thus if the point determined is the Instantaneous Centre, the particle is at rest for the instant, while the space-pwint has a velocity. The two aspects — with respect to body and to space — of the Instantaneous Centre we shall distinguish by the notation /j and /„ respectively. Similarly, the body centrode will be denoted by C^, and the space centrode by C,; and, in general, the suffix b will be used to denote anything relating to the body, while s will denote the corresponding thing for space. 31. Velocities of /^ and /,. The velocity of 7^ is, of course, nothing. The velocity components of 7, are the time- rates of increase oi x andj/ in equations (8) and (9) of Art. 29. Thus . . d ,^. .. are the velocity components of 7,, expressed in terms of the motion of a definite particle, G, of the body, and the angular velocity of the body. The velocity components of I, can also be found by differen- tiating equations (i) and (2) of Art. 29, on the supposition that r is variable and that — is not the angular velocity of the body — since it is not the rate at which the, line joining two definite particles revolves. Thus But so that i: = = d-|-cos0 3- at — rsinfl 1^. dt r cos 5 = - J3 - — > (0 rsin5 d CD x = rco dr dt ade u>dt' Body Centrode and Space Centrode. 43 which must, of course, be equal to the value in (i). Similarly for j); and the student may show that both methods lead to consistent results. 32. Equations of the Body Centrode and Space Centrode. The equation of the Space Centrode is obtained by eliminating the variable (usually the time, t) from equations (8) and (9) of Article 29 and thus obtaining a relation between X, y, and constants. To obtain the Body Centrode, i.e., the curve traced out in the body by those particles which have had, or will have no velocity at some time during the motion, draw two rectangular axes G^ and Gi\ at G, and let each of these axes always contain the same row of particles of the body — i. e., let them be fixed in the body. Let the line G^ make an angle <^ with the fixed- space line Ox. Then if f and r/ are the co-ordinates of any point with reference to the axes G^ and Gr\, we have f=(j\;— a)coS(^ + (>'— /3)sin(^, (i) r; = (_;/— /3)cos^ — (.a;— a)sin^. (2) Now -^ obviously measures the angular velocity of the body at any instant ; and if we give to x and j/ the values in (8) and (9), Art. 29, we have f = — - cos is constant) when BI is least, and this is so when the bars lie both in the position AC. At any instant the component of velocity of any particle of the rod BD in the direction BD is of . IC, or to x perpendicular from A on BD, yvhich is the same for all particles. In practice, BD is a piston which works up and. down in a 4^ Displacement and Velocity. cylinder capable of revolving round a fixed axis perpendicular to the plane of motion. To find the C^ntrodes of the motion of BD (those of AB both reduce, of course, to the point A). Space Centrode, C,. As explained in Art. 32, this curve will be known if we find a relation between the radius vector Al and the angle CAI. Let AB = a,AC = c,AI=^p, LCAI= 0, LACB = ^. Then in the triangle AIC we have p cos (fl + <^) = c cos <^, and in the triangle ABC flsin(0 + <^) = f sin(^. Each equation gives a value of tan ^ ; and equating these, p cos B—c asmd \ psind c— a cos 6 .•. p + a = . ccosd—a Now the polar equation of a hyperbola referred to one focus is a(e^-t) e cos — 1 where a is its semi-axis and e its excentricity. Hence if we construct a hyperbola having A for focus, its semi-axis in the direction AC and of length a, and its excen- . . c , tricity - ) we obtam the Centrode C, by diminishing each radius vector of the hyperbola by the length a, since we see that p=-r—a. Body Centrode, C^. As explained in Art. 32, this curve will be known if we find a relation between the radius vector BI and the angle CBI. Now let BI= p, LCBI= 6, LacB = <^. Then (/) + d) cos d = c cos 0, asmO — csva.^; ' cos fl Examples. 47 We might practically con struct this curve thus : take a point, K, 'va.BD such that BK= '>/c^—a^; at jTdraw a line perpendicular to BD ;, take any point, P, on this perpendicular ; at P draw PQ perpendicular to BP and equal to a ; draw BQ ; take QP, equal to a, along QB towards B, and QP', equal to a, along BQ remote from B, and finally measure oif BP' and BP" along 5P, equal to BP and 57?', respectively. Then P' and /^' are points on the body Centrode. The curve consists of two infinite branches on opposite sides of the line HTP, the branch towards B corresponding to positions of B on the lower portion of the circle round A between the points of contact of tangents to it from C; and the branch remote from B corresponding to the upper positions of B. The curve C, passes through the point C, which is the position of / when the directions of the bars coincide ; and the curve Cj, cuts the bar BD at two points distant c—a and c+a irom B. These are evident (J /mn". The machine whose principle we have here described is one of those whose object is ihe conversion of continuous circvlar moUon into alternating rectilinear motion. Examples. I. A bar, AB, moves in one plane with given angular velocity round an axis fixed at A, while at B it is freely jointed to another bar, BC, whose extremity C is constrained to move along a fixed groove, AD; find the Telocity of C in any position. [Crank and Connecting Rod.] The instant.aneous centre, /, for the bar BC is the point of intersection of AB with a perpendicular at C to AD. Now if a is at any instant the angular velocity of BC, the velocity oi BSsai . BI, and that of C is (u . CI; therefore Telocity of C _CI _PA velocity ofB~BI~AB' if CB is produced to meet AP, the perpendicular to AD at A, in P, If n is the angular velocity of the bar AB, the velocity oi B is Ci . AB ; therefore the velocity of C is H . PA, and this velocity will be a maximum when PA is a maximum, if il is constant. Let us find the position in which the ratio of the velocity of C to that of -£ is a maximum. Now when AP is a maximum, it is equal to its next consecutive value 48 Displacement and Velocity. (as far as the first order of small quantities), i. e., the point /" is for a moment stationary, or in ■'•■^ other words, P is at the moment the point of con- tact oi SC with its en- velope ; i. e. (Art. 33) -P is the foot of the perpen- dicular from / on £C. To determine this po- sition, let /JBAC = ^, Z.BCA = e, AB = a, BC = c. Then PC^ACsscB; '~-.. ,.'' but, the angle at P being Fig. 23. right, \PC = C/. sin 6, and C7 = ^C tan ; therefore AC s£cB = AC sine tan ^ ; or cot <^ = sine cose. (i) Also osin^ = csTne, (2) from which equations

, respectively ; that is to say, the portion of the curve taken will serve for all crank and connecting rod systems that can occur. Now take the point C on Ox, such that OC = ^, and construct an ellipse, ABQ, whose semi-axes, CA and CB, are f V 5 and j-V^S, respectively. _ 'V^27 Then for any given crank and connecting rod, draw DH = ■, ', draw HP parallel to Ox; from P draw the ordinate PJ, meeting the ellipse in Q, and the length QJ will be, on the scale adopted, the cosecant of , the angle corresponding to the maximum velocity ratio. The proof of this may be left as an exercise to the student. 2. If any two points of a plane figure are guided, in any manner, along any two right lines in the plane of the figure, the Space Centrode and the Body Centrode are circles. For, let F (fig. 25) be the moving figure, two points, P and Q, of which are guided along the fixed lines 0^ and 0^. Then /, the instantaneous centre, is the point of intersection of two perpendiculars, PI and QI, to OA and OB. Describe a circle round the quadrilateral IPOQ. Then PQ is a given length, and the angle, POQ, which it sub- ''2- ^^^ tends at the circumference of this circle is given ; therefore the length of the diameter, OT, is given. That is, the point / is always at a given distance from the fixed-space point 0. Hence C, is a circle whose centre PQ is and radius -; — =Tr-L . smPOQ To obtain Cj express a relation between / and some invariable particle, or line of particles, in the body. Now PQ is such a line of particles, and / is a point in the body such that PQ subtends a constant angle at it ; i. e., the body locus of / is the circle QIPO, whose diameter is half that of C,. 3. One point of a plane figure is guided along a fixed right line in its plane with constant velocity, while the body rotates with constant angular velocity ; find the Centrodes. Ans. C, is a right line parallel to the given one, and Cj is a circle having the guided point for centre. E 50 Displacement and Velocity., 4. One point of a plane figure is carried round a fixed circle in its plane with constant velocity, while the body rotates with constant angula? velocity; find the Centrodes. Ans. If V is the constant velocity of the tracing point G, a the angular velocity of the figure, a the radius of the given circle whose centre is ; then C. is a circle with O for centre and radius 3; while Cj is a circle V with G for centre and radius — 03 5. One point of a plane figure has harmonic motion in a right line in the plane of the figure, while the body rotates with constant angular velocity ; find the Centrodes. Ans. If G is the tracing jioint moving along the line OG, O being a fixed origin, and if OG = asvaij.t while is the angular velocity of the crank, the angular velocity of the connecting rod is j • ] 11. If a lamina moves so that two lines fixed in it always touch two circles fixed in space, prove that Ci and C, are circles. [This is at once reducible to example 9.] 12. If a lamina moves so that a right line fixed in it always touches a parabola fixed in, space, while another right line fixed in the lamina at right angles to the first always passes through the focus of the parabola; prove that C, is a parabola, and that Cj is the curve whose equation is mr=^, where r is the distance of any point on the curve from the point of intersection of the above two right lines, and p is the perpendicular from the point on the first right line, 4OT being the latus rectum of the given parabola. 13. One point, A, of a lamina is carried with uniform velocity, v, along a right line Ox fixed in space and in the plane of the lamina, while the lamina performs small angular oscillations about A, defined by the equa- tion r i^y\ this dsc dx component becomes — + A — j the gain of velocity parallel to dx the axis of x being therefore A — ; and as this gain is made in . . dx the time A/, it is made at the rate of — r — units of velocity per unit of time. This in the limit is di' Similarly the acceleration parallel to the axis oiy is — • The time-rate of increase of any quantity whose value changes with the time may be found by the following simple rule which covers all possible cases — Write down the value of the quantity at the point t; then write down the value of the quantity at the end of a very short interval of time, A// subtrcu:t the first value from the second, and divide the difference iy At. Acceleration, 53 The limiting value of the fraction thus found, when A/ is in- definitely diminished, is the time-rate of increase of the quantity at the tim£ t. Thus, if the temperature is rising continuously at a certain place, and we wish to know its hour-rate of increase at 12 o'clock, we shall get a good idea of it by subtracting the temperature at 12 o'clock from the temperature at 5 minutes past 12, and multiplying the result by 12 (or dividing by -^); but we should get a better approximation still by subtracting the temperature at 12 from the temperature at i second past 12, and multiplying the diflference by 3600 (or dividing by 360o)- 36. Proper denomination of Acceleration. Acceleration is, then, velocity added (or subtracted) per unit of time; so that if length is measured in feet and time in seconds, velocity will be feet per second, and acceleration will be (feet per second) per second ; or, as we shall write '-A, feet per second per second. A minor inaccuracy in speaking of velocity and acceleration must be corrected. It is not at all uncommon to hear that 'velocity is space described in the unit of time,' and that ' acceleration is the amount of velocity accumulated in the unit of time.' This is quite an erroneous way of speaking. The unit of time may be of any magnitude — a day, a month, a year, or a century — and in the unit of time the motion con- sidered may have passed through several positive and negative stages of velocity, and at the end of the unit of time the velocity may be exactly the same as at the beginning. The truth is that velocity and acceleration have each reference to a particular instant, and each is time-rate of increase at that instant. Now if a train is moving past a station at a definite instant at the rate of 50 miles an hour, it by no means follows that 50 miles will be the distance travelled over actually in the next hour — for the train may stop in this time. If we regard velocity (as, of course, we must) as a rate of describing length, we shall get perfectly consistent and correct results in speaking of its velocity whether the unit of time is one hour, one second, or one century. 54 Acceleration of Velocity. And similarly for acceleration — and, indeed, for every other rate. Beginners are exceedingly prone to speak of acceleration as 'feet per second.;' i.e., as velocity. The example of the above train will show the meaninglessness of such language. Thus, suppose the above train to be moving faster and faster. Now it is a perfectly intelligible question — at what rate per minute is its velocity increasing f Supposing that its velocity is measured in miles per hour, the question is simply — how many miles per hour are being added to its velocity per minute .' And this is tantamount to saying that we are here measuring acceleration as miles per hour per minute, and not as miles per hour. Accele- ration, then, must be spoken of zs, feet per second per second, or miles per hour per hour, or by some other equivalent expression. No concise names are in common use for the unit velocity and the unit acceleration. Professor Lodge in his Elementary Mechanics proposes to call them a speed and a hurry, re- spectively; so that the acceleration g would be concisely described as * 32.2 hurries,' (about). 37. Acceleration of motion of a falling body. The simplest acceleration is, of course, an acceleration of constant magnitude ; and of such the simplest example is that afforded by the motion of a body falling near the earth's surface in vacuo. The force with which the earth attracts a small body (the weight of the body) does not sensibly vary if the body is carried some few hundreds of feet above the surface ; so that if such a body be let fall in an exhausted tube, it will all through its fall be acted on by a constant force, there being no air to resist it with a force which would vary with the velocity. ■ The effect of this constant force is to produce a constant acceleration, which is always denoted by the letter g, the acceleration being roughly at every place 32.2 feet per second per second; but the exact value depends, of course, on the position of the place on the earth's surface. 38. Diagram of accumiilated velocity. If at any instant a is the acceleration of the velocity of a moving point in any Acceleration along the Normal. 55 direction, the increment, ^v, of velocity acquired in the time A/ is given by the equation Av = aAi' and the whole velocity accumulated in any time in the given direction is given by the equation the integral being taken throughout the given time. Hence if we take two rectangular axes. Ox and Oy {Hg. i, Art. 2), and if the successive values, OM, &c., of t are laid down on Ox, and the corresponding values, MP,' Sec, of the accelera- tion a drawn perpendicularly to them, we shall have a curve, AB, of accelerations, and the area aABd of any portion of it will represent the velocity accumulated in the given direction in the time interval represented by a6, 39. Acceleration along the Wormal. When a point moves along any curve it has at each instant an acceleration along the normal, which is best found by the rule just given. Let P be the position of the moving point at any time, /, Q the position which it occupies -after the interval Ai, PN and QN the normals at P and Q, PT and QT' the tangents at P and Q, V the velocity a.\.P,v+Av the velocity at Q, Ad the angle be- tween the tangents, or normals, at P and Q. We wish to find the ac- p-jg. 26. celeration along PN'. Then, velocity along PN at time / = smce -t; = ». •p At p dt 5^ Acceleration of Velocity. Students sometimes feel a difficulty in understanding how a point P moving along any path can have an acceleration along the normal at P since at no point of its path has it ever any velocity along the normal. A familiar example ought to remove this difficulty, by pointing out the fact that — it does not follow that because a moving point has at a particular moment no velocity along a certain direction, it has no acceleration in this direction. Take the case of a stone thrown vertically up. When it is at its highest point, it has no velocity; yet, not only then, but at every instant during its upward (and, subsequently, downward) motion, is velocity being generated in it in the downward sense at about the rate of 32-3 feet per second per second. Hence even at the instant at which the velocity of the stone is zero, it has a downward acceleration, g. Similarly, when the moving point is at P (fig. 26) it has no velocity along iW; but when it gets to Q, it has acquired a component of velocity along PN — not along QN, of course. 40. Acceleration along the Tangent, The acceleration of a moving point along the tangent to its path at any instant is di) d^ s -J- > or -j-^ , where s is the leftigth of the arc of its path measured from some fixed point on it up to the position, P, of the moving point (fig. 26). For, velocity along PT at time t+At is (w + A») cos AO, or v+Av; and velocity along PT at time t is v; gain of velocity in time At Av dv ■■• AT = a7 = ^"'^ *^ ^™^'- The acceleration along the tangent may also be written dv "Is' dv dv ds dv ., . smce i'^ = "^=j~^^ = ^j~' A.ISO It may be written d-'s ^' ds . dv d^s smce « = -, and- = -. Resultant Acceleration. 57 The student must not imagine that the expression — for the acceleration along the tangent is evident without calculation d'^x because the a.cceleration along the axis of x is —r-^ • For, the latter expression holds only because the axis of x is a fixed line in space, whereas the tangent is a line whose direction is perpetually changing"; hence an expression which holds for acceleration along the first would not necessarily hold for acceleration along the second. The application of the rule of Art. 35 shows that the expression for acceleration along the tangent happens to be the same as that for acceleration along a fixed direction ; but, in general, it is not true that the accele- ration along a changing direction is the second time-rate of increase of the co-ordinate of the moving point in the ditection. CoR. r. It therefore appears that if the point travels along its path with constant velocity, the acceleration along the tangent is constantly zero, and the resultant acceleration is along the normal. Cor. 2. In general, the resultant acceleration of the moving point is inclined to the normal at an angle whose tangent is p dv V ds 41. GrapMe construction for Besultant Acceleration. Through any point, P (fig. 27), draw a right line, PT, parallel to the direction of the velo- city of the moving point P in fig. 26, Art. 39, at the time /; and let the length PT repre- sent the velocity, v, of the moving point at Fig. 27. this instant. At i' draw also i'r' parallel to the direction, QT" (fig. 26), of velocity at the time /+ A^, and let the length PT' represent the velocity, v + Av,ztQ. Then TT' is parallel to the direction of the resultant accele- ration of the moving point. For, the component of TT' in any direction is obviously the component oiPT' — the component of /Tin that direction; 58 Acceleration of Velocity. but the difFerence of these components, divided by A/, is, according to the rule of Art. 35, the component of the resultant acceleration. Hence TJ^ is the direction of this latter, and the magnitude of the resultant acceleration is the limiting value of the ratio XT' ~Kt' The vector XT' a7' in its limiting value, is what Newton means by the term change of velocity. 42. Acceleration in a changing direction. Let KH (fig. 28) be a right line whose direction alters continuously according to any assigned law, and let it be required to find the expression for the component of the acce- leration of the moving point P along KH. Let -^ be the angle, Fig. 28. PTH, between the direction of motion at /"and the line KH; let KH become KH' in the time A/, while the direction of motion of the point has become QT' ; » = velocity at P; w + A» = velocity at e ; i// + Ai/r = Lqt'h' ; Ax = LlfKH, X being the angle which KH makes with a fixed right line. Adopting the rule of Art. 35, take the component of .» + Ao at Q. along KH; this is (w + Aw) cos (t|^ + A\/f + Ax) ; and the component oiv 2A P along KH = w cos ■\|f. Hence the required component of acceleration is the limit of (w + Az')cos(t/^ + Ai/f + Ax) — w cos t/t a7 (w + Aw) cos (i/f + A\//) — w cos \/f — w sin i/f Ax = -^^ A(w cos\/^)— w sin i/fAx " a7 (a) Central Acceleration. 59 in which reduction we have rejected the squares and products of the infinitesimals Aw, Ai/r, Ax- 43. Acceleration along and perpendicular to the Badius Vector. The expression (a) of last Article can be directly applied to find the components of acceleration of P along and perpendicular to the moving line OP (fig. 2, p. 3). Identifying OP with the line ICS in fig. 28, we have x = ^ = dr the angle between OP and a fixed initial line ; also v cos \jf= — dd (Art. 4), and wsin>/f = r — • Hence the acceleration along the radius vector OP is, by (a), --ri^Y. iy) dt^ ^dt' ^'^' Identifying the perpendicular to OP at P with the moving direction JCff, we have X = ~ + ^ > ^ <^°s '*/' = velocity perpen- ,. , de ^. , , . . dr dicular to OP=r —; —z)smYf = velocity along OP = -j- Therefore the acceleration perpendicular to OP is d, dB. drdd i-(r''-). (8) r dr- dt' ^ ' 44. Central Acceleration. The particular case in which the resultant acceleration of a moving point is always directed towards a fixed point or centre is deserving of special notice on account of the part which it plays in kinetics^ — as, for instance, in the theory of planetary motions round the sun. If the resultant acceleration is towards the fixed point 0, there will be no component perpendicular to the radius vector ; so that from (6) of last Article we have — ( r" — ) = o, dt^ dt' ' or .^f=A, (I) where h is some constant. To give this equation a geometrical interpretation, observe that if P and Q are two close positions 6o Acceleration of Velocity. of the moving point, separated by the small time-interval A/, and if A9 is the circular measure of the angle QOP, the ex- pression \r^N9 will be the area, QOP, traced out round in the time A/; so that \r^-r is the area traced out per unit of ' . ^ dt time in the position P of the moving point. Hence equation (i) gives as the essential feature of motion in which the resultant acceleration is always directed to a fixed centre the property that — the time-rate of description of area round the fixed centre is constant in all positions of the moving point. The equation (i) can be put into another form which is useful. If Aj = length of the elementary arc PQ, joining two very close positions of the moving point, and if/ is the perpen- dicular from the pole on the right line PQ. (which is ulti- mately the tangent to the path of the moving point at P), the area QOP'\% \p^s- Aj , so that the time-rate of description of area is \p -r- > or \p>v, where » is the velocity along the tangent at P. Hence (i) is equivalent to py = k, (2) where h is the constant expressing double the area traced out per unit of time about the pole. This result can be deduced by another method. We know dv that the resultant acceleration of P can be broken up into v — along the tangent and — along the normal; and since the resultant is directed towards 0, these components have equal and opposite moments about 0. Hence dv v^ pv— p cot

; ^ = LtP0. Then if w = ve- locity at /", /} = radius of curvature Fig. 29. at P, the acceleration of P towards is — cosec A, since the 9 r h resultant is along PO. But cosec = -> and w = - j therefore p p the resultant acceleration is (3) AV h'' dp -5—) or — r-;— • p^p />* dr Of course the expression (y) of last Article holds whether the acceleration is central or not. Again, the expression (y) of last Article may be put into a slightly different form which is useful in the Lunar Theory. If we denote - by u, we have -=- = -— ■; and if (as is r ■' dt u^ dt the case in all central acceleration) r constant, ' d ^ d , ,. dr , du -rr = hu^-T^; therefore -7- = —h^-, dt dQ dt do de Yt h, where ^ is a d^ dt' '' ifl tr — : d6^ so that the expression (y) becomes — A'w'^fM H-^ttIt')) which is the acceleration measured outward along the radius vector OP, i. e., from to P. Hence A'«H« + ^") (4) 62 Acceleration of Velocity. will express the 'acceleration of the moving point P measured from P towards the centre O. de If the acceleration is not central, r^ — - will not be constant at throughout the motion ; but we may still put h for it, observing that h is variable, and the component acceleration along the line PO from P towards is expressed as 45. Case of Direct Distance. Let the central acceleration be directly proportional to the distance, PO, between the moving point, P, and a fixed point, 0. Suppose the acceleration to be equal to it.r ; then if {x,y) are the co-ordinates of P referred to two rectangular axes through 0, we have dfi = -lix, d^y ■ = -ijy. Now each of these denotes a simple harmonic vibration the period being (see p. 26) — =; and, as shown in p. 19, the re- suiting motion is motion in an ellipse having for centre, the time of revolution being, of course, the same as that of each constituent harmonic vibration. 46. Hodograph. It appears from Art. 4 r that if through any point (fig. 30) we draw two right lines. Op and Og, parallel to the tangents, PT and QT', to the path AS of a moving point, at two very close points, P and Q, the lengths of Op and Og being proportional on any scale to the velocities at P and Q, the line />g will be parallel and proportional to the acceleration of the moving point at P. Suppose this process to be continued— in Fig- 3°- Hodograph of a Central Orbit. 6"^ other words, suppose that at we draw an infinite number of radii vectores, such as Op, each being parallel to a tangent to AB and proportional to the corresponding velocity of the moving point P; then we shall thus trace out a curve, ab, whose tangent at any point, p, is parallel to the direction of the resultant acceleration of the moving point P at the correspond- ing point on the curve AB. The curve ab thus obtained from AB is called (rather inappropriately) the Hodograph of the given pq motion. By Art. 41 it appears that ^ , in the limit, is the magnitude of the acceleration of P, where A/ is the time of moving from P to Q. Hence the velocity with which p describes the Hodograph is the acceleration of P. 47. Hodograph of a Central Orbit. By a Central Orbit is meant an orbit described by a moving point whose resultant acceleration is in every position directed to a fixed point or centre, 0. The characteristic property of such an orbit is (Art. 44) expressed by the equation pv = h = 2. constant, where/ is the perpendicular from on the tangent to the orbit at any point, P. Now about as centre describe a circle whose radius is */h; let fall the perpendicular 07'(fig. 29) on the tangent, PT, at P; and produce or to G so that OQ.OT=h; then the locus of Q. is the polar reciprocal of the given orbit with respect to the circle of radius '/h. But OT=p, therefore OQ = V, so that the polar reciprocal constructs, by its radii vectores drawn from 0, the velocities of the moving point P in directions always at right angles to those of the velocities. Hence if we turn the polar reciprocal round through a right angle, it will be the Hodograph of the motion. In this particular case, therefore, of central acceleration, the Hodograph and the polar reciprocal of the orbit about O are virtually the same. Of course no such result holds when the motion of P takes place without acceleration towards a fixed centre. If r, p, p denote the radius vector, perpendicular from on tangent at P, and radius of curvature at P, and /, /, p' the ©4 Acceleration of Velocity. corresponding things for the Hodograph (ot polar reciprocal) of a central orbit, it is at once seen that h h , hr' r = -, p = -, p= -— 3. / r '^ pf Observe that the tangent at Q to the polar reciprocal is ob- viously perpendicular to OP, and as by rotation this becomes the tangent to the Hodograph, we have the tangent to the Hodograph parallel to the radius vector OP, i. e., parallel to the direction of resultant acceleration oiP — as we know (Art. 46) it must be in all cases, whether of central acceleration or not. 48. Case of Inverse Square of Distance. Let the accele- ration of the moving point P be always directed to a fixed point, (fig. 29), and let it be equal to — ^j where /x is a constant r and r = OP. It is required to find the orbit. Describe the hodograph about (fig. 30). Then r^~ = h, therefore the ac- u.dQ celeration = t -vr > but the acceleration of P equals the velocity of ^ and it is directed along the tangent to the hodograph at p — which is parallel to PO; therefore — is the angular ve- locity of tangent at p. Hence we have velocity of * u ; ; — : 1: = -7 = constant : angular velocity of tangent at/ h in other words, the radius of curvature of the hodograph at / = T = constant, therefore the hodograph is a circle. The orbit of P is then (last Article) the polar reciprocal of the circle with respect to O — i. e., it is a conic having for focus. (See Hamilton's Elements of Quaternions, p. 720.) 49. Accelerations of I^ and /,. Let a, /3 be the co- ordinates, with reference to fixed axes, of any point, G, of a body moving with uniplanar motion. Then, as in Art. 29, the^*' co-ordinates {x, y) of any other point, P, in the body will be ■ given by the equations x = a^-rzo%6, j' = /3 + rsinfl. Accelerations of /j and I^. 65 Differentiating these twice with respect to the time, on the supposition that r is constant, we have X = a—(i?r cos 6— iarsva 6, (i) j' = i3 — to'/- sin 5 + air cos 9, (2) denoting, as in Art. 29, time-rates by dots. Now if P is /j, we, know that r cos 9 = — -. r sin = - CO CO (Art. 29) ; and substituting these in the above expressions, we obtain ;.^ . (3) (4) x='d d + £oS, CO j't=)3 j3 — coct, for the components, in fixed directions, of the acceleration of /j. The components of acceleration of /, are obtained by differ- entiating the velocities of /, in Art. 31. Thus ^-'^-ttAJ' (5) -=^+^0- (6) For clearness we collect in a table the co-ordinates, velocities, and accelerations of /(, and /, , in terms of the motion of some one point, G, in the moving body. ^ /. Co-ordinates. ■, a-S. CO CO CO Velocities. ■ ■'A Accelerations. . CO ; d a-}- CO/3, . 1 CO 8 iS— coa. CO -J.6. ^-^.& 66 Acceleration of Velocity. The components of acceleration of 7j, can also be written a,[^+^(j] and - a> [d - |, (f )] , and comparing these with the velocity components of /,, we obtain the result that — The direction of acceleration of /j is at right angles to the direction of velocity of I^, and the magnitude of the acceleration of /j, is equal to the velocity of I, multiplied by the angular velocity of the body. 50. Instantaneous Acceleration Centre. We have seen that in one-plane motion there is at every instant a body-point, /,„ with no velocity, and that the velocity of every other point, P, takes place at right angles to the line PI-^ and is proportional to the length of PI-^,. We proceed to show that there is also at every instant a point, J, of no acceleration, and to deduce analogous results for it. Putting- x = o,y = o'vi\ equations (i) and (2) of Art. 49, and using f and tj for r cos 9 and rsin^, we get u?a—k'^ d)a + a)°/3 for the co-ordinates, referred to the arbitrary point Gj of the point, y, of no acceleration. In uniplanar motion there is, there- fore, always such a point, since we have obtained definite values for f and rj. The expression for the acceleration of any point, P, may now be simplified by choosing y as the point G of reference. Equa- tions (i) and (2) of Art. 49 will become (if a = o, ^' = o) J?= —ay'^—car], y = oi^—at^r). Hence Vx^'+y^ = Va^ + w^ . Py, which proves that the magnitude of the acceleration of every point is proportional, at each instant, to its distance from the point y, which is called the Instantaneous Acceleration Centre. Again, if we put —^ = tan <^, and '/a)*-fa)= = k, we have X = —h{^cos^ + ri sin<^), y= h{^ sin <(>—r] cos <^). Instantaneous Acceleration Centre. 67 Denoting /"yby r and its direction angle by 0, these b3come jf = — ^r cos(^— <^), y = —kr sin(5— (^). These values show that the di- rection of acceleration of the particle at P is the line PA drawn at the angle ^, whose tangent is — ^i with yp. The sense in which o) is mea- sured determines the side of the line JP at which we draw tan'^ -^ . If m -^'S- 31 ■ a? is measured in the sense denoted by the arrow, while o) is nega- tive, PA will be drawn at the under side of JP in the above figure. Now ^ is at each instant the same for all particles. Hence — in all uniplanar motion there is at each instant a point having no acceleration (the instantaneous acceleration centre), and the acceleration of every other point is in magnitude proportional to its distance from the acceleration centre, while its direction makes with the line joining the point to the acceleration centre an angle which, though varying with the time, is at each instant the same for all points in the moving body. If we are given, in magnitudes and directions, the accelera- tions of any two points, P and Q, the point J can be found. For, let the directions, PA and QB, of these accelerations meet in O, while the accelerations are p and q. Then the angles OPJ and OQJ are equal ; therefore the circle round the triangle OPQ passes through J. ■b fP Again, - = '^—; therefore J lies on the circle which is the 9 7Q locus of points whose distances from P and Q are in the given ratio — ■ One point of intersection of these two circles is y. CoR. If the angular velocity of the body is constant, the accelerations of all points are directed to one point. The whole system of accelerations of the various points of F 2 68 Acceleration of Velocity. the body at any one instant we shall call an acceleration system of the moving body. There will, of- course, be a fixed-space locus and a body-locus of J, which we shall call the Space Acceleration Centrode and the Body Acceleration Centrode, respectively. 51. Theorem. Any two possible acceleration systems in uni- planar motion are superposahle in a single acceleration system. To prove this we shall show that if at any point, P, there act two forces, PA and pa', of magnitudes \i. . PJ and /x' . PJ', respectively, and making angles, JPA and y'PA', constantly equal to a and a, wherever P may be, y and y' being two fixed points, the re- sultant of these forces will, in all positions of P, be proportional to the distance of P from another fixed point, JC, and will make a constant angle with PK. Replace PA and PA' by their components along and per- pendicular to Py and Py', respectively. Then we have two forces, fj. cos a. Py and fj.' cos a' . Py', along Py and Py', respectively ; and also two forces, /i sin a . Py and jj.' sin a' . Pf, perpendicular to Py and Py', respectively. Now (see Statics, Art. 20) the first pair give a resultant equal to (jj. cos a + i/ cos a') . PG directed from P to G, where G is the point dividing yy' so that yG _ fJ.' cos a yd Fig. 32. - ; and in the same way the second pair of forces fi cos a give a resultant equal to {y. sin a-\-y.' sin a) . PCf acting perpendicularly to PG', where C' is such that yc fi' sin a' yG' fj, sin a Denote /n cos a-\-y.' cos a by 2ju cos a, and ft. sin a + iif sin a by 2/x sin a. Now take G as origin and GG' as axis of x, and Accelerations of different orders. 69 let X and Y be the components of these forces along and perpendicular to GC'. Let x and y be the co-ordinates of P, and let C(f equal D, then X = x'2,\i. cos a —y^n sin a, K= (j;— Z>)2jii sin a+j'SjU cos a. Now a force consisting of components m cos (^ . P/^ and »2 sin (^ . /"jT along and perpendicular to PJiT would give components « cos . (^-f)-w? sin (j) . {y-v), m sin (j) . {x —^ + m cos . {y—v), in these directions, if the co-ordinates of IT are f, tj. Identifying these with X and y, we have m cos (p = 'S,n cos a, m sin (^ = '2^ sin a, f = Z> sin" (j>, 7j = Z> sin (^ cos <^. Hence the reduction is possible and the point HT is found. To construct ^, we see that we must describe a circle on GG' as diameter, and from G' draw a line making the angle with the positive side of GG' (if tan ^ is positive) ; this line cuts the circle in J^. Also we have for the constants m and (p m^ = {Sfi cos af + (Sjm sin a)', Su sin a tan(^= =^-^ Sju cos o Of course we can regard PA and /M' as two accelerations, and since JiT is not dependent on the point P, it follows that an acceleration system with constants /J. and a whose centre is y and another with constants j/- and a' whose centre is y' com- pound a single acceleration system with constants m and (j> (given above) whose centre is J^. In the general, or three-dimensional, motion of a rigid body the superposition of two acceleration systems, each of which is separately possible, is not possible. 52. Accelerations of different orders. The resultant of d^x d^v —rr and -^ has been called the acceleration of the point '^^ ^^ ■ d^x , x,y. In the same way we may take the resultant of -^ and d^V -jj ) and regard it as the acceleration of the acceleration, or 7© Acceleration of Velocity. an acceleration of higher order ; and similarly the resultant of — T- and — ^ is an acceleration of still higher order. For if through any point, P (fig. 27, p. 57), we draw FT and PT' parallel and proportional to the accelerations of a moving point at the times t and t+^i, respectively, the line XT' will, for the reason explained in Art. 41, be parallel and proportional to the change of acceleration, and the direction cosines of XT' will d^x d^y be proportional to -j-^ and —fj- 1 since the co-ordinates of T ** *^ d'x ^ O'y (referred to axes through />) are proportional to -j^ and -^ ) and those of T' are proportional to d'^x d^x . , , d'^y d^y ^ ^+^A/, and J + ^A/. 53. Theorem. For acceleration of any order there is in uniplanar motion of a rigid body at each instant a centre or point of no acceleration of that order ; and the accelerations of all points are related, in magnitudes and directions, to the centre exactly as the velocities and ordinary accelerations are related to the instantaneous centres of velocity and acceleration. For if we express the motion of every point in terms of the motion of some one point, we have, as in Art. 29, x-a-^r co%Q, _>» = y3 + r sin 9. d'^x Hence, denoting.-—- by .arW, &c., jf («) = a (") + mr cos Q + m'r sin Q, yW = /3(")-|-otV cos e—mr sin e, . d^'icose) ^ . , . . denotmg — ^— — by mcos0 + m sm 5, and observing that sin5= — cos(- + 0)- Now xW and yW will be zero for the values mam+m'Bm ^ mBn-m'aW ; 7? — and —!—— -, m^ + m'^ m^ + m'^ of r cos and r sin 0. Hence there is a centre of accelera- tions of this order, and by taking it as point of reference we may put, as in Art. 50, aW = /3(") = o, and Rolling on a Fixed Surface. 71 and it follows, exactly as in the case of the ordinary accelera- tion centre, that the resultant of x ("^ and _y W is proportional to the distance of the point x, y from the centre, and that it m' makes with this distance an angle whose tangent is — > which is the same for all particles in the body. 54. Boiling on a Fixed Stirface. When a body, M, rolls on a fixed surface, AB, the point, 7, of the body which is at any instant in contact with the fixed surface has an acceleration which we proceed to find. We suppose the rolling to be unaccompanied by slipping, so that at the end of the element of time A/ the point L of the body will be that which is in contact with AB, and it will occupy the position // on AB, such that the length of the arc IL measured on the surface of the rolling body is equal to the length of the arc 11! measured on the fixed surface. The figure represents sections of the body and sur- face by the plane of motion. Consider the point I of the body which at the time / is the point of contact, and which at the time /+ A/ comes to /' such that IL = fL'. The point L' is the instantaneous centre at the end of th^ time A/, so that /' is moving at right angles to I'l', its velocity being w . i'l!, where co = the angular velocity of the body at the instant under consideration. Let us find the acceleration of the point, /, of the body along the normal /A'' to the, surface of contact. This normal becomes i'n' by rotation. Draw the normal at Z,', and let it meet I'N' 7 a Acceleration of Velocity. in C and IN in C', Then C is ultimately the centre of curvature of the contour of the rolling body at /, and c' is ultimately the centre of curvature of AB ; so that if p is the radius of curvature of the body at 7 and p' that of the surface at /, we have p = tc, p' = icr. Denote the angles I'CL' and IC'l' by M and M. Then the angle through which the body has turned in the time A/ is 69 + 80', since it is the angle between INz.-aA i'n'. Now, velocity of / along 7/V, at time /, = o, velocity of /' along IN, at time t+Af, = m.I'l'. cos {86 + be') = a>.l'l.', nearly, =o).p85; gain of velocity bd dd . ^, .. . .: — T- = top -r- = top -TT » in the umit. A/ ^ A/ dt 8/94- 8^ But to = limit of — — — i and pbO = p'bO', since the arc i'l' = arc IL'-, , p.dd d6 a> therefore the acceleration of the point /of the body along IN\s — I I or hta^, if we put -r for - + -,• h p p If the concavity of the fixed surface is turned upwards, or in the same direction as that of the rolling body, the acceleration of the point of the body in contact is — P P' for it will then be obvious that bd—bff is the angle turned through by the body in the time A/. There is no acceleration of the point / along the tangent, for gain of velocity along tangent u).i'L' . sin (60 + 89') 2W " AV o^pbd{bO + bff) A/ = o, in the limit. Examples. 73 This expression for the acceleration of / holds only when the motion is pure rolling, i.e., when the point / of the body does not slip along the surface during the rotation. The student has already seen that the effect of slipping accompanying the rotation is to throw the instantaneous centre above or below the point of contact to a distance V (O along the normal, where v is the velocity of slipping (i.e., where » A/ is the amount of slipping which takes place while the body turns through an angle coA/). Examples. I. A lamina moves in its own plane so that two points in it describe two fixed right lines with given accelerations; find the acceleration of the instantaneous centre. (Wolstenholme's Book of Mathematical Problems, p. 416, 2nd ed.) Let P and Q (fig. 34) be the two points in any position, ..40 and BO the lines along which they move. At P and Q erect perpen- diculars, PI and QI, to OA and OB ; then / is the instantaneous centre for the lamina. Describe a circle round OPQ; it will pass through /. Also if J is the ac- celeration centre, _/ will be on the circle, since (Art. 50) the angles OPJ and OQ/axe equal, and each is tan~' — , where '' sin''' o s and >' being the accelerations of P and Q. Of course J" and s' are not independent. We have s = k.PJ, r = k.QJ, where k = -^aP^a^; and also PJ^-iP/ .QJxos,a-i-Q_P = a', .•. P— ii's' cos a + y = k'^a'. ' i. If two points in a lamina are guided along any two right lines fixed in space in the plane of the lamina, with accelerations, 's, s', satisfying any fixed relation of the form /(— ) = o, the acceleration centre is an in- variable point in the body, and also the acceleration of any one particle is always in the same direction in fixed space — the direction being different for different particles. ' Firstly, J (fig. 34) is always the same particle of the body. For, the accelerations, s and s', of P and Q being connected by a fixed homogeneous relation of the form /(— ) =0, we have the distances PJ and QJ con- nected by the equation /( ^^) = o, which determines a body-locus of J. But the circle POQ is an invariable circle in the body, since it is one having a fixed body-line PQ for a chord, and this chord subtends a constant angle at the circumference. Hence, in the case supposed, J will be a point of intersection of this circle with the other body-locus; it is therefore always the same particle of the body. Secondly, let M be any particle whatever. Draw MJ meeting the circle in m. Since Mani/ are two invariable particles, the lint MJ is invariable in the body, and so therefore is the point m ; so likewise is the line Pm ; and therefore the angle POm which Om makes with a fixed-space line OP is constant because it is that subtended in the circle by the invariable length Pm. Now at any instant the angle which the direction of accele- ration of M makes with M/ is equal to LjPO, which is equal to the angle JmO; therefore the direction of the acceleration of Mis parallel to mO — Examples. 75 which, as we have just proved, makes a constant angle with a line fixed in space. Therefore, &c. For the particular case in which s = s' see Wolstenholme's jBeok of Mathematical Problems, ex. 2430, p. 417, second ed. 3. Two right lines in a lamina are kept constantly passing through two points fixed in space, while the displacement of the body is made with constant angular velocity; it is required to find at any instant the accelera- tion of any point of the lamina, and also the Acceleration Centrodes of the motion. Let P and Q (fig. 25, p. 49) be the two fixed-space points (which may be two small swivel rings) through which two lines, OP and OQ, of the lamina F are guided. Now the points P and Q being fixed, and the angle POQ being given, the fixed-space locus of / (or C,) is the circle POQ ; and since the diameter PO of this circle is -; — ^-— - , the distance 01 is fixed : so that the centrodes sm/'OQ C, and Cj of the present problem are exactly the reverse of those in p. 49. Now the rotation of the lamina ^ in a small time A? is the angle between OP and OfP, where 0' is a position of on the circle POQ after the time A A The angiUar velocity of F being a, it therefore appears that the angular velocity of about the centre of the circle POQ is 2 a;, and if r is the radius of this circle, the velocity of is 2 ra, so that this velocity is constant. Hence the acceleration of is (Cor., Art. 50) wholly along the diameter of this circle passing through — i. e., along the line 01. Also since w = o, the acceleration centre, J, must be on the line 01 (Art. 50) ; and since the acceleration of is (Art. 39) 40)'?-, we have a''./0 = ^ar'r, This shows that the Body Acceleration Centrode is a circle described round as centre with radius equal to four times that of the circle POQ. Also the distance between y and the centre of the fixed-space circle POQ is 3>-, so that the Space Acceleration Centrode is a circle concentric with POQ. The acceleration of every point in the body is directed towards J (since , ol its resultant acceleration is always a tangent to another curve ab ; 76 Acceleration of Velocity. Fig- 35- find a relation between the length of the tangent from P to the curve ab and the angular velocity of this tangent. Let the line, Pp, of resultant acceleration at P touch ab'mp; let Q be the position of P after the time At, and Qq the line of acceleration at Q touching ab ztq; let fq = da; t = i^; and let dtfi — the angle between Pp and Qq. Then since Pp is the direction of resultant accelera- tion at P, the component acce- leration perpendicular to Pp is zero. This component we pro- ceed to calculate by the Rule of Art. 35. Let V = velocity at /"; v + dv = velocity aX Q; <)> = angle between Pp and tangent to AB at P; , velocity perpendicular to Pp at time t+dt is (o + dv) sin (if + (/^ — dtfi). „ gain of velocity perp. to Pp _ (v + dv) sin(

—dili) — vsin di ^ di ■ and since there is no acceleration in this direction, we must have (v + dv) sin (

= o, or d (p sin Now by drawing from Q a perpendicular to Pp, we have da—dr cot

dt^^^ J But if {x, y) are the fixed-space co-ordinates of 7, or co- ordinates of /,, we have x = a 'J' = 0-l--; therefore to ft) or the tangents to C, and C^ are the same dx d^ line. The lengths, ^dx^->rdy'^ and Vd^^ + drf, of their arcs between two successive instantaneous centres are also the same. Hence Cj rolls without sliding on C, ; or — If a rigid body moves, in any way whatever, parallel to one plane, the motion may be in all respects produced by causing the Body Centrode to roll, without sliding, on the Space Centrode. . Geometrical proofs of this proposition will be found in many works. (See Cliflford's Kinematic, p. 137.) This is the fundamental theorem of Epicycloidal Motion, the consequences of which it is proposed to develop in the present Chapter. 66. Elliptic Compass. If a right line, AB, of fixed length have its extremities carried along two fixed rectangular grooves, OA and OB, and carry a tracing pencil at any point P in its length, this pencil will trace out an ellipse of which the axes are PA and PB. This instrument is called an El- liptic Compass. Let us now replace this motion by the epicycloidal motion of the Centrodes, C^ and C,. The instantaneous centre, /, is the point of intersection of perpendiculars to the grooves at A and B. Also 01 = AB = a constant ; therefore I is always at a constant distance firom a fixed point in space ; again, if M is the middle point of AB, MI =3. constant; therefore the distance of/ from an invariable body-point is constant ; and hence C, is a circle with centre and radius AB, while C^ is a circle of half the radius. The groove motion may therefore be replaced by the rolling of the circle A OB on that of double the radius, the latter being fixed in space, and the point P (rigidly connected with the small circle) will trace out an ellipse. It is not merely one point, P, rigidly connected with the Fig- 36. Oblique Elliptic Compass. 8i rolling circle which traces out an ellipse ; every point, such as Q, also traces out an ellipse. For, draw QM, meeting the small circle in A' and E^, and draw two fixed-space lines OA' and 0^'. Then the particles (or body-points) at A'' and ^' are moving at right angles to A' I and J^I, respectively, 1. e., along OA' and O^ ; hence the line A'^ of invariable length is moving so that its extremities are describing two fixed rect- angular lines, OA' and (7^, therefore the invariable point Q on this right line traces out an ellipse. If the extremities A and ^ of a moving line describe two non-rectangular right lines, the motion may be otherwise pro- duced, as above, by the rolling of one circle inside another of double the diameter; for the student will easily see that C^ and C, are in this case also circles, one rotund ABO, and the other with O as centre — as in the above case of two rect- angular lines. The theorem of the Elliptic Compass may, of course, be otherwise stated thus — if two points of a lamina are made to describe two fixed right lines, every point in the lamina describes an ellipse. 57. Oblique Elliptic Compass. Given two right lines, OA' and OB" (fig. 37), it is required to trace out by continuous motion an ellipse of which OA' and OB^ are semi-conjugate di- ameters in magnitudes as well as in directions. q Ip A' Suppose a triangle P'(^0^ to move with the extremities, P' and (^, on the '^' ^'' lines OA' and OB^ ; then the locus of its vertex, (f, is an ellipse whose centre is O. If we take OA' and OB' as axes of xsaAy; if LS^OA' = a ; if the sides <^cf and P'o' are m and n respectively; and if a+<^(fP^ = 00, the equation of the ellipse is easily found to be x^ 2xy . y sin^o) H ^ cos o)-l r? mn rr sin'' a 83 General Theorem of Epicycloi^dt Motion. Hence OA' and O^ will be conjugate diameters va. directions if o) = -; and the lengths of the semi-conjugate diameters will be and sm a sm a Hence if we wish 0A\ OB' to be the semi-conjugates, we must take m = OA' . sin a, n= OB', sin a. At erect OC perpendicular to OA' and equal to it. Take OP = perpendicular from B' on OA' = OB^ sin a ; draw /"g parallel to B'C. Then the triangle BQO is the one whose motion traces oiit the required ellipse, as is very easily seen. A piece of paper cut into the size of this triangle can be ■ Bsed with great ease for describing the curve. The ordinary elliptic compass — that in which the lines OA' and OB' are the axes of the ellipse — is a particular case of the above ; for then BQ becomes the difference of the axes and the tracing triangle becomes a right line. 58. Area of any Rou- lette. If any curve roll without slipping on a fixed curve, the curves traced out by all points rigidly con- nected with the rolling curve are called Roulettes. Thus, if a circle roll Fig. 38. " along a right line, any point on the circumference of the circle traces out the particular Roulette called a Cycloid. Let any plane curve, C (%. 38), roll on any fixed plane curve, ^B, and let the rolling curve carry with it a point P which traces out a roulette in fixed space. Let / be the point of contact of the two curves at any instant, let L' be a point on C which after a small motion comes to Z on the fixed curve, an4 let Q be the point to which P comes at the end of this motion. We shall regard the area IPQL as the area of the portion of the roulette generated by this motion, and the area of any portion whatever of the roulette will, defined thus, be the area Area of any Roulette. 83 included between any two normals to the roulette and the corresponding portions of the fixed curve and the roulette. This elementary area -area Z/P+area LPQ; and since the distance between L and l! is an infinitesimal of the second order, these points may be regarded as coincident, so that area LIP = area if IP = dc, where dc signifies an elementary polar area of the curve C traced out round P as pole. Again, if dm is the angle between the tangents to C at 7 and' L', and dm the angle between the tangents to AB at 7 and L, the angle between PL (or PL') and QL is d, the angle of rotation of the rolling curve ; so that if PI is denoted by r, the area LPQ = \r^ {dm + dm') = ^r^il, say. Hence, if d^ denotes the element of area of the roulette, we have d^^dC+ \r^ da, ( r). .-. S=C+4/r^^i2. (2) We may, indeed, put dC=^\r^dB, where 6 is the angle made by PI with a fixed line, so that we can write 'S., = \fr^{dd + dQ), and the integration may extend over the whole curve C, or between any two points on it. It is necessary, however, to be more explicit as to what we mean by the area of a roulette, and to point out the fact that portions of the area may sometimes require to be understood as negative.. Supposing that the lines, PI and QL (or QL!), joining two con- secutive positions of the tracing point to the corresponding positions of the instantaneous centre intersect each other at a point, O, in their imf reduced lengths, the element of area which we consider is area 7'GO- area OIL. In fact, area PQL = | rVn, and area PIL (prdC) ^Ir'dB, and to avoid taking the portion POL twice over, we must take the arithmetical difference between these areas; the algebraic sum of them would do, however, and our formula (i) may be regarded as applicable, since if we take any fixed point, P. in the plane of G 2 84 General Theorem of Epicycloidal Motion. any curve, the radius vector from P to the curve must be regarded as tracing out triangular elements of area (such as PIL', or dC) of one sign so long as it is approaching a tangent to the curve from P (by rotation in one direction), and as tracing out elements of area of the opposite sign so long as it is receding from the tangent (by an opposite rotation). In estimating thus the area of a plane curve it is not necessary that we should have a tangent, or tangents, in the ordinary sense of the word ; it is enough that we should have positions of the tracing radius vector in which this line changes the direction of its rotation. The curve may, in fact, be a polygon with sides crossing each other in any complicated manner. The vectorial area of any curve (closed or not) must be understood with this reference. And the rule for estimating not only the element, (f 2, but any integrated portion, S, of the area of a roulette, as used above, is this — Start from the first position, P„ of the tracing point ; follow the roulette round to any point, P„; then move along the normal, P^In, to the roulette as far as In, which is the point on the fixed curve corresponding to the point Pn ; then move along the fixed curve through the points, IN inverse ORDER, which have been instantaneous centres, until the first position of the instantaneous centre, /,, is reached ; finally, move along the normal I^P^ back to the original point, /*, . The vectorial area of the complex path thus traced out, estimated round any point as pole, is the area of the roulette. This rule applies to every case that can arise ; and special attention is directed to the fact that the contour of the fixed curve must be passed over in the inverse order in which its points have been instantaneous centres, because we may have to deal with cases in which the roulette is a closed curve (possibly with several loops), so that we have to move along it until we again reach the original point, P,, before moving along the iixed curve at all. In this case the area 2 may obviously be regarded as the area proper, A, of the roulette itself— without reference to the fixed curve, and taken with positive and negative portions according to the contrary rotations of a radius vector — di- minished by the area proper, S, of the fixed curve estimated in like manner. That is, 2 = A—S. We may now regard the complex function dd + dQ, as a quantity varying from point to point on C, and having for each point on the curve a peculiar value associated with the point, so that the idea Fig. 40. °^ rolling may be discarded, and the curve C may be considered fixed. 59. Areas of different Roulettes compared. Take Areas of Roulettes compared. 85 any other point P', (fig. 40), and consider the area of the corresponding portion of its roulette. We shall have for ^ d^'^d'C^Y^dO., where r = P'l, and d'C is the polar element of area round /^. Draw any two rectangular axes, Ap, ^, at i'; let x, y be the co-ordinates of P' with reference to them, and let PP' make an angle 6 with Px. Then instead of dc (area IPl') we may use \pds, where / = perpendicular from P on tangent at I, and instead of d'C (area IP^l') we may use \p'ds, where J>' = perpen- dicular from /^ on tangent at 7. But obviously, if to = angle between p and axis of x, p' -p—x cos m—y sin co,. .-. d'C = dC—\xQO% and E. Denote them by a and 5 ; then area DP'E = area BPE—iax—^iy. Also r'^ = t^ + x''+y'^—2r{x cos0+y sin d), .: //•'dQ,=fr'da + Q,{x''+y^)-2x/r cos OdD. — 2y.fr sin Qd£L, Q, being fdO, between the points £> and E. Hence I,'=C-lax-i6y + ifr'd£l+iQ.{x^+y^)-gx-hy, (if we denote fr cos ddQ, and frsiaddQ. by g and h), or • S' = S-t-^ii(j;»-f-y)-Cr-l-ia)^-(A + i%. (i) From this expression we derive the following theorem — All points which trace out Roulettes of the same area lie on a circle. For if 2' is constant while P' varies, the above equation (i) shows that the locus of P' is a circle. Also, by varying the area of the Roulette, we get a series of concentric circles. For, only the constant term of the above equation between X and> will change if S' is varied. 86 General Theorem of Epicycloidal Motion. Precisely the same theorem holds for the areas of the pedals of a curve taken with respect to different poles, as has been proved by Steiner. (See Williamson's Integral Calculus, p. 202, third ed.) A direct consequence of the above is Kempe's theorem — viz., if one plane, sliding upon another, start from any position, mov^ in any manner, and return to its original position after making one or more complete revolutions, every point in the moving plane describes a closed curve, and the locus, in the moving plane, of points which describe curves of equal area is a circle ; and by varying the area We get a series of concentric circles. (Williamson, ibid., p. 210.) According to the principle of epicycloidal motion, the motion of the plane may be conceived as produced by the rolling of C^ on C,; every point describes a Roulette, and the areas of the Roulettes are connected by our theorem above. It is not of course necessary for the truth of the theorem that the moving plane should be brought back to its original position; the only effect of bringing it back is to bring the points £ and Z> (fig. 40) into coincidence, i.e., to make a = b = o in equation (i). The centre of the system of circles is necessarily the point which traces out the Roulette of minimum area. If we use it as point of reference instead of /", and if Ji" is the area of its Roulette, the area of any other is given by the equation 2 = ;r+il2(^2+y). If the rolling takes place through the entire length of the curve C, a and b will both be zero, and the centre of the system of circles is such that frcosdd£l = o, frsmddil = o, i.e., this point is the centre of mass of a distribution of matter over the curve C, the density at each point being proportional to dil. 60. Theorem of Holditch. Two points,^j and A^, at a constant distance apart describe two closed curves of areas (c) and (Cj), the line A^A^^ making a complete revolution so as to 'Theorem of Holditch. 87 return to its original position; it is required to find the area of the curve described by any point P on the line A^ A^. The motion may be regarded as produced by the rolling of the Body Centrode on the Space Centrode. Let the. areas of these centrodes be denoted by (Cj) and (C,), respectively, let A^ A^ = /, and let P divide A.^ A^ so that PA^ = nL Then the area between the roulette of A^ (which is the curve C^ and the Space Centrode is (Cj)— (c,), and if r^ denote the distance between ^1 and any point, 7, on the Body Centrode, we have by Art. 58, (Ci)-(c.) = (c»)+i/r,va (i) Similarly (C,)-(c,) = (Ci)+|//-,V12, (2) {X)-{C,) = {c^)+i/r^da, (3) where X = area of curve described by P, r^ = A J, r = PI. Now obviously r^ = {i—n)r^Jrnr^—n{i — ti)P. (4) Multiply (i) and (2) by i— w and n, respectively, and add; then -(C,) + {i-n){C^) + n{C^) = {C^) + \fr-da + iQ.n{i-n)P; therefore from (3) {X) = (i -«) (Ci) + n (C,)-if2« (i -n) P. (5) If the line A^A^ returns to its original position, its total rotation, Q., is 2 ir, and then (;sr) = (i-«)(q)+«(c,)-7r«(i-»)/2, (6) which is Holditch's Theorem. It is to be observed that the area traced out by P is sus- ceptible of a minimum value, the corresponding value of the ratio of the segments into which it divides the line A^ A^ being ■nP + A' where A stands for (C^)— (C^). The theorem of Holditch is proved otherwise thus by Mr. M<=Cay. Let a consecutive position of the moving line^i^^ be A^ A^. From A^ draw A^JO equal, and parallel to A^A^. Then the elementary area ^i^j^/^i' = on A^A^, and if A^Q is equal and parallel to A^P, we have d{{x)-(C^ ^H'^A^Q+ln^PdO.. But nxll"'.A^D= irA^ Q ; therefore nd-{C^)-nd{C^-\nPda = d{x)-d{C^~\n^PdQ,. Integrating this, we obtain equation (6). 61. Extension of Hol- ditch's Theorem. If two vertices, A and B, of a trianglcj ABC, of given magnitude are displaced along two given curves, the third vertex, C, will trace out a curve the relation of whose area to the areas of the two given curves we pro- pose to find. Denote the sides by a, b, c, and the areas of the curves by (a), (b), (C) ; let ^ denote the sum of the areas of the Space Centrode and the Body Centrode (what in our previous nota- tion is C»+ Cj)) ; let / be any position of the instantaneous centre ; AI=r,B/= r\ CI=r"", LlAB = B. Then = >r+ ^/ {r" + 3' - 2 ^r cos (^ - 0)} ; .: c{C) = acosB{A) + 5 cos A (B) + iraic cos C — 3f sin ^/r sin ddil. Now observe that r sin 5 is the length of the perpendicular from I on AB, that the foot, £, of this perpendicular is the point of contact of AB with its envelope, and that if /' is a consecutive position of / and /'^ the new perpendicular on AB from the instantaneous centre, the element of length of the envelope is IPdQ.-\-EBf, or r%\a.QdQ.-^EEf (see Art. 68). If, then, we denote the whole length of the envelope of AB by E^, and by A the area of the trianjgle ABC, we have, since "^EEf = o (the line AB returning to its original position) c{C) = a cos B{a) + 1) cos A {B) + iraic cos C— 2 AS^, (2) or (c) = (/') + w/» _/,£,, (3) where (/") is the area traced out by the foot of the perpen- dicular from C on AB, and p is the length of this perpen- dicular. Equation (3) is deduced in the following way more siortly l.y Mr. M°Cay. It is evidently a theorem relative to a line, CB, of constant length, /, whose extremities, C and B, trace out curves (C) and (/"). Take then a consecutive position, CB', of this line, and at the points B, B' draw two normals to the curve {P), which of course meet in /, the instantaneous centre. Also at P and B' draw two lines BE and B'E perpendicular to BC and B'C, respectively, and meeting in a point E, which is obviously the foot of the perpendicular from / on AE. Finally from B' draw B'D equal and parallel to BC. Then the element of area CBPC, which is denoted by d [(C) -(-^)]. consists ultimately of the parallelogram BCDP' and the triangle B'DC, the area of the latter bemg \^dn, where, as before, dO, = z between /"Cand B'C; and the integral of this in a complete revolution is it^. The area of the parallelogram = px PB' sia BB'D = pti. lEdn, and, exactly as above, the integral of this in a complete revolution = length of envelope of BE, or E„ . Hence {C)-(,P) = iif-pE^, as" before. Mr. Mc Cay observes that equation (2) proves Kempe's Theorem. 9° General Theorem of Mpicycloidal Motion. For if we seek the locus of the point C so that (C) shall be constant, the equation is of the form /(a2+c2_33) + »i(^2 + c!_a2) + «(a;H3''-c^) + '4/ = const, (4) since (a), {b), c, E^ are constants. Referring the position of C to the line AB as axis of x and a perpendicular to it at its middle point as axis of j', we have therefore the locus of C as given by (4) is obviously a circle. Equation (2) gives a result which serves as a verification — viz., if Ea, E^, E„ are the lengths of the envelopes of the sides a, b, c, aEa + bE^ + cE„ = 4irA, (5) which is, of course, evident so long as./ is inside the triangle, since if ^, q, -r are the perpendiculars from / on the sides apdCI.-\-bqd^ + crdtl — 2Adn. If / is outside the triangle, it is customary to regard ap + bq + cr as still equal to 2 A, with the convention that the perpendiculars are not all of the same sign ; and the portion of the arc of the corresponding envelope must be regarded as negative. In fact, though a right line may continuously revolve in the same sense (say clockwise), the radius vector, OP, from a fixed pole, 0, to its point of contact with its envelope may have contrary rotations — as, for example, the tangent to a cusped curve. Using the polar formula for the element of arc, ds, we see that dd will change sign when OP becomes the radius vector to the cusp, and so, therefore, will the element of arc. 62. Generalised Rotilette. We may here investigate the area of a curve of which a roulette is a particular case. Suppose that a plane lamina of any shape is displaced in any manner in its plane ; then a point fixed with reference to the lamina describes a roulette; but a point moving, according to any assigned la.w, with reference to the lamina, and also partaking of the absolute motion of the lamina, describes in fixed space a curve which, in the absence of a better name, may be called a generalised roulette, , . , To find an expression for the element of area of a generalised Generalised Roulette. 91 roulette, suppose the motion of the displaced figure to be pro- duced by the rolling of a curve C on a fixed curve AB; let ah be the curve described in the moving figure by the tracing-point P; and during an indefinitely small rolling motion, dQ., which carries the point P to q, perpen- dicularly to PI, and L' to L, let the tracing-point travel over a length Pp of the curve F'g- 42- ah; then since the absolute motion of the tracing-point is obtained by compounding, according to the parallelogram law, the motions Pq and Pp, the element, PQ, of the space-path described by the tracing-point is the diagonal of the parallelo- gram whose adjacent sides are Pp and Pq ; and the element of area of the generalised roulette is PQLT. This is equal to Pql+ qQL'l (since L and L' are coincident to the first order of small quantities) ; and if PI= r, and we denote qQL'l by dC^ , we have d'S.^dCr + ir^dO,, where d'2 stands for the required element of area. The integral of dC,. will be the sum of the quadrilateral elements of area which are obtained by connecting the ex- tremities of all elementary arcs of the curve C with the corre- sponding extremities of the corresponding elementary arcs of the curve ah. If the curves C and ah are closed, the integral of dCf will be generally the area of the space included between them ; and if the tracing-point merely oscillates along an arc, ah, of any length, the whole figure returning finally to its original position, the integral of dC^ will be generally the area of the curve C. We may illustrate this by an example, due to Mr. E. B. Elliott, slightly more general than Holditch's theorem. (See William- son, Int. Cat., p. 209.) Suppose AB, fig. 41, p. 88, to be a right line, one extremity, A, of which travels round a curve (w), 93 General Theorem of Epicycloidal Motion. whik the line itself is carried round in any manner and inter- sects another curve, {b), in a point, B, at a variable distance from A. It is required to find the area of the curve traced out by a point, P, taken always on AB and dividing AB in a constant ratio, the line returning exactly to its original position. Let the motion be produced by causing Cj to roll on C, (Art. 59). Here the moving figure consists of a right line, AB, and P and B are two tracing points, each describing a generalised roulette, and each describing with reference to the line AB a curve of no area. Moreover, for each of the points B and P we have fdCr^{C^ = area of Body Centrode. Suppose / to be a position of the instantaneous centre; let AI=r, BI=p; AB = l, AP=nl, and denote by (X) the area of the path of P. Then, as in Art. 60, (^)-(C,) = (Cj) + i>Vi2, (5)-(cJ = (C6)+i/pV12, (^x)-{c,) = {c^)+ifppda. But {i—n)r^ + nf? = n{i—n)P + PI^; therefore {X) = {i-n){A) + n{B)-in {i-n)/Pda, and yPd^ is simply the area of the relative path of ^ and B. This result verifies in the particular case in which the line AB is always kept a tangent to the inner curve. 63. Amsler's Planimeter. The theory of some Plani- meters is very simply and. appropriately deducible from the preceding principles. Let us, for example, con- ,,--''/'•, sider Amsler's Planimeter. (Compare also Williamson's Integral Calculus, where dif- ferent discussions of the in- strument are found.) Let CA and AB (fig. 43) be the two arms jointed at A ; C the fixed end of one arm, and B the extremity of the other arm which traces out ,the contour D whose area is to be measured. Drawing Fig- 43- Amskf's Planimeter. 93 the normal to the circle locus of A at A, and the normal at B to the curve D, we get I (their point of intersection) which is the point of contact of the Space and Body Centrodes. If (a) denotes the area traced out by A, and (X) the area of D, we have, with the previous notation, But in the working of the instrument A describes no area about C, .-. (^) = o. Let fall IN perpendicular to AS ; then since, if AB - /, BI^—AI^ = P—2l. AN, we have (X) = I /2 fi - If AN. da. (fi) But i2 = o, since, on the whole, the arm AB does not rotate. Now take any point, -P, on the arm AB, and at P let a graduated roller be fixed with its plane always perpendicular to AB. This roller, after any amount of motion, will indicate the total amount of motion of B perpendicular to AB. Now when / is the instantaneous centre, the elementary motion of P is /B.da, and the component of this perpendicular to AB is IB . cos IBN. dil, or BN. dil. Hence if (n) stands for the reading of the roller, (B)=/PN.dil, and, disregarding sign, equation (^) gives which is the result used in the working of the instrument. We may observe that nothing depends on the curve along which A moves, since equation (J3) will hold if (A) = o, what- ever the curve described by A may be. It is requisite only that A should oscillate back to its original position. Let us obtain the result for this instrument on the supposition that the end A is allowed to trace out a curve of any area {A). . Suppose the graduated roller fixed at a point P in AB at a distance c from A. Then if for a small motion d{Ji) is the 94 General Theorem of Epicycloidal Motion. reading of the wheel [(i?) being, of course, the amount of motion of P perpendicular to AB\, d {R) = IP . cos /PB . da = lA . cos /A3 . dOi—cdO,. Now, as before, (x)-{4) = iP Q.-lfIA . cos TAB . d€l If AB is allowed to rotate through 2 it, we have iX) = (^)-/(^) + TsP- 2'ncl. By taking c = ^l, we should obtain {X) = {A)-l{R). We may also observe that the graduated roller may be fixed anywhere on an arm attached rigidly to AB at any point, /", the axis of the /roller being fixed parallel to AB; for if R is the point on yflie attached arm at which the roller is fixed, the elementatry motion of R along RP is IR . sin IRP. dil, which is Pjy\ dH and is independent of the positioi; of R. Although not coming directly under the general theorem of epicycloidal motion, it may be well to give here the theory of another instrument, the principle of which has been employed by Amsler, and the purpose of which is to find the ' moment of inertia ' of any plane figure aiout a right line in its plane. Suppose Ox (fig. 44) to be the right line about which the moment of inertia of the area of the curve represented is required. From any point, A, on the line let an arm, AB, of constant lengthi a, be drawn to a point on the curve; let be any fixed F point on the line, and lef 6 denote the angle BAO. Let AB receive a slight displacement along the line and along the given curve, so as to come into the position A'B'; let AA'=dx, z.B'A'0=e + de, OA=x. Now the moment of inertia of the quadrilateral SAA'B" about Ox may be found by drawing from A' a line parallel and equal to AB, so that the O A A' Fig- 44- Amsler's Planimeter. 9^ quadnlateral ia broken up into a parallelogram and a triangle. The moment of inertia of the parallelogram is ja'sin'e.ffe; and the moment of inertia of the triangle about Ox is Ja'sin'Sf/e. Hence if 7 is the required moment of inertia, -^= |oYsin' 6dx^ |aVsin' fl dB (a) = Jo Vsin edx- ^a'/sin 3 e (fe + |aVsin' B dS. (/3) Now if a perpendicular, BM, is let fall from B on Ox, and ^ is the area of the given figure, A=fBMd{OM) =/a sin 9d{x—a cos 9) = a/sin 9 ?->rb[i?-\-cv^-k-2h\\).-\-2g\v-\-2f\i.v, (2) if we observe that a = \fx'^dSl is a quantity simply depend- ing on the nature of the Body Centrode and the Space Centrode, and in no way depending on the variable line (i). Now if (X) is constant, equation (2) is the tangential equation of a conic (see Salmon's Conic Sections, chap. XVIII.) ; hetiee all lines which give Roulettes of equal area are tangents to the same conic. Moreover by varying (JST), equation (2) denotes a series of confocal conies (Salmon, ibid^. This theorem is proved in the following elegant manner by Mr. M^Cay. It is shown in Salmon's Conic Sections (p. 339, 6th ed.) that \i L^ = o, L^ = o, ... are the equations of six right lines which all touch the same conic, we have kL^^l^L^^ ... =0, where \, /j, ... are certain multipliers. Hence if /i, /j, .../, are the j)erpendiculars from any point on six lines which all touch the same conic, «iA' + «2A'+ -+«6A' = o, (3) where a^, a^, ... a^ are constant multipliers. Moreover these multipliers are connected by the relation a^ + a^+ ... +a, = o, (4) as may be seen by considering the perpendiculars as drawn from an infinitely distant point. Now take any six lines (such as P£>, fig. 45), which all touch a conic, and which envelop roulettes of areas (A^), {a^), ... {a^); then, calling the perpendiculars from / on them p^,p^, ... and putting .ST for (5') + (c) in equation (o) of last Article, we have" {A,) = jr+i/A'dil; (^,) = JS'+|/AVI2.. .. Circles of Inflexions and Cusps. 99 Multiplying these by ) is the area of the roulette traced out by D, we have from Art. 60 (Z>) - (C.) - (Cj) = \flD'^ . dO,. Hence and as all tangents to the conic chosen give (x) constant, it- follows that {£>) is constant. Hence all points on the director circle of the conic traceout roulettes of the same area; i.e., the director circle is a Kempe circle ; but its centre is the same as that of the conic, and Kempe's circles are all concentric; therefore so are the conies. . The common centre may be regarded as the centre of mass, of a certain distribution (Art. 69)- ■ 67. Curvatures of Point-Roulettes and Line-Boulettes. Circles of Inflexions and Cusps. If in fig. 38, p. 82, the lines PI and QL are produced to meet in the point 0, the radius of curvature at P of the, roulette traced out by P is the ultimate value of PO. Now if ^x = ^^OQ, we have PQ = PO.dx; but PQ;= PI.dQ., since PI is at right angles to PQ, and dQ,, the angle through B 2 loo General Theorem of Epicydgidal Motion. which C has revolved, is PIQ (see p. 83). Denote PI and FO by r and R, respectively. Then Rd\ = rdil. (i) ■„.,., , -^^ ^0 .^ ,, But m the tnangle /£0, we have -r- = -; — -— ; or if the ^ dx sm Z/0 angle made by IP witji the normal at / is i/f, we have ds R—r dx cos i/r Again, dil^ds{--^ — >) = — (Art. 54), where , -5 is put for on' P P " Hence from (i) P + P r—A cos-yjr The value of R will be infinite if P is such that r—k cos i/f = o, i.e., if P is taken anywhere on a- circle described at the upper side of AB, passing' through /, and -having its diameter (of length i5) coincident with the normal at /. Points on this circle are therefore points of inflexion on the roulettes to which they give rise ; and the -circle is hence called the Cmie of Inflexions. Similarly for the ■ I,ine-Roulette (fig. 45, p. 96). The lines PO and QO are two very close normals, therefore each is ultimately equal to the radius of curvature of the roulette at P, and the angle between them is shall be, able to express a relation Examples. 103 between the length PI and the angle, <^, which PI makes with the tangent at / to AB; but this tangent is also a tangent to the required rolling curve. Hence, if r is the radius vector PI froni P to the required curve, we shall have some such equation as r=f{^\ which defines the curve, and can be converted, if necessary, into the usual polar form of equation. Examples. I. Prove that the length of an epi- or hypo-trochoid is expressed in the same manner as the length of an elliptic arc. The case is that of a circle rolling on or inside another circle. Let be the centre of the rolling circle (whose motion we may entirely neglect), P the tracing point, / any point on the circumference, PI = r, PO = ( ; then (Art. 68) the length of the roulette is/rdCi, where da = de+de" = (-±\)ds. where a and b are the radii of the circles, and ds the element of circular arc at 7. If ^ = LPOI, ds = adij) ; therefore the length of the roulette is (^±j)/rd.l> = (i + j)/V'^^— 2 <^<2 cos

+ p'. [The locus of a point P carried in this way is properly called a GHssette. Thus the glissettes of all points P are essentially the same curve, placed in different positions.] 3. Prove that the lengths of the epi- or hypo-trochoids traced out by any two inverse points with respect to the rolling circle are to each other as the square roots of the distances of the tracing points from the centre. (Observe that for two inverse points -j is constant, where r and / are the distances of any point on the circle from them. Then use Art. 68.) 4. If a curve (Cartesian oval) whose equation is Ir + mr' = k, referred to two fixed points, A and B, roll on a right line, prove that for a complete revolution the lengths, L, L', of the roulettes traced out by A and B are connected by the equation //; ^ j^jj = T,k. g. Two parallel right lines at a constant distance apart are moved about in any manner in their plane and brought back to their original position ; prove that the difference of the areas traced out by two points on them which are on any common perpendicular is constant, whatever perpendicular be chosen. (It is vh'—hE, where ^= distance between them, and £= length of envelope of either.) 6. A system of points at constant distances apart is displaced in any manner in its plane ; prove that the mean of the areas of the curves traced out by the points is equal to the area of the curve described by their ceutroid, increased by Ji^'^Q, where k is their radius of gyration about the centroid, and CI is the whole angle of displacement. (Write down the equations (i), (2), and similar ones, of Art. 60 for the points and for their centroid, and add.) The same result holds if the moving figure is any continuous plane area. 7. Prove that if a graduated roller whose plane is parallel to a moving bar has its centre rigidly attached to the bar by an arm of length h perpen- dicular to the bar, the reading of the roller after any motion which brings the system back to its original position will be E—2irh, where E is the length of the envelope of the moving bar. 8. When two points of a lamina are guided along two fixed grooves, show that — (a) The locus of points which describe right lines is the circle Cj ; {b) There is only one point which describes a circle (the centre of Cj) ; {c) All points at the same distance from this point describe equal ellipses. [Mr. M" Cay. See Art. 59.] Examples. 105 9. Prove that Holditch's Theorem follows from equation (3), Art. 6i. [Mr. M<=Cay.] (Take any point, R, on the line PC; then the perpendiculars to PC at P and R envelop parallel curves the difference of whose lengths is 2tr./'i?.) 10. If any plane figure be displaced in any manner in its plane and return to its original position after any number "of complete revolutions, prove that all points which have described roulettes whose centroids (' centres of gravity ') are on a given line lie on a conic. CHAPTER IV. Mass-Kinematics of Solid Bodies. 70. Definitions. The momentum of a moving particle in any direction is defined to be the product of the number of units of mass in the particle and the number of units of velocity in its component of velocity in that direction ; so that if m and V are the mass and velocity component of the particle, its mo- mentum in the assumed direction is mv. If instead of being a particle, the body is a solid of any magnitude moving so that all its particles are moving with the same velocity, v, in magni- tude and in direction, its momentum is Mv, where M is the mass of the body. The resultant momentum of a moving particle is, of course, the product of its mass and resultant velocity. The moment of motnentum of a particle moving in any plane about any point in the plane is the product of its momentum and the perpendicular from the point on the line along which the particle is moving. If / (measured in the same units of length as those employed in measuring v) is the length of this perpendicular, the moment of momentum of the particle about the point is mpv. Just as in Statics when we speak of the moment of a force about a point, we mean, in reality, its moment about an axis through the point perpendicular to the direction of the force, so the moment of momentum of a particle is, in reality, the moment of the momentum about an axis through the point perpendicular to the direction of motion. Since the motions which we consider in this work are all in one plane, or parallel to one plane, we may adhere to the expression 'moment about a point,' which saves the circum- p" Definitions. 107 locution 'moment about an axis through the point perpen- dicular to the plane of motion.' Hh& force of inertia of a moving particle, in any direction, is the product of its mass and its component of acceleration in cPx that direction. Thus, if —^ is the acceleration of the particle in the direction of the axis of x, the force of inertia in this ^x direction is m The resultant force of inertia of the particle is the product of its mass and its resultant acceleration. Thus, with the no- tation of Chap. ii. p. 65, the resultant force of inertia of a /~ ^4 particle of mass m moving in any path is m A / (s)*-| and its component forces of inertia along the tangent and normal to its path are ms and m — • P The quantity which we have here defined as force of inertia is one which is already known in this country by another name. It is spoken of as above by Newton {Principia, Definition iii, Book i). Newton says : ' The Tiis insita, or innate force of matter, is a power of resisting by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest or of moving uniformly forward in. a right line.' And in his remarks on the definition he says that ' this vis insita may by a most significant name be called vis inertia. But a body exerts this force only when another force impressed upon it endeavours to change its condition ; and the exercise of this force may be considered both as resistance and impulse ; it is resistance in so far as the body, for maiuT taining its present state, withstands the force impressed,' &c. This terminology has been wholly ignored by English writers, and, as a result, the fact that a body exerts a kick {if we may use the expression for clearness of illustration) against any agent which acts on it by direct contact or through a medium for the purpose either of deviating its motion from a rectilinear course or of accelerating its velocity, has been lost sight of. The student must carefully observe that the force of inertia of a moving particle is not a force acting on the particle, but one exerted by it on some agent direct or indirect — a kick against change of motion. D'Alembert, in enunciating the kinetical principle known by his name, speaks of force of inertia as effective force, and this deviation from Newton's definition and the physical idea contained in it has been followed J)y English writers. io8 Mass-Kinematics of' Solid Bodies. Most (if not all) of the modem French writers have adhered to Newton's definition, and some (such as Delaunay) very clearly emphasise the nature of force of inertia. As the momentum of a particle was defined to have a moment about any point, so the moment of its force of inertia may also be taken about any point. The energy of a moving particle is half the prodiict of its mass and the square of its velocity, i. e., energy = \ nafl. Of the quantities which we have defined, it will be observed that the last alone is defined without reference to direciion. Any quantity which has direction as well as magnitude is called a directed quantity, or a vector quantity; and any quantity which has merely magnitude but not direction is called an undirected, or scalar, quantity. Thus a volume is, like energy, a scalar quantity. A force, a couple, a velocity, is a vector magnitude, the vector representing any couple being a right line drawn in the direction and sense of its axis, its length being, on some conventional scale, proportional to the moment of the couple. An element of area must also be regarded as a vector quantity, the vector being drawn parallel to its normal and proportional to its magnitude. A vector, however, as defined in general in Quaternions is not localised, i. e., so long as its magnitude, direction, and sense are not altered, it may be drawn at any point in space. We shall be concerned chiefly with localised vectors, such as those, for example, which are drawn at each point of a moving body to represent the corre- sponding velocities or momenta in the actual lines in which theii velocities or momenta take place. Finally, the student is assumed to be familiar with the theorem of mass-moments, which expresses the distance of the centre of mass of any body, or collection of separate particles, from a plane, in terms of the masses of the constituent particles and their several distances froni the plane (see Statics, p. 252); and with the ordinary elementary facts cdncerning moment of inertia (see Routh's Rigid Dynamics, chap, i, or Williamson's Integral Calculus, chap. x). 71. Momentum-System of a B,igid Body. To reduce to Momentufn-System of a Rigid Body. 109 its simplest form the. momentum system of a. rigid body moving with uniplanar motion. Consider first a plane lamina moving in any manner in its own plane. Let G (fig. 18, p. 39) be the centre of mass of the lamina ; let m be the angular velocity of the lamina at any instant ; let P be any point of the lamina, dm the element of mass at F, and / the instantaneous centre. Then the velocity of the particle dm at P is at right angles to IP, and its momentum is lit .IPdm. Imagine at each point P a vector drawn to represent the momentum of the piarticle at, the point; and, exactly as in the reduction of a system of forces in Statics, at /introduce two equal and opposite vectors, oj . IPdm^ each parallel and equal to the vector at P. This will give a system of vectors at /whose type is 0) . IPdm, and a sys^tem of couples whose type is o) .IP'^dm. The sum of all the couples is (oflP^ dm, or Men (F+IG^), where k is the radius of gy-ration of the lamina about an axis through G perpendicular to its plane. Also (see Statics, p. 16) the re- sultant of a system of vectors whose type is co .IPdm, if each were directed from / to P, would be a vector caM.IG directed from /to G; hence the resultant of the vectors when (as here) each is perpendicular to IP is a vector M .tn .IG perpendicular to IG. The momentum^system is therefore equivalent to a mo- mentum vector M.isi.IG at /perpendicular to IG, together with a momentum couple M. to (W-\-IG'^'). We may combine, exactly as in the reduction of forces, the vector at / and the couple so as to give a single localised momentum vector, by altering the arm of the couple and making each vector in it equal io M .a> .IG. If IG is produced through G to ^' so that M . and v = GI .u>, therefore pv = oi . PG^, GI and therefore by (2) the moment of momentum about P is M{P + PG'').o), or Mi'^o}. Hence if /"denote the sum of the moments, about an axis, of any system of impulses applied to a body, the equation loi^P, where / denotes the moment of inertia of the body about the axis and ni the angular velocity generated by the impulse-system, holds round any point on the circle described on GI as diameter. 72. System of Forces of Inertia. To reduce io its simplest form the system of forces of inertia of all the elementary particles of a rigid body with uniplanar motion. Firstly, let the body be a plane lamina moving in any manner in its own plane. Let G be its centre of mass, J the accele- ration centre at any instant, P any point in the lamina at which the element of mass is dm, a> and to the angular velocity and angular acceleration of the lamina at the instant considered. Then if Voi^ + «* = f, the acceleration of the particle at P=i.JP and its direction makes tan~^ —z with JP. The re- sultant force of inertia of this particle = e.yp. dm. Let it be represented by a vector drawn at P; and pursuing the same system of reduction as in the last Article, introduce at y two vectors equal and parallel to this vector, and opposite to each other. We shall thus have the vector t.JP.dm at J, and a couple whose moment = (i) . yp^dm. The resultant of the system of vectors at ^ is a vector e .yc .M parallel to the direction of acceleration of G; and the sum of the couples = i&fy^dm = M{J^-\-y(?').ia, where ^=the mass of the lamina and k is its radius of gyration about an axis through G perpendicular to its plane. Just as before, change the couple into one in which each vector is t.M. yG, and in which the arm = -{yG+ — )• Produce the line yc through G to y' so that Gy.Gf^li', (a) and the single vector f.M.yc, or M.a (if 5. denotes the t^orcies of InerAa. tij acceleration of G), to which the whole system reduces, will be at Y parallel to the direction of acceleration at G. Hence the whole system of forces of inertia ds the same as if the lamina were concentrated into a single particle of mass M at the point j' having the acceleration, d, of the centre of mass. CoR. I. If ^ is the perpendicular from any point, 0, on the line of acceleration of G, the sum of the moments of the forces of inertia about is M(k^!i> +^d). CoR. 2. The sum of the moments of the forces of inertia about an axis fixed in the body as well as in space is where U is the radius of gyration about the axis. Secondly, let the body be a solid of any shapes let G be its centre of mass ; let JK be the acceleration axis at any instant, J being the foot of the perpendicular from G on this axis, and, as before, divide the body into a number of thin plates per- pendicular to this axis, any one of which is met in the point j by the axis, the centre of mass of this plate being g, and its mass hp. Take the line JK as axis of z, JG as axis of x, the length JG being denoted by h, and let {x, y, 2) be the co- ordinates of ^. Then reduce the system of forces of inertia of the plate to a couple inl^^hp about the axis JK and a vector e.jg.hp a.t j\ ff being the radius of gyration of the plate about JJC. The sum of the couples for all the plates is where k is the radius of gyration of the whole body about an axis through G perpendicular to the plane of motion. Also the vector e .jg has for components parallel to the axes of co- ordinates (— ft)'x— wj', Mx—a>^y, 0)] so that the system of vectors of which e ^jg . 8/ is the type gives rise to a vector at J whose components are {^—oi^Mh, inMh) along and perpendicular to JG — i.e., to a vector e.M.JG at 7 parallel to the line of acceleration of G — together with two couples with axes parallel and perpendicular to JG, their moments being — /w + wa)^ and — /to"— WW. I 114 Mass-Kinematics of Solid Bodies. The whole system, therefore, reduces to a vector Mo. at J, together with a couple y!/ (^'^ + A'') to about the axis JK, and the two couples just mentioned. Consequently if the axis through G perpendicular to the plane of motion is a principal axis, the case becomes the same as if the body were a plane lamina, and the system of forces of inertia reduces to a single force of inertia localised at a point j' deterT mined by equation (a). On the line JG describe a circle, the angle, JPG, in one segment being 7r— tan— =1 and the angle in the other segment tan~^ —^ ; then, assuming the to' Fig. 46. body to be a lamina, or that (as ex- plained above) the case is the same as if it were a lamina, the sum of the moments of the forces of inertia taken about any point, P, on the circumference of this circle is Mk'^a), where U is the radius of gyration of the body about P. If (^ = tan~^ -5 ) the angle yPQ = A, therefore the acceleration to of P is along PQ. Also the acceleration of G is along the tangent GT to the circle, since LjGT^^. The diameter of the circle ■■ sin<^ 1 therefore the perpendicular, p, from P on io.PC^ hence substituting in (o) ^^. PG^ . ^ PG^ in GT IS sm d) = . - JG ^ JG i we have the moment of the forces of inertia = .(5/ (i''+i'G^)cb, or Mk^^ui^ as above. It is an immediate consequence from Newton's Third Law that the system of forces of inertia is completely equivalent to the system of external forces acting on any body or any material system '. Hence the sum of the * This principle was subsequently enunciated by D'Alembert, and since his time it held undisputed sway as D'Alembert' s Principle, until Thomson and Tait pointed out the fact that it is clearly contained in Newton's Axiom. Indeed, assuming Newton's principle of the equality of action • and reaction between every pair of mutually influencing bodies, it is sur- prising that the necessity for D'Alembert's Principle could h^ve been felt. Energy. 1 15 tooments of the forces of inertia about any axis is equal to the sum of the moments, L, of the external forces about the axis ; and we have just proved that for an axis represented by any point on the circle JPG the expression for the moment of the forces of inertia is la, where / is the moment of inertia about the axis. Round all points on this circle we have therefore the equation /a = Z 73. Energy in XJniplanar Motion. Let dm be the element of mass at any point, P, in the body; from P let fall a perpendicular, r, on the instantaneous axis. Then if (a is the angular velocity of the body, the velocity of P is cor, and the energy of the particle at P is \a?r^dm, so that the energy of the whole body is \tt?fr''dm, or \Mk"^(i^, where U is the radius of gyration about the instantaneous axis. But since — is the distance between G and the instantaneous axis (p. 40)^ — 'v'^ where » = velocity of G, we have K^ = l^-\ — -j. Hence the energy is \M{^^^k^^^). " (a) Hence the whole energy may be described as consisting of two parts; one is \Mv'^, which is the energy of the body on the supposition that it is condensed into a single particle movitig with the velocity of the centre of mass ; and the other is \Ml^u^, which is the energy due to the rotation round the centre of mass. It is not true, in general, that the energy of the body is equal to the sum of that due to the rotation round any point and that due to the velocity of translation of the particle at this point ; i.e., if w is the velocity of translation of any particle, P, and A. the radius of gyration about an axis through P perpendicular to the plane of motion, it is not in general true that the energy is \M {;v^ ■'r'i^ u?). Let us see whether there are any particular points for which it is true. Let / be the instantaneous centre, v Gl=h= -» PI^p, Lpig^'^; then w, the velocity of the particle at P, = top, and >? = k^-\-P(i^ = k^^p^—2h()Q.o^^-irW'; therefore v'+'K^u? = (k^-\-h^-ir2^—2hpzo^^)u?; and if this = » 2 4. ^2a,2 = (^2 ^. ^2) a,'', we must have p^—hp cos i/f = o, or p = h cos y\r ; I 2 ii6 Mass-Kinematics of Solid Bodies. i.e., the energy can be put into the form ^iil/(w^ + \'a)'^ if P is any point on the circle described on GI as diameter. Hence, in uniplanar motion — the total energy of a body is •equal to the sum of the energy due to the velocity of translation of any point on the circle described on GL as diameter and the energy due to the rotation round this point. There is an analogous property in the general, or three dimensional, motion of a rigid body, the proof of which is left to the student s viz., the total energy of a rigid body is equal to the energy due to the velocity of a point P added to the energy due to the rotation about this point, if P is any point on the surface of the second degree whose equation, referred to the principal axes at G (Jhe centre of mass), is A.' + /*' + v" + oX + i^ +^v = o, where (o, b, c) are the component-velocities of G parallel to the axes, A.^o'jZ— <"»/, /i=i«jX— aiiZ, K = 01,^—0)2 jr, and (a;,, oij, Wj) are the angular velocities of the body about the axes at G. 74. Energy of Bapid Vibratory Motion, Suppose a very small particle of mass dm to have a velocity which at any instant is represented by ecoszir-) where t is an extremely small period of time — say something like xwtnrth part of a second. Then within even such a small period as one second the velocity has run through all its values 20,000 times, and the ■ value of the energy of the particle will have varied between ^e'dnt and zero 40,000 times. Hence we may look upon the energy of the particle as constant and equal to its mean value e'dm C'' t during the period r, i. e., / cos'' 2v-dt, or i-i^dm. 2r Jo r We have in the previous Article spoken of the total energy of a body as being of the form \Mv'^+\Ml^ u>^; and this would be correct if the body were perfectly rigid, or, in other words, if the velocity-components, u, v, of every point in it could be represented by the equations u = a—(ji>y, v = b+(i>x, (a) (as in p. 41). The energy thus expressed is often spoken of as the energy of visible motion — an expression which is, of course, Energy oj Vibratory Motion. 117 logically indefensible * — by which is signified the energy of the body considered as a strictly rigid body. The total energy of a natural solid in a state of motion may not be expressible in the form \Mv^+\Mk^m\ (p) ■which results from the supposition that the velocity com- ponents of every particle are given by equations (a) ; for the molecules (using the term in its chemical sense) of the body may be executing rapid vibrations of very small amplitude about certain mean positions, while the whole system is moving through space. We do not know the precise forms of the vibratory velocity components of a molecule, but we may assume them to be of the type e cos 2 zr - > so that the com- plete expressions for the velocity components of a molecule will be of the forms t , ■ t , , K = a— wj/-f-ecos 27r-) » = e + a)j;-f-7j sm 2w-> \y) if all the motions are uniplanar; and in three-dimensional motion the expressions for the velocity components u, v, w, which hold for a rigid body (viz,, in the usual notation, u = a + u>^z—w^, etc.), must be corrected by the addition of periodic terms of very short period, as in (y). Now, taking the expressions (y) — the method of dealing with which is precisely the same for the case of three-dimensional motion — the total energy of the body will be |y[^(ar— aj_y+ecos 2-n-Y + {b + (ax +11] sin 27: -)-]dm) (8) ■which will contain such terms as a/e cos 211 -dm, /ey cos 211 -dm, fi^ cos" 2 w - dm. But, T being extremely small, cos 2ir- will, before the co- ordinates X and J/ have time to alter sensibly, run through all possible values, the mean of which will be zero ; so that the 1 Visible to whom ? Instead of this expression, we shall use ' energy of the firet order.' Ji8 Mass-Kinematics of Solid Bodies. first two of these three integral types vanish ; and, as before explained, the third is practically equal to \fi^dm. Hence (6) becomes \J\{a - wyY + {d + co xY] dm + \f{f + tj") dm, (e) the first part of which — energy of the first order, or rigid- body energy — is equivalent to the form (;3), and the second — energy of the second order — is evidently the energy due to the molecular vibrations, which we may denote by H; so that the total energy of a natural solid is, in general, of the type The energy of molecular vibration, H, is exhibited, in general, in the form of heat or sound. The subdivision of the whole energy of a natural solid is often carried farther than two terms — one expressing energy as if the body were rigid, and the other expressing energy of vibratory motion of molecules as if they were little rigid bodies — for it may happen that the atoms which constitute the molecules are disturbed in such a manner that the molecules cease to behave as rigid bodies. Such is , probably the case in a solid incandescent body moving in space and emitting rays from ultra-red to ultra- violet. In this case the terms introduced into the expressions for the velocity components («, ») at any point would still be periodic, and we may regard H in the expression (Q as including the energy of atomic vibration. If the .^-displacement of a molecule from its natural position is of the form a sin ztt--) the amplitude, a, is extremely small, 2 11 a t but the velocity of this motion is ■ cos air — , and the co- T T - . ^ 2'7Ta eiiicient > or e, may, on account of the extreme smallness of T, be very great, so that /e^dm may in the case of a hot body moving in any way be several hundreds or thousands of times greater than the body's energy of the first order. Thus, the very moderate amount of energy (of the second order) which is known as one thermal unit has been found by Energy of Vibratory Motion. ii^ Joule to be equivalent to 772 foot-pounds— the thermal unit being the quantity of heat necessary to raise i lb. of water at 60° F. one degree more in temperature. Adopting gravitation units, the energy of a body of weight w lbs. moving with a translational velocity v feet per second is foot-pounds ; so that if w = I, and this energy = 772, we have » = 224 feet per second (about) ; so that in order to possess the energy measured by one thermal unit in the form of energy of the first order, the pound of water should have a velocity of translation (common to all its particles) about three times that of the fastest express railway trains. If the mean square of the resultant vibratory velocity of any molecule of a body is denoted by i}, its energy of the second oxAtx = \f{^dm; and, in analogy with moment of inertia, we may write this in the form 1 ;,, 2 M being the mass of the body, and \i? being obviously the mean square of molecular vibratory velocity. If B is the temperature of the body, c its capacity for heat (referred to Water), and J Joule's equivalent, this expression for the energy must be equal to MJcQ; so that In a material universe every body is directly in contact with some other body and in mediate communication with every other body (by means of air or ether), so that it is impossible for a body to move without communicating motion — or transferring some of its energy — to other' bodies. Hence it is manifestly impossible to construct a material system which is energy-tight, i.e., which maintains undiminished a quantity of energy which it has once received. Even if we could construct such a. system, it would be impossible to preserve its energy in the form of energy of the first order — i.e., rigid body energy — for, collisions or friction between its parts would inevitably set molecules in vibration ; and since in such a system we have assumed the expression (f ) to be constant, there would be a continual transformation of energy of the first order into energy of the second order (molecular energy). lao Mass-Kinematics of Solid Bodies. Examples. 1. A lamina moves in any way in its own plane ; show how to stop its motion completely by fixing one point 2. A lamina moves in any way in its own plane ; if it be required to. fix a point in such a way that the angular velocity about it shall be given, find the locus of the point in the lamina. Ans. A circle. If the new angular velocity = n, lu = angular velocity of the lamina before the fixing of the point, v = velocity of centre of mass, G, before the fixing, the equation of the circle is r>--r sine + *"('l-^) = o, G being the origin, and the line of motion of G before the fixing being the initial line, and k the radius of gyration about G. 3. Show that the fixing of 1 point entails a loss of energy (of the first order). If « is the velocity of the point before fixing, h its distance from G, p the perpendicular from it on the line of motion of G before fixing, the loss, of energy of the first order can he written 4. A lamina moves in any manner in its own plane ; find the locus of a: point the direction of whose velocity coincides with that of its acceleration. Find the locus also if these two directions are at right angles. Ans. In each case a circle passing through the points /, J. 5. Prove that the surface locus in the theorem at the end of Art. 73 is a' circular cylinder having the axis of resultant angular velocity at G and the axis of instantaneous screw motion for diametrically opposite generators. (Prof. Townsend.) [The theorem of Art. 73 was published by the author in the Edttcational Times for July, 1882, and a discussion of it by Prof. Tovmsend appears in the number for August.] 6. If V is the resultant velocity of G, ^ the angle between the direction of V and the axis of resultant angular velocity («) in last question, show that the diameter of the cylinder = — sin i(<. a ^ 1. Find (firom Arts. 71 and 72) the Virials, with respect to any point, of the momenta and forces of inertia of the particles of any lamina moving in its own plane. CHAPTER V. Analysis of Small Strains. 75. Kature of Strain. When a natural solid (such as a mass of iron or wood) is not acted upon by any external force, its molecules assume certain distances from each other, which are called their natural distances ; but when some external force, or impulse, acts on the body — as, for instance, a pressure, whether continuous or sudden — some alteration of the mole- cular distances, however small, must result. These alterations may be so small as to be invisible to the eye, and yet they may produce effects which are otherwise very readily appreciated — e. g., when minute and rapid molecular alterations result in ,the production of sound, as in the case of a bell or a telephone. Any alteration of the natural distances between particles of a body is called a strain. Every motion of a body is not ac- companied by strain,- for the motion may be one which is consistent with perfect rigidity. Thus, if a solid is slightly moved parallel to the plane xy so that the small changes (u, v) of the co-ordinates of every point {x, y) are given by the equations u = a—(iiy, tr-i + cax, (i) where a, 5, a> are the same for all points, no strain results since these equations express the displacement of a rigid body (Art. 29), a and d being the components of a motion of translation common to all points in the body, and 00 a motion of rotation about the axis of s common to all points. 76. Changes in Relative Co-ordinates. Let fig. 47 repre- sent a section of a body made by the plane {xy) of dis- placement ; let Ox and Oy be two fixed axes, of x and y ; let laa Analysis of Small Strains. P be any point in the body ; and Px, Py two lines parallel to Ox, Oy. . , We confine our attention to smau strains, and we suppose the displace- ments to be the same at corre- sponding points in all sections of the body parallel to the plane of the figure. Suppose, then, that the body is strained in such a manner that P is brought to P', and Q, a point very Let the components of PP' parallel to Ox and Oy be u and v. Then the strain is produced according to some law which assigns u and v as functions of the position of P. Suppose that « =/^(^, j,), » =/^(^, y). (i) Again, if^ the co-ordinates of Q with reference to Px and Py are (^, t)), the displacements, «', v', of Q (i. e., the components of ee') are «'=/,(^ + f,j/ + 7j), if=A{x->r^,yA-'<\\ Fig. 47- near P is brought to G'. or i/—i ,.du du dx dy , ,.dv dv dx dy (2) If the whole body receive a motion of translation equal and parallel to P^P. so that the point P' is brought back to P and e' brought to C", no further strain results, and the expressions (2) are those for the co-ordinates of Qf' with reference to Q. Denote the changes t/ —u and v'—v in the relative co- ordinates of Q with respect to P by Af and Arj, and we have . ^ ^ du du ^dv dv ■■^Tx^''Ty' by a and h; and denote ^ du , dv Denote -r- and ^- dx dy dv and — by f + 0). Then these equations can be written du dy (3) by J— 0), (4) Af=af+(j— w)7j; A7j = 37j-|-(f-f-a))f. From these expressions we deduce the following results, (i) All points in the immediate neighbourhood of P which in the Transformation of Co-ordinates. la^ unstrained state of the body lay on a right line, continue after strain to lie on a right line {with altered direction). For if (f, 7j') are the co-ordinates of any point )^+(i+^),,, ^^' so that (f, 7j) are also linear functions of (f, rf) ; and if the first satisfy a linear equation, A^+jonj-l-i» = o, so must the second. (2) Two parallel lines in the immediate neighbourhood of P before strain become two parallel lines {with altered direction) after strain. This follows by causing v alone to vary in the equation of the above line. Hence a parallelogram becomes another parallelogram. 77. Transforination of Co-ordinates. Given the values of a, b, s, 0} with reference to two rectangular axes, Ox, Oy, to find their values with reference to two other rectangular axes, Ox', Oy. Let Ox' (fig. 47) make the angle ^ with Ox ; let two axes, Px', Py, be drawn at P parallel to Ox', Oy ; let {x',y') be the co-ordinates of P with reference to Ox', Oy' ; and let («', v') be the values of («, v) with reference to these directions. Then x = 3, y = x' sin (p +_/ cos ^, 1/ = u cos <^ + z) sin ^, v' = —us,m. + vcos^. du' du' dx dt^ dy , . , j- ,. . Also -r-/ = ^-3-'+ j--T^' wliich from the above dx dx dx ay dx equations gives — 5 = acos'' —, by a', J/, we have 124 Analysis of Small Strains. -^—,= -r-? by a', 5', we have dx dy ' a'+i' = a + 6. ' (2) Also if -7-7 — --7-> is denoted by 2 cb', we have 0)' = o). (3) The relations (2) and (3) being independent of (p, are invariant relations, showing that the numbers a + 5 and to are the same whatever be the two rectangular lines, Px, Py, at P with respect to which they are calculated. Other invariant relations may be found by eliminating <^ from the equations (a), ... (8) in pairs. From (i) we see that / can be made zero by properly choosing the lines Px', P/ ; and that the directions of these lines are given by the equation 2 S tan2^ = — — > (4) showing that the two lines in question are the axes of the conic discussed in the next Article. Cor. I. The equations (a), (/3), and (i) expressing the components of the given strain with reference to a new set of axes, Px', P/, constitute the resolution of strain. CoR. 2. Two strains, denoted for shortness by (a, b, s) and {a', b', s'), one expressed with reference to one set of axes, Px, Py, and the other expressed with reference to another set, Px', Py', are said to be equivalent when either produces the other- Hence we may replace one by the other. Thus we see that any strain {a, b, s) can always be replaced by one of the form («', h', 0). 78. Elongation in any Direction. The elongation, pro- duced by the strain, in the direction PQ is defined to be the Elongation. 125 ratio which the change in the length of PQ bears to the original length PQ. Denote this elongation by e ; then ^ PQ"—PQ _ APQ PQ ~ PQ ' Now PQ^ = $^ + rf, .-. PQ.APQ = ^A^+-nAri = «f +2j^7j + ^5)' by (4), Art. 76. If Z QPx = , this gives e = acos^(^+2f sini^cos^ + 3sin"' denote the angle Qf'Px, the rotation of the line PQ will be - From , V - . , , ,, f + <<)+ (i+3) tan d> equations (5) of Art. 76 we have tan a> = ) ■■ ■. > ^ ^ ' i+a+(j'— a))tan

. But tan <^'— tan (^ is A tan ^, or sec^ <^ . A<^, where A^ = + scos2(j) + u). (i) We may put s = o, and then A(^ = i(?2—«i) sin 2^ + 0). (2) If W) = o or -> i.e., tf PQ is either axis of the elongation 2 conic, the amount of rotation is oa — which may serve as a definition of the invariant w. It may happen that some line at P is not rotated. Putting A6 = o, we have sin 2 = > which gives either two values of or none, according as 2£o< or >i?i— ^j. 80. Change of Inclination of Two Lines. If two lines in the unstrained state make angles <^, (/>', with Px, the change in the angle between their directions after strain is A = o, we see that /Ae number 2s is the cosine of the angle between the strained positions of the axes of reference Px, Py. 81. Signification of the number s. The meaning of the quantity f can be presented in another way. /* Let Px, Py (fig. 49) be the unstrained posi- / tions of two rectangular axes at P. By ,'j!; a: strain these lines will become two right lines including an angle whose cosine is 2s. I P x Fig. 40. Suppose that after strain P is brought back to its original position, and also that by a rigid-body rotation (which produces no strain) the line Px is brought back to its original position; then the line Py will occupy the position Py', where Z xPy = cos"''- 2 s. Also any line, /A, parallel to Px will become tA', also parallel to Px (Cor. 2, Art. 76). From /' let fall I'p perpendicular to Px; then /' has advanced in front of P by the distance pp, which = 2s . Pi'; but pr = {i+b)P/, Pp .: — = 2s{i+b)= 2s, approximately, (a) or in other words, every point in the line PA has advanced or slid forward parallel to PA through a distance which is 2j times the distance between PA and Px. A practical example of such motion is afforded by a book with flexible binding. If one cover is placed on a table and kept fixed (this cover being represented by Px above), the successive pages above it may all be slid forward through distances proportional to their distances from the table (any The Strain Ellipse. 129 page being represented by lA above). Such a strain is called shearing strain— which is formally defined thus — ■ When one plane is held fixed in a body^ and all planes in the hody parallel to it are slid parallel to it, those at one side in the same sense, and each through a distance proportional to its distance from the fixed plane, the strain so prodtcced is called shearing strain. The fractional amount of sliding, i. e., the ratio of the actual amount of sliding of any plane to the distance between this and the fixed plane, is called the amount of the shear. Since we might equally well have brought the line Py back to its original direction, leaving Px rotated and strained, the number is expresses equally the, amount of shear of lines parallel to ly — it may therefore be described as the shear of the axes Px and Py. Similarly the number 2 cr given in (2) of last Article is the: amount of the shear of the lines (<^, ^ + -)' 82. The Strain Ellipse. Any small circle in the neighbour- hood, of P becomes an ellipse by strain. If we express (f, rj) in terms of (^, rf) from equations (5), Art. 76, we obtain approximately ^=(i_a)f'-.(j-co)r)', ^=_(^ + „)^+(i_3),,'. (i) Consider now a small circle of radius r in the unstrained state, with P as centre. Its equation is f' + r?*=r', which by (i) gives the relation (i-«)r-2^.r7,'+(i-3)r,'^-ir= = o (2) between (f, 77'). The ellipse denoted by this equation is called the Strain Ellipse. It is obviously co-axal with the elongation conic. Every pair of equal rect- angular lines through P, in the unstrained state, become con- jugate semi-diameters of the Strain Ellipse. _. Let PA and PB (fig. go) be " ^' two rectangular radii, each- equal to r, of a circle described 130 Analysis of Small Strains. about P; let C be any point on the circle ; and let fall the per- pendiculars QM, QN on PA, PB. Let p, a, b, q, m, n be the displaced positions oi P, A,... . [The point / is in reality considered as coincident with P; but for clearness of figure they are represented apart.] Then the figure pmgn is a parallelogram (Cor. 2, Art. 76). Also if e is the elongation in the direction PA, we have pa = {i + e)PA, , PM pm , . ., , PN pn and pm = (i + c)PM; or — =^—; and similarly — ='—;• ■^ ^ ' ' PA pa' ^ PB pb But PM^ p^ ^ pm^ pn^ PA^'^PB^'^' ''' 'p^^ '^ pb^ ~ ^' the second of which equations is- that of an ellipse referred to two semi-conjugate diameters, pa, pb, as axes. [This, of course, furnishes another proof that a circle becomes an ellipse by strain.] In particular, the semi-axes of the ellipse must be the strained positions of some two rectangular radii of the circle; hence a/ every point, P, there are two particular rectangular lines which are strained into two rectangular lines, these latter being the axes of the strain ellipse at P; and in general these latter are rotated from their original positions. The principal axes of a strain at any point are those two rectangular lines which become the axes of the strain ellipse at the point; and the principal elongations at the point are the elongations along the principal axes. If the axes of reference at P are taken in the principal directions of the strain, the equation (2) of the strain ellipse becomes (j_^^)^!! + (j_^^)y_i;.i! = o, x^ f which is, of course, directly evident. 83. Pure Strain. The strain at a point is said to be pure strain if the principal axes (axes of the strain ellipse) are not rotated by the strain. Now the invariant &> is the amount of rotation of these par- ticular axes (Art. 79). Hence the condition for a pure strain is Pure Strain. 131 or expressed analyticaHy with reference to any pair of axes dv du dx dy ' which is the condition that the function udx + vdy = o should be a perfect differential. Fig. 51 represents the nature of the dis- placements round P when the strain is pure. J'A' and PB' are the semi-axes of the strain ellipse into which the circle of radius J'A is converted, the extremities A and B of two rectangular radii moving along these lines to A' and S'. ^'S- 51- The general, or rotational, strain would be represented by turning the ellipse round P through an angle whose circular measure is co, the lines PA' and PA being, of course, no longer coincident in direction. A pure strain is also called an irrotalional strain. The values of f and ?j', equations (5), Art. 76, show that every strain can he supposed to result from a pure strain followed by a rotation as of a rigid body ; for writing them in the forms Af=(«£+J7j) — wij, since to belongs equally to all points near P, the changes o»ij and ft)^ in f and r\, respectively, indicate (p. 41) a motion of rotation of a rigid body about P — ^which is unaccompanied by strain; and the other terms indicate a strain in which ^e principal axes are not rotated. If the axes of reference are the principal axes of the strain, s = o, and the parts of A^ and Arj which are due to strain are Af=^if, A7? = ff2Jj, (a) Hence the strain proper is produced by lengthening the co- ordinates of all points with respect to the principal axis of x in the ratio i + e^: i, and the co-ordinates parallel to the principal axis oiy in the ratio i+e^: 1. Now a simple elongation of a body in a direction normal to a plane consists in the drawing out of each particle from the K 2 13a Analysis of Small Strains. plane through a distance proportional to the natural distance of the particle from the plane. Hence equations (a) express two simple elongations parallel to the principal axes of the strain ; so that — Every small strain at a point can he produced by two successive simple elongations, followed by a rotation, as of a rigid body, about the point. Considering only the strain proper, i. e., the pure portion, or neglecting the mere rotation, the strained position of any point, Q, on the circle (fig. 51) is easily found. For, if r is the radius of the circle, and the angle made by PQ with PA, the co- ordinates of Q are ^=r cos <^, ij = r sin (^ ; and if (f is the point on the strain ellipse to which Q comes, the co-ordinates of (f are (f ' = ( i + «i) r cos (^, tj' = ( i + ijj) r sin <^ ; and as the equation of the ellipse is . r-^ + -r-^ — ^ = r^ we see that . <^ is the excentric angle of the point (f. Hence producing PQ to R so that PR = pa', a perpendicular from R on PA meets the ellipse in e'. The essential elements of a strain, therefore, are elongation {a, b) and shear {s) ; and they are not to be confounded with the other magnitude (to) which enters into the expressions for the changes of relative co-ordinates (Art. 76), this last being mere rigid-body-rotation. 84. Case of no Expansion. When the invariant a + b is zero, the areas of all small figures in the neighbourhood of P are unaltered by strain. The elongation conic becomes in this case a rectangular hyperbola, and it is, of course, accompanied by another rectangular hyperbola of compression (Art. 78). Now we know that the equation ax^+2sxy—ay^ = k^ be- comes of the form a (x'^—y''^) = k^, if the axes of the curve are those of co-ordinates, and a = -/a^-F j''. But this latter form of the equation of the conic corresponds to an elongation, a, along one axis, accompanied by a compression, a, along the other. Again, if instead of the axes of the curve we take its asymptotes as the axes of co-ordinates, we know that its equa- tion becomes 2 a/oM^ .xy = k^. Shearing Strain. 133 which indicates a strain consisting' wholly of shear, whose amount (Art. 81) is 2 ■/«" + *". Hence (see Cor. 2, Art. 77) any non-expansional strain (a, —a, s) can be replaced by a non-expansional strain (o, —a, o) with no shear, with reference to axes properly chosen; or by a simple shearing strain (o, o, 2a), with reference to axes properly chosen. Conversely, a simple shearing strain is not attended with, expansion. Let us consider geometrically the nature of the strain pro- duced by an elongation of any amount, a— i (no longer assumed to be small) along a line Px (fig. 52), and an equal compression along the perpendicular line Py. Consider the line PA which is drawn making LaPx = tan~^a. ^ „/ 0. Jil Draw Pa at an angle of - with Px. B'S/\ Then if a parallel, Aa, to Px be Z/. drawn from any point, .4, on the line ^" '^ PA, the point A will, by the elon- gation, viz., a— I, parallel to Px, be drawn out to a. Again, by the compression in the direction jiP, the point a will be brought to A', such that aA' is parallel to yP and PA' = PA. For if {x, y) are the co-ordinates of A, y lax,y) are the co-ordinates of a, and {ax, -) are those oi A'; but y = ax, .: A is (x, ax) and A' is {ax, x), .: PA' ■= PA^ and all lengths measured along PA remain unaltered. Evi- dently pa' and PA are equally inclined to Pa. Again, considering the analogous line PB in the left hand quadrant, i.e., such that LBPy= LAPy, the length PB is un- altered, and B comes to B', such that PB^ is perpendicular to Pa. Introduce a rigid-body-rotation to bring PA' back to FA, and this will bring B^ to E^' ', and it is easy XO see that BB^' is parallel to PA, so that the point B is slid forward parallel to the plane PA through the distance B^'. Since all parallel lines are elongated or shortened in the same ratio, it 134 Anafysis of Small Strains. follows that all lines parallel to PA remain unaltered in length. Hence if Q is any point on BB", and G" the position of Q after strain, B"(^' = BQ, so that Q and all points on the line BB^' slide through the same distance. The strain is therefore equivalent to a shear. If / is the perpendicular distance bb" between BB" and PA, the amount of the shear = — — » which P is easily proved to be equal to a-\- (I) a Neglecting the effect of pure rotation, the strain could other- wise be produced by holding fast the other unelongated line, PB, and sliding all lines parallel to it. The lines bisecting the angle between the lines, PA and PB, of zero elongation are called the axes of the shear. They are Ox and Oy in the figure. If the strain is small, a = i + J, where j is a small quantity, and - = I —s, so that the amount of the shear a = i+J— (i— j) = 2J„ which agrees with previous results. The expression for a shearing strain if the axis of x is taken coincident with one of the unelongated lines (held fixed) becomes u = 2sy, v = o. 85. Strain Potential. If at every point of the strained body the rotation, a>, is zero, the components, u, v, of the absolute displacement of every point are such that udx + ndy is the differential of some function of the co-ordinates, x,y, of the point (Art. 83). Denoting this function by ^ i^x, y), Or simply by <^, we have at all points when the strain is pure udx -f- vdy - d=C are called curves of equal potential. Into this subject we shall not further enter here, because it will be fully discussed in the next chapter. Examples. 1. Prove analytically that the shear of any two rectangular lines inter- secting at any point is equal to the difference between the elongations along the internal and external bisectors of the angle between them. z. In any case of strain what is the line at any point which is most rotated by the strain ? [The rigid-body-rotation is to be discarded.] 3. Represent graphically the shear of any two rectangular lines { is great. V These two characteristics require us to divide fluids, for practical purposes, into two classes — incompressible and com- pressible, respectively — the first class being in strictness (like perfect fluids themselves) ideal, but closely approximated to, when the intensity of pressure applied to them is not enormous, by liquids. We shall therefore use the term liquid in the sequel to denote an incompressible fluid. To the compressible class belong gases. In a liquid, therefore, the density is not altered by pressure, since by such means it is impossible to squeeze a given volume of it into a smaller space. 88. ITuiplanar motion of a fltiid. In accordance with Statistical and Historical Methods. 139 the plan of this work, we shall deal almost exclusively with fluid motion which is exactly the same in parallel planes, how- ever much it may vary from point to point in any one of these planes. We shall therefore assume that the motions which we consider take place in the plane xy — there being, therefore, no displacements in the direction perpendicular to this plane. 89. Statistical and Historical points of view. Con- sider a fluid of infinite extent, i. e., occupying the whole of the plane xy. /Q^ ■■■ Then there are two ways of regarding .,-^p =f(x -1- udt, y + vdt, t+dt); Relative Velocity of two close points. 145 which is the rate of change of density of the partide. We shall denote -^, for conciseness, by D^p. In the case of a liquid, therefore, the fact of its incompressibility is expressed by the equation D,p = o. 95. Equation of Continuity for a Liquid. As we have just said, for an incompressible fluid Z)jp = o. Now the equation of continuity, (a). Art. 93, is obviously the same as dp dp ^ dp ,du dv . ^ dt dx dy '^ ^dx dy ' 1 ^ du da , ^ -p''^<'^rx+dy'°- <^) Hence the equation of continuity becomes for a liquid DfP = o, (o) du dv ,„. d^ + dy-^- (^) 96. Components of Relative Velocity of two close points. Let the components, parallel to the axes of x a.ndy, of the velocity at P (fig. 53, p. 139) be « and v at the time /, and let ^ and rj be the co-ordinates of a very close point Q with reference to axes drawn at P parallel to the fi-xed axes. Then u and V will be functions of x,y, t of the forms « =/ {x,y, 4. » =A kx,y, i); (i) and if at the same time, i, «' and »* are the components of the velocity at Q, t^ =fAx + ^,y + r],f), r>'=/,{x+i,y+ri, fj. , ^du du Hence «^-^=^^+'' :^' , ,^ dv ^ dx dy It is to be observed that u and v are not now (as they were in the theory of small strains) small duplacemenh of a particle, I. 146 Kinematics of Fluids. but velociiies of any magnitudes. The equations just written are to be put into the forms (4), p. 122 by assuming, as previously, du do , dv , du dv du dx dy dx dy dx dy so that denoting the components of relative velocity of Q with respect to P by d, ;8, we have d = af+f»j— £017, (2) ^ = jf+3ij + a)f. (3) Now (Art. 29) the terms —m\ and a>^ in these equations are the components of velocity due to a rigid-body rotation, with angular velocity (o, round an axis through P perpendicular to the plane of motion; and the terms a$+sj] and s^+i^ are the components of velocity due to a pure strain at P (Art. 83). Hence if we consider what happens in any small element of the fluid surrounding P, we see that, no matter hmo the motion of the fluid is produced, — The relative velocity of any particle in the element, with respect to P, at any instant, is compounded of velocity resulting from two catises, viz. — (a) Rotation of the element, as a rigid body, about an axis through P, and ib) Pure Strain of the element about P as a fixed point. Observe that this is stated with respect to the relative velocity of every particle in the element with respect to P. To get the absolute velocity of each point, Q, in the element, we must com- bine with this relative velocity the absolute velocity oi P; so that the components of absolute velocity of Q are a+d, w+jS — as is obvious from the definitions of d and p. 97. Expansion. The expansion at any point of a fluid in motion is defined to be the ratio which the time-rate of increase of any small volume at ihe point bears to that volume. If the I dv small volume selected is V, the expansion is — -j- » which will vary from point to point, and also (unless the motion is steady) from time to time. Taking the small volume at P (fig. 55, p. 142) to be one Vortex Motion. 147 standing on the rectangle dxdy with any height, it is easily seen r dV du dv . . " V~di '^ dx'^'^' ^"^ The expansion is, then, —+ — , which we shall denote by 0. dx dy ■' The meaning of the equation of continuity, (/3), Art gg, for a liquid is, therefore, that at every point the expansion is nothing — which, of course, is necessarily the case since a liquid is incompressible. The above equation (a) is exactly the same as (i). Art. 95; for since the mass of an element does not alter, Fp is constant, so that ^nv=—lz>.p. V P 98. notation. Vortex Motion. If m, the angular velo- city, or, as we shall call it, the rotation or spin ^ of the -fluid element at P is not zero, the motion is said to be vortex motion at P. It is also called rotational motion. It may happen that all through certain limited regions of the fluid the rotation is not zero, while throughout the remainder of the fluid it is zero; then the regions in which a> exists are called vortex regions, i.e., the motion at each point of such a region is vortex motion. It is of fundamental importance that the student should clearly understand what it is that constitutes vortex motion. Such motion does not necessarily result from revolution about an axis. For example, a mass of water may be conceived to whirl round an axis in such a way that though every particle of the water describes a circle round the axis, the rotation to which constitutes vortex motion at each point is zero all through the liquid. Uhis will be proved a little farther on. The essential thing to observe now is that the 'rotation' which constitutes vortex motion is not any angular velocity of the fluid particles about an axis fixed in, space, but an element- ' The term spin is used by Clifford, With him the spin, is a vector indicating at once the axis of the rotation and its magnitude. L 2 148 Kinematics of Fluids. rotation at each point, i.e., a rotation of a small element of the fluid about an axis drawn through a mean point inside the element. Vorticity at any point does not depend on velocities, u, v, at the point, but on rates of increase of these velocities— its measure being ^ -3 ^just as the sign or amount of the electrification at any one point of a conductor acted on by any inducing charges cannot be inferred from the value of the potential on the surface of the conductor, but depends on the rate at which this potential changes (both in magnitude and in sign) as we move outwards from the conductor along the normal at the point. It will be observed that we are taking positive rotation (positive vorticity) in the sense from +j; to +j'; and this sense we shall, for shortness, frequently denote by the notation X ' 99. Irrotational Motion. At every point in the fluid at which a> = o, the motion is called irrotational motion. . , . . du dv At such pomts smce -j — j- = o, the expression udx + vdy is the differential of some function of x^y, and /. We shall denote this function by <^, so that in the non-vortical regions we l^ve udx+vdji = d{x,j', t), or =d(^, simply. Of course if the motion has become steady, (f) will not involve /. The function is called the velocity potential of the motion; aiid we see that it always exists in those regions of the fluid where the vortical spin is zero. If (/>! is the value of ^ at P, we may construct the curve, PA, whose equation is (^ = . This is called an equipotential curve. We may at each instant map out the whole irrotational region by drawing a series of very close equipotential . curves. Theorem. 149 PA, F^B, P"c, ... and if the motioa is steady this map will not alter with time. 100. Theorem. The com- ponent of velocity at any point of an irrotational region along any direction is the line-rate of in- crease of the velocity potential in this direction. This is at once evident by taking the axis of x parallel to the direction ; or it may be seen thus. Let P be the given point, the direction being PP^, and p' very close to P. Let PP' = ds, and let the projections of PP' along the axes be dx and dy. Then the velocity at P resolved along PP' is Fig. 56. dx dy but u dx dy therefore this component is d4> dx d dy dx ds dy ds -V) or d^ Ts' which is the line-rate of increase of (^ as we go from P to y. Observe also that the sense of the component velocity along any line at a point is the sense in Which the velocity potential increases along the line. Cor. I. The resultant velocity at any point in an irrotationdl region is in the direction of the normal at the point to the equi- potential curve passing through it. For if P is the point (fig. 56), and PA the eqnipotential curve ±rough it, the component velocity in the direction PQ is zero, since <^ is the same at Q as at P. Hence the resultant velocity at P takes place along the normal, PP', to the curve PA. If ^p denotes the value of ^ along the curve PA and ^pi the value along I'B, the resultant velocity at P is approximately p'~p PP' ' 150 Kinematics of Fluids. and the approximation will be closer the nearer the curves PA and I^B are drawn to each other. If dn denotes the element of normal to the equipotential curve at P, the resultant velocity is d4, dn and the sense of its vector is that in which increases. CoR. 2. The component of velocity normal to any curve drawn through /■ is ^> where dv is an elementary length measured along this normal. CoR. 3. The fluid moves from regions of lower to regions of higher velocity potential. Consequently, as we follow the course of a stream-line (see next Article) in the direction of motion the velocity potential continually increases; and if it should happen, as it may in some cases, that the stream-line returns to the point from which it started, forming a closed curve, it will follow of necessity that the potential at this point has more values than one. [This always happens when there are vortices present.] 101. Lines of Plow and Stream-lines. If at P we draw an element of length PP' in the direction of the resuhant velocity at P, and then at P^ continue the line in the direction, P^P'', of the resultant velocity at P^, and so on, we get a con- tinuous curve, pa', such that at every point on it the resultant velocity at the point is directed along the tangent to the curve. Such a curve is called a Line of Flow. Similarly at any other point, Q, we can draw a line of flow, Q^ ; aind we can in this way map out the whole region by drawing lines of flow. Every line of flow cuts every equipotential curve which it meets at right angles ; for at each point the resultant velocity is along the tangent to the line of flow and along the normal to the equipotential curve. Lines of flow exist in all parts of the fluid, whether vortex regions or irrotational regions ; for every particle of the fluid must at any instant be moving in some definite direction. When the motion becomes steady, each line of flow becomes Differential Equations for^ and ^. 151 the actual path of a fluid particle, which is called a stream-line. If the motion is not steady, the map of lines of flow will change from instant to instant, so that the actual path of any particle which lies at the time / on a line of flow merely touches the hne of flow at the point without coinciding with it *, If at any instant u and v are the components, parallel to the axes, of the velocity at the point x,y, the differential equation of the line of flow is ^ ^ ■J- = -1 or ax u ' udy—vdx = o. (a) In the case of a liquid the left-hand side of this equation is the exact differential of a function, i/^ {x,y), of x sxiAji ; for the condition that it should be is 7- + "j~ = °> which is the equa- tion (/S) of Art. 95. ^ The function ■^ is called 'Cor function offlonii). Thus, thenj if the moving body is a liquid, and if its motion is steady, there exists at every point, whether in a vortex region or not, a flow function, i/r ; and in the irrotational regions there exists at each point a velocity potential function, ^, in addition ; and the equations rf) = c C and C' being any constants, denote a system of orthogonal curves, i. e., mery curve of one system cuts orthogonally every curve of the other system which it meets. 102. DifFerential Equations for ^ and ^. In the case of a moving liquid, whether homogeneous or not, ^ exists in irrotational regions and ^ throughout the whole. Now putting ^ and — for u and v in equation O) of Art. 95, it becomes dx dy 1 CUfford (^Kinematic, p. 199) gives a good illustration of the relation and distinction between lines of flow and stream-lines in a case of unsteady motion: '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 pomt, and consequently 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.' 152 Kinematics vf Fluids. rf* d^ where V stands for —■, + j-j" AgaJDj since « = -^ and v = —-— > we have -T-^ + -r^ = T 7- =—20) (Art. 96), almg an orthogonal curve at the -point. For (see fig. 56, p. 149), assuming now that PQ is any curve 1 54 Kinematics of Fluids. at P, and PP' at right angles to it, the time-rate of flow across PQ is -j^; but it is also v, the normal velocity at P, which is * I ds -^,, where ds' is the element of arc of PP', (Cor. 2, Art. 100). as Hence at every point d(j> d\j/ ds' ds where ds and ds' are elements of any curve and an orthogonal curve respectively; and each of these expressions is the time- rate of flow across the first. From this we see that the velocity at any point in any direction may be determined from the flow-function yj/ instead of from the velocity potential ^ ; and its determination from the former is always possible, whereas the latter function may not exist at the point. [It will not exist when the motion at the point is vortical.] TAe velocity in any direction {ds") at a faint is the line-rate of increase of the flffvo-function in the perpendicular direction (ds) — the sense in which the diiferential coefiScient of the I low-function is taken (sense in which ds is measured) is determined, by the sense in which the rotation is measured* Thus, if P is the point considered, and the rotation, m, at P is measured in the sense of the arrow (Fig. 57), the velocity in the direc- F'g- 5?- tion Pa is ~ (when Pli, which is perpen- dicular to Pa, is diminished indefinitely) ; and the velocity in the direction /"iJ is -^ > i.e., — -^ (-P« and Pc being dimi- nished indefinitely). We have had at the outset a particular example in the equations «= -^> v= — -r^> the sense of . When the curve is closed (or B coincides with A) the flow round it is called the circulation in it (Sir W. Thomson, Voriex Motion, Edinburgh Trans., 1869). We shall calculate the value of the circulation round any small closed curve surrounding any point P (Art. 89) in the liquid. Let C be a point on the curve; then the circulation round the curve is, with the notation of Art. 96, taken over the same curve. Now observe that k, », -^-1 -3- dx ay belong to the point P, and do not vary with Q ; they may there- fore be taken outside the integral sign ; and since in the closed curve /^d^= o, fridr] = 6, the circulation is But, observing that we have taken the circulation round the curve in the same sense as that of the rotation at P, i.e., from X toy, it is clear that f^d-q^ — yij .hS, i. e., twice the product of the area of the curve and the vortical spin inside it. We can now prove that the circulation round any curve A is equal to twice the surface-integral of the rotation taken over its area (fig. 58). For, inside A draw any other closed curve, a', very close to A at all points. Draw arbitrary lines aci, bb', c/, ... very close to each other across the curves, and apply the last result to each of the little areas abb'd, hcc'b', ... ; and we have 156 Kinematics of Fluids, circulation round abb'a' = 2 . area a^3V x rotation within it, „ „ bcc'b' = „ bcc'b' X „ „ If we add the left-hand sides together, a glance at the figure shows that the portions of circulation in all sides, bb', f/, •■- which are common to two little areas destroy each other (as shown by the arrows); the total result of the summation is, therefore, circulation in ^4— circulation in ^' = 2 X surface-integral of rotation in the space between them. Similarly the circulation in A' is equal to the circulation in any internal curve close to it, plus twice the surface-integral of the rotation over the space between tliem. We may thus go on contracting the internal curve until it disappears, and we have the required result, circnalation round A = 2j'm.dS, where dS is any element of area within the curve, and o) the vortical spin at the position of the element. This Article and Art. 103 enable us, therefore, to attach physical meanings to the fi\nctions ^B—^A and ■^B—'^A which appertain to any two points A and B of the liquid. From the theorem of this Article we can prove also that if a liquid is enclosed within a boundary every point of which is at rest, either there is no motion whatever of the liquid inside, or, if there is, the motion must be vortical. For the liquid particles at the boundary (if they are in motion) must move along the boundary, which is therefore a closed stream-line. And if there is no vortex inside, such a stream- line cannot exist, since the circulation along it "would have to be zero by this Article, whereas the flux across every Fig. 59. section of a tube of flow is the same, so that the circulation along it cannot vanish. Non-vortical motion is therefore impossible inside a boundary which is kept at rest. Theorem. 1 57 • 106. Theorem, If any closed curve, PQR, in the plane of motion be described in an irrotational region of a liquid ; and if <^ is the value af the velocity potential at any point, -^ the normal rate of increase of ^ measured constantly outwards from the curve, and ds the element of arc at the point, v?e have / rft'^' = °' ^''^ assuming that the velocity is not infinite anywhere inside the area. For, (a) asserts that the line-integral of the normal velocity taken over the curve is zero. Now if A (fig. 58) is the closed curve, its area may be broken up into indefinitely small closed curves, such as abh'a', ... ; and the line-integral of normal velocity taken over each of these is zero. For, if any small closed curve be described round P (fig. 53), the line-integral of normal velocity round it is /K . du dus , /• , . (fz) , dv\ ,yi dx dy' ^ dx dy' ... tdu dv^ . „«. , „ and, as m Art. 105, this is ( j- -1- -~) . 6.?, or QlS-^ but ^ = o since the fluid is a liquid. Hence (a) follows exactly as in Art. 105. This assumes that neither u nor v is infinite at any point ; for if u is infinite, we must not assume ufdt] = o ; i. e., the closed curve A must not include a source or a sink. Equation (o) holds whether the curve includes vortical regions or not ; for all that we have assumed is that at every point inside 5 = o, and the velocity is finite. But if the curve cuts through a vortical region, the function <^ does not exist at any point of the portion of the curve in- cluded in this region, so that we cannot represent the normal velocity at such a point by ^; this normal component is «cosflH-»sin^, where Q is the angle which the normal makes with the axis of 158 Kinematics of Fluids. x; and hence for the portion of the curve included in the vortical region we must write the corresponding portion of the line-integral in the form y(a cos 5 + f sin &) ds. 107. Tubes of How. Assuming the motion to be steady, the flow-lines are stream-lines, and the particles at P and Q (fig. 56, p. 149) actually travel along the paths PA' and QE^. No liquid ever crosses any of the stream-lines, PA', QB', — Hence the space between the stream-lines PA' and QB' is a channel through which liquid is flowing, none of the liquid within the channel ever leaving it. Hence also it is obvious that the mass which flows across the section PQ of this channel in any time is the same as that which flows across any other section, P'Qf, etc., in the same time ; and this quantity is measured by the product of the time and p(^p~^fl)> or P^'^- Another way of expressing this fact is this — if <&j and ds^ are any two normal sections of a channel, where (-r-), means the value of -^ at the first section. ^ an' an In the case which alone we consider (uniplanar motion) we imagine the channel PQB^A' to be a section of a column of indefinite height, which is obtained by erecting perpendiculars to the plane xy at all points along PA' and Q.^, so that we are really considering a mass of liquid moving between two in- definitely high walls standing on the stream-lines. In the general case (in which motion is not all parallel to one plane) the channels of never varying liquid are tubes — not columns between indefinitely high walls — and such a tube of flow may be constructed by describing any closed curve in the fluid and at each point of its contour describing the line of flow. Although treating only of uniplanar motion, we shall use the term tube offlmja as equivalent to the channel included between two lines of flow, such as the lines PA' and Ql^, Closure of Equipotential and Stream Curves. 159 108. Closure of Equipotential Curves and Stream- lines. We now proceed to inquire whether the equipotential velocity curves, <^ = C, or the stream-lines, ^=C, can be closed curves, in the case of a liquid. To begin with the ktter, it is clear that no tube of flow can begin at one place and end at a different place within the liquid. For the flux across each end of the tube would be zero, and we know that the flux across all sections is the same ; therefore there is no motion at all inside the tube. Hence a tube of flow, if it terminates, must do so at the boundary of the liquid, and the boundary will form part of it ; but a tube of flow may obviously be a closed tube, i. e., one returning into itself. The stream-lines may, therefore, be closed curves within the liquid ; but they cannot be so unless the liquid included contains a vortex. In a fluid which is devoid of all vortex motion no stream-line could be closed ; for the circulation round a stream- line can never be zero (see Art. 100). Next, assume an equi- potential curve, (^ = C, to be a closed curve. Then it must be possible to find immediately inside this curve another (necessarily) closed curve on which has a constant value, C', differing infinitely little from C, so that if dn is the normal distance between these curves at any point, the value of -^ is — ; — » ' *^ dn dn which, though it has different values at all points on the curve, has everywhere the same sign ; and it is easy to see that this will be impossible unless a source or a sink is included within the area of the curve. For if there is no included source or sink, we must have I -^ ds = o, the integral being taken over the contour of the curve; and this integral cannot vanish unless C' is the same as C. If there is a source or a sink inside, the integral is not required to vanish and the equi- potential curve may be closed; but if there is not, C' and C must be the same ; and continuing the same process, we see that ^ must be constant throughout the space -included by the curve, i.e., no motion is taking place inside. Hence if an equipotential curve is closed, there must be inside it either a source or a sink. (In illustration see example 5 following.) i6o Kinematics of Fluids. 109. Velocity System. By a velocity system ■within any region of a fluid we shall understand a diagram in which at every point of the region there is drawn a right line repre- senting both in magnitude and in direction the velocity at the point. Unless the motion is steady, this diagram of the simultaneous velocities at all points will change from time to time. It will be the same thing if we understand by the velocity system analytical expressions for the components, u, v, of velocity at each point, as functions of the co-ordinates of the point (and of the time, if the motion is not steady). For conciseness denote a velocity system by the notation [«, »]. 110. Theorem, If on any closed curve enclosing, in the plane of motion, a portion of a fluid (liquid or gas) the velocity potential has at each point of the curve an assigned value, and if also at eaeh point of the enclosed fluid the values of the expansion and vortical spin are assigned, there cannot be two distinct velocity- systems satisfying the assigned conditions. For, let (^ denote the assigned value of the velocity potential at any point of the curve (the value of <^ being, of course, different for diflFerent points on the curve) ; and, if possiUe, let there be two velocity-systems, [«, vl and [a', a'], (a) each of which will give the same value of ^ at the same point of the curve, and die same values of & and a> at the same point inside. The exact reverse of any given motion is, of course, possible. Reverse everything in the second system, and then superpose the first on it. We shall thus have a constant (zero) velocity potential at each point on the given curve, and no ex- pansion or vortical spin at any point inside. But (Art. io8) this requires complete rest at every point of the enclosed fluid (which with no expansion may, of course, be regarded as a liquid). Hence at every point inside the velocity-system [«—«/, v—v'] is null, i.e., the two systems (o) are identical — which proves the proposition. u = ^ + dp Zl = 17 Uniqm Velocity-System. i6i This theorem is of fundamental importance ; fof if we can find out any one velocity-system producing a certain distribution of velocity potential over a closed curve and of expansion and vortical spin inside it, we are now enabled to assume that this is the only system fulfilling the conditions. We now proceed to determine (for the particular case of uniplanar motion) the velocity components, u, v, at each point when the values of d and co are assigned. Assume, as in Art. 102, dp _ dN ,^ dnr Zc' <''> where P and JVare (as yet) unknown functions of x,y. Then V^P=e, and V«ivr=-2a). (6) But in the theory of attractions it is proved that if V is the potential, at any point, of matter attracting according to the law of the inverse square of distance, and p the volume-density of the matter at the point, V^F= — 47rp, where d^ d" d^ ~ dx-"^ df^ dz^ If the distribution of matter is the same at all points in space whose co-ordinates x, y are the same, differential coefficients with respect to z disappear, and in this case V^ = -5-3 -h -j^- Hence equations (8) show that— (a) the value of P at any point, A, of the fluid may be regarded as the potential (per unit mass) produced at the point by a distribution of matter extending infinitely above and below the plane {x, y) of motion, the density of this matter at any point, Q, being > where is the expansion at Q ; 41T (b) the value of N at any point. A, of the fluid may be regarded as the potential (per unit mass) produced at the point by a distribution of matter extending infinitely above and below the plane {x,y) of motion, the density of this matter at any point, Q, being — > where m is the vortical spin at Q. 2Tr M 1 62 Kinematics of Fluids. It may happen (as will be exemplified in a subsequent section) that there is expansion or spin at only certain isolated points of the fluid. In this case the distributions of matter which we imagine for the purpose of calculating P ox N will be confined to infinitely long slender cylinders perpendicular to the plane of motion of the fluid. Now, as we know that in the theory of attractions V can be expressed in the form / — > where dm is the element of mass at Q, and r = the distance of Q from A, we see that J\r= — -dxdydz, (^ which give the values of P and JV, and therefore the values of « and V, when the values of and and (3) becomes dF dF Since -^ and -r- are proportional to the direction-cosines dx dy of the normal to the boundary, this equation is obvious ; for if V is the resultant velocity at the boundary and x the angle which its direction makes with the normal, (4) is simply V cos X = o. Any flow-line may obviously be taken as a boundary of the field since it satisfies the condition of having no flow across it. Examples. I. If the particles of a fluid describe a series of concentric circles, find the equation of continuity in its simplest form. ' Let be the common centre of the circles ; P a point on any circle whose radius, OP, \&r; 6 the angle which OP makes with any fixed right line; Q a point on the circle indefinitely near P; 6 + d9 the angle of M 2 D 164 Kinematics of Fluids. direction of OQ; describe with centre a circle of radius r + dr, and produce OP and OQ to meet this circle in P' and C. Also let p be the density at P and 9 the angular velocity of the fluid particle at P about O, at the time t. [Observe we do not assume the fluid to be a liquid.] Then the normal velocity at the face /"/" is ri ; therefore the flux into the area PP'Q'Q in the time dt is pridrdt, and the flux out of it through QQ! is friJr ^ d6 drdt ; therefore the loss of fluid in time dt is ri^drdOdt. do Again, the quantity in the area at time i is p X area, or prdrdff; and the quantity in it at time i + di is (j>r + r -^ dt)drd6; therefore the gain is r-j-drd6dt. Equating this to the former expression for the gain, we have ^p d{pS) dt^^r = °' which is the required equation of continuity. [Of course this equation could have been otherwise obtained by transforming the Cartesian equation of continuity to polar co-prdinat^s.J. i. Find, generally, the equation of continuity in polar co-ordinates. At any point, P, whose polar co-ordinates are r, 9, let the fluid velo- cities, at the time t, along and perpendicular to OP be a and /S, re- spectively, and at this time let p be the density at P. Draw a radius vector OQ equal in length to r + dr, and making an angle dB with OP; from /"draw Pp perpendicular to OQ, and from Q draw Qq perpendicular to OP. (Fig, 60.) In the time di the flux into the area by the face Pf is. po x i^ x di, or pardSdt; and the flux out of it by the face Qq is [par+'^-^^^dr\,adt; therefore the loss of fluid by the motion along OP is iSi^drdOdt. dr Again, the flux into the area by the face Pq is p&drdt ; and the fiux out by the face pQ is [p)3 + -^ de\ drdt, therefore the loss by this motion IS ^-^drdOdt. de But the mass covering the area at the time / is prdrdS; and the mass ftt time t-k-dth (p + -j dt)rdr dS. Hence we have Examples, 165 df d(par) d(p$) _ ,. Observe that p, the density at a fixed-space point /", is evidently in general a function of the time and the position of the point ; so that P =/(»-. e, t). At the end of the time t + dt, the fluid element which was at P will have polar co-ordinates (j' + adt, 6 + — dt), and will be at a point P' (which will not of course be Q unless PQ is so chosen as to be an element of the line of flow at P) ; and the density at /" is, by the above formula, o /(r+adt, S+ —dt, t+dt). But the particle at P' is the one which was at P, and if the density of this particle has changed in the time dt to p + Dtp. dt, we have Here of course Vtp is thie ;ofei/ time-rate of change of />(see Art. 94). Hence (i) can be written I _, I rd(,ar) dfi-i so that the polar equation of continuity for a liquid is d(a.r) dB _ dr ^ de~°' 3. Transform the equations vV = °> V"^ = A/> from Cartesiaii to polar co-ordinates. If " = '^' ^-rde- Hence equation (3) of last example becomes I ^ d'

I d' id d' id' [Of course it can be easily proved that V* = - ^ + ^a + ^ ^2 , by transformation,] 4. Transform, by elementary principles, the equation v°^ = -^<>> to polar co-ordinates. Let P be the point at which the value of the flow function is ^ and the rotation a, in the sense opposite to that of watch-hand rotation ; let O be the origin of polar coordinates, OP=r, OQ = r + dr, lQOP = dB; let fall O^ Pp and Qq perpendicular to OQ and OP; and Fig. 60. ■express the fact that the circulation round the area PqQp'nia-x. the area. 1 66 Kinematics of Fluids. The velocity along Pq is £i = ii , .-. flow along Pq = ^-^.dr; PP rde ^ rde dip the velocity along Pp is — -j- (see Art. 104), df .'. flow along and in the sense of fP = + j" '"'^^ 5 , . , ^ d\b d'f , veloaty along qQ = - -^ - -^ dr, .: flovir along qQ = - (-^ + -^'*') (.r+dr)d$; velocity along fQ = :pig + -p -J^'^^' Hence the total circulation counter-clockwise is , d'xf, dtp 1 d'lf/y. ~ V"d^ '*"d^ "^r dev drdS, neglecting infinitesimals of the third order. Also the rotation multiplied by the area = a.rdrdO; hence d^f T_dip I d'f 1? '^ r ~dr '^ r" dB"" 5. If the stream-lines of a liquid are right lines all passing through one point, find the law of irrotational motion. Let be the point of convergence (or divergence) of the lines of flow OP, OQ, ... (Fig. 60); then since the velocity at P perpendicular to OP is zero, we have ^^ dd, ^ being the flow function and tp the velocity potential (which by supposition exists at every point). The second equation gives, by example 3, d'^tb I d(b -d^*r-dr^°' '•'■' Examples. 169 dn being an element, PQ, of the normal at the point drawn inwards (Art. 104), and Q being a point on a consecutive ellipse of the series (on which the flow-function is ^ + dtf/) ; then, / being, as above, the perpendicular from the centre on the tangent at P, we have where 6 is the angle made by the tangent with the axis of x. The tangent at Q is parallel to that at P, so that dn = —dp=— —dn (since 6 is the 2p same for both tangents). Hence the velocjtjf, V, at P is d^ -'^djl' °' ^ = ^. ... ^ = ^=^^. (9) so that the velocity really takes place in the sense 1 . For points far removed from the centre the ellipses tend to become circles, and for such points p is approximately equal to »/n. Hence at infinitely distant points V = o, i.e., the infinitely distant particles are at rest. At points taken on the same stream-line the velocities are directly pro- portional to the central perpendiculars on the tangents at the points. The potential function, Now if Vc is the semi-major axis of the confocal hyperbola through P, we have , _ /"' . ,^ _ it^-^jc'-y) .'. dx = — \/-dv; ds^i V ,^7"^ ''. (d/i being zero along ds) ; hence Substituting these values in (9), we have d _A I .-.

the velocity-function, they will simply interchange their previous values, so that we have >' ; heiice at all such points there is no velocity. At each focus the velocity is infinite as in the previous case. At points between a focus and tie centre the velo- cities are perpendicular to the line FF', and they vary from 00 at i^ to zero at the centre. At points on the line ^00, the velbcities (being along the hyperbola) are all along this line, and of magnitudes varying from 00 at F to zero at infinity, since for all such points v = c^ and ti, = x'. The infinitely distant parts of the liquid are at rest, as in the previous case. Since along the line i^oo the velocities are along the line itself, this line may be considered as a fixed smooth wall round the edge, F, of which the liquid flows, • those particles which have travelled along one side of the wall in the sense oaF turning back, when they reach F, with infinite velo- city and travelling off along the other side of the wall in the sense F 00. 8. Determine the law of motion of a liquid in order that the stream-lines may be a series of confocal ellipses of Cassini, the motion being irro- tational. Let F, F' (fig. 61) be the foci, the centre, / and /' the distances. Examples. 171 PF, PF', of any point P in the plane of motion, from the foci, and let the equation of the Cassinian through P be ff=k\ (I) k being the constant belonging to this ellipse. Then if OP=r, lPOF=B, the above equation is equivalent to y»_ 2£»r''C0S2e + ^*— <5* = o, (2) where c = OF = iFF'. 2/ \p \ ^~.,^^ r^^F' \ ^ ^\\ ^ X X ^=^ ^"~~^^, - 1 p , Fig. 61. Now we have, by hypothesis, some (as yet) unknown function of k ; and by example 4, dr" * r dr * r'dW ~ °' (3) (4) Transform this into an equation in which k is the independent variable. Now from (2) we have dk t^—c'rcas29 ,„. . ^. -, {0 bemg taken as constant) (5) dr /6' d^_ <:Vsin29 de ~ ^ ' (r Ijeing taken as constant). and (4) transforms into d^itflMl' I dl% df „, .d'lp d^ _ in which we may reject the arbitrary constant B. (6) (7) (8) 173 Kinematics of Fluids. Now the components of velocity along and perpendicular to OP are given by the equations dip A dk Ac^r . „ , . Vdrk^M^-W'"^^^' ^9) diji Adk Ar , , . „ , . Squaring these and adding, we have for the resultant velocity, V, V-% (") in virtue of (a). On any one ellipse the velocity is directly proportional to the distance from the centre. Also at infinitely distant points r^f=f' = k, ap- proximately, therefore at all such points V = o,ot the fluid is at rest at infinity. To find the equipotential curves. Let denote the velocity potential at P. Then (ex. 3) we have d-'— 2 c^ r^ cos 2 fl + c* ' and integrating with respect to r and adding an arbitrary function of 6, we have a 1 " n . -^ ^ ,/>■'-<: cos 2fl\ .,„^

- as a constant), and equating the result to the value of -^ in (12), we have ad f{e)=^-A, .: /(a)=-A0, rejecting an arbitrary constant. Hence A^ , . ^, /r'—c' cos 2B\ Multiply both sides of this equation by 2, divide it out by A, and take the tangents of both sides, and we have /-' (cos 29— sin 2 9 tan-^) = c^. >^COS2(9+|)=^COS?^. (15^ Examples. 173 Now observe that for any one eqnipotential curve ^ is constant ; hence the curve denoted by this last equation is a rectangular hyperbola, whose major axis is inclined at the angle ~z to the line OF. By varying ^ we get a series of rectangular hyperbolas having for centre and all passing through the foci, F, F'. The foci are points at which V = given in (14) may be put into a simpler, form. From it ■we have ,rf, ^ cos 29-,:' A r^ sm 2 9 But if e, and e, denote the angles PFca and PF' 00, we find easily r' sin 2 9 tan (e, + 62) = 'cos 2S—C' Hence ^ = ^_(e, + e,), (l6) so that e, + ^2 is constant along any one equipotential curve (rectangular hyperbola). The circulation roimd any curve enclosing one focus is easily found. For, let the line yy form part of the curve, the remaining portion being one half of a very distant stream-line cut off by this line. Such a distant stream-line may obviously be considered as a circle, of radius R suppose, since the ovals ultimately become circles. Now at each point of the line yy the velocity is normal to the line, and the flow along this line is therefore zero. Also the velocity at each point of the semicircle is A . R A — ^-^ by (11), or — (since k'^ = R^, nearly); therefore the flow round it is 11 A, which is therefore the circulation round every curve enclosing F; and the same value holds with reference to the other focus, F'- These ellipses of Cassini and rectangular hyperbolas will be arrived at more simply from another point of view in a subsequent section. 9. Determine the circumstances of motion so that the stream-lines in the irrotational motion of a liquid may be a series of equiangular spirals having the same pole and the same constant angle. The equation of any one of the spirals vrill be r = ae*^, (i) where k is the cotangent of the common angle of the spirals, and a is a constant varying from one spiral to another. Assuming f =/(a), and using the polar differential equation for tf/, we obtain d'ti dti .: yji^A log a, (3) or expressing i^ in terms of the co-ordinates of a point,

-sm c The potential function, ^, is found from (5) and (6); and since dib d-ii d= -A{klogr + e). (8) The curve ^ = const, is also an equiangular spiral, _*. _*- r =ce k, \c = rkA-\ whose angle is the complement of the constant angle 6. The velocity at the pole ' is infinite. To find the circulation along any curve surrounding the pole, describe a circle of radius r roimd it, and take the circulation round this circle. It is r/0d6, where is given by (6). Hence the circulation is . 2-nA. It may be well to call the attention of the student to a possible error. Suppose it proposed to find the nature of the motion so that the stream- lines shall be a series of equiangular spirals of different constant angles all having the same initial radius a ; that is, the number k in (i) is now the changing constant which determines each spiral. If we seek to form a differential equation for ^i in terms of k, analogous to (z), we find that such is possible, and the equation is (/^ifr d^ /I r\ (1 + ^°) -73 + 2^ -^ = o ; therefore 111 = A tan-'*^ = A tan-' ( ^ log-) • dir dk ^0 a* Now tan-' k has an infinite number of values, so that an infinite number of these spirals vrill pass through any one point, P, in the .plane of motion. This signifies that at every point the resultant velocity takes place in an infinite number of directions. But such motion is possible only at a source or a sink, and we should be compelled to conceive what Clifford calls a ' squirt ' as existing at every point in the plane. Such is clearly impossible. We must take care, then, that the form which we assume for the flow- function does not at every point give us an infinite number of stream- lines. The spirals assumed in the example which we have just solved — ^those in Examples. 175 ■which a is different and k the same for all — do not labour under this, objection, since, except at the pole, no two of them can ever intersect. 10. Determine the circumstances of motion so that the lines of flow- shall be a series of parabolas having a common focus and coincident axes. Ans. If r, are the polar co-ordinates of any point in the plane, referred to the focus and axis, 6 9 \p = Ariwa — ; tp = Ar^cos — . The equipotential lines are another series of confocal parabolas, having the same focus and axis as the first set. Let i^oo be the axis through the focus F from which 6 is measured, and Fx' this axis produced in the opposite sense through the focus. Then the velocity at every point on i^oo is along this line and towards F; the velocity at every point on Fzo' is perpendicular to this line ; the velocity at F is infinite, so that there is a stream along the line 00 F, the liquid thus coming escaping down through F, but there is no escape through F of fluid coming in any other direction, so that F is not a sink ; and if we calculate the total quantity of liquid passing per unit of time into any circle described round F, i.e., fvds (Art 92), we find it to be zero, since only an infinitesimal amount escapes from the line cdF aX F. 11. If pi and Pi are two variables defined by the equations Ci = /i {x, y), P2 = /a {x, y), such that the curves pi = const., pz = const, are orthogonal, transform the equation v"^ = o iito one involving differential coefficients with respect to pi and P2 only. At the point /"(fig. 60, p. 165) draw an element, Pp, of the curve for which pi is const., and an element, Pq, of the curve for which pj is const. ; draw also the close curves, qQ and pQ, for which the values are p, + rfp, and p2 + dfi, respectively. Regarding (^ as a liquid velocity potential, the equation which is equivalent to v'^ = ° will simply express that the total quantity of liquid passing, per unit of time, out of the contour of the little area PqQp is zero; and we shall obtain the transformed equation by expressing this fact. If {x, y) are the co-ordinates of P, and {x + dx^y + dy) those of jf, we have pi + (/pi = /^{x -v dx, y -^ dy) and p^ =/,(« + <&, jz-K^), iPl^+iEldy^dpr, dx dy -^ dpi I -Pdx + dx Solving these for dx and dy, and denoting £p, dx ' dpt dx' dpi dy dp, dy by /, and dpi puttmg ^ dpi dy - '*' ' dx \dp. dy ■■ h^, we have Jy. Pq = Ihdpi ; 176 Kinematics of Fluids. and similarly by determining the co-ordinates of/, we have Jx Pp = h^dfi- Denote Pq by ds-^ and Pp by (/r, ; them _/&, = h^dft ; JdSi = Aji/p, . But it is easy to prove that J ■= h^h^; hence dsi = 7- dh^ \ , , T^ir/^'^'- -Q j^.n/^''^^'' **' \\-d^-h-d^,d^)'^^''^i^- Expressing in the same way the total flow through the sides Pq and pQ, and adding, we have ■wad'^fp ,2 d'^^ h\ dtp dhz kz dip dhi ' dp^^ ^ dpi hi dpi dpi hi dp^ dpi ' which is the required transformed equation. 12. Determine the circumstances of motion (irrotational) so that the stream-lines shall be the series of curves obtained by varying a in the equation ?-» sin «9 = o". \%. Determine the motion so that the stream-lines shall be a series of rectangular hjrperbolas having the same centre and asymptotes, and varying only the magnitudes of their axes. Ans. f = A ix'—y") ; ip= —2Axy. 14. Is it possible to determine a velocity-potential function (or a flow- function) of the form j;'/(— )? Ans. Yes ; such a function will be of the form A {y' — x") + Bxy, where A and B are arbitrary constants. [Putting v' •■*■"/() = °> ws obtain the equation (I +».»)/"- 2X/'+ 2/= o, where ^ is used for — , and /', /" stand for -^ , ^^ ^ , re- spectively. To integrate this, observe that/"=- — -jr- Q^f'—f) ; so that the equation becomes — - — — Q^f'—f)— 2 Q^f—f) = o), which gives ^{^^A. Again. X/'-/-4,({), ••• ,4 (^) = ^ (-.;^)' whence/is at once found.] Examples. 177 15. Show that the equations «= —5 — ^, o = -^ — ^ express an irro- jc^+j/" x'+yr "^ tational liquid motion, and that the velocity-potential function is obtained by combining the well-known functions log r and 6 in the form ^ = a\ogr-{-b9, a and b being constants. 16. A liquid is whirling round a. fixed centre so that each particle describes a circle round the centre with an angular velocity varying as the m"" power of the radius ; find the ratio of the vortical spin of the particle to its angular velocity about the centre. Ans. . [Use the equation v^ ^ = — 2 <" in polar co-ordinates, and observe that ^ = o, -f- = r9 =^ ir^+^, &c. Otherwise, put u — — 6y, V = Bx, &c.] I y. Round any point, P, in the plane of motion of a gas moving with uniplanar motion, a small closed curve is described ; prove that the mean tangential is to the mean normal velocity on the curve as 2 0; is to 6, where [) contained between siu- cessive pairs of them is the same and equal to a (the values of' the flow function for these, lines being -^j^ i/f + a, t/c + 2 a, . . . ) ; suppose also that another cause, acting independently of the first, would give a series of stream lines denoted by 1/^', 1// + a, ■^If' +2a, ... so constructed at intervals- that the time-rate of flow across their channels is also a (the values of their flow functions being 1^', i/^' + a, i/r'+aa, ... ); it is required to construct frpjn this dual system the series of stream lines which will result if both causes of motion simultaneously agitate the fluid. [The stream lines of each svstenj are supposed, to be con- structed at very small intervals.] Superpositi&n of Stream Lines. f]^ Let a be the point at which the two lines ^ and i/r' intersect, and b, c, d the remaining intersections of the lines i/^, ■^■\-a, \j/', yjr'+a. Then abed may be considered as a smatt parallelogram. Also let the velocity at a due to the first cause be from ato d (of course along yjf), as indicated by the arrows, and the velocity at a due to the second from a to b. Denote these velocities by Wi and »2. respectively. Then if « denotes the length of the perpendicular from a on the curve t/t+o, we have (Art. 104) a ' n Also if m' denotes the perpendicular from a on the curve V^' + a. a V W Hence — = — . But the area of the litde triangle v^ n ^ ^ abc^if/xab, and also it =inxad: .: — = ^: hence n ab Wj ad »2 ab and therefore the resultant of v^. and »j is directed in the diagonal ac. Similarly at c the resultant velocity is directed in the diagonal f/"; and that at /"in the diagonal y^. Hence ac/g... is the resultant stream line at a. All other resultant stream lines are similarly constructed, by drawing the diagonal of any other little parallelogram in the map and continuing it as above. We shall now prove the followirig theorem — In the stream line map the successive differences of the flow function from each resultant stream line to the next consecutive is equal to the constant difference of flew function which was assumed in drawing the map. Let f*'= resultant velocity at a = velocity along ac; N= normal distance between the stream line acfg and the next consecutive = perpendicular from b on ac; (n ^^ angle bad. Then n^ + n'^+ 2 nt^' cos u> , N 2 Also ah n smo) F= ac . siniu — ; — . a 1 80 Kinematics of Fluids. ad= — ) .■. ac^ = — ; > sin to sin^ ft) and ac x iV^= 2 . area adc nn = ab .ad sin co = -^ : therefore sin CO a N „ difference of flow function ■ ,.„. <. But K= > .•. the dmerence of two N consecutive resultant flow functions is equal to a, the difference assumed for the two superposed systems in drawing the map. Exactly the same theorem holds with reference to the super- position of two separate systems of equipotential curves ; but the diagonal of each little parallelogram must in this case be taken in a way different from the preceding. For, let •\//, i/r + a, ... i/f', i/f' + o, ... be now two sets of equi- potential curves. Then the velocity at a due to the first cause is at right armies to ad and equal to - (Art. 1 00) ; similarly the second velocity at a is at right angles to ab and equal to -77-- These components are, as before, directly proportional to ad and ah, respectively, therefore, by the triangle of velocities, the resultant velocity at a is at right angles to hd; and therefore, if the intervals are suflBciently small, the resultant velocity at every point on hd is at right angles to hd, which is therefore an element of an equi- potential curve. This element continued from point to point by the same rule becomes an equipotential curve. Moreover, exactly as before, the successive potential differ- ences along the resultant curves are all equal to a, the difference assumed in drawing the map of the component systems. This graphic method of superposition is found in Clerk- Maxwell's Electricity and Magnetism (Vol. i. p. 265, 2nd ed.) ; it has also been employed by Professors G. Carey Foster and O. J. Lodge for the study of the effects of a multiple system of sources and sinks in the flow of electricity over a uniform plane Energy in non-vortical motion. i8i conducting surface. (See the Proceedings of the Physical Society, Feb., 1875.) 113. Energy of non-vortically moving Iiiquid. Let ADBC (fig. 63) be any closed curve containing non-vortically moving liquid. [Of course we do not assume this curve to be a boundary of the liquid. See Art. III.] We propose to find an ex- pression for the energy of the liquid enclosed by it. Divide its area into an indefinitely large number of narrow stream channels, """^ and consider the liquid in one of Fig. 63. them, AB. At any point P draw the normal section of the channel, and in the time A/ suppose that the mass of liquid crossing the section at P is that contained between the normal section at P and the normal section at a close point, Q. Let this mass be A/k; \t.\. PQ = t^s ; = velocity potential at Q. Then the energy of the mass A« is ^(#)^A^, or i#-9A«.. ^^ ds' ^ ds ds But since the particles in the section at Q have gone over Af in As . dd> , . . . , the time At, we may put — for -^ ; and smce Am is the quantity which crosses every section of the channel in time At, . , , , . , Am „ d(b . the energy, A£, m the channel is f -— 2, — Aj, the sum- mation extending from B to A ; i.e., A-ff = |-(<^^ — (#)i) -^• But -r is the time rate of flow across the section of the dt limiting curve at A, which (Art. 92) = density multipUed by normal velocity multiplied by element of arc at A = p (-p ) . ds^ . av ^ Similarly, measuring the element of normal outwards at B, dm ,\ T dt dv ' s Hence AE = \p<^J^-^^ ds^^\p<^J^f^ ^ds^; l82 Kinematics of Fluids. so that if we take all the channels of flow, and measure dv, the element of normal, consistently outwards all over the curve ADBC, we have for E, the total energy contained, = *P A'i*. w the integration extending over the whole contour of the curve. From (a) we derive another proof of the result given in Art. 105, viz., that irrotational motion is impossible inside a boundary which has at no point a velocity along the normal to it; for E = \pj'{ti' + 7^)ds, and if at every point on the boundary d ,. .... -— = o, we must have « = o, v = at every point inside. Hence if we take a liquid at rest ; then set its boundary in motion in any way whatever, altering its shape to any extent ; and then suddenly stop all motion of the boundary, the whole of the fluid inside must instantly come to rest. (Thomson, Vortex Motion, Edin. Trans., 1869.) 114, Green's Equation. Let U and V be any two functions of x, y, the co-ordinates of a point P inside any d^ d^ closed curve in the plane xy; let V^= -^-5 + -7-=; let f^i" be dx' dy the element of area at P; let ds denote the element of arc at any point of the curve, and dv the corresponding element of normal drawn outwards; then will dUdV dUdV. dxdx dy dy> ' ^ ' First let us see how the line-rate of variation of V (any function) along the normal at any point A of a curve -/(: is deduced from the Let / and m be the line-rates -t- dx and — Fig. 64. (^ cosine and sine of the angle which the normal ^C makes with the axis of X. Then ii AC = dv and (a, ^) are the co-ordinates of A, those of C are {a + ldv, p + mdv); so that if y=/(x,y), we have y^ =/(«, ^), Energy in vorficid moiion. 183 dV dV Vc =/(a + Idv, /3 + mdv) ^^^+{^J^+ ^Jal ^^ > . Vc-V^ dV ,dV dV /r r d^v d^v UV^ VdS = 1 ^{-j-^ + -jt) ^^ dy- Pirst take /d'^V U j-^ dx, and perform the integration between B and A, the points in which a line through P parallel to the axis of x meets ^, , dV. , dV. CdUdV , the curve. Thus we get {p-^ - {u-)-^ -^Jx. Multiply this by dy, and observe that at A we have dy = Ids, and ai. B,dy= —Ids; taking therefore the terms relative to A and B in one, we get Similarly, JJc^-^dxdy^fmC/^^ds-JJ^^-^dxdy. Adding, jjaV^Vdxdy=ju{l^^+rn -)ds-JJ {-^^+ ^^)dxdy. which is identical with (a) above. CoR. Let Z7and rbe the same; then /.v.«../.f.-/,f|Vff,*. m 115. Energy of Vertically moving Iiiquid. Let the liquid contained within the contour ADBC (fig. 63) be in vortical motion, and suppose ^ to be the flow function at any point P. Divide the area into elements of the type dxdy,\. e., small rectangles with their sides parallel to two fixed rect- angular axes. Then the square of the resultant velocity at P is ( -i) 4. (Jlj , .-. the energy of the element of mass at P is 3 Hence E^ipJJ'C^J + dfr djy dx] dy }dxdy. }dxdy; 184 Kinematics of Fluids. and this formula may be put into another shape by means of Green's equation. From (;3) of last Article we have But V^y\r= —2(a (Art. 102), therefore The expression for E given in Art. 113 for the case in which dd) (f> exists follows at once m the same way, smce — d(f> dy the square of the velocity at I', so that we get the result from (i) by writing ^ for \j/ and putting o) = o. In non-vortical motion, therefore, the term in the energy consisting of a sur/ace-integial^ over the area disappears, but not so in vortical motion. If the liquid is at rest over the boundary -j- = o, and we have simply E = p/u^i^dS; and if there is not vortical motion throughout the whole area, but only local vortices, this integral will reduce to a simple sum of terms equal in number to the number of vortices. ue. Resistance of a Mow Channel. Suppose AB, fig. 63, to be a channel of flow bounded by two very close stream- lines. Then, in analogy with Ohm's Law in the theory of electric current flow, the resistance of this channel to the flow is defined to be the difference of potential of its extremities divided by the time-rate of flow across any of its normal sections. Hence if A>/f is the difference of the flow function for its bounding stream-lines, its resistance is Consider now any area, ABCD, fig. 65, bounded by two equipotential curves, AB and CD, on which the velocity ' This is in reality a ziff/awze-integral ; dS stands for a volume with unit height standing on the area dS; and similarly ds is really an area with unit height. Lagrangian Method. 1 85 ■potentials are <^i and ^^, respectively, and by two lines of flow, AD and BC, on which the stream functions are ■^^ and i/^j, respectively. Then to find the resistance of this area, we may consider it as broken up into an in- definitely great number of flow channels, or \^ — an indefinitely great number of equipo- ''^ tential strips. Adopting the latter method, 'V;.:'---'-'--- let pqrs be one of the strips, the potential I along rs being ^, and that along /y being —4 + A^. The flow takes place normally to this strip, and its amount (Art. 103) is '^' ^' 1/^1—1/^2 ; hence the resistance of the strip is Also the liquid encounters the strips in succession, so that their combined resistance, Ji, is the su7n of their resistances ; therefore „ 4>-, ,. The same result would have been obtained if we had chosen to break the area up into flow channels. In this case the liquid does not encounter the strips in succession but side by side ; so that, other things being the same, a multi- plication of such channels diminishes the resistance to flow, since a wider channel is thus provided for the flow. Their combined resistance is found from the fact that its reciprocal is equal to the sum of the reciprocals of their separate resistances. Integrating the reciprocal of (i), and taking the reciprocal of the result, we get (3). The energy of the liquid contained in the area ABCD is at once obtained from (i) of last Article. The portions of the integral on the right-hand side contributed by the lines AD and BC are zero; that contributed by AB is ip<^i(^2— '<|'i)j and that contributed by CD is ipx-^^- (4) 117. Lagrangian, or Historical Method. In this method, as already explained (Art. 89), we follow the course of an 1 86 Kinematics of Fluids. individual fluid particle, P. At any timCj /, let {x,y) be the co-ordinates of this particle, at this time let p be the fluid density at P, and {a, b) be the co-ordinates of the point, A, occupied by P at the origin of time. Then x and jf are supposed to be expressed as functions of a, b, and /. Suppose x^/{A,b,i), y-g{a,b,/), (I) where / and g are symbols of functionality. We shall now find the form which the equation of continuity takes. Let S he a. particle very near A when / = o, the co-6tdinates of B being {a + a, b + 0); let p^ be the density at A at this time ; also at this time let C be another particle very close to A, the co-ordinates of C being a + a, b + 13'; at the time i let the particles B and C occupy the positions Q and i?, respec- tively, which are both very close to P; and let the co-ordinates of e and ^ be {x + ^, j + rj) and {x + ^, j'+r]'), respectively. Then we shall express the fact that the maifs covering the triangular area QPJ! is equal to the mass covering BAC. Now the area QP/l is hence and that of BCA is + a',0' ■Po a, 13 a',/3' constant. {^) Again, x + ^=/{a + a, b + fi, t) ^/+a^+^^, where / stands for /{a, b,i); so that we have (3) (4) ^=4+4{' '=4-4' and similar values of ^ and rf. Substituting these in (2), we get P df df da' db dg dg da* db = Po. or 7p = f a> (5) (6) Equation for Vortical Spin. i &'] where y is used for the determinant, which is the Jacobian of the functions _/" and ^ with respect to the co-ordinates a, b. Equation (6) is, therefore, the Lagrangian form of the equation of continuity. For a liquid p (the density of an invariable particle) is constant and = Po, so that y= I, for a liquid. (7 118. Lagrangian Equation for Vortical Spin. We shall now calculate the vortical spin, or rotation, ca, at the particle P. It may be calculated by transformation of variables ; but the following method relies on elementary principles. The method consists in .calculating the components of relative velocity of any very close particle, Q, with respect to dx J dy P. The component velocities of P are, of course, -3- and -3- , at at or f and g, respectively ; and since /, a, and h are completely independent variables, the order of differentiations with respect to them is interchangeable, so that the component velocities of Care /(a + a, 3 + /3, ^), #(a-|-a, 3 + ^, /), (C being originally at B, whose co-ordinates are a + a, 3-I-/3). Hence the components for Q are so that the components of relative velocity are The components of strain at P are obtained, of course, by expressing these in terms of the co-ordinates, f, j/, of Q relatively to P. Now from (3) and (4) of last Article, we have ^« = f|-'f' <3) Kinematics of Fluids. Substituting in (i) and (2), we get for the components d/dg dldj_,_d£d£_d£d£ J^^dadb db da' ' da db db da ' lv,d±dg_ J^^da db (5) (6) db da' ^ da db db da' We must now put these into forms showing a pure strain and a rotation (Art. 96), i.e., we must put them into the forms Xf+jij — a)»), (7) where 2 s and to will be the shear and vortical spin, respectively. The values of to and s are at once given by the equations (9) I ^_ df_ da db I dg dg da ' db ^ df df da ' db ~^7 da db I df df dg dg da' db I da' db df df da' db 27 dg dg da' db (10) while for the dilatation, or expansion, k + n, or 0, we get (") This last we might at once have deduced from (6) of last Article; for (Art. 95) 6= £>tp. r 119, Generalised Co-ordinates. We have assumed that x,y, the co-ordinates of an invariable fluid particle, are expressed as functions of / and the initial rectangular co-ordinates of the particle, a and b. This latter restriction is not necessary. Instead of a and 3, the initial rectangular co-ordinates, we may express x and y in terms of / and any two constant quantities whatever, K and Z, which serve to identify the particular particle ; so that we may take X = F(K, L,t); y^ Gi^JC, L, t). (i) For ^ and L will each be some function of a and b, so that Expansion, Shear, Spin. 189 we shall, if we choose, be able to express x.^n&y in the very same forms, (i), Art. 1 1 7, as before ; and the results (6), Art. 117, and (9), (10), (11), Art. 118, will still hold; but we must express them in terms of IC and Z. Now it follows at once that df df dF dF dK dK da db dJc' dL d^' Tb dg dg dG da X dL dL da ' db dK' dL da ' db (2) or y„ suppose, coming out with similar values of the determinants in (9) and (10) of last dJC dK . . , , 1 . 1- '^ db Article, the multiplier jt jj ' or y„ suppose, co da db in all; so that equation (6) of Art. 117 will be simply 79 = 7^9^, where ynow stands for the first of the two determinants on the right side of (2); and the values of to, s, and 6 of precisely the same forms in K and Z as in a and b. Hence we may regard a and b as any two constants whatever identifying a fluid particle — e.g., its initial polar co-ordinates. 120. Graphic representation of Expansion, Shear, and Vortical Spin. Expressing the rectangular co-ordinates, x,y, of any particle, P, in terms of any two co-ordinates (simple or generalised), K and Z, in the forms (i) of last Article, take a point, i^, whose rectangular co-ordinates are dF , dF -— and -j-\ dK dL' and also take a point,' /"j, whose co-ordinates are dG , dG — - and -J- ■ dK dL Then if is the origin, 7 is obviously equal to double the area of the triangle P^OP^; so that equation (11), Art. 118, in- forms us that the expansion at P at any instant- is equal to the areal expansion of the triangle i\ OP^. Also if h^ and h^ denote 190 Kinematics of Fluids. double the areas swept out per unit of time by />, and P^ round O, we have by (9) and (10) of that Article If the fluid be a liquid, it follows that the two derived points determine with the origin a triangle of constant area. 121. Invariability of Vortices. On account of its im- portance in the general theory of fluid motion, it is thought advisable to introduce here a proposition which is not kine- matical but kinetical. The proposition is this — If any fluid in which the density at any particle is either constant or a function of the intensity of pressure at the particle, moves under /& action of external forces which havi a potential, then, if at any time whatever there was vortical spin in any particle, this particle will always continue to have vortical spin ; and if at any time the particle had no vortical spin, it can never acquire it. Suppose that V is the potential, per unit mass, of the external forces, at the point x, y, where there is a particle of density p, and where p is the intensity of fluid pressure, the equations of motion of the particle are d^_dV_ idp^ dt^ dx pdx^ d^y _ dir_ i dp dt^ dy p dy Hence if p =f(f), these equations can be written d^x _ dSL d^y _ dil dt^ ~dx' IF''^' '^' where i2 is a function of x, y. Multiplying these last by ^-- and -J- ) respectively, and adding, and also regarding x and y as given in terms of the constants. ^ and L by equa-- tions (i) of Art. 119, we. have- Invariability of Vortices. 191 dF ■■ dG dQ, dJT^ dK dK ■■dF ■■ dG dQ. " , . d^F F denoting — -, &c. Similarly multiplying the expressions (i) by -j-^ and ^, and adding, we get • • dF .. dG dQ. dL dL dL ^^' Since IC and L are independent quantities, we get identieat results by differentiating (2) with respect to L and (3) with respect to JC. Doing so, we get by subtraction dF^ dF dFdF dG dG dG dG , dklL~ lllk^ JkTL~ 'dl'dk.^ °' ^"^^ But the first two terms of this equation are obviously d dp dF dFdF, dh, , ^ , Jt^TK-dL-TLT^' °' ^ (Art. 120); dh and the last two are — ^; so that (4) is the same as ••• ^(M = o„ (5) .-. ya> = yo'»o. (6) which proves the proposition. For, if a)„ (which may be taken as any previous value of the vortical spin of the particle) is not zerOj o) can never become zero unless J becomes 00 ; and if ft)„ = o, ft) is always zero. Thus, supposing the fluid: to be compressible, so that y is not constant, the vortical spin of an element will get quicker as the element begomes. more compressed, and slower as the element becomes less dense, since " = ^. (7) P Po in virtue of Art. 119; andif'the fluid! is a.liquid the vortical spin of each element remains constant throughout the motion. 193 Kinematics of Fluids. 122. Acceleration at a point. Supposing, as in Art. 96, that the components, u and v, of velocity at a point, P, are expressed by the equations « =/i(-^. y^ ^). "" ^Aix, y, t), (0 then at the end of the time-interval A/, the fluid particle which at the time /was at /"will be at the point {x-\-uA.t, jy + v^f); and by (i) its components of velocity will then be /^{x + u^{,y + v/^t, i+Ai) and /^{x + uA(, j/ + vAt, t+Ai); so that the gain of .a?-velocity by the particle is puAi+pvAi+^^Ai; dx ay at , du du du, . ^"^ + "^ + ^)^'' and the gains, ci, ^', of velocity parallel to the axes, per unit of time, i.e., the components of acceleration of the particle, are therefore .. Section II. — Multiply-Connected Spaces. 123. Single-Valued and Many-Valued Function. A function of one, two, or any number of variables is said to be a single-valued ftmction if it can have only one definite value when definite values are assigned to the variables. Thus, sin - is a X y single-valued function of x and y; but sin ^ - is a many- valued function, because there are several angles each of which has its sine equal to -• The velocity potential at any point in the case of a whirl (p. 167) is a many-valued function ; for it is of the form A B, or A tan~^-; but in this case the stream function, Alogr, X is single-valued. The reverse takes place in the case of a squirt. We have seen (Art. 105) that the flow from a point A to another, B, is a definite quantity independent of the path pursued if the velocity potential function is single-valued ; but if it is many-valued, the flow is not definite, but depends on the particular patii, drawn from A to B, along which it is estimated. Generally, whenever there are vortical regions anywhere in a fluid, the velocity-potential function (which exists, of course, only in the non-vortical regions) is a many-valued function; but its diff'erential coeflBcients — which express components of velocity at any point — are necessarily single-valued. 194 Kinematics of Fluids, Hence theorems, such as Green's (Art. 114), require modi- fication when they are concerned with a function whose value at one and the same point is ambiguous. In this case there is an artificial method of removing the ambiguity from the function ; and this method, after a few pre- liminary definitions, Tve proceed to explain. 124. Simply and Multiply- Connected Regions. Let DEF (fig. 66) be a contour enclosing any portion of a moving fluid. We may speak of the whole of this space as a region. Within this region there may be several smaller regions, such as A, B, C, within each of which the nature of the motion differs in some essential particular from that of the motion in the space outside it. For example, the motion may be vortical within A, B, and C, and non-vortical in the space outside them. By the region DEF we shall now understand only that portion of the space hounded by the contour DEF which is not included in any of the sub-regions A, B, C. Now in the region take any two points, P and Q, and connect them by any path, FrQ, -every point of which lies in the region. This path FrQ is said to be recon- cileable with any other path, PqQ, connecting F and Q if we can imagine the closed curve formed by the two paths to be capable of shrinking up into a point without requiring any of its points to leave the region — or (which comes to the same) if the second path, FqQ, can be imagined to change into FrQ by a gradual motion of all its points without any of its points ever leaving the given region ; and two paths connecting F and Q are said to be irreconcikdble if this cannot be done. Thus the paths FrQ and FcQ are irreconcileable, because in order to bring the second into coincidence with the first by gradual changes, it would be necessary to bring a portion of it through the separate region C. Multiply-Connected Regions. 195 • Similarly, the paths PaQ and FbQ are irreconcileable with PrQ, and with each other. Hence our figure represents the case in which four irrecon- cileable paths can be drawn between any two points in the region. This region is therefore said to be quadruply connected. Similarly, if in a region n irreconcileable paths can be drawi; between two points, the region is said to be ' »-ply connected.' A simply connected region is one in which all paths drawn between the same two points are mutually reconcileable. Thus, if the sub-regions A, B, and C all vanish, the region DEF will be simply connected. The term ' circuit ' will be used to signify any closed path. Now it has been proved (Art. 105) that the circulation round any circuit is equal to twice the surface-integral of the vortical spin over its area. Hence if this surface-integral over the region A \s\k^, measured in the sense of the arrow, the circu- lation along the path PrQa is k^. Denote the flow from P to Q along the path PrQ hy PrQ; then obviously, since QaP= —PaQ, we have PrQ^PaQ+k^. (i) But the flow (being the line-integral of the velocity) is /(udx + vdy), and this integral is what we have denoted by <^, the velocity- potential. Suppose that we assign an arbitrary value, h, to the velocity potential at P, and let (^g be its value at Q. Then PrQ = ^Q—h, and equation (i) gives PaQ = -potential at any point in the given region is therefore multiple -valued, or indeterminate to the extent of a quantity of the form n^k^->rn^\-\-n^k^, where »i, %, «s are integer numbers. In a simply-connected region no such ambiguity exists, since /{udx+vdy) is the same along all paths drawn from F to Q — i.e., <^ has a definite value at every point. 125. Method of Barriers. The potential at any point can be made single-valued by making the region simply connected ; and this is efifected by drawing barriers of arbitrary shapes across from the regions A, B, C to the region DEF. Suppose these barriers to be drawn across to the points a, /3, y (fig. 67), and assume that loth sides of a barrier are portions of the boundary of the region DEF, the other portions of the boundary being the contours of the regions A, B, C, and the contour DEF itself. Then Starting from any point (suppose Z>) on the boundary it is possible to travel continuously over the whole of the boundary, thus completed by the barriers, and to come back to the starting-point; one portion of this motion will consist, for example, of a motion from a along the left-hand side of the barrier up to the surface of A, then a motion round A, and a return motion from A to a along the right-hand side of the baraier. Supposing, then, that we take any two points, F and Q, within this modified region, and that we estimate the flow from Fio Q along any curve (not represented in the figure) belonging strictly to the region, i.e., any curve, however complicated, which crosses none of the barriers ; this flow will be the same Method of B((yriers. 197 for all such paths-; for if we draw two such paths, they will form a circuit which does not include any vortical region, and therefore the flows along them from P Xo. Q are the same. Hence we do not get a multiple-valued potential at Q. Moreover, the circulation along any circuit, PQRSTP, drawn in the unmodified region, i.e., a circuit which cuts the barriers any number of times, is easily Jbund. T£us, the circulations ^ij ^21 ^3) being estimated in the senses ©f the arrows, the- circulation PQRSTP = k^-^k^; iox PQR = PQRP+PR= -^k^+PR; and the flow from R to S along the ^ath = k^ + RS; then the flow along the path from S to T=\ + ST, so that PQRST = kj+Pr (since PR + RS+ST=PZ); finally, the flow from T to i' along the path -i^—PT; therefore the whole circulation. in the circuit = ^j+ji,. The same method applied to any other path shows that the- circulation in it is «iiJi + «j^ij + «,^5+ ... (e); there being any number of vortical regions,. ««rf n^will be the number of times thai the path crosses the corresponding harrier, a, in the direction of the circulation round the corresponding region, A. Thus, in the path drawn in the figure, a point travelling along it from P in the sense assumed above crosses the barrier a twice, first in the sense opposed to k^ and afterwards in the sense of k^, so that «, = o. Supposing, then, that we artificially modify the region by drawing barriers, the velocity-potential has one definite value at every point ; and the values of (ji at two points indefinitely close to each other but on opposite faces of a barrier, a, differ by the corresponding circulation, k^. For, take two such points, iV, J^, on opposite sides of the barrier a. Then PJ\r—PN' = ii; but PJ\r=(l)jv—4>pi and P^ = N'—4'j'> ■'• 4>N—N' = K- Hence when we draw the barriers, we may select any one point, O, in the region and assign the value zero to the velocity- potential at it ; and then the value of the potential at any other 198 Kinematics of Fluids. •point, P, in the region is the flow from to P along any pain •which does not cut any harrier. In this way all ambiguity is removed from the value of '^. For example, take the case of the whirl, example 6, p. 167. Here =^9; but e, as determining the direction of a line OP, is multiple- valued,, the same direction being given by e + 2»7r. However, we may draw any line Ox, which we can regard as a barrier, assign a zero value to ^ at any point on the upper side (or aspect) of this line, and then at any point P, the value of ^ wiU be ^ X circular measure of L.POx, this angle being always less than 2 ir. The quantities k^,k^,k^,...2tXQ called the cyclic constants of 0. 126. Thomson's Modification of Green's Equation. Green's equation (a), p. 182, fails in definiteness if the function U is multiple-valued. Consequently the ambiguity in U must be removed by the method of barriers. Confining our attention to the equation which is derived from Green's general equation, and which expresses the energy, at any instant, of the liquid contained within a given closed curve, DEF (fig. 67), and assuming to be multiple-valued — or, in other words, that there are vortical regions within the contour — the equation will be applicable without ambiguity only when the region is modified by barriers, and its boundary consists of the contour DEF, the contours of the vortical regions, and both sides of every barrier. Selecting any arbitrary point in the modified region, and assigning a zero potential to it, the value of /r=— aoj* (i) and in those regions (if any such exist) in which there is no vortical spin, there is, in addition, a velocity-potential function, <^, which satisfies the equation V> = o. (2) Motion due to a Single Vortex. 301 We have shown (Art. no) how in the case of a compressible fluid the components, », v, of velocity are found at every point — ■ by the determination of two quantities, P and i\s— when the law of ejspansion and that of vortical spin are assigned for each point. In the present section we shall confine ourselves to the case in which the fluid is incompressible and without inertia; aaid under this head we shall include the case of the flow of elec- tricity — to which this section is specially devoted. Pursuing the same analogy with the case of a gravitation potential, we write the equation (i) in the form V2ilf= _47r. — , which shows that i^ may be regarded as the gravitation potential at any point produced by a distribution of matter along right lines perpendicular to the plane {x,y) of motion, and of practically infinite length above and below this plane, in such a manner that the density at any point in space is — times 2 w the value of the vortical spin at the orthogonal pryection of the point on the plane of motion: Now we are not always concerned with the value of \/r itself at any point, but we are concerned' with the velocity at the point, or with its two components (i», z)) ; and this analogical method of regarding ^ as an attraction potential, enables us to find the velocity components without calculating i/r itself; for there are methods in Statics for calculating the resultant attrac- tion of matter, or its components, without finding the potential. Now the components of the attraction at any point, parallel d^ to the axes of x andy, of matter arranged as above are -j- and -^, which we know to be, respectively, —V and u, the velocity components at the point. Hence these latter are at once given from the attraction, 128. Motion due to a single Vortex. Suppose that at a point A there is a very small circular area, ds, in which the aoa Kinematics of Fluids. mean value of the rotation is m; then zmdS is defined to be the strength of the vortex, so that the strength of the vortex is the circulation round any closed curve enclosing the vortex once. Denote it by m. To calculate the components, u, v, of velocity at all points in the plane we are to imagine an infinitely long solid cylinder described on ds as base, whose mass per unit length is represented by — • Now if we calculate the attraction potential of this cylinder at any point, P, in the plane, we shall find its value to be infinite (see Statics, p. 403). The attraction on a unit particle at ^ is however easily found. It is (see Statics, p. 417) directed from P along the perpendicular to the cylinder, i.e., along PA, and is equal to (I) 21! .PA If the co-ordinates of P are (x,y) and those of A are (a, /3), this attraction is m And we have just seen that the x and y components of this attraction are, respectively, — » and u. Hence, denoting PA by r, 2TS T m x—a , . Now d^ = udx + vay=^^''-''^'^^-y-^'>'^'' 2 It r^ = ^rftan-iJLZ^. (5) 2it X — a ^ ' Hence if 6 denotes the angle made by the line PA with the axis of X, the velocity-potential at P due to the vortex msX A is given by the equation which is 2 7rr infinite at A. This resultant is at right angles to the line PA, and in the same sense as that of the rotation in the vortex at A. Again, 27r r^ 217 r ' therefore ^ = — — Ibff r . 2ir ° omitting a constant*. Hence the stream lines are a series of concentric circles having the vortex for centre, and the equipotential lines are right lines diverging from A, so that we are led to the case already discussed (p. i«7), which we now see to be a case of motion due to a vortex of small area placed at the origin. The plane of motion may either extend to infinity in all directions, or be bounded by any circle having A for centre. (See Art. iii. No other boundary would leave the motion unchanged.) If the stream lines are drawn so that the values of the stream function for them proceed with a common difference, they will be a series of circles with radii in G. P. 129. Electrical Equivalent of a Vortex. The magni- tude and direction of the lesultant velocity at P due to the vortex at A answer exactly to the magnitude and direction of the action, on a magnetic pole of unit strength, of an electric current transversing the infinitely long wire, or cylinder, wTiich we have imagined to extend from A perpendicularly to the plane, above and below it ; the strength of the current being proportional to the strength, m, of the vortex. Hence all our results for vortices in a liquid are directly applicable to electro-magnetic phenomena, if we imagine indefinitely long straight currents to replace our vortices, the strength of each current being proportional to the strength of the corresponding vortex; and the plane of motion must be ' In reality the value of f, calculated as the attraction potential of an infinite slender cylinder, contains an infinite constant term. See Lamb's Treatise on the Motion of Fluids, p. 162'. 204 Kinematics of Fluids. considered as a field at any point of which, instead of a liquid particle, we imagine a magnetic pole. Or again, for the vortex at A we may substitute a magnetic pole of strength proportional to m ; and at each point, P, in the plane imagine a little element of electric current running perpendicularly to the plane. 130. Motion due to two Vortices. Let there be two vortices of strengths m^^ and m^ at two points, A^ and A^, in the plane of motion ; then if ^ is the velocity-potential resulting from both together, and 0i, ^^ those due to each separately, we know that ^ ^ £*l + ^, ^^ d^^d^^d^^ ax dx dx dy dy dy since the resultant j;-velocity is the sum of the separate x- velocities, etc. Hence (^ = (/)j+<^j; and similarly \/f = 1/^1 + ^2, where i/r is the stream function. We have, therefore, by last ^=-^logr,-^logr2, (2) where 0, and 6^ are the angles made with any fixed right line by the lines, PA^, PA^, joining the point P, to which <^ and i/r refer, to the vortices A^, A^.;. and where also PA^ = r^, PA^ = r^. The stream lines are curves given by the equation r^"'i . r^'"i = C, where C is a constant. If m^ = m^y or the two vortices are of equal strength, this becomes r^r^ = k'^, which gives the ovals of Cassini already discussed. Also in this case d^^-^; 6^=* — ^ , .: the curves = const. are rectangular hyperbolas, as found in p. 173. If we wish the plane of motion to be of limited: extent, we must make its boundary one of the Cassinian ellipses. The case in which m^= —m^ is interesting. In this case the curves i/r = const, are given by the equation Motion due to Two Vortices. 205 i.e., they are a series of circles, loci of vertices of tiiangles standing on A^A^ as base and having their sides in a constant ratio. A limited field must be bounded by one of them. The equipotential curves become ^1— ftj = const. =a, i.e., a series of circles, loci of vertices of triangles having A^A^ as base and a constant vertical angle, a, which has dififerent values for the different loci. Every one of these equipotential circles passes through both vortices ; and observe that also in the last Art. every one of the equipotential lines passes through the vortex. This is a general property of all equipotential lines. (See ex. 8, p. 173). The case in which OTj = ot^ , i. e., in which there are in the field two vortices of equal strengths and the same sense, is realised by making two holes, A^, A^, in a flat plate and passing two very long straight wires through these holes, the wires conveying the same electric current in the same sense, i.e., both up through the plate or both down through it. The magnetic action of this current system, i. e., its action on magnetic poles lying in the plane of the plate, is represented by the system of ellipses of Cassini (as lines of electro-magnetic force) and the system of rectangular hyperbolas (as lines of equal potential). The Case in which m^= —m^ requires the wires passing through the holes A.^ and A^ to convey the same current in opposite senses — i.e., one up and the other down through the plate. It will be good exercise for the student to map out the resultant electro- magnetic fields, which correspond to the two cases discussed in this Article by means of the graphic method of superposition explained in Art. 112. Observe that in the second case {ot, = — m^, if we draw the velocities along the lines of flow (circles) corresponding to the vortex Ai from right to left, those along the lines of flow (circles) corresponding to A, must be drawn from left to right. If from a vortex Ai we draw any number of rays dividing the whole angle frr into equal parts, and then from any other vortex, Ai, we draw a number of rays on which the velocity-potential has 3o6 Kinematics of Fluids. successively the values which it has on the rays of the first pencil, the strengths of the vortices are proportional to the numbers of rays in their respective pencils. 131. Any ntunber of Vortices. Let there be any number of vortices, A^, A^, ... A^,oi strengths m^,m^, ...m„; then, as shown in last Article, their functions of potential must be added to produce the resuhant potential at any point, and similarly for their flow-functions. Hence 2'n.) are the co-ordinates of the variable point /', and (^1, J'l) •■. those of ^j . . . , the equation is the same as OTitan-i ■^~^^ +Watan^^ -^~-^'' + ... = const., which is manifestly satisfied by x = x-^, y =_y^, since we may take all the terms except the first together, and write the equation tan"-' = V, or jy—j/i = {x—x^) tan f» Similarly for all the other points. In the particular case in which the strengths m^, m^, ... are all equal (whether with or without the same sign), every com- plete curve of flow is a curve of the «* degree (there being n centres); and in general the degree of a complete flow-curve is 2»?, taken arithmetically. If the poles .are unequal, each stream-line passes more than once through the stronger ones. It passes through each as often as its strength contains the G. C. M. of the set. 133. Infinite Velocity. We shall find in all cases (see example i, following) that the velocity at each point-source and also at each point-sink is infinite. It is not at all surprising that our investigation of the motion should lead to such a result ; for one explanation, at least, appears to be that we have all through regarded a source as a mere point, in the strictest geometrical sense; and we have assumed that through, this point (which is a mere extentless entity of the imagination) a finite quantity of something measurable (fluid, electricity, etc.) is discharged per unit of time. The ' two conceptions are manifestly contradictory of each other; and we are compelled therefore to admit some one of three things — viz., (i) that our sources and sinks are not geometrical points (and that there- fore none of the quantities rj, r,, ... can ever possibly be zero); or (2) that their strengths, OTj, m^, ... which are measures 3o8 Kinematics of Fluids, of the quantities they discharge, are all infinitely small, so that, for example, when we meet such an expression as — j we must not assume it to be infinite when r = o, because m is also zero; or, lastly, (3) that if a finite quantity is discharged through a point, an infinite velocity is required at the point. The first of these choices is, of course, the one to be adopted — every source and every sink is in every actual and physically possible case a little equipotential area — not a point. 134. Plow in a given Sounded Flape. We have already mentioned (Art. iii) the case in which the plane or field of motion is limited in extent and bounded by a given invariable contour, while inside the field there are given sources and sinks. The values of <^ and ■\/a given in the previous Afticles are not at once applicable here, because they suppose the field to be either unlimited or bounded in a very particular manner — viz., by some one of the lines of flow due to the given causes of motion; whereas in the present case the boundary is to be taken at random. Undoubtedly the boundary, whether taken at random or not, is in all cases a stream-line of the fluid ; but the values of ^ and ■^ just given and calculated from the given causes alone will not make it so. Hence the problem to find the velocity, etc., at each point in this bounded field, due to the given causes (sources, sinks, etc.), is much more difiicult than it would be if the field were unlimited. Still, the two problems are solved on the same lines. What we do is this. Imagine the field to be unlimited. With the given causes (/4i, A^, ...) of motion we then combine certain others {B^, B^, ... ), which we must completely determine in such a way that, taking them all together, they would, in the infinite field, make the given boundary curve a stream-line. The velocity-potential and stream-function, at any point either inside the given bounded field or outside it, will then be the sums {(p^ + s and yfr^+ylrg, respectively) of those due to the given causes (a) and the determined causes (B). Of course the whole diflSculty consists in finding out the system (b) — which is called the image of the given system (a) Flow in a Bounded Plane. 309 in the given boundary, in analogy with the language of Optics, because the image-system (B), if it existed, would produce the same effect wiihin the given limited field 2& the boundary does — 'just as,' to quote Professor Lodge*, = z«i log r, + ff«j log ?s + OT3 log ?-a + ... (1) and the direction of flow is that of the normal to the surface = const. Now the normal to this surfdce is found by measuring from P along the lines PAi, PA„ PA,, ...lengths, Pa^, Pch, /"a,, ... proportional t6 —1 — ^. -^, ... aad finding the resultant, PG, of these lines. If Ai is a source, the length Pa^ will be measured from P in the sense PAy , and if it is a sink, the length will be measured from P in the sense A^P. For the proof of this theorem See Statics, p. 76, 2nd ed. The points a,, a^, a„ ... may be practically determined in various ways. One way is this : round P as centre and with any convenient radius, k Exampks, ^11 describe a citcle, C; tHrongh A^ dia* a circle, C,, cutting (? (WthogOnally ; thett if A4, nieets C. in *„ we havd A = - ; hfenc6 by measuring a line Pdi equal to »z, . Pi^ from ^ eitlier towards or from A^, according as Ai is a source or a sink, the point o, is found. Similarly for all file othef points. The velocity at P may be regarded as the resultant of the velocities Which wbtild be produced by A^, A, A^, ... Sep&ratrty, i.e., the reSullant o» -rr- -T"'"-! *°^i^ t^^rSare * centres the r63ttltantof/'i«i,/^, ... '1 '1 is « . PG {Statics, p. i6). 2. To investigate resistance to eleetrical flow in an unlimited sheet of tinfoil containing a source and an equal sdnki let * be the strength of -the gtfufee. A, and -m that of the Sink, B (fig. 68). Then the distances of aiiy point frorti A and B being j* and /,

. The difficulty i4 avoided by drawing found A ail equipotential circle, /, Of very small radius, and another, j, round B and considering them as electrodes^ Tha P 2 312 Kinematics of Fluids. expression for the resistance of the strip terminated by these circles will not be indeterminate. Let « be the value of the ratio —, on the circle j>, and — r ^ its value on q ; also let the value of 9— fi' on C be a, and on D be /3. Then the resistance of the stjjip is 1 loge loff p + log p' . More generally, the resistance will be — , if we take as elec- trodes any two equipotential circles (one round each pole), -yyhere p and p'. are the values of —, for the circles. r One stream -line is AB, and for it 6—0 = v, so that the resistance of the area included between the circle D, the line AB, and the circles/, q is c log e For the remainder, I/, of the circle D, the value of 9—9' is it—&, and if R is the resistance of the whole area of the circle D, excluding the portions within the small circles /, q, 1 11—0 it—(n—0) It R 2 log c 2 log e 2 log € ' " .-.R^'-^. (I) IT , which is independent of the circle, D, selected, • In the same way the resistance of the area included between any two equipotential circles, P, Q, may be calculated. The resistance of as much of this strip as is bounded by the circles C and D is ^Q—p _ p i/'D—'pa ^— « and if iS and a are nearly equal, so that P— o = da, the reciprocal of the resistance of the interval between P and Q bounded by the upper side of the line AB is da It r *o — ^p 0a — 0p since for points on AB between A and B the value of a is it, and for points on AB beyond B the value is o. Hence the resistance of the upper half of the strip between P and Q is — (0g— 0p) ; and the resistance of the lower half being the same, the total resistance of the strip is — (Q—'l>p) 21t r— i.e., one ha^ the resistance of the half strip, since the strips are aireast and not in series. BxampleS, 213 Hence the resistance of the area between P and ^ is — (log log A), 29r € if \ is the value of -; on /"; or — log — . If we put \ = i, /" will be the right line bisecting AB perpendicularly, and the resistance of half the infinite sheet, leaving out the area of q, is — log - , The other half 27r 6 is, as regards flow, arranged in series with this half, so that their resultant resistance is their sum, or }(,„ ^ IT' (neglecting sign) ; and this is half the resistance in (i), which denotes the resistance of the area of any circle of flow. To find the velocity at any point, P, we may regard the actual motion as a superposition of two separate motions, one due to the source and the other due to the sink. If PA = r, PB = r", the first motion would give a velocity — ■•— along ./J/', and the other a velocity — ' -y along PB\ hence the resultant velocity at P is m. AB 27r rr' The curves of constant velocity are therefore ellipses of Cassini. The arrows in the figure indicate the senses of flow. If A and B were both sources, or both sinks, the velocity at any point would be — • -T- , where p is the distance of the point from the middle v rr point of AB. The system of eqnipotential circles, P, Q, ... may be drawn by taking any point on the line AB (produced) as centre and describing a circle with radius equal to the length of the tangent from the point to any one, C, of the circles of flow. 3. Given two sources, A, B, each of strength m, and a sink C in the right line AB, also of strength m, show that one sti-eam-line consists "i-C" "x p^ partially of a circle. / \^^^ v^- Let A and B be the two sources / ^>C "'• (fig. 69) and C the sink; let P be f /^\]h' any point in the plane, APAx = 0, \ .1 7\^^ LPBx = &, /-PCX = e". Then j / \ if the flow-fiinction is i^, we haye ^ ■■•.,_■ \~yB C x ^=^(«H..'-n. ""\ Consider the value 5^ = 0. Then jrig_ gg, LPAx = LBPC; therefore if a circle is described through A, B, P, Jte line CP is a tangent to it at P. Hence if ^ = o, P lies on the locus of points of contact ZI4 KinemaHes of Fluids. uf t^pgents from C tp circles passing through 4 w4 B, i.e„ tfie locns of /" is a circle, PQ, with C as centre and radius = a/CA . CB, which is the required result. But the right line AB is also part of this stream-line (^ = o), ?ince at all points on it e = Q, $' => 9" = t. Also thf line ^C is part of a stream-line for which ^ = —\m\ the production of BA beyond A is part of a stream -line for which f = \m; and the production of BC through C is part of a stream-line for which f = q. Each complete stream-line is a curve of ^e third degree (Art. 11%). Now, in addition to this arrangement, let there be at two points, A', B', which are collii;eaT with C, and which lig on any circle passing through A and B, two sinks, each of strength m, and at C a source pf strength m. Of this new system the circle PQ will ^Iso be a strefim-line ; ■ hence it will be a stream-line if we superpose the systems ; and, as C is thus obliterated, the superposition gives two sources, A> B, and two siiiks, A', ff. We can regard this as produced by bifurcating (he wire from the + pole of a battery and connecting it with A and B, and sunilarly bifurcating the wire from the — pole and connecting it with A' and ff ; apd adjusting th? resistances in the circuit so that tlie current splits equally between the branches. Of course the circle PQ is only a part of the stream-line, because if we measure S from the line Ax as an initial line and let the angles made with it by the lines PA, PB, PA', and PB' be, respectively, S,, 0,, 9,', St, we ''^^^ m 1^- jy («.+9.-e.'-e.'). and the curve f = const, is (Art. 132) of the fourth degree. To find the value of ip on the circle PQ, let LPCx be x. 3"°^ 4/4C/4'r== o; theny(== -^(9, + 9a-"X +X-fij'-'0 ; l"it fli+flj— x^°> and X— *i'— ^2' = <*. a ^ 2ir all over the circle PQ. Moreover this is also the value of ip at all ppints on the circle passing through ABB' A', as is easily seen from the property that angles in the same segment of a circle sre cquai. Hence the complete stream-line consider©! i^ the systejn of tiue circle? represented in the figure. The other stream-lines will each consist qf twee s.eparate portions^ — one (more or less oval-shaped) passing thrpiigji B and B' and connpletely contained within the circle PQ, and the other passing through A and A' and wholly external to the circle PQ, since, except at the pples, no two different stream-lines can intersect. A drawing of these curves is given in the paper by Professors Foster and Lodge {Proceedings of thf Physical Society, Feb., 1875). It is evident that the velocity at P, and also at the other point of Examples. JS15 inteisection of the circle PQ with the circle of polies, is zero. For the velocity at P along the arc PA' is the line-rate of variation of ^t along the arc PQ (p. 154), and this is zero. Also the velocity at P along the arc PQ is zero for a similar reason. The velocity at P may be graphically repre- sented (examplf i) as the resultant of two forces from P towards 4 ajid B, inversely proportional to PA and PB, and two forces frpm A' and B', in- versely as PA' and PEf . Siict forces are therefore in equilibrinn>, DD' AA' Again, we see (example 2) that the quantities ^ and PB.PB PA. PA' are equal, since we may regard the sfate of affairs bX P as, due to a super- position of the effects of the source and sink fJB, B') and the source and PM PM' sink (A, A ). And again, p2~PB ^ PA' PB" ^^""^ ^' ^' ""^ '^'^ middle points of AB and A'B', respectively. But if r, s, /, / denote the distances pf any point frpm A, B, A', B'^ respectively, ^ ^^ Hence the potential ^t P = — log . The equipqtential curves ate given by the equation -r-, = h, k being a constant (Ranging from one curve to another. They are therefore pf the fourth degree, except the curve for which k = i. The equipotential curve through P is one having P for a double point. If we produce A A' and BB' to meet in a point, D, the circle with D as centre cutting the circle of poles orthogonally is part of an etjuipoteotial curve ; for (last example) the source and sink {A, A') and the source and sink {B, B') would separately give this as an equipotential curve. This is easily seen otherwise ; for since A, A' are inverse poin ts with regard to TA *JDA this Wfle, if /i§ any point on the circle, •^z ?= - -- ; similarly /-<* yDA' je - ^dW • ■■ M' J^ ^ PA' ■ i^B' '=°°'*-' so that is constant all over the circle with D as centre cutting the circle of poles orthogonally. The remaining portion of this equipotential locus is easily seen to be the (imaginary) circle which cuts the circle of poles orthogonally and has for centre the point, jy, of intersection of the lines Aff and BA'. For, by expressing the ratios of the lines involved in terms of the sines of angles, ^ . ^ ,., DA.DB jyA.I/B , : ' we at once obtam the equality _^, -. ri> = 1T£~WW'' * circle with D as centre and radius equal to '/—D'A .DB' will be a locus (imaginary) of the vertex of a triangle having AS for base, with the 3i6 Kinematics of Fluids, constant ratio V — j^rsi between its sides; similarly for the pair of points B, A'. The imaginary circle with If as centre cutting the circle of poles orthogonally is then, by last example, an equipotential circle for the source and sink {A, B) and for the source and sink ^B, A'). When k = \, the equipotential curve becomes a circular cubic, and it obviously passes through C and through the centre of the cirde of poles. Now in the case of the four poles consider the nature of the equi- potential curve on which the value of the potential is a very great positive quantity. If the distances of any point P from A, B, A', B' are s, r, /, /, respectively, we have ^ — — log -7—, ; and if this is a very great positive quantity, we must have -7-7 = j, where k is very small. Now — will be very small if s' is very small, i. e., if P is very near A'. By taking A' as origin of polar co-ordinates (/, S), expressing s, r, / in terms of J, 6, and constants, and neglecting such small quantities as /' and ks', the equation sV = ksr z^wei A'B'.i^ = k.A'B.A'A, which shows that P must lie on a small circle with A' as centre, its radius A'A A'B being k — -j^, — . Similarly,

J^ d radius = k ^-j, — . Hence the complete equipotential curve for which ^ has the constant large value — log -r consists, approximately, of two small circles surrounding the poles A and B ; and the curve on which ^ m has the constant large negative value — log k consists of two (srcles sur- rounding the poles A and B. Let us now make a boundary along the whole circumference of the circle PQ ; i. e., suppose that we have a circular sheet of tinfoil with the positive pole of a battery connected with it at B, and the negative pole at B. The circumstances of flow are to be calculated from the above case of four poles. The resistance of the sheet may be thus calculated. Since the circumference of the sheet is a stream-line, no electricity crosses it, and the whole of the electricity discharged from B flows into B. The quantity discharged by B in all directions and absorbed by B, per unit of time, is m. Consider then the whole area of the sheet, with the exception of the two small circles, one of large and the other of small potential, sur- rounding B and B . If the radii of these circles be each A, we shall have on thfe circle surrounding B m , AB.BB *^^^^°S K.A'B' and for that round B m ^ AB '^"^^^^''^A'B.BB' Examples. %if and the resistance of the whole sheet, leaving out the two small circles, is B I , AB'.A'B.Bff^ ~^r-' °' ^^°S X'.AB.A'B' ■ 4. In example 3 show that the velocity at any point, /, on the circle of poles and very near P is equal to (aa'— P/S') .9, where 6= /.sub- 47ra tended by PI at the centre of the circle of poles, a = radius of this circle, and a, a', P, $' are the cotangents of the halves of .the angles made with the di- ameter through P by the radii drawn to A, A', B, B', respectively, these angles being all measured round in the same sense (that of watch-hand rotation). [The consideration of this case gives the following theorem : If from any point C outside a circle a line be drawn cutting it in A, A', and a tangent touching at P, the sum of the cotangents of the halves of the angles made with the diameter through P by the radii drawn to A, A' is constant.] 5. In an unlimited sheet of tinfoil containing any number, «, of equal sources, Ai,Ai, At, ... , and the same number of equal sinks, Bi, Bi, B,, ... , prove that the resistance of the whole portion included between the equi- potential curve passing through any point P and the eguipotential curve {)assing through any point Q is I PAi . PA, ... PA„ . QB, . QB, ..: QjB„ 2»ir °^' PBt.PB,...PB„.QAi.QAi... QA„' fitis^tr^.j i- nm -■ 6. Find the expression if the strengths are not all equal. [Consider unequal poles as superpositions of equal ones. See Prof. Lodge's paper on the Flow of Electricity in a. Plane, Phil. Mag. vol. i. 1876.] 7. An indefinitely long rectangular strip of tinfoil of breadth b is taken, and the electrodes of a battery are connected with it at two points A, B, one on one edge and the other on the opposite edge of the strip, the line AB being at right angles to the edges ; prove that the resistance .of the strip is of the form ^b k log — , where ;! is a constant depending on the thickness and specific conductivity 01 the foil and A the radius of each (circular) electrode. (Prof. Lodge, ibid.) [Use the (infinite number of) images both of A and B in the edges of the strip, and apply the result in example 5.] 8. Find the resistance of an infinite rectangular strip when the two electrodes from a battery are connected with two points on its middle line, in terms of their distance apart, the breadth of the strip, and the radii of the (small) electrodes. 9. The velocity at P (fig. 69) being zero, explain how there can be flow along thepaths PQ, Pff, and QB, 51 1 8 Kinematics of Fluids. (See Art. 133 ; phy§ic' = «**.] 16. Find the resultant force at any point in the last ejcample. [If AB = p, LAPB = a, the force , ^ /'"-'^,^^'°° + ^' . 1 - *- 27r pr -I 17. Show front elementary considerations that the resistances are th? same for all strips opening out at the same angle from a pole and termi- nated by the same equipotential curve. [If successive stream-lines are drawn proceeding by a constant difference between their stream-functions, then infinitely near any pole they will form an equiangular pencil, since at such points the distant poles produce no effect.] 135. Theory of Linear Plow. The resistance of a portion pf any channel of flow bounded by two equipotential curves is (Art. 116) ;'- y , or in other words— the potential difference of its extremities divided by the quantity which flows, per unit of time, across p,r& section of the channel^ (a) The boundary walls of the channel, i.e., the flow curves i/'u 1/^2, may be very close together throughout the whole length of the channel ; and, all other things being the same, the effect of narrowing the channel is to increase its resistance to the flow. TheQry of Linear FloB—^At between its ex- tremities (by an electrometer),- and at the same time the time- rate of flow, C, across a section of it (by a voltameter and watch, or a galvanometer), the ratio "^ is always con- stant. The time-rate of flow, C, is called the strength of the current, and Faraday proved experimentally that this is the same at all sections of the wire (Jamin, Cours de Physique, vol. 3). If, then, the resistance of the wire between A and S is denoted by R, we have .expressing the current strength in the wire in terms of the potential difference of its ends and its resistance. This equation assumes that there is no sudden alteration of potential at any point of the wire between A and B^ Suppose, however, that at any one point, P, there 'S^~ ? P" <2 Q ■^ occurs a sudden change jpig ^^ in the potential; or that sudden changes occur at two or more points. Fig. 70 takes the case of two points P and Q at which sudden changes occur, and, for greater clearness, the places of sudden change are spread out; the points P and P^ are in reality coincident, as also Q and G'. 220 Kinematics of Fluids. Let the resistance of BP be />, that of i'V being p', and that of QA being p". Then, by Faraday's Law the current strength will be the same all through. Hence we have C = ^^ ^— = ; = 7? — ; '"cre- P P P fore C — ; / , // ■ \'/ The resistance, ^, of the whole is, of course, p + p' + p", and if we denote the sudden changes of potential by Aj and A^, these being, as we see, measured in the same sense, we have The sum of the sudden changes of potential, Aj + Aj, measured in the same sense (that of the flow) is called the Electromotive Force between A and B. Denote it by ^ ; then R which is the modification of (/3) when sudden changes of potential occur. If the extremities, A and B, of the wire are brought together, there being still sudden changes at P and Q, the wire forms a closed circuit, and ^b—^s-°'i then («) becomes ■ ^=1' b ^""e the potentials at A and S, we have by last Article Similarly for tlie branch AE^ B, Heftce Cj^2~'^3^s = ^2~^s' If we estimate the currents and electfoinotive forces all in the sense of watch-hand rotation, we can write this equation I!j^C^+Il,{ — Cg) = E^ + {-^E^), ift which form it is included in the general analytical expression of the second Corollary, viz., for any closed circuit whatever. 137. Current Power. The expression already given for the power«of a flow (Art. ii6) becomes in the case of linear flow in an interpolar between two points, A and 3, C{^i—j). The power Of the battery P'Q' (fig. 70) is C(^p,— ^^); this added to the power in the interpolar gives C{A^+A^ or CE, as the total power. If B is the resistance 6f the battery, and Jl that of the interpolar, the Whole power is {B+Jl)C^. Examples. I. Stow that if between any two pdiats. A, B (fig. 71), ttere are inserted » wires of resistances ri,r\,, ...r^, their resultant resistance, p, is given by the equation — =— + — h... + — . Let the current, C, of any battery break up at A into currents c^ c^, ... c„ along the wires, respectively. Then ? = Ci + f ,+ ,,.+ f^ . Alsfr ^•^~^^ =: f , ; writing down n sucli equations and adding, we have (^^ —s)S— = C. But as C is the total current passing between A and B, we must have •^.^-^^C; therefore! = 2'-. Examples. 323 4. Calculate the separate iBagilitudeS of the beaftch eurreiflts, «■,, ti...t„. Take the closed circuit forcfted by the KareS n and ^j ; then since there are no electromotive forces in the Wires, Kirchhoff's second corollary gives <^\i'\—Cii^i = o, .'. f, :C2 = — : — J and similarly ?-, r, III I C\'.Ci\S\\ ,,, i€^ = — ; —\— : , , . : — , n Ti n Tn I Hence Cx = — i- . C, etei [Branch wires inserted in this way between si two points are said to be arranged in Multiple Arc^ 3. Show that the battery current at A breaks up in such a way that thq sum of the energies in the branch Wires is less than it Would be if the same total current were broken up in any other mannen The energies in the bfttiches are c^r^, c}i>t, ... no matter What the values of ^1, c^, ... are; therefore if Xis the sum of the energies, X^c^r.,-^c}r,-\- ... +i:,ff„; and if ^is made a minimum by the variables c^, c,, ... , we must have Also we have dei + dc,'t' ... -^i/e^ ==,0, (af) since Ci + c,-¥ ... +e„ is given. Multiplying the second ty A, Subtracting from the first, and equating to zero t!he coefficients of n&i, elc„ we have »-,Ci— \ = o, »-jfj— A. = o ... >-„t„— \ 3 o; .". c.ic,-. ... = —•. — : ... which shows that the current actually breaks up in the required manner. 4. To represent grajihically the fall of potential along a. wire, being given the potentials at its extremities. Let A and B be the extremities, and ^^, ^^ the potebtials at them; let P be any point on the wire between A and B ; let 0p be the potential at P. Then the current strength, C, being the same all thrdugh the wire, arid the resistance (ti any port ibn of the wife being propdrtioUal to its length, we must have A-f-A + -^-^xB AP ~ BP •■■'''' AM which shows that if We dVaw any right liue, AiS,, to represent the re- sistiulce of the given wlte, at A, and ^, ereut tWo perfendisnlars, A^ A' and Bi B/, to represent ^^ and ^ j, resjpectively, take betweeb A^ and B-^ a A t* AP point Pi dividing AtB^ so that -^-^ = -^p, and at /", erect the perpen- dicular /',/' meeting the line A'B' in /", then the ordmate P^P' repre- sents the potential ff, , .. . . 234 Kinematics of Fluids. 5. Two wires, AP3 and AQB, connect, in multiple arc, two points, A and B ; the current from any battery enters at A and emerges at S ; the ends of a wire are attached at P and Q to the two first mentioned ; find the condition that there shall be no current in the wire PQ. It follows at once from last example that we must have resistance AP resistance AQ resistance BP resistance BQ ' since if no current flows through PQ, the current strength is constant through APB, and also constant through AQB, and moreover (pf must be equal to "a); (^'s; i^s)) •■■ of the remaining vortices, we have? at once miU^ + m^u^ + m,u,+ ... =0, or 2w?a = o, (3) tn^Vi + m^v^ + m^v^+ ... =0, or 2»?o = o. (4) Now the strengths m^, m^, ... all remain constant throughout the motion (p. 191), and these equations hold at every instant. Take at any time / the point, G, which is the centre of mean position of the points {a^, )3,), (a^, 0^), ... for the system of ' A fuller account of this subject is given in Mr. Greenhill's papers {Quarteriy Journal) and in his article on Hydrodynamics {Encycloi. Brit.) ; and also in Lamb's Treatise on the Motion of. Fluids, chap. vi. Motion of Plane Vortices. 335 multiples m^,m^, ... . If its co-ordinates (referred to the fixed axes of co-ordinates) are (jk, y)., we have 'x'Stm = m-ia^-\-m^a^+ ... , j'Sm = mj^Pi + m^^^+ ... Take the position of this point, G'.'again at the time /+ A/; let its co-ordinates be (3e', J''). Now at this time the co-ordinates of /«i will be {fflj + KiA/, /3i-f-0iA/), and similarly for the other vortices. Then x'^m = mi{a^ + Uj^A^ + m^{a2+U2Af)+ ... , = S2M-f-A/S»2«, = x2im; hence x' = x, and similarly 3^= Ji, so that G' is the same as G. Hence — the ' vortical centre ' of any system of freely moveable and muttmlly influencing vortices is fixed in space Mroughout the whole motion — a result identical with that which holds for the centre of mass of any material system which is left to its own mutual actions. [The student must be careful to observe that the fixity of the vortical centre (or of the centre of mass of a mutually attracting system) does not imply that the velocity of the fluid which exists at this point is always zero (or, in the second case, that the resultant attractive force at the point is always zero).] The signs of the quantities m^,m^,... depend on the direc- tions of their spns. Examples. 1. Find the motion of two mutually influencing vortices. Ans. Each describes a circle about their vortical centre, with constant velocity. TS mi=— m^, G is at infinity, each describes a right line, and the line joining them retains a constant direction. 2. Find the motion of a single vortex, m, in front of a fixed smooth virall perpeijaicular to the plane of the fluid motion. [Take the image of the vortex, which will be another vortex, —m, at the optical image of the first. Thus we have the last case ; and the vortex moves parallel to tie wall with constant velocity.] Q 3Z6 Kinematics of Fluids. 3. Prove that a single vortex moving inside the space bounded by two TT plane walls enclosing an angle — describes the Cotes spiral r cos « S = a. (Mr. Greenhill.) 4. Prove that a single vortex of strength m inside a circular cylinder of radius a, at a distance c from the centre will move with velocity due to an image of strength — »2 at a distance — from the centre, and that it will describe a circle of radius c with velocity = — 5. (Mr. Greenhill.) ' lit a'—c' 5. Prove that the energy of any system of moveable vortices is , -^ Ml m, (OTi + m^ „ "^ «h m^ m, A' ^ where ra means the distance between the vortices m^ and m^ at any instant, and Aja means the area of the triangle formed by w?,, wjj, and m,. Section IV.^ Conjugate Functions. 139. DeflnitioH of two Conjugate Functions. If ^ and 1^ are two functions, each, of them a function of x and y, such that <^ + i|^'v/— I is a function of x+y/ —i, then (/> and ^ are said to be two conjugate /unciions of j; and j/*. In the case of two such functions, then, we have 4> + ^^~~i=/{x+y'/^i). (i) We may, of course, convert the proposition and say that if we take any function oi x+yZ—i, suppose /{x ^-yV — i), and express this function as the sum of a real and an imaginary term in the form /f = o. (r) (2) The following relations hold between two conjugate functions as defined by equation (i) of last Article— ^ = ^. ^ + ^ = 0. (2) dx dy' dy dx ' For, using a, for a moment, instead oi x+yV—i, and diflferen- tiating (i) of last Article with respect to x, we have differentiating it with respect to j;, we have .hence from these two equations . d<^ d\jf , d^ * J rfi/r dx dx dy dy which, by equating the real and the imaginary parts, gives the equations (3) ; and these two equations may be taken as giving a new form of definition of conjugate functions, since equations (i) both of this and of last Article obviously follow from them. (3) V ^ "■''^^ ^ '^''^ "■''& ^° conjugate functions of x andy, the curves d^ d<^ d\lr < intersection is ^ ^ 2 "^ ' and by (2) the numerator of this is zero ; therefore, &c. The foregoing properties prove that we can regard ^ and yfr Q a 238 Kinematics of Fluids. as a velocity-potential function and a function of flow, re- spectively, or vice versa, since it — = u, -^ = v, \fe have ux uy di^r d-\\> dx ' dy ' (4) J/ = const. Regard (j) and i(f as the velocity ftinction and function of flow in the irrotational motion of a liquid^ and the equality enun- ciated is exactly the equation of Art. 104. Of course all other properties which have been proved for equipotential and flow functions hold for any two conjugate functions. {5) -If and -^ are any two conjugate functions of x and y, and if^ and f\ are any two other conjugate functions of x and y; then by putting ^ and ij instead of x and y in the Values of (p and ^|/, we get two new conjugate functions of x andy. For, let <^ =fi{x,y), f =fn{x,y). In these put f for x and ij for J/, and let ^' =f {$, tj), ^ =f (f; ij) ; then dj^^^di^df^dn^ d^ ^dfd^ df^dr)^^ dx d^dx dr\dx^ dy d^ dy d7\ dy' But since f^ = ^, and 4^ + ~=^o, it is evident that dx dy dy dx de dr,' ^"** dri^ Tr Also by supposition -? = ^ , and ^ + ~ = o. Hence dx ^ dy dx d^' d^' , . ., , d^' dyl/ ■J— = ~ J and similarly — r- + —r- = o, dx dy ' dy dx ^ which are necessary aijd sufiicient to define two conjugate functions of x andj". This is a very important property, which we may enunciate thus in general terms, considering it with reference to fluid motion — Properties of Conjugate Functions. zzg If the circumstances of ar^ irrbtaiional fluid motion, which takes place over the plane xy, are at any point {pc, y) expressed by the velocity and flow functions and \j/, the circumstances of another possible state of irrotoHonal fluid motion oner Me same plane and at the same point (Xty) will be expressed by veiocHy and flow functions derived from (j) and yjf by replacing, in their expressions, the co- ordinates x,y by any two conjugate fumtions of x andy whatever. Instead of regarding the matter from the point of view of fluid motion or electrical current flow, we may regard it from that of lelectrostatic distribution. The functions and i^ will then be the attraction potential and force-direction functions; and we may say that if [0, ■v/r] expresses a possible electrostatic distribution at any point x,y in a given plane, we shall get another possible electrostatic distribution at the same point x, y, by writing for X and y, in the expressions for (j) and \}r, any two conjugate functions ) If ^ and-q are two conjugate functions of x and y, then, conversely, x .andy are two conjugate functims of^andr\. For, \tX.f{x +y v'— i) = ^ + t/ •/ — i . Differentiate this equa- tion first with respect to f and then with respect to tj, and divide the one result by the other. Thus we get ,dx , dy. , dx , dy which involves dx dy ^ dx dy the necessary and suEBcient conditions that x and y should be conjugate functions of f and tj. (7) Let $ and tj be any two conjugate functions of x and y, the latter being the co-ordinates of any point, P, referred to two rectangular axes Ox and Oy ; draw any two new rectangular axes, (/f and O'tj, and in this new system lay down the point, P', whose co-ordinates (f, 1)) have reference to the values of x and J/ at P; and let this be done for all the points P in the first figure. Then, corresponding to any curve or continuous space 330 Kinematics of Fluids. in the first figure we shall have a curve or continuous space in the second figure. [Were it not for the confusion of figures,,, we, might lay off the values of f and tj along the given axes Ox and Oy ; but it will be better to imagine the axes Ox, Oy and the points P laid down on one piece of paper, and the axes Of, 07) and the corresponding points P' laid down on a sepa- rate piece of paper.] Now any two curves passing through P in the first figure will transform into two curves passing through p' in the second figure, and the angle between the curves atP' will be the same as that between the two corresponding curves at P. For, let A he 3, point very close to P on one curve, and B a point very close to P on the other; let A' and P' be the corresponding points on the corresponding curves through P' in the second figure. Let {x, y) be the co-ordinates of P ; {x + a,j/ + 0) those of A; {x + a',y + ^) those of ^; and (f, 1) those of y. Then the co-ordinates of A' are similar expressions holding for the co-ordinates of P'. Also, if the lengths PA and PB are dsj^ and ds^, and the lengths p'a' and P'b' are ds^ and ds^', we have at once ds.'^^ds ds: = Kds} ^3; where jT" — ^ dx ' d_i dy dy dn' dx Again, the direction-cosines of P'a' (with reference to O'f, O'rj) are ^l^af+fif) and -i-(af^+^^), /Cds^ ^ dx dy' Kds^ ^ dx dy ' similar expressions holding for the direction-cosines of F'b^. Hence, in virtue of equations similar to (2), p. 227, holding between f and tj, we have , ' cos Z ^P'A' = ""f^f^ = cos Z BPA, (4) Vhich proves the proposition. Again, if dS is any small element of area in the first figure, the Properties of Conjugate Functions. 23t area of the corresponding small element in the second figure is X^dS. For, if P is any point inside dS, the radius vector, r, from P to any point on the contour of ds transforms into IC . r drawn from F^ to the contour of the corresponding area, dS" ; and the angle between two radii vectores from P remains unaltered; hence ds' = A^.dS. (5) It follows, in particular, that any two curves 'orthogonal at P transform into two curves orthogonal at P' ; and that any small closed curve at P transforms into a similar small curve at P'. [It is scarcely necessary to observe that the axes of reference (/f, o'lj are not the transformed equivalents of the axes Ox^ Oy. These latter transform, in general, into curved lines in the second figure.] (8) Let f{x, y) denote any quantity having reference to a point P, whose co-ordinates are x, y.. Transform this quantity by substituting _/j {x,y^ for x and fi{x,y) for_y, where y^ and_/^ are any two conjugate functions of x and y; so that the new function connected with P is or, briefly, P(x,y). It is required to find a value for , d'' d^. ^, X " ^• in terms of the original form, f, of the function. We have dP ^ df\A{x,y),f(x.y)'\ ^ df{f,f,) df ^ df{f,f) df dx dx df dx df^ dx where we have used f and f, for abbreviation, instead of \ ^E ^ ^B k .. — E' 'Df f{x,y) and/j(j?^;, d^F JY(J^,A)dA dx'' ■ df^ dx Hence ■ , d-f{f„f) df ^ dfi '^- dx + 2 , df{f„f) dV, ■^ df dx" "^ d^f{f,A) dAdA df df dx did dVi.A,A) d% df dx"" ' "^ dy and ^p ax \dA dy 333 Kinematics of Fluids- Writing down, in the same way, the value of -— -> and adding the two results, we have -^ ^ + ^ = ^1 dA- + dA- ]' ^''^ in virtue of the relations just proved between the conjugate functions y^ OfiiA, and using JC^ to denote the common value of ^^ dx Now let F^ be the point whose co-ordinates (deduced from those of/") are /i{x,y) and /^{Xyy); then the quantity in brackets in (6) is the value of {-fr + -3-^ f{xj y) ^^ ^• Hence — transformed value of V^/{x, y) at P=JC'^x untransformed value of V^/{x,y) at F^, (7) where P' is the point 'corresponding' to F in the sense explained in No. 7. Again, if ds denotes any small area surrounding F, and dS^ the ' corresponding ' small area surrounding F', we proved in No. 7 that ds= —^ ds'. Hence the product of the transformed value of V^/{x,y) at F and a small area surrounding F is equal to the product of the untransformed value at F' and the corresponding small area. Again, lei it be required to find the line-rate of variation of the transformed quantity, F{x,y), at P in any direction, FQ, in terms of the original form, f of the function. dF Let the elementary length FQ be ds ; then — is the line- rate of variation of F along FQ. But d^_ df(f,f) df ^ df(f,f) df ds df ds df ds Let g* be the point corresponding to Q, as ^ corresponds to P. Then if F'(^ = ds', we have from equation (3) ds' = Kd', so that dF df Properties of Conjugate Functions. 2^^ or transformed value of tine-rate at P= K {f4n(ransformed value of line-rate at P'). The application of these theorems to electrostatic distribution is obvious. Suppose that at all points on the same right line parallel to the axis of z the electrical -density, force, &c. are the same; then clearly if V is the attraction-potential function, the general d^V d^V d^V ^ d^V d^V equation _ + _ + _ =0 becomes — + — = o; and the volume-density, p, at any point, P, is — ' — V^V, i.e., pdS is the quantity in a cylinder of unit height standing on the small area ds surrounding the point P, the base ds being parallel to the plane x,y. Now transform, the whole distribution by puttifig for x andj/, in the expression for V, any two conjugate functions of x and y, and let P' be the point ' corresponding ' to P. The result is a law of a new possible distribution. TJien the quantity of electricity in a cylinder with unit height standing on any small area at P in the new distribution is the same as the quantity in a cylinder standing on the corresponding small area at P^ in the old distribution ; and the new volume-density, p, at P is connected with the old volume-density, p', at P^ by the equation p = ^y. (9) It may happen that the electrical density, force, &c. are the same not only at all points on the same line parallel to the axis of z, but also at all points on the same line parallel to the axis of X J and in this case we have simply d^V ■^ = °- This latter is the case when we have two infinite (or prac- tically infinite) electrified plane* surfaces held parallel to each other at a small distance apart, the origin of co-ordinates being at the middle of one plane, and the axis of ^ perpendicular to the planes. Our supposition will then hold good for all points in the region between the plates which are not very near the edges. 834 Kinematics of Fluids. In like manner, we may suppose /{x, y) to be the flow function, y\t, at a point, in a given motion of a liquid, so that the spin at P is —\V^^ ; and we have at once the strengths of vortices at all points in a new motion given as equal to the strengths of vortices at corresponding points in the old motion. The second theorem, expressed by equation (8), gives a result with regard to the transformation of the sur/ace-dL&a.s\iy of a given electrostatic distribution — just as the first theorem relates to the transformation of volume-Aensiiy. For, supposing the function /{x, _y) to express electrostatic potential at {x, y), which is constant along a surface whose section by the plane of xy is the curve PD (fig. 72), and that PQ is an element of the normal to this surface at P; then if o- is the surface-density at P when the potential is transformed into /{^f^, f^, or F{x, y), we know (see Statics, p. 440) that _ ■ J_dF 477 ds And as_/(_/^,_^) is the value of the potential at the point {/i,/^, i- e., at P', in the old distribution, in which let are obviously conjugate functions, and if (f, rj) are any two conjugate functions of x and J/, so will i^—ri^ and 2^ jj be (No. 5). Hence if (^', ?j') are any other two co*ijugate functions of x and y, so will be the Emanants ^^'-^m' and ^' + fi? (proved otherwise in Clerk-Maxwell, vol. i., Art. rSy). (11) If (^ is any function of {x, y) satisfying Laplace's equa- tion V^rf) = o, then will -^ .and — -^ be two conjugate dx dji functions of x and y. For they satisfy the (necessary and sufficient) equations of No. z. Hence the components of velocity of a, liquid at any point of a non-vortical region — the sign .of one component being altered — are two conjugate functions of the co-ordinates of the point. 141. General Formulae of Transformation. In the old distribution let {v', F', is constant; then draw two axes of (f, 7/) ; eliminate y from the two relations between {x,y) and (£ tj) ; suppose that this gives /"(f, r\, x) = o; now trace out a series of curv.es /(e n, a) = o, /(£ r,, dO = o, /(f, 7,, d') = o, (a) obtained by making x constant and successively equal to a. a', a", ... ; again, eliminate x between the given relations and suppose the result to be ^(f, n,y^ = o ; liien trace out a new series of curves g (£, r,, b) = o,g (e V, b') = o, ^ {e, n, n = o, (^) obtained by making j/ successively equal to b, b', b", .... Now take the point (or points) of intersection of the first of the series (a) and the first of (j3), and we obtain a (f, if) point (or points) corresponding to the given value of ^. Similarly the intersection of any curve of the series (a) with the cor- responding curve of the series ((3) will give another required (f. l) point ; &c. This method applies, of course, whatever be the natures of 338 Kinematics of Fluids. ^i f> 'J ; but it is especially useful in the case in which <^ is a potential (or flow) function, and (f, tj) are two conjugate func- tions of {x,y). In this case (^ will also be a potential (or flow) function of the new variables (f, rj), and the curve thus graphi- cally traced out will be a transformed equipotential (or flow) curve. 143. Betermiuation of Conjugate FTinctions. An in- finite number of forms of two conjugate functions of (x,y) can be found from the fact (Art. 139) that the real part and the co^ efiicient of v^— i in the expansion of /{x+jf^—i) are conjugate functions, whatever be the form of _/! Thus v^ Jt and y/'^x)--^r'{x)+ =r,, {2} are two conjugate functions of {x,jy). The following are some examples of most frequent occur- rence. (a). Let/{x) = x^. Put .ar+j''/— i for j;, and we get f = x^—y ; ri = 2xj>. 0). Let /{x) = log - . Put X +y ■j/^ for x ; then put x = rcosd, y = r sin Q, where r = V^^+y, Q = tan~* — ■ X Then log ^ ^ ~^ = log - + log (cos Q-V V~i sin fl). But cosfl+V'^^sinfl = tf*^-^; therefore log ^ = log- + fl^/-i; f = log -= log ^ ; jj = fl = tan- which are the well-known conjugate functions which play such an important part in the cases of whirls and of sources and sinks (see Section I). Determination of Conjugate Functions. 239 If we put - = ?'', we may write x = ae'' cos 0, y = ae'' sin ; then p and 6 are conjugate functions of x andj/; and con- versely, X and,;* are conjugate functions of p and 6 — as may be dx dy , dx dv seen at once, smce -= -^ and — + ^=0. dp dO do dp Cor. I. Since (No. 11, Art. 140) 3^ and % are con- dx dy jugate functions of {x, y), if ^ is any function satisfying V^ 71 =• ;; -„> and we have x^-k-y^ x^-\-jr so that if we use this value of ^in the equations of Art. 140, we get the complete connection between the two distributions. D O Fig. 73- Thus, let ABCD (fig. 73) be any figure at any point on the contour of which the potential has a value _/"(.y, j/) ; and let the a* potential be transformed by the inverse substitution — and r Electric Inversion^ 241 — 6 for r and 6. Then over the same contour ABCD in the new distribution the potential at any point will have the same value as, in the old distribution, it had at the corresponding point of the contour A'B'c'lf, which latter is obtained by taking the reflection, in Ox, of an inverse of the contour ABCD. If in the previous distribution there is, for example, a quantity e of electricity at the point if, there will in the new distribution be a charge e at the point Z», and no charge at if. This charge is not an electric 'displacement,' or surface- density, but the product of a volume-density and an element of volume. Again, if in the first distribution there is a charge e at Z>, this in the new distribution will be transferred to i/, and there will be no charge at D. In both distributions there will be equal electric displacements, while 7-7 = ^j we have ^ =4wcr: and smce dv dv dk b '. the charge at C plus the total charge on the inner side of the circle is zero, 2 wo- 5 = —e; .: V=V^ + 2e\ozY ^') We must express this in terms of OP{ = r) and the angle POC{ = 6), for the purpose of inversion about O. Then V=K^+2e\og ^ ^=-, (2) Yr^—2cr cosd + a' where c^CO. Now consider the points, C' and P', corre- sponding by inversion about to C and P. In the new distribution there will be at C whatever charge was at C' in the old— i.e., none; and in the new distribution there will be at (f whatever charge there was at C in the old — i. e. there will be « at c'. Again, at every point P in the new distribution the potential will be what it was at P' in the old ; a' and as this latter is obtained by writing — and — for r and 6 in (2), we have 5r V=V^ + 2e\og (,\ Electric Inversion. 343 expressing the potential at any point due to a charge « at C and an electrification over the metallic circle AB. This is evidently the same as K=F„+2.1og^^. (4) In the second distribution the potential at any -point of the circle AB is equal to the potential, in the first, at the corre- sponding point on the circle /f^ — which latter is the inverse of the forrner with respect to 0. Hence, in general, the circle AB ceases to be a curve of constant potoitial in the new distribution. But if we arrange so that the inverse, A'lf, is the circle AB itself, then AB will remain a curve of constant, potential. This of course is done. by taking a equal to the tangent from to the circle = »/) The first equation (fi) gives u^ + v'^= a function of x and j/ only ; while (y) and the second equation (/3) show that u and v are conjugate functions of x and_>'. To satisfy these conditions, assume u = A cos/(z) — B sin/(3), (6) w = ^sin/(a) + >ffcosy(z), (e) where A and B are functions of x andj/ only. Using equation , . , dA dB , dA dB ... (y), we have j~ + ;7~ = ° ^°" ~i — j~ = ° > which prove that A and .5 are conjugate functions of x, y in the same way as u and v. Equations (6) and (e) can be put into the forms u = Va^ + B^ cos {/{z) + Tj} , v= ^A^ + B^ sin {/{z) + »}} , » where rj = tan"^ — • Now we know that log Va^ + b^ and rj are conjugate functions of x,jy. Hence if we put f for log Va^ + b^, u = e^ cos {/{z)+ri}, (i) zi = «*sin {/{0) + r)}, (2) express a liquid motion in which the axis of vortical spin at any point coincides with the line of resultant velocity at the point — all such axes being parallel to the plane of x_y. This is a screw motion at every point. 2,^6 Kinematics of Fluids. Examples. I. Two large plane metallic surfaces are electrified, and kept parallel to each other at a small distance apart, each, being at a given potential; to determine the distribution of potential in the space between them, and to deduce therefrom other possible distributions. , Let one plate, Ax, be at potential K, and the other, SC, at potential V,. '^ Take the origin! at the middle of the first plate, and the axis oi y perpendicular to the plates. Let j/ be the distance of any point P (not near the edges) between the A plates from the first plate. Then if V is the potential cPV Fig. 75. at /", K is a function of ^ alone ; hence -^ = o, and if OJ = «= distance between the plates. The electric displacement, a, at any point is — - ' B 4ira Now if in (i) we substitute any function, <^, satisfying the equation/ v'0 = o we get another possible distribution. Thus if we put 2 xji for y, the surface Ax becomes two rectangular planes, and the surface BC a rectangular hyperbolic cylinder to which these planes are asymptotic. Again, if for j/ in (i) we put r sin 6, where 6 is the angle POx, and then write -jj^j for r and » 8 for 6 (Art. 143), r=K, + (r,-r0^sin«9 (2) will express a new distribution at P. The line BC in the first distribution will correspond to the curve r^ sin n6 = a" (3) in the second ; and the line Ox in the first will correspond to Ox in the second, while OA in the first will correspond to a. line, 0I>, making an IT axigle, DOx, - with Ox in the second. In the new distribution, therefore, the potential has the constant value Vj over the curve (3), and the constant value Vi over a system of two right lines intersecting at 0. This system of two lines is, of course, in reality two planes perpendicular to the plane of the figure, and the curve (3) a cylindrical surface. Since the equipotential lines in the space between the planes Ax and BC were lines parallel to Ax, the lines of force were lines parallel to 01, their equations being >■ cos 9 = const. Hence in the new distribution the lines of force are ^« cos « 9 = const. (4) and the equipotential lines ^n gin „e = const. /-n Examples. M^ It is well to observe that all this remains true if we have only one infinite metallic plane, Ax, electrified to the potential K, . The value of V at any point would still be a function of ^, so that -r-j- = o, and we should have V = Ay + F; as the integral instead of (i). The value of VaXP cannot be completely determined unless we know its value, r^ , at some one point off the plane ; and our assumption of two me- tallic planes the potential on each of which is assigned serves this purpose. We have now the law of variation of potential corre- sponding to the electrification of two infinite metallic planes Ox, OD (fig. 76) intersecting in an edge at O ; and we have found LDOx = - . Hence if a is the angle, we n must have ir « = -, a and K= r, + ( V^- V,) (-Ysin - 6, where V^ is the potential on the equipotential surface whose equation is Fig.. 'j&. IT (-)sm-e = x. (7) At any point P on the system of planes the surface-density a is the I dV value of y. , with r = OP, - o; i. e., 47r raB The total charge on the portion of the plate Ox between and P and having a breadth (perpendicular to the plane of the figure) equal to b is i I adr. 2. To discuss the ellipses and hyperbolas of example 7, p. 167, from the point of view of conjugate functions. It appears from (7), p. 239, that the functions' (c* + « '')cosiJ' and 2X (e<('_e-*)sinf are conjugate functions of and ^. Denote them by — and —, so that c 2.x: = f («* + «"*) cos (t, 2j|/ = <: («'''-«-'') sin ^. (l) ' It would, perhaps, be better to use ^ and ^ instead of (f and ^, where .4 is a magnitude of the same kind as ^ and ^ — for the sake Of homogeneity. 348 Kinematics of Fluids. Conversely, then [(6), p. 229], "^ and f are conjugate functions oix,y. Hence in fluid motion and electrical flow we may take <^ and 'fi as potential and flow-functions, and vice versa; and in electrostatics as potential and induction (or force) function, and vice versa. Let us then find the curves over which ip is constant. Eliminating f, we obtain which denotes an ellipse when ^ is constant. Eliminating to be the potential function and \\i that of induction, we may imagine FF' to represent a plane strip of metal of infinite length (of course extending perpendicularly to the plane of the paper both above and below it). It is at potential zero; and we can calculate the charge on it per unit length (measured perpendicularly to the paper). If a is the surface-density at any point of FF', we have 410- = -, since dy is the dy element of normal to the eqnipotential surface FF' \ .-. ±-ita = — .-. the dx charge on the length dx (and unit height) is — dii, .-. the charge on the 4" portion between any two points on FF" is — ('f'i—'/'0> where t/i^ and >/ij are induction functions at the points. Now at F' we have 'p = ir, and at Fff = 0; .: the charge on the strip FF' with unit height = J of a unit. The ellipses are the eqnipotential curves (sections of elliptic cylinders) in the dielectric due to the electrification supposed. Again, we may take if> as the potential function and f — "■ must be numerically < c. i^ _ ffi ■' „, . „ r ". ' Hence since for every point i = 5 ^ ,p = o 7, ij, = ir given by (5), i = tee", i// = 3 it follows that x must lie c, .'. y is given by those values in (5) which make cos ^ positive, and X must lie between o and + 00. Hence _y = o, tb-n, ^bti, (6) and the required points lie on the lines 00, /, 00, ... . The points at which in the new distribution ^ has the value -n are seen, in the same way, to lie on the lines /, 00, /a 00, ... . Let us now find the points which in the new distribution correspond to the axis oiy in the old, i. c, points at which \p = —. The points P' lie on the axis of y, i. e., bv sbir (2M + l)6iT ^ = T'— '- ^' (« no restriction being placed on the value or sign of x. The results of this transformation are applied by Clerk Maxwell to two cases in Electrostatics {Electricity and Magnetism, Vol. I, pp. 276, &c., 2nd ed.). Of the first of these we propose to give a sketch here. Take ^ as an electrostatic potential function, and ^ as the force (or J^duction) function. Consider now the limited line 00, along which tp = o, and the two unlimited lines above and below it, along which ip = -. Let these lines become indefinitely thin metallic planes electrified to these potentials. Take any point. A, on the middle plane. Ox, at a distance x from 0. Then, since if = o at 0. Now the transformation which is effected by equation (4) is expressed by the equations X 2e ^ cos I =(«* + £-*) COS ^t, (8) X 2e'i sia^ = («*-«-*) sin it, (9) so that, since for the point A we have y = o, ^ = 0, (8) gives — /~IE. ^ = log(«* + V^T_i) ^j^^ Examples. 351 (the + sign being given to the square root, since when x becomes in- definitely great, must not be zero). Assume now that b is very small, i.e., that the parallel metallic planes are at a very small distance apart. Then for points. A, at a considerable distance from 0, x will be a very large multiple of b, so that e —i is 2X very nearly the same a.% e " ; and for such points, we have X •p = log 2e * = — + log 2. (11) Hence at all such points on the middle (limited) plane, - = 2ccos^v'-i = 2«cosh0, and /->- = zrcosf, so that

^/~I), may be exhibited in terms of the distances of the point {x,y) from three collinear points A, B, C. (Mr. Greenhill.) Examples. 253 [If k is the modulus of the elliptic function, take A as origin, take have AB = c, and AC = ^^•, then denoting PA, PB, PC by r, /, r", we r = csn|(iJ + (^V'-l)sn|((«'-.fV'-l). »^ = <:cn|(^ + -" sin«fl = a", where a varies, prove that the displacement at any point 00 ;-""'. 8. Verify that the equation in example 3, p. 165, is equivalent to df' '^dB^~°' where p = log — . NOTES. Note A. Centrifugal Force. When a particle of mass m moves in a curved path, it has at each point an acceleration along the normal, towards the concave, or inward, side of the path, equal to — (p. 56). To produce this acceleration there is required a component, N, of force along the inward normal equal to • This force will be in dynes if m is in grammes, v in centimetres per second, and f in centimetres. The particle itself exerts a force which is the exact equal and opposite of this normal force, N, on the agent which produces N, the action of the agent on the particle and of the particle on the agent being propagated by a medium intervening between them. Indeed, sim- plicity might be gained by saying that the force N is produced on the particle by the medium immediately in contact with it, and the particle reacts -on this medium with a force exactly equal and opposite to N. Thus the medium experiences a force N, or , along the normal drawn out- wards, or towards the convex side of the path. Similarly, of course, for the component, m -^-, along the tangent in the sense in which v at increases, the medium experiencing the same force reversed. Thus, then, the external force acting on a moving particle, i.e., the force produced on it immediately by the surrounding medium, though mediately, perhaps, by something else — as when a vortex or a source at any one point is a cause of motion throughout the field — and the reaction of the particle against acceleration, or its force of iTiertia (p. 107), are merely the two aspects which every stress necessarily has {Statics, p. 4S6). aS^ Notes. The object of the present note is to guard the student against erroneous notion which is contained in the expression ' centrifugal for which is applied to the force , — viz., that a particle moving in a i P cular path is acted upon by a force outwards from the centre, (centre-flyin The very reverse is the case ; the force acting on the particle is whc towards the inward side of the curve ; the force exerted by the particle the external agent is exerted in the opposite sense ; and similarly for i other curved path. Take the familiar instance of a particle tied to one end of a string, other end of which is held in the hand, this latter being kept fixed oi smooth horizontal table, along which the particle is projected. Here only force acting on the particle is the tension of the string, which directed inwards towards the centre ; the hand experiences an equal fo outwards ; and the numerical value of each is in absolute measure, P — in gravitation measure, M w = weight of particle. Hence will be seen the fallacy of the expression — a very common ont ' portions of the revolving solar atmosphere thrown off by centrifugal fori Given a normal component, N, of force, of definite magnitude, and a mi ni revolving in a circle of radius r ; then the velocity of the body must /yvv equal to A^ — ; and if from any cause the velocity is increased abc this limit, while the normal force remains the same, the result is that 1 body must widen its orbit to suit the circumstances. But there is no sii thing as ' flinging off by centrifugal force.' The term ' centrifugal force ' is a very unfortunate one, and it ought be abandoned by physicists. Note B. Strain Invariants. It may be well to add a few remarks on the general, or three-dimension strain of a body. With the notation of Statics (p. 461), let u, b, c he t elongations along any three rectangular axes (x,y, at a point /" in the body ; zj,, 2S2, zs, the shei with reference to these axes; and ai,, lu^, eu, t components of the rotation. At P draw another < of rectangular axes whose direction-cosines wi respect to the first set are given by the usual schei here indicated. Then if a' s^',.,,a:^ denote the componente of the displacement produced by the strai X y 2 I m n I" m' k' ^ yi' Conjugate Functions. 257 with reference to the new axes, we have the following types of trans- formation ; — «,' = /(»i + ««^i^'c+{m'ti" + m"n')h+ (n' 1"'+ 1' ti') s, all these tesulting &oni the relations x! = Ix + my + nz, with similar Values of y, z', (o') a' = /a + wiO + ^OT, with.,siniilar values of »', oi'. (a") Now it can be proved from elementary considerations with i^ard to the elongation quadric and the line of resultant rotation that the following quantities are invariants : — S S ai + ic + ai—s^—s^—s^, A = aic + 2SiSii,—as^—is^—cs', * = aai' + ia,' + ca' ^- 2Sia2»',+ 2Jji)Ujiu, + 2J-,w,ei^. fif$ d^ d^ Also the vector whose Gomponents alone the axes are -; — > -r- ) -r— ao), floJj rfajj is an invariant; for these quantities transform according to the type (a). d^ Again, -j— , &c. transform according to the type (j3). ait In uniplanar strain these invariants are (a + b), (fibs''), and oi. In the general case the elongation along any line is inversely proportional to the square of the radius' vector of the elongation quadric in the direction of the line ; and it is easy to see that the shear of any two rectangular lines is -=-=>: cos «, where R and J? are the lengths intercepted from the k^ k'' i' lines by the quadric the squares of whose axes are '-^-± , — ^ , —-=■ ' «,, e,, e, being the principal elongations, k any constant, and a the angle between the tangent planes to this quadric at the extremities of Ji and Ji'. Note C. Conjugate Fmtotions. The whole of this work was in type when 1 tecame acquainted with a very important and elegant paper on Conjugate Functions by Mr. Routh in the Proceedings of the London Mathematical Society (Nos. 170, 171). Mr. Routh proposes to employ the method of Conjugate Functions to solve hydrokinematical problems which are commonly Solved by the method of Images. S 358 Notes. ■The main features of this paper, so far as it deals with fluid motion, as follows ;— (1) Imagine two different motions of a fluid over a plane, the derived from the other, with the dependence of corresponding poi P, P, explained in (7), p. 229, and let ABCD and A'SCU (fig. p. 240) represent generally any two corresponding boundaries. Then p. 235, shows that velocity of P = JiTx velocity of /", by taking i/(f, ij) a velocity potential or a current fiinction. (2) If a source or a vortex exist at P', there will be a source or a voi of equal strength at P. If there is a vortex at P", there will also, be on( P, but the vortices do not necessarily continue to move so as to oca corresponding points. However, given the motion of P', the motion o can be found thus. Let the components of velocity of P" parallel to axes be expressed as ^}^' ^' and ^T^^' ^° ^^^ x' Q, v) ' current function for the vortex P — but, of course, not the current fnnct whose differential coefficients give the components of velocity at ev point of the fluid in which P' is moving. Then the function, x(-*'.J')j x,j/ whose differential coefficients will give the velocity- components of vortex P is x'(£> v) log ■*', where m is the strength of the vor P' and A' is the modulus of transformation (p. 234). [Of course xC-"^. J') is not the current, or flow, function which belongs every point of the fluid in ABCD, but merely that which defines 1 motion ofthevortex P.] (3) Suppose now that we require the motion of a fluid, in which there i Vortex, /"i, over an infinite area, part of whose (unclosed) boundary is ^j (fig- 73)- Imagine the vortex removed, and find, if possible, a steady acyc irrotational motion of the fluid. Suppose the velocity and stream functic of this motion to be ^ and ifi ; then the boundary being a stream line, has a constant value, k, along it. Now use f and >fi as the £ and r/ transformation, i.e., the co-ordmates of P' are (^, ^). Hence every stres line of the motion within ABC transforms into an infinite right li parallel to the axis of x ; and, in particular, the boundary ABC correspon to the right line y = k. Now replace the vortex at the point P, inside ABC, and there will b( corresponding vortex at the corresponding point, P,' ; and the motion . is already known (ex. 2, p. 325\; P( moves parallel to the axis oi x\n -a velocity , where n is the ordinate of/",'. Hence X (£.')) = :;— log 1? = — log ^ ; so that x (.x.f) = — log ifi- ^ log Jir, and thus the function determinmg the motion of Pi is found. Vectors and tiieir Derivatives. 259 The same method applies if there are more vortices er sonrces than one within the area ABC, and the solution of the problem will depend on a solution for the case of a rectilinear boundary — i. e;, on the simplest case of the method of Images. Apply this to a particular case. Let there be two infinite right lines, OA, OB, the first being taken as initial line, and let o be the angle between them. Then ^ = r" cos nO, i/z^r" sin n9, where « = - will express a possible steady irrotational motion in a fluid contained between the lines (see p. 246). Hence when a vortex is moving in this fluid we have X C-^, y)= — log (r« sin n6)-~ log K, where - - (f )■* ($)'- m* {%■)- '""-■ - - ."-.. ••• X (■«, y)= ^ log (r sin nO), 2 IT neglecting the constant — log «. The path of the vortex is found, of 27r course, by putting x equal to a constant, so that it is the Cotes Spiral »- sin »9 = a constant. In this very simple manner Mr. Routh solves Mr. Greenhill's problem (see ex. 3,, p. 226). Note D. Vectors and their Derivatives. The theorems of Arts. 105 and 106 are particular cases of general results in the theory of three-dimensional displacement. On account of the importance of the general results in the theory of Electromagnetism, it is thought desirable to present them here for the consideration- of the more advanced student. Let {i,j, k) be Hamilton's system of rectangular unit vectors, and let vr denote any vector identified with a point, P, in space. Let the com- ponents of OT in the directions of i, J, k be «, v, w, respectively, so that w^ ui+vj + wk. (i) . d . d d L«t V denote the operator « j— + J -7- ^ ^ ~r < where {x,y, 2) are the co-ordinates of P referred to fixed axes in the directions of i, j, k. Then, from the fundamental relations between i,j, k, we have -^ -h-r + -7-1 + ( dv dw\ fdw dv \ . rdu dw- ^dx dy dz' \dy dzJ Wz dx^ S 2 26o Notes. Thus, when a vector is operated upon by v, the result is a quatem ■whose scalar portion is called by Clerk Maxwell {Electricity and Mag tism, Tol. i. p. 29) the convergence of the given vector, and whose vec portion is called the rotation of the given vector. Let 9 be put for du dv dm dx dy dz' and '2 a for the vector portion of the right-hand side of (2) ; then ■VOT = —6 + 201. The scalar, 9, and the vector, 2 a, thus derived from a given vector the operation V are the same whatever be the system of axes of referea so that if we take any new axes {xf, }/ , «') and calculate the componei — -; TV, &c., of the roitation with respect to them, from the tra dy dz formation types (o'), (a") of Note B, we shall find these to be sim -equivalent to a cos \, &c., where eo is the resultant rotation, a/ 01^ + 01,' + a and A. is the angle between the assumed axis of x" and the line round wh (u takes place. It will be observed that if «, v, w are the velocity-components at P a moving fluid, 6 is whatt we have called the expansion, while a is spin-vector at P. Thus, then, in the case of any moving fluid, the condensation at i point is the convergence of the velocity ; and the spin is the rotation of •velocity at the point. Operating -on ot a second time with Vj i.e., operating with v on bi sides of (2), we have d^ d^ d" •where on the right-hand side — V^ stands for -r-i + -r-s + -r^. But c dx' dy' dz' at the right-hand side V' may be understood to mean ,. d . d d .^ V-Tx*^^*^^)' as it does at the left-hand side ; for, it is at once found that (.d .d ,d^' fd' d' d''s VTx^^Ty^^di) " = -te + ^ + ^)«- Hence (4) shows that the result of the operation v" on any vector purely a vector, and (5) that the result of the operation v" on amy scalai purely a scalar ; or, in other words, the operator v^ is essentially scalar. Now imagine any curved surface bounded by an edge of any form. " shall pi-ove that if there be a vector magnitude drawn at every point on surface and on the edge, its value being any function of the position of point ; and if at each point of the edge the vector be resolved along tangent to the edge, the line-integral of this tangential component tal Vectors and their Derivatives. 261 all over the edge is equal to the surface-integral of the normal component of the rotation of the vector taken all over the given surface ; in other words— t/ie line-integral of the tangential component of a vector along any closed curve is equal to the surface-integral of the normal component of its rotation over any sttrface whatever having the curve for an edge. (o) 1° fig- 58, p. 155, let A be the given edge — ^no longer, of course, a plane curve — and break up the surface, or cap, bounded by it into an indefinitely great number of small areas, such as abVd, which we may assume to be plane areas. Then, obviously, if we prove the theorem to hold for any such small area, we can, exactly as in pp. 155, &c., extend it to any surface. At the mean point, P, al the area draw the normal (axis of f). and any two rectangular lines (axes of f and ij) in the plane of the area ; let the com- ponents of the given vector, ■nr, at P along these axes be p, q, r, so that •s! = pi + gj+r&, if {i,j, k) denote unit vectors in the directions of the axes; and let Q be any point on the contour of the little area, its co- ordinates with reference to P being (f, rj). Then the values of p and q at Qaiep-t-f^+f]-^ and q + t-r-+V-r-i and the sum of the resolved ax dy ^ dx dy parts of these along the tangent to the curve at Q., multiplied by the element of arc, is the integral of which, exactly as in p. 155, ds {-— — ™) i^^S", where dS =Sidr\ = the area of the small curve. Now from (2) we see that -y- ^ is the normal component of the rotation of is ; therefore the dx dy proposition is proved. In particular, if the vector ■as is the velocity -of a moving fluid, the line integral over any closed curve is what we have called Ihe circulation round it ; and the result is that the circulation round any curve is equal to twice the surf ace-integral of normal spin taken over any cap having the curve for an edge. This is the general case of Art. 105. Again, imagine any closed surface. We shall prove that if at every point on and inside the surface there be drawn a vector whose value is some function of the position of the point — the surface-integral of the normal component of the vector, taken all wer the closed surface, is equal to the volume-integral of its convergence, taken all through the enclosed volume. (0) Break up the enclosed volume into an indefinitely great number of small closed surfaces, or cells, of any shapes. Then, observing that the normal component of the vector at every point on the surface is supposed to be a6a Notes. measured constantly inwards along the normal, if we prove the propositi to hold for any one small closed surface, it will hold for any closed surfa however large, since by addition, exactly as in pp. 155, &c., the parts the surface-integrals belonging to the portion of surface common to t adjacent cells will cancel. At the mean point, P, of any small volume draw rectangular axes £, j;, f ; let the vector OT at Pbe pi + qi^-rk; let (f, rj, te the < ordinates, referred to P, of any point, Q, on the closed surface ; then if is the surface element at Q, and (/, m, n) the direction-cosines of 1 outward normal at Q, the element of the surface-integral contribu' f dr dr fdr\ Now observe that ldS=drid.^, while jff'^dtidC = dV == volu enclosed by the surface -^Jfrididi =JJXdidi), and that all the otl integrals, such as JTiidi\d^, vanish. Then the surface-integral comes tc \ dx dy dz I which is the convergence of the vector OT at /" multiplied by the elementi volume. Hence, as observed above, the theorem enunciated is true for any clo! surface and its. enclosed volume. When the vector, OT, is one deduced from any scalar function, ^, by ' operation v, the theorem is at once deducible from Green's equat dA dd> d(b J , . (Siaiics,-p. ^i) ; ior p, q, r are then —^, —=- , —y-, and the ci vergence becomes — v''*?- Making, then, the second function in Gree equation constant, we deduce the result. In particular, if "HS is the velocity of a moving fluid, the surface-integ pf outward normal velocity o^er any closed surfcu:e ii equal to the volu-i integral of the expansion taken throughout its volume ; or if v denote outward normal velocity at any point on the surface where the element area is dS, and S the expansion at an internal point where the elemenl volume is dF, fvdS=fedV. The theorem of Art. 106 is the particular case of this equation uniplanar motion ; for the closed surface is in that case a cylinder of u (or any) height described on the curve (fig. 59) in. the plane of motion, i the element, dS, of surface becomes the element, ds, of arc of the bound: curve multiplied by the height of the cylinder ; while the element, d V, volume becomes the element of area (represented by dS in Art, k multiplied by the height of the cylinder. Current Power. 265 ^y assigning different forms to the vector in theorems (a) and (fl), we arrive at properties of the motion of a fluid and of the strain of a solid. Thus, for irrotational fluid motion, let ts = pip{ui + vj + wk), where /> is the density at any point, ip the velocity potential, and «, v, w the com- ponents of velocity. Then the component of ot normal to any surface is dip p

and comparing this latter form with the exp sion of Ohm's Law, we see that when external work is being done, resulting current-strength is the same as it would be with a diminis] IV electromotive force equal to £ — ^ in ^ circuit ef the same total res ance, />, in which no external work is being done. If the sole object aii at is the production of external work, the portion p C of the powe: merely waste power. The equation (a) may be put into another useful form. Suppose, definiteness, that the battery is used to decompose an electrolyte. Ijt current of unit strength decompose f grammes of zinC, or other substar in the working battery, and let 9 units of heat be -evolved in the battery the decomposition of i gramme ; then 6 f units of heat will be evolved unit current, and O^C units of heat by the passage of a current of streni C. If, then, J (Joule's equivalent) is the number of ergs equivalent to 1 heat unit, the power of the battery in ergs per second is JB^C. Simila if the decomposition of i gramme of the substance set free in the electrol requires the absorption of 9' heat units, and if a weight f grammes decomposed per unit of current, the power absorbed in the electrolytic ( is J a ? C when a current of strength C passes. Hence (a) becomes therefore ^= /(?fr^. P Thus the electromotive force of a battery can be expressed in the fa J 8^, and if its current is used to effect decomposition in a series electrolytic cells, the resultant electromotive force of the whole arrans ment is /(sf-e'f'-e'T'-...). INDEX. Acceleration along normal, etc., 55. — along changing direction, 58. — of any order, general theorem of, 70 — resultant, when tangential to a curve, 76. ^- in fluid motion, 19a. Amsler's planimeter, 92. — integrometer, 94. Barriers, method of, 196. Boundary condition, 162. Central acceleration, 59. Centrifugal force (note A). Centrodes, body and space, 39. equations of, 43. — acceleration, 68. T- rolling of, 80. Change of velocity, 58. Circulation, 154. — theorem on, 155. Clerk Maxwell's graphic transform- ation of fluid motions, etc., 237- Clifford, on harmonic motion, 20. Compressibility, measure of, 1 38. Conjugate functions, 226. ' Continuity,' equation of, 142. in polar co-ordinates, 164. Crank and connecting rod, 47. Current, electrical, strength of, 219, — power of, 222, (note F). Pusps, circle of, 99. Diagram of space described, 2. — accumulated velocity, 54. Dilatatiai^ 209. M<=Cay, W. S., on Holditth's theo- rem, 87. — Kempe's theorem, 89. — theorem of, 99. — on lengths of roulettes, loi. Molecular velocity, mean square of, 119. Momentum, 106. system of rigid body, 109. Multiply-connected spaces, 193. Newton, on force of inertia, 107. Ohm's Law, 22a. Oscillating cylinder, 45. Plates, two parallel electufied, 246. — any number of parallel, 249. Potential of strain, 134. — velocity, 148. — in multiply-connected space, 195. Relative motion, 26. Resistance of a flow channel, 184. ■ — sheet of tin -foil, 212, etc. Rolling, acceleration in, 71. Rotation, pure, 37. — produced by strain, 127. Roulette, area of, 82. — generalised, 90. Rouleftte, curvature of, 919. — length of, loi. Routh, E. J., on conjugate functii (note C). Screw motion in a liquid, 244. Shear, graphic representation 189. Source, strength of, 167. Spin, vortical, 147. graphic representation of, 1 ' Spirals, equiangnlat, 173. Squirt, 167. Steady motion, 140. Steiner on pedals, 86. Strain, resolution of, 124. — invariants of, 124, 136, (note — ellipse, 12Q. — shearing, 129, 133. — pure, or irrotational, 130. Stream Unes, 1 50. closure of, 159. — ■ — graphic superposition of, i — function, 151. Surface-integr^, 140, (note D). Thomson, Sir W., tidal clock, 13 — on Green's equation, 198. Townsend, Prof., 120, 1,36^ Tubes of flow, 158. Velocity, fluid, in terms of poten and stream-functions, 154. — systemdeterminedfromexpans and spin, - 160. • — infinite, in liquid motion, 207 Vibrations, composition of, 10. — elliptic, 19. — resolution of, zi. Vortex motion, 147. ■ invariability of, 190. — motion in liquid due to, 201. — electrical equivalent of, 203. Vortical centre, 225. Vortices, plane, motion of, 224, (n C). Wave disturbance, 16. Wheatstone's bridge, principle 224. Whirl, 167. Wolstenhohne, Prof., on motion lamina, 73. THE END. Works in Mathematics and Physical Science, &fc., recently published by the Clarendon Press. A Treatise on Statics. By G. M. Minchin, M.A., Pro- fessor of Applied Mathematics in the Indian Engineering College, Cooper's Hill. Second Edition, Revised and Enlarged. 1879. 8vo. doth, 14?. A Treatise on the Kinetic Theory of Gases. B7 Henry William Watson, M.A., formerly Fellow of Trinity College, Cam- bridge. 1876. 8vo. clot%, IS. 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